UNIVERSITY OF MINNESOTA

This is to certify that I have examined this copy of a doctoral dissertation by

Fei Xu

and have found this is complete and satisfactory in all respects and that any and all revisions required by the final examining committee have been made.

Prof. Peter Webb

Name of faculty adviser

Signature of faculty adviser

Date

GRADUATE SCHOOL

Homological Properties of Category Algebras

A DISSERTATION

SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL

OF THE UNIVERSITY OF MINNESOTA BY

Fei Xu

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Prof. Peter Webb, Adviser

June 2006

Acknowledgements

I wish to thank my thesis advisor, Professor Peter Webb, for leading me into this area, for numerous helpful conversations and comments and for much encouragement during my years in Minnesota.

I also would like to thank Haiying, my wife, for her endless encouragement in my life.

Contents

1 Introduction 1

2 Category Algebras: Definitions and Examples 7 2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Basic Homological Properties 12 3.1 Cohomology theory of small categories . . . . . . . . . . . . . . . . . 12 3.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Overcategories and a reduction theorem . . . . . . . . . . . . . . . . 19

4 EI-categories, relative projectivity, vertices and sources 28 4.1 Projective modules and simple modules . . . . . . . . . . . . . . . . . 30 4.2 Restrictions of projective modules . . . . . . . . . . . . . . . . . . . . 34 4.3 Relative projectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4 Vertices and sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.5 Structure of the vertex of the trivial module . . . . . . . . . . . . . . 63 4.6 Categories with subobjects . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Resolutions and their applications 79 5.1 The minimal projective resolution of an RC-module . . . . . . . . . . 79 5.2 Computing Ext_{RC}^{n}(S_{x,R}, S_{y,R}) . . . . . . . . . . . . . . . . . . . . . . . 83 5.3 Homological dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 Cohomology rings of finite EI-categories 89 6.1 The cohomology ring is not finitely-generated . . . . . . . . . . . . . 89 6.2 The cohomology ring modulo a nilpotent ideal . . . . . . . . . . . . . 91 6.3 The existence of a cohomology ring with non-nilpotent elements … 93

7 An example 94

Chapter 1

Introduction

In this thesis, we introduce and study the notion of a category algebra, denoted by RC, where R is the base ring and C is some small category. This is a type of associative algebra, which simultaneously generalizes several important constructions in representation theory and combinatorics, notably the path algebra of a quiver, the incidence algebra of a poset, and the group algebra of a group. Representations of small categories arise when we consider any diagram of modules, or take inverse or direct limits. We are greatly motivated by the representations of certain categories constructed from subgroups of a group, which have been intensively studied in recent years and are currently the subject of active investigation, see Broto-Levi-Oliver [6], Dwyer [10], Grodal [17], Jackowski-McClure-Oliver [24], Linckelmann [27] and Symonds [37]. As an example we mention the work in which classifying spaces of groups are approximated by homotopy colimits of certain diagrams of spaces, indexed by categories made of certain subgroups of the groups. The homology of such a homotopy colimit may be calculated using the Bousfield-Kan spectral sequence, whose E_{2} page is expressed in terms of higher limits. We mention also as a motivation the constructions of Ronan and Smith (described in Benson [5]), in which they construct irreducible modules for finite groups from diagrams of other modules.

Throughout this thesis, we assume R is a commutative ring with an identity. Let C be a small category. A representation of C over R is defined to be a (covariant) functor from C to R-mod. In other words, a representation of C over R is an object of the functor category R-mod^{C}, which is well-known to be abelian and has enough projectives and injectives. There is a result, due to Mitchell [28], saying that if C has finitely many objects one can identify the category of representations of C with the category of (left) RC-modules (Proposition 2.3.2). Because of this, any representation M : C → R-mod will be called an RC-module. There is a prominent module, the trivial module or constant functor \underline{R} ∈ RC-mod, which plays an important role in our research, sending every object of C to R and every morphism to the identity. When we study the representations of a category C, it is natural to think of the homomorphisms Hom_{RC}(M, N) between two modules M and N. The computation of the values Ext_{RC}^{∗}(M, N) of the derived functors of Hom_{RC}(M, −) at N (or equivalently the values of the derived functors of Hom_{RC}(−, N) at M) will be a focus of this thesis. When M = \underline{R}, the groups Ext_{RC}^{∗}(\underline{R}, N) can be identified with \underleftarrow{lim}_{C}^{∗} N, the higher inverse limits of N over C. The identification reveals the connections between the results in this thesis and those in the papers mentioned in the first paragraph, where computing higher limits is one of the key ingredients. The higher limits of a functor N over a category C can be thought as the Baues-Wirsching cohomology groups H^{∗}(C, N) of C, because of the well-known isomorphism H^{∗}(C, N) ≅ \underleftarrow{lim}_{C}^{∗} N. It’s worthy of noting that when N = \underline{R}, we have \underleftarrow{lim}_{C}^{∗} \underline{R} ≅ H^{∗}(C, \underline{R}) ≅ H^{∗}(|C|, R), where |C| is the classifying space of C.

The usual way of computing Ext_{RC}^{∗}(M, N) is to take a projective resolution of M, i.e. P → M → 0, and then the homology of the cochain complex 0 → Hom_{RC}(P, N) gives rise to the desired Ext groups. Similarly one can also use an injective resolution of N to obtain a cochain complex and to compute these Ext groups, but this is less used in practice under our circumstances. In general, the explicit computation of the Ext groups is very hard, and thus many researchers are looking for principles of reducing the computation of the Ext groups over C to that over some simpler category D, which is often a (full) subcategory of C. In light of this, searching for reduction formulas is also our direction on computing Ext groups in this thesis. We comment here that when C is a category associated to a finite group (e.g. an orbit category), some nice reduction formulas have been achieved by various authors, see for example the work of Grodal [17]. In that case, group action is heavily used throughout the calculations, and this is not the approach of us because we consider general small categories. In the general context, there is a result of Jackowski and Słomińska [25] (Corollary 3.2.7) which establishes an isomorphism between Ext_{RC}^{∗}(\underline{R}, M) and Ext_{RD}^{∗}(\underline{R}, Res_{μ} M) for a functor μ : D → C satisfying the conditions that the overcategories associated to μ (Definition 3.2.2) are all R-acyclic (that is, the reduced homology groups of the classifying spaces of the overcategories with coeffecients in R vanish).

We begin with the observation that Jackowski and Słomińska’s result can be established within a purely representation-theoretic setting, because their proof uses the left Kan extension of μ : D → C, which can be identified with the induction ↑_{D}^{C} = RC ⊗_{RD} − (when RC is a right RD-module) since they both are the left adjoints of the restriction Res_{μ} : RC-mod → RD-mod. From this new angle, the R-acyclicity of the overcategories associated to μ : D → C can be obtained by requiring the functor RC ⊗_{RD} − to be exact (Lemma 3.3.3) (i.e. the right RD-module RC is flat) and having an isomorphism RC ⊗_{RD} \underline{R} ≅ \underline{R}. As an effort to get the flatness of RC, one can choose suitable D and μ : D → C so that RC becomes a right projective RD-module. Sorting out the projectivity of RC as an RD-module is easier than examining the R-acyclicity of the overcategories associated to μ : D → C, especially when D is a (full) subcategory of C. There is a lot of information about RC-mod in the book of tom Dieck [9], where many results are attributed to Lück [26]. When C is a finite EI-category, Lück has classified the simple and projective modules of RC, while tom Dieck has listed equivalent conditions for RC being projective as an RD-module.

One who knows the Eckmann-Shapiro Lemma (see Benson [4]) in the cohomology theory of algebras would compare the isomorphism of Jackowski and Słomińska with the well-known isomorphism Ext_{RC}^{∗}(M ↑_{D}^{C}, N) ≅ Ext_{RD}^{∗}(M, N ↓_{D}^{C}), assuming that RC is a right flat RD-module (in representation theory, the restriction is normally written as ↓_{D}^{C} instead of Res_{μ}). Intuitively, by replacing M by \underline{R} in the Eckmann-Shapiro Lemma one gets immediately something similar to the isomorphism of Jackowski and Słomińska. Furthermore if the RC-module \underline{R} can be expressed as \underline{R} ↑_{D}^{C}, an induced module from the RD-module \underline{R}, then one has exactly the the isomorphism of Jackowski and Słomińska. As a matter of fact, we show that every indecomposable RC-module is an induced module, and as a special case we always have \underline{R} ≅ \underline{R} ↑_{D}^{C} for some subcategory D. This is a result from our theory of vertices and sources for category algebras, which tells us a way to parameterize the indecomposable RC-modules according to their vertices. From the same theory one can show that every group Ext_{RC}^{∗}(M, N) is isomorphic to Ext_{RD}^{∗}(M ↓_{D}^{C}, N ↓_{D}^{C}) for some subcategory D, if RC is a right flat RD-module.

The layout of this thesis is described as follows. In Chapter 2, we define category algebras and give some examples. In Chapter 3 we introduce the cohomology theory of small categories H^{∗}(C; M), where C is a small category and M is an RC-module. The definition is followed by a proof of the fundamental isomorphisms \underleftarrow{lim}_{C}^{∗} M ≅ H^{∗}(C; M) ≅ Ext_{RC}^{∗}(\underline{R}, M). Then we collect some interesting properties of the cohomology and representations of small categories, as well as existing isomorphisms that reduce the computation of the groups Ext_{RC}^{∗}(M, N). We spend Section 3.3 to explore the isomorphism of Jackowski and Słomińska, and then give it an algebraic interpretation as we mentioned above.

From Chapter 4, we focus on the so-called EI-categories, in which every endomorphism is an isomorphism. The most important results in this thesis are obtained when we work with EI-categories. We need to point out that many categories occurred in the representation and cohomology theory of finite groups fall into this class, and we may readily list posets, groups, the Frobenius categories and the orbit categories associated to finite groups as major examples. At the beginning of Chapter 4, we provide a complete description of the projective and simple RC-modules, which was achieved by Lück [26] (also see tom Dieck [9]). The simple RC-modules are usually denoted by S_{x,V} because each of them is determined by a simple R Aut_{C}(x)-module V for some object x ∈ Ob C. Consequently, the projective cover of such a simple is written as P_{x,V}. Note that in order to invoke the Krull-Schmidt Theorem, we normally assume the base ring R is a field or a complete local ring. With the description of indecomposable projectives, we can show when \underline{R} is projective (Proposition 4.1.5). The next step is to tell when the restriction ↓_{D}^{C} : RC-mod → RD-mod preserves (left or right) projectives, where D is a subcategory of C. The main results were obtained by tom Dieck [9], though we wrote out these results before knowing his. The core of Chapter 4 is the development of our theory of vertices and sources, which is divided into two sections. The first one gives some general results about the theory of relative projectivity for category algebras, and the second one deals with full subcategories only. Let M be an RC-module and D ⊂ C a subcategory. Then M is D-projective if M is isomorphic to a direct summand of M ↓_{D}^{C} ↑_{D}^{C}. A key fact is that if M is projective relative to a full subcategory D, then M ↓_{D}^{C} is indecomposable and M ≅ M ↓_{D}^{C} ↑_{D}^{C}. This enables us to prove that the category of RC-modules which are projective relative to a full subcategory D is equivalent to the category of RD-modules. We also prove that for an indecomposable RC-module M it’s possible to find the smallest convex subcategory relative to which M is projective. This subcategory, V_{M}, will be called the vertex of M. Centered at this fact, we develop our theory of vertices and sources, which functions in a similar way as its counterpart in group representation theory. At the end of Chapter 4, we consider two things that are of particular interest to us. Firstly we want to know more about the vertex of the trivial module \underline{R} because we always have higher limits in mind. Let D be a full subcategory of C such that \underline{R} is D-projective. We obtain a characterization of D through the overcategories associated with the inclusion. As an alternative description of D, we also show D must contain the so-called weakly essential objects of C (Corollary 4.5.5). Secondly, we realize that when C is a poset the subcategory V\underline{R} is totally determined by the weakly essential objects. Since every EI-category has an underlying poset, we want to know whether this statement remains true for other EI-categories. Since the proof for the statement for posets uses a theorem of Bouc (Proposition 4.6.6), we generalize his theorem to certain types of EI-categories before we can make further observations. The categories in which we are interested are the categories with subobjects, defined in Section 4.6. Any category with subobjects is written as (C, I), where I is a special subcategory of C controls many homological properties of C. We prove any finite category with subobjects (C, I) is an EI-category so that it makes sense to study V\underline{R}. We show when (C, I) is a finite category with subobjects (e.g. posets), then I is a poset and the full subcategory of C consisting of weakly essential objects of I is the smallest full subcategory, relative to which \underline{R} is projective (Corollary 4.6.9). Thus the vertex of \underline{R} is the “convex hull” of the full subcategory made up of all weakly essential objects of I.

In Chapter 5 we investigate some particular projective resolutions for RC-modules, which leads us to certain aspects of the representation and cohomology theory of categories, as well as some supplementary methods for computing Ext groups other than those shown in Chapter 4. These methods work very well when dealing with Ext_{RC}^{∗}(M, N) for two atomic modules. The most significant example is the calculations of Ext_{RC}^{∗}(S_{x,V} , S_{y,W}), where C can be replaced by the full subcategory consisting of objects “between” x and y. When V and W are equal to R, these Ext groups can be identified with cohomology groups of some topological object. Also in Chapter 5, we use minimal resolutions of the simples to show when a category algebra has finite global dimension. As a vehicle to study higher limits, we investigate the cohomology ring, Ext_{RC}^{∗}(\underline{R}, \underline{R}), of an EI-category C in Chapter 6. We find that the cohomology ring Ext_{RC}^{∗}(\underline{R}, \underline{R}) is not finitely generated in general. Instead we manage to prove that the cohomology ring modulo a nilpotent ideal is isomorphic to a subring of the cohomology ring of a finite group. We conjecture that this subring is finitely generated, and this is still an open question. Finally, we put an example in Chapter 7 to illustrate our main results obtained in this thesis.

This thesis is typeset by L^{A}TEX 2ε, while all the commutative diagrams are drawn by using the package XY-pic.

Category Algebras: Definitions and

Examples

Throughout this thesis, the base ring R is always a commutative ring with an identity. A module will be a finitely generated left module, if it is not otherwise specified.

2.1 Definitions

Definition 2.1.1. Let C be a category and R a commutative ring. The category algebra RC is the free R-module whose basis is the set of morphisms of C. We define a product on the basis elements of RC by

f ∗ g = \{ \begin{matrix}f ∘ g, & if f and g can be composed in C \\ 0 , & otherwise \end{matrix}

and then extend this product linearly to all elements of RC. With this product, RC becomes an associative R-algebra.

A category C is said to be finite if its morphisms form a finite set, denoted by Mor(C). Note that this implies the set of objects Ob C is finite. Also note that RC is of finite R-rank if and only if C is finite.

Remark 2.1.2. If Mor(C) is finite, then ∑_{x∈Ob C} 1_{x} is the identity of RC, where 1_{x} is the identity of Aut_{C}(x) for every x ∈ Ob C.

We say a category C is connected if C as a (directed) graph is connected. Every category C is a disjoint union of connected components C = ∪_{i∈J}C_{i}, where each C_{i} is connected and J is an index set. Consequently the category algebra RC becomes a direct sum of ideals RC_{i}, i ∈ J, and thus in order to study the properties of RC it suffices to study the properties of each RC_{i}. For simplicity we make the following assumption.

Convention In this thesis, we assume C is connected.

We shall show that a fundamental property of a category algebra RC is that it provides a mechanism for discussing representations of C, in a sense which we now define.

Definition 2.1.3. Let C be a small category. A representation of C over a commutative ring R is a (covariant) functor M : C → R-mod.

The following result was discovered by Mitchell [28].

Proposition 2.1.4. For any category C with finitely many objects, its representations can be identified with unital RC-modules.

Proof. Assume M : C → R-mod is a representation of C. We construct an R-module M̃ = ⊕_{x∈Ob C}M(x), and show it has a natural RC-module structure. For any m ∈ M(x) and any morphism f ∈ Mor(C), we define f · m = M(f)(m), if f ∈ Hom_{C}(x, x′) for some x′ ∈ Ob C. Otherwise, we define f · m = 0. By extending this operation linearly to all m ∈ M̃ and f ∈ RC we establish an RC-module structure on M̃. On the other hand, if M̃ is an RC-module, we may construct a representation M of C by defining M(x) = 1_{x} · M. We observe that if f ∈ Hom_{C}(x, x′) and m ∈ 1_{x} · M, then f · m = (1_{x′} ∘ f) · m = 1_{x′} · (f · m) ∈ 1_{x′} · M and from the axioms for M̃ to be an RC-module we see that M : C → R-mod is a functor. We may check these two constructions are inverse to each other, up to natural isomorphism. It follows that the category of representations of C and that of RC-modules, are equivalent.

Because of the above proposition, we will mostly talk about RC-modules throughout this thesis, which is very common in representation theory. There is a distinguished module that plays an important role in the representation and cohomology theory of small categories. The trivial module (or the constant functor) \underline{R} is defined by \underline{R}(x) = R for all x ∈ Ob C and \underline{R}(f) = Id for all f ∈ Mor C. We will see shortly that it is a generalization of the trivial module for group algebras to category algebras.

2.2 Examples

We provide three examples which give us the inspiration of defining category algebras. Our first example asserts that quiver algebras are category algebras.

Definition 2.2.1. A quiver 𝔮 = (Γ_{0}, Γ_{1}) is a directed graph having a set of vertices Γ_{0} and a set of arrows Γ_{1}.

The path algebra of a quiver may be defined in terms of the free category generated by the quiver.

Definition 2.2.2. (Mac Lane [29]) Any directed graph 𝔤 = (Γ_{0}, Γ_{1}) may be used to generate a category C_{𝔤} on the same set Γ_{0} of objects; the morphisms of this category will be the strings of composable arrows of 𝔤. It is called the free category generated by 𝔤.

Example 2.2.3. For the directed graph 𝔤 : x→^{α}y→^{β}z with objects {x, y, z} and arrows {α, β}, C_{𝔤} is the category with the same objects {x, y, z} and with the six morphisms {1_{x}, 1_{y}, 1_{z}, α, β, βα}. The category algebra RC_{𝔤} has R-rank 6. In fact, RC_{𝔤} is isomorphic to the algebra of 3×3 upper triangular matrices, because we can pose the following assignments: 1_{x} ↦ E_{33}, 1_{y} ↦ E_{22}, 1_{z} ↦ E_{11}, α ↦ E_{23}, β ↦ E_{12} and βα ↦ E_{13}, where E_{ij} is an elementary 3 × 3-matrix. Since these elementary matrices multiply in the same way as the basis elements of RC_{𝔤}, we see that these assignments extend linearly to an algebra homomorphism.

Definition 2.2.4. The path algebra of a quiver 𝔮 is the category algebra RC_{𝔮} of the free category C_{𝔮}.

The above definition is equivalent to the usual definition of the path algebra of a quiver. There is an extensive literature on the role of quivers in representation theory, and we mention only Auslander-Reiten-Smalø ([2] III.1) for background. The following result is a direct consequence of Definition 2.2.4.

Lemma 2.2.5. The modules of the path algebra of a finite quiver can be identified with the functors from the free category generated by this quiver to R-mod.

It is necessary to point out that, given a category, its category algebra is usually different from its path algebra. Nevertheless, there is a relationship between these two algebras for a category.

Proposition 2.2.6. Let 𝔮 be a category. We may regard 𝔮 as a quiver and form the free category C_{𝔮} over 𝔮. There is a natural functor ϕ : C_{𝔮} → 𝔮, which extends to a surjective homomorphism, still denoted by ϕ : RC_{𝔮} → R𝔮, from the path algebra of 𝔮 to the category algebra of 𝔮, such that its kernel I is generated by {←^{α_{1}}←^{α_{2}} − ←^{α_{1}α_{2}}}, where α_{1} and α_{2} are arrows of 𝔮. This epimorphism induces a natural isomorphism of R-algebras RC_{𝔮}/I and R𝔮.

Proof. The functor ϕ is defined as follows. For each x ∈ Ob C_{𝔮}, we let ϕ(x) = x. For each α ∈ Mor C_{𝔮}, we let ϕ(α) = the composite of the maps in the string α. The functor can be extended linearly to an epimorphism Φ : RC_{𝔮} → R𝔮 with a kernel equal to I so the statement follows.

Our second example says the incidence algebra of a poset (partially ordered set) is a category algebra.

Definition 2.2.7. Let Γ be a finite poset and R a commutative ring. The incidence algebra of Γ, consists of all incidence functions

A(Γ) = {f | Γ × Γ → R with f(x, y) = 0 if x ≰ y}

by taking point-wise summation and scalar multiplication of such functions as sums and scalar product

(f + g)(x, y) = f(x, y) + g(x, y)
(αf)(x, y) = αf(x, y), α ∈ R

and by defining the product h = f ∗ g as

(f ∗ g)(x, y) = ∑_{x≤z≤y} f(x, z)g(z, y)

Note that the Kronecker delta function δ(x, y) is the two-sided identity of A(Γ).

Let Γ be a poset. We can define a category, still named Γ, whose objects are the vertices of the poset. For any two objects x and y in Γ, there exists a (unique) morphism in Hom_{Γ}(x, y) if and only if x ≤ y. We show the incidence algebra A(Γ) of Γ is naturally isomorphic to its category algebra RΓ by defining an incidence function for every morphism in Γ as follows. If α ∈ Hom_{Γ}(x, y) is the unique morphism with domain x and co-domain y, then we define an incidence function f_{α}(∗, ∗) ∈ A(Γ) by requiring f_{α}(x, y) = 1 and f_{α}(∗, ∗) = 0 otherwise.

Proposition 2.2.8. The incidence functions f_{α}, α ∈ Mor(Γ), form a basis of the incidence algebra A(Γ) of Γ. There exists a natural isomorphism between the incidence algebra A(Γ) and the category algebra RΓ.

Proof. Let f ∈ A(Γ). We can define a map sending f to ∑_{α:x→y} f(x, y)α. This is an isomorphism.

The last example shows that the notion of a category algebra also generalizes the notion of a group algebra.

Example 2.2.9. Let G be a (discrete) group and R a commutative ring. The group algebra RG is additively a free R-module with a basis the elements of G. Let Ĝ be the category with only one object ∗ whose morphisms are the elements of G. We can see the group algebra of G is a category algebra, i.e. RG = RĜ. By definition a left RGmodule M is an R-module equipped with a group homomorphism Φ : G → Aut_{R}(M). This can be interpreted as a representation of the category Ĝ, which sends ∗ to M ∈ Rmod and AutĜ(∗) = G into Aut_{R}(M).

It’s easy to see that \underline{R} is exactly the trivial module of the group algebra RG = RĜ.

Basic Homological Properties

In this chapter we introduce the cohomology theory of small categories which can be used to define and calculate the higher (inverse) limits of an RC-module M over C. After giving the definition, we will establish the connection between the cohomology theory of small categories with that of category algebras. Then we will provide formulas for computing these cohomology groups.

3.1 Cohomology theory of small categories

The cohomology theory of small categories has been discussed in various places in the literature, and readers are referred to Baues-Wirsching [3], Generalov [15] and Oliver [31] for more details. One can also find in Gabriel-Zisman [14] and Hilton-Stammbach [21] the homology theory of small categories.

Definition 3.1.1. Let C be a small category and R a commutative ring. We define the n-th cohomology group H^{n}(C, M) of C with coefficients in a covariant functor M : C → R-mod to be the n-th homology group of the following cochain complex {C^{∗}(C, M), δ}, where

C^{n}(C, M) = {f | f : N_{n}(C) → ∏_{[x_{0}→x_{1}→···→x_{n}]} M(x_{n}), and f([x_{0} → ⋯ → x_{n}]) ∈ M(x_{n})}

for all n ≥ 0, with N(C) = {N_{n}(C)} the nerve of C, and where for any f ∈ C^{n}(C, M),

δ(f)([x_{0} → ⋯ → x_{n}→^{ϕ}x_{n+1}]) = ∑_{i=0}^{n}(−1)^{i}f([x_{0} → ⋯ → x̂_{i} → ⋯ → x_{n+1}])

+(−1)^{n+1}M(ϕ)(f([x_{0} → ⋯ → x_{n}])).

The next result was originally proved by Gabriel-Zisman [14] and J.-E. Roos [36]. Our proof for it is taken from Oliver [31].

Proposition 3.1.2. Let C be a small category and R a commutative ring. If M : C → R-mod is a functor, then

\underleftarrow{lim}_{C}^{i}M ≅ Ext_{RC}^{i}(\underline{R}, M) ≅ H^{i}(C, M),

for any i ≥ 0.

Proof. The first identities can be proved by using the definition of the inverse limit of a functor M (see Mac Lane [29]). Let H = Hom_{RC}(\underline{R}, M) and ΔH be the diagonal functor from C to R-mod such that ΔH(x) = H for all x ∈ Ob C. In order to show H is the inverse limit of M, we have to construct a natural transformation μ : ΔH → M and show it’s universal. Firstly, we define μ_{x} for any x ∈ Ob C by μ_{x}(f) = f_{x}(1) for all f ∈ Hom_{RC}(\underline{R}, M) = H = ΔH(x). It’s easy to check that for any σ ∈ Hom_{C}(x, y), μ_{y} = M_{σ} ∘ μ_{x}. Hence μ is a natural transformation. Secondly, for any other H′ equipped with a natural transformation μ′ : ΔH′ → M, we need to show there exits a unique group homomorphism τ_{H′} : H′ → H which is compatible with the natural transformations μ and μ′. For each h′ ∈ H′, we define τ_{H′}(h′) = g^{h′} where g^{h′} ∈ H = Hom_{RC}(\underline{R}, M) is defined by g^{h′}(1_{x}) = μ′_{x}(h′) for all x ∈ Ob C. We can verify that τ_{H′} is well-defined and that it is the unique homomorphism making the following diagram commutative.

[figure: H′; μ′_{y}; μ′_{x}; τ_{H′}; H; μ_{x}; μ_{y}; M(x); M_{σ}; M(y)]

Thus we proved \underleftarrow{lim}_{C} M ≅ Hom_{RC}(\underline{R}, M). Since the kernel of δ : C^{0}(C, M) → C^{1}(C, M) can be identified with Hom_{RC}(\underline{R}, M), we get H^{0}(C, M) ≅ Hom_{RC}(\underline{R}, M).

Now let’s prove the isomorphisms for higher groups. Since \underleftarrow{lim}_{C} − ≅ Hom_{RC}(\underline{R}, −) is left exact and \underleftarrow{lim}_{C}^{i} − is its i-th right derived functor, we have \underleftarrow{lim}_{C}^{i} M ≅ Ext_{RC}^{i}(\underline{R}, M) and so it remains to show Ext_{RC}^{i}(\underline{R}, M) ≅ H^{i}(C, M). Because Ext_{RC}^{i}(\underline{R}, M) can be computed by using any projective resolution of \underline{R}, we can construct a particular projective resolution named (P = {P_{n}}_{n=−1}^{∞}, σ) to calculate it. For each n ≥ −1, we define a functor P_{n} : C → R-mod as follows. For any x ∈ Ob C, let P_{n}(x) be the free abelian group with a basis the set of all sequences [x_{0} → x_{1} → ⋯ → x_{n} → x] of morphisms in Mor(C) ending in x. For any morphism f ∈ Hom_{C}(x, y), P_{n}(f) is defined by its action on base elements: [x_{0} → ⋯ → x_{n}→^{ϕ}x] ↦ [x_{0} → ⋯ → x_{n}→^{fϕ}y]. Next we define the boundary map σ = {σ_{x}} : P_{n} → P_{n−1} by setting

σ_{x}([x_{0} → x_{1} → ⋯ → x_{n} → x]) = ∑_{i=0}^{n}(−1)^{i}[x_{0} → ⋯ → x̂_{i} → ⋯ → x_{n} → x],

on base elements. It’s easy to check that, for each x, the chain complex

⋯ → P_{2}(x) → P_{1}(x) → P_{0}(x) → P_{−1}(x) ≅ \underline{R} → 0

is split by the maps [⋯ → x_{n} → x] → [⋯ → x_{n} → x→^{Id}x], and hence is exact. In order to complete our proof, we have to show that (P, σ) is a projective resolution of \underline{R}, and that \{Hom_{RC}(P_{n}, M), σ^{∗}\} is isomorphic to the complex \{C^{n}(C, M), δ\} for any functor M. In fact we shall establish the following isomorphisms of complexes

{Hom_{RC}(P_{n}, M)} ≅ {C^{n}(C, M)} ≅ { ∏_{x_{0}→···→x_{n}} M(x_{n})},

and then we can see Hom_{RC}(P_{n}, −) is exact and hence P_{n} is a projective RC-module. Since the second isomorphism is natural, we only prove the first here. For any natural transformation θ ∈ Hom_{RC}(P_{n}, M), we define a f ∈ C^{n}(C, M) by f([x_{0} → ⋯ → x_{n}]) = θ_{x_{n}}([x_{0} → ⋯ → x_{n}→^{Id}x_{n}]). On the other hand, for any f ∈ C^{n}(C, M), we can construct a θ ∈ Hom_{RC}(P_{n}, M), such that θ_{x}([x_{0} → ⋯ → x_{n}→^{ϕ}x]) = M(ϕ)f([x_{0} → ⋯ → x_{n}]). One can easily check that these two constructions are mutually inverse, and commute with the differentials.

The resolution {P, σ} = {P_{n}, σ_{n}}_{n≥0} appeared in the proof is called the bar resolution of \underline{R}.

In general, if P is an arbitrary projective resolution of \underline{R}, then P(x) is always an exact sequence of R Aut(x)-modules beginning with R for any object x ∈ Ob C, although the modules in P(x) do not have to be projective. These modules will be projective if R Aut_{C}(x) is semi-simple, or if Aut_{C}(x) acts freely on Hom_{C}(y, x) for each y ∈ Ob C with Hom(y, x) ≠ ∅ because then each R Hom_{C}(y, x) is a free R Aut_{C}(x)- module. For example if P is the bar resolution of \underline{R} then P(x) becomes a projective resolution of the R Aut_{C}(x)-module R for any x ∈ Ob C. One might think that when this happens Ext_{RC}^{∗}(\underline{R}, M) would be related to Ext_{RC}^{∗}(R, M(x)), at least for certain modules with simple structures. But it is not true as we can see from our first example on cohomology of categories.

Example 3.1.3. Let R = 𝔽_{2} be a field of characteristic 2 and C the following category

[figure: x; 1_{x}; g; α; β; y; 1_{y}; h] ,

where g^{2} = 1_{x}, αg = β, βg = α, h^{2} = 1_{y}, hα = α and hβ = β. Suppose M is an atomic functor concentrated on y such that M(y) = 𝔽_{2}. Then if we use the bar resolution P → \underline{𝔽_{2}} → 0 we can get H^{n}(C; M) ≅ \underleftarrow{lim}_{C}^{n} M = 0 for all n ≥ 0. On the other hand, the bar resolution P → \underline{𝔽_{2}} → 0 evaluated at y becomes a projective resolution of 𝔽_{2} Aut(y)-module R, i.e. P(y) → \underline{𝔽_{2}}(y) = 𝔽_{2} → 0, which gives H^{n}(Aut_{C}(y), 𝔽_{2}) = 𝔽_{2} for all n ≥ 0. We omit the tedious calculations here because after we know more about the projective modules we can compute these groups in a quite neat way, see Example 4.6.11 at the end of Chapter 4 which discusses the same category.

The functor M appeared in the above example is a so-called atomic functor.

Definition 3.1.4. A functor M : C → R-mod is called atomic, concentrated on an isomorphism class of objects [x] ⊂ Ob C if M(y)(= 1_{y} · M) ≠ 0 if and only if y ≅ x.

For simplicity, we will just say M is concentrated on x, instead of [x]. We also call an RC-module M atomic if the corresponding functor is. The atomic functors are important to us because every functor admits a certain filtration by its subfunctors with the property that every subquotient is an atomic functor. Thus using exact sequences or even spectral sequences consisting of higher limits of subquotients from such a filtration of a functor, one may find the higher limits of the functor. This method for computing higher limits was introduced and studied by Oliver [31], and is frequently used by others, see for instance Grodal [17].

For any M ∈ RC-mod, Ext_{RC}^{∗}(M, M) has a ring structure with the product given by the Yoneda splice, see Benson [4]. When M = \underline{R} the ring is of great importance to us. As a vehicle for studying higher limits, we make the following definition.

Definition 3.1.5. We call the ring Ext_{RC}^{∗}(\underline{R}, \underline{R}) = ⊕_{i≥0}Ext_{RC}^{i}(\underline{R}, \underline{R}) the cohomology ring of the category algebra RC. The product in this ring is defined by the Yoneda splice of Ext classes.

3.2 Basic properties

We prove some fundamental and important homological properties of category algebras in this section.

Definition 3.2.1. Suppose μ : C′ → C is a (covariant) functor. We define Res_{μ} : (Rmod)^{C} → (R-mod)^{C′} to be the restriction along μ. Given a functor M ∈ (R-mod)^{C}, we have Res_{μ} M = M · μ ∈ (R-mod)^{C′}.

Proposition 3.2.2. Let C and C′ be equivalent small categories. Then

1. RC and RC′ are Morita equivalent, under an equivalence which sends the constant functor to the constant functor; and

2. the nerves NC and NC′ are homotopy equivalent.

Proof. We prove the first assertion, and the proof for the second can be found in Baues and Wirsching [3].

We want to show the two functor categories (R-mod)^{C} and (R-mod)^{C′} are equivalent, because it implies the module categories RC-mod and RC′-mod are equivalent and hence RC and RC′ are Morita equivalent. In fact if μ : C′ → C and ν : C → C′ are equivalences, we have Res_{μ} Res_{ν} ≅ Id_{RC′} : (R-mod)^{C′} → (R-mod)^{C′} because of the following diagram

\xymatrix{M(νμ(x)) = (Res_{μ} Res_{ν} M)(x) \ar[r]^{≅} \ar[d]_{M(νμ(α))=(Res_{μ} Res_{ν} M)(α)} & (Id_{C′} M)(x) = M(x) \ar[d]^{(Id_{C′} M)(α)=M(α)} \\ M(νμ(y)) = (Res_{μ} Res_{ν} M)(y) \ar[r]_{≅} & (Id_{C′} M)(y) = M(y)}

where M ∈ (R-mod)^{C′}, α : x → y ∈ Mor C′ and Id_{RC′} is the identity functor. Similarly we can show Res_{ν} Res_{μ} ≅ Id_{RC} : (R-mod)^{C} → (R-mod)^{C} so we know Res_{μ} and Res_{ν} provide the equivalences between the functor categories. Clearly the constant functor restricts to the constant functor always.

It’s well-known that if a functor μ : C′ → C has a left adjoint ν : C → C′, then Res_{ν} is also the left adjoint of Res_{μ}. Under the circumstance, one can show Ext_{RC}^{∗}(Res_{ν} M, N) ≅ Ext_{RC′}^{∗}(M, Res_{μ} N) for any M ∈ RC′-mod and N ∈ RC-mod, because both Res_{μ} and Res_{ν} are exact (see for example Jackowski-McClure-Oliver[24] II, Proposition 5.1). If μ is indeed an equivalence, we have a stronger result.

Corollary 3.2.3. Let μ : C′ → C be an equivalence of two small categories. Then we have

Ext_{RC}^{∗}(M, N) ≅ Ext_{RC′}^{∗}(Res_{μ}M, Res_{μ}N),

for M, N ∈ RC-mod. In particular there is a ring isomorphism

Ext_{RC}^{∗}(\underline{R}, \underline{R}) ≅ Ext_{RC′}^{∗}(\underline{R}, \underline{R}).

Proof. The thing is, RC and RC′ are Morita equivalent by the functor Res_{μ}, which takes \underline{R} to \underline{R}.

In [3] Baues and Wirsching showed a special case of the above result. They proved Ext_{RC}^{∗}(\underline{R}, N) ≅ Ext_{RC′}^{∗}(\underline{R}, Res_{μ} N) by establishing a homotopy equivalence between the cochain complexes used to define and calculate H^{∗}(C, N) and H^{∗}(C′, Res_{μ} N) (see Proposition 3.1.2). We will discuss a result of Jackowski and Słomińska [25] in the next section, in which they use some “weaker” equivalence between C and C′ to achieve the reduction Ext_{RC}^{∗}(\underline{R}, N) ≅ Ext_{RC′}^{∗}(\underline{R}, Res_{μ} N). Their result will be generalized in Chapter 4 as an application of our theory of vertices and sources.

Corollary 3.2.4. Given a small category C, we have Ext_{RC}^{i}(\underline{R}, \underline{R}) ≅ H^{i}(|C|, R). Here |C| stands for the classifying space of C. Following this result we can also show Ext_{RC}^{∗}(\underline{R}, \underline{R}) ≅ Ext_{RC′}^{∗}(\underline{R}, \underline{R}) as algebras, provided C′, C are equivalent.

Proof. By Proposition 3.1.2, we just need to compare C^{n}(C, \underline{R}) with Hom_{R}(C_{n}(|C|), R), where C_{n}(|C|) is the group of n-chains of |C|. By Hom_{R}(C_{n}(|C|), R) ≅ Hom_{R}(N_{n}(C), R) and Hom_{R}(N_{n}(C), R) ≅ C^{n}(C, \underline{R}), we know C^{n}(C, \underline{R}) and Hom_{R}(C_{n}(|C|), R) are isomorphic. It’s easy to see that the two differentials can be identified with each other. Therefore we have the isomorphism of these cohomology groups.

We may proceed either topologically or algebraically to prove the second part. When C and C′ are equivalent we know that |C| and |C′| are homotopy equivalent by the maps induced by functors between the two categories. So H^{i}(C, \underline{R}) ≅ H^{i}(C′, \underline{R}). Alternatively, because RC and RC′ are Morita equivalent under an equivalence which preserves the constant functor, the Ext algebras are isomorphic.

By Proposition 2.3.2 given a category with Ob C finite, we can identify the functor category (R-mod)^{C} with the module category RC-mod. Thus given a functor μ : C′ → C, the naturally induced functor Res_{μ} : (R-mod)^{C} → (R-mod)^{C′}, can be considered as Res_{μ} : RC-mod → RC′-mod. On the other hand, since μ : C′ → C extends linearly to a natural map of R-modules μ̄ : RC′ → RC, it’s reasonable to ask if μ̄ is an algebraic homomorphism and if it is, whether or not its induced functor, the so-called change-of-ring or restriction, ↓_{RC′}^{RC}: RC-mod → RC′-mod, coincides with Res_{μ}. The answer to the first question is no, and here is a simple example. Let C′ be a category with two objects and only identity maps, and let C be a category with one object and the identity map along with the unique functor μ : C′ → C. Then the map μ̄ : RC′ → RC is not an algebra homomorphism for the product of the two morphisms in C′ is zero while the product of their images is not. We show when μ will be an algebra homomorphism.

Proposition 3.2.5. A functor μ : C′ → C extends linearly to an algebra homomorphism μ̄ : RC′ → RC if and only if μ is injective on Ob C′. When this happens, the induced functor followed by 1_{C′}, 1_{C′}· ↓_{RC′}^{RC}: RC-mod → RC′-mod is exactly Res_{μ}.

Proof. We know μ(βα) = μ(β)μ(α) for any pair of composable morphisms α, β in C′. The injectivity of μ implies two morphisms α, β ∈ Mor(C′) are composable if and only if μ(α), μ(β) ∈ Mor(C) are composable.

If μ is injective on Ob C′, then we define a map μ̄ : RC′ → RC as the linear extension of functor μ, i.e., μ̄(∑_{i} r_{i}α_{i}) = ∑_{i} r_{i}μ̄(α_{i}) for any r_{i} ∈ R, α_{i} ∈ Mor(C′). This μ̄ is indeed an algebra homomorphism because our previous observation of μ implies μ̄((∑_{j} r_{j}β_{j})(∑_{i} r_{i}α_{i})) = μ̄(∑_{j} r_{j}β_{j})μ̄(∑_{i} r_{i}α_{i}) is always true.

On the other hand if the linear extension μ̄ : RC′ → RC is an algebra homomorphism then we must have μ̄(0) = 0 and then μ̄(1_{x})μ̄(1_{y}) = μ̄(1_{x} ·1_{y}) = 0 unless x = y. This suggests that μ is injective on Ob C′.

When μ̄ : RC′ → RC is an algebra homomorphism, we show 1_{C′}· ↓_{RC′}^{RC}= Res_{μ} : RCmod → RC′-mod. Let M be an RC-module and γ ∈ Mor(C′). Then M = M_{M} for some M ∈ (R-mod)^{C}. The RC′-module structures of M = M_{M} given by Res_{μ} and 1_{C′}· ↓_{RC′}^{RC} coincide for Res_{μ} M_{M} = M_{Res_{μ} M}, γ · M_{M} = μ(γ)M_{M} and 1_{C′} kills the elements of M ↓_{RC′}^{RC} that aren’t supported on any objects of C′.

In Chapter 4, we will take C′ as a full subcategory of C. Then the restriction Res_{ι} : RC-mod → RC′-mod is determined by the algebra homomorphism ῑ: RC′ → RC, hence by ι : C′ → C. For this reason we won’t distinguish Res_{ι} and ↓_{RC′}^{RC} from now on, and will write ↓_{RC′}^{RC} and Res_{ι} as ↓_{C′}^{C} which is common in representation theory.

There is another cohomology theory we can apply to category algebras, namely the Hochschild cohomology. In general this is not the same as H^{i}(C, \underline{R}) and examples where the two cohomology groups are different can be found when C is a group (Benson [5], Theorem 2.11.2). However, when C is a poset these cohomology theories do coincide, and this is a theorem of Gerstenhaber and Schack [16]. One can find a description of the Hochschild cohomology theory in [16], as well as in Mac Lane [30] (X.3) and Happel [19].

Theorem 3.2.6. [16] Let Γ be a poset and R a commutative ring with identity. Let |Γ| be the topological realization, i.e., the order complex, of Γ. Then H^{i}(|Γ|, R) ≅ HH^{i}(RΓ, RΓ), where RΓ is the incidence algebra.

If Γ, Γ′ are equivalent as categories, then from a standard result in the Hochschild cohomology theory we have HH^{∗}(RΓ, RΓ) ≅ HH^{∗}(RΓ′, RΓ′) since RΓ and RΓ′ are Morita equivalent. Hence H^{∗}(|Γ|, R) ≅ H^{∗}(|Γ′|, R) which matches the statements of Corollaries 3.2.3 and 3.2.5.

3.3 Overcategories and a reduction theorem

We have seen in last section that a certain pair of adjoint functors μ, ν between D and C leads to a reduction Ext_{RC}^{∗}(\underline{R}, N) ≅ Ext_{RD}^{∗}(\underline{R}, Res_{μ} N) because of the existence of another pair of (induced) adjoint functors Res_{μ}, Res_{ν} between the corresponding functor categories. The truth is, a single functor μ : D → C is enough to induce a pair of adjoint functors on the level of functor categories. This time the left adjoint of Res_{μ} is called the left Kan extension of μ, and is not necessarily induced by any functor from C to D. Whenever there exists a pair of adjoint functors on functor categories, one may find suitable conditions on the triple ι : D → C so that one can establish isomorphisms between the cohomology groups Ext_{RC}^{∗}(\underline{R}, N) and Ext_{RD}^{∗}(\underline{R}, Res_{ι} N) (or something similar). This consideration leads us to an important reduction theorem, Corollary 3.2.3, which is due to Jackowski and Słomińska [25].

First of all, let’s review the definition of the left Kan extension, which uses the so-called overcategories ι ↓_{y}, y ∈ Ob C.

Definition 3.3.1. Let C and D be small categories along with a functor ι : D → C. For each y ∈ Ob C, the overcategory ι ↓_{y} consists of objects (x, α), where x ∈ Ob D and α ∈ Hom_{C}(ι(x), y). The morphisms in the overcategory are determined via completing commutative diagrams as follows

\xymatrix{ι(x) \ar[dr]^{α} \\  & y, \\ ι(x′) \ar[uu]^{ι(β)} \ar[ur]_{α′}}

where (x, α), (x′, α′) ∈ Ob ι ↓_{y} and β ∈ Hom_{D}(x, x′).

When C is a poset and D is a subposet along with the inclusion ι : D ↪ C, the overcategory ι ↓_{y} is isomorphic in a natural way to D_{≤y}, a subposet of D consisting of all x ∈ Ob D which satisfy the condition that Hom_{C}(x, y) ≠ ∅.

It’s easy to see that for a fixed functor ι : D → C between two small categories, we can naturally define two functors ι ↓_{?}: C → sCat, the category of small categories, and |ι ↓_{?} | : C → Spaces, the category of topological spaces. For any y ∈ Ob C, we use C_{∗}(ι ↓_{y}) to denote the simplicial complex coming from the simplicial set associated to the small category ι ↓_{y}, which can be regarded as the chain complex of the space |ι ↓_{y} |.

Definition 3.3.2. The left Kan extension of a functor ι : D → C is defined by

K(M)(y) = \underrightarrow{lim}_{ι↓_{y}}M ∘ π,

where M ∈ RD-mod, y ∈ Ob C and π : ι ↓_{y}→ D is the projection (x, α) ↦ x.

Since K is the left adjoint of the exact functor Res_{ι}, it preserves projective modules. It is not exact in general, but when it is it sends any projective RD-resolution to a projective RC-resolution.

In Proposition 3.1.2, we have seen an explicit way to construct, for an arbitrary small category D, the bar resolution P of \underline{R} = P_{−1}. In fact it’s not hard to see that the exact sequence P → \underline{R} → 0 evaluated at each x ∈ Ob D, i.e. P(x) → R → 0, is isomorphic to the reduced simplicial chain complex of the classifying space | Id_{D} ↓_{x} |, where Id_{D} ↓_{x} is constructed via the identity functor Id_{D} : D → D, see Gabriel-Zisman [14] and Grodal [17]. Since every Id_{D} ↓_{x} has a terminal object (x, 1_{x}), by Quillen’s Theorem A [33], the space | Id_{D} ↓_{x} | is contractible and hence we get another proof of the exactness of the bar resolution P → \underline{R} → 0. It was observed by Dwyer and Kan [11] that K(P) ≅ K(C_{∗}(Id_{D} ↓_{?})) → K(\underline{R}) → 0 is isomorphic to the complex of RCmodules C_{∗}(ι ↓_{?}) → K(\underline{R}) → 0, in which every C_{∗}(ι ↓_{?}) ≅ K(C_{∗}(Id_{D} ↓_{?})) remains projective. When K(\underline{R}) happens to be isomorphic to \underline{R} the complex evaluated at any y ∈ Ob C becomes the reduced simplicial complex C_{∗}(ι ↓_{y}) → R → 0 of |ι ↓_{y} |. It was shown by Jackowski and Słomińska [25] that this complex of RC-modules C_{∗}(ι ↓_{?}) → K(\underline{R}) → 0 will become a projective resolution of the trivial RC-module \underline{R} if we assume that all overcategories associated to ι : D → C are R-acyclic (a small category is said to be R-acyclic if all reduced homology groups of its classifying space with coefficients in R vanish).

When we turn to our representation-theoretic settings, the left adjoint of the restriction Res_{ι} : RC-mod → RD-mod is normally written as the induction ↑_{D}^{C} = RC ⊗_{RD} −, provided RC is a right RD-module. By Proposition 3.2.6, RC becomes an RD-module if ι : D → C is injective on the object set. When this happens we must have K ≅ ↑_{D}^{C} as two functors, and particularly must have K(P_{D}) ≅ P ↑_{D}^{C} = RC⊗_{RD}P. Obviously the complex K(P) will be exact if RC is a flat RD-module. Let’s put these important observations into the following lemma. The first statement in the lemma was known to Dwyer and Kan [11], and the second is new, though a special form of it has appeared in Symonds [37]. The conclusion is essentially obtained by Jackowski and Słomińska [25].

Lemma 3.3.3. Let ι : D → C be a functor and P be the bar resolution of the RDmodule \underline{R}. Then

1. there are isomorphisms of complexes of projective RC-modules

K(P) ≅ K(C_{∗}(Id_{D} ↓_{?})) ≅ C_{∗}(ι ↓_{?}).

For each y ∈ Ob C, K(P)(y) is isomorphic to the simplicial chain complex of ι ↓_{y}. If furthermore ι : D → C is injective on objects, then the above three complexes are isomorphic to RC ⊗_{RD} P; and

2. if every ι ↓_{y} is non-empty and connected, then K(\underline{R}) ≅ \underline{R}.

Thus if for every y ∈ Ob C the overcategory ι ↓_{y} is R-acyclic, we obtain a projective resolution

K(C_{∗}(Id_{D} ↓_{?})) ≅ C_{∗}(ι ↓_{?}) → K(\underline{R}) ≅ \underline{R} → 0

of the trivial RC-module \underline{R}, which, evaluated at y, is isomorphic to the reduced simplicial chain complex of |ι ↓_{y} |.

Proof. Part of the proof to (1) is more or less explained in the two paragraphs after Definition 3.3.2. In order to show K(P) ≅ K(C_{∗}(Id_{D} ↓_{?})), we just need to prove P ≅ C_{∗}(Id_{D} ↓_{?}), which is clear if we write out these two complexes and their evaluations at each object of C.

Let y be an object of C. We can show for each n ≥ 0 that C_{n}(ι ↓_{y}) is isomorphic to \underrightarrow{lim}_{ι↓_{y}} C_{n}(Id_{D} ↓_{?}) ∘ π, by the universal property of \underrightarrow{lim}. Let (w, α) ∈ Ob Id_{D} ↓_{y}, (x, β) ∈ Ob Id_{D} ↓_{y} and γ ∈ Hom((w, α), (x, β)), i.e., α = βι(γ). Then we have the following commutative diagram

[figure: C_{n}(Id_{D} ↓_{w}) = C_{n}(D ↓_{?}) ∘ π(w, α); γ̄; ᾱ; μ_{w}; C_{n}(Id_{D} ↓_{x}) = C_{n}(Id_{D} ↓_{?}) ∘ π(x, β); β̄; μ_{x}; C_{n}(ι ↓_{y}); τ_{M}; M]

where ᾱ, β̄ and γ̄ denote the composite by α, β and γ, respectively. Let M be another R-module that fits into the commutative diagram. We want to construct a map τ_{M} from C_{n}(ι ↓_{y}) to M so that we can say C_{n}(ι ↓_{y}) is isomorphic to \underrightarrow{lim}_{ι↓_{y}} C_{n}(Id_{D} ↓_{?}) ∘ π. In fact, if for any n-chain in C_{n}(ι ↓_{y}) ending with (w, α), say

(w_{0}, α_{0})→^{γ_{0}} ⋯ → (w_{n−1}, α_{n−1}) →^{γ_{n−1}}(w, α), · · · · · · · · · (∗)

we can find a preimage c of it in some C_{n}(Id_{D} ↓_{z}), z ∈ Ob D, then we define a map τ_{M} from C_{n}(ι ↓_{y}) to M by τ_{M} = μ_{z}(c). If we can show this τ_{M} is well-defined and unique, then by the universal property of \underrightarrow{lim}, we get an isomorphism

C_{n}(ι ↓_{y}) ≅ \underrightarrow{lim}_{Id_{D}↓_{y}}C_{n}(Id_{D} ↓_{?}) ∘ π.

Now we verify the validity of τ_{M}. First of all, there exists a preimage of such an element (w_{0}, α_{0})→^{γ_{0}} ⋯ → (w_{n−1}, α_{n−1}) →^{γ_{n−1}}(w, α), because we can always construct from it an element of C_{n}(Id_{D} ↓_{w})

c = (w_{0}, α_{0}/α)→^{γ_{0}} ⋯ → (w_{n−1}, α_{n−1}/α) →^{γ_{n−1}}(w, 1_{w}),

where α_{i}/α denotes γ_{n−1} ⋯ γ_{i} since α_{i} = αι(γ_{n−1}) ⋯ ι(γ_{i}). It’s easy to verify this is a preimage of (∗). Second of all, we shall prove the definition of τ_{M} is independent of the choice of the preimage of (∗). The thing is, any preimage of (w_{0}, α_{0})→^{γ_{0}} ⋯ → (w_{n−1}, α_{n−1}) →^{γ_{n−1}}(w, α) must lie in some C_{n}(Id_{D} ↓_{z}) and must be in the image of some

map from C_{n}(Id_{D} ↓_{w}) to C_{n}(Id_{D} ↓_{z}). Suppose c′ = (v_{0}, θ_{0})→^{σ_{0}} ⋯ → (v_{n−1}, θ_{n−1}) →^{σ_{n−1}}(v, θ)

is a preimage of the given (∗), where θ_{i}, θ ∈ Hom_{D}(v, z). Then there exists some η ∈ Hom_{C}(ι(z), y) inducing a map η̄ : C_{n}(Id_{D} ↓_{z}) → C_{n}(ι ↓_{y}), which sends c′ to (∗). Thus from

η̄[(v_{0}, θ_{0})→^{σ_{0}} ⋯ → (v_{n−1}, θ_{n−1}) →^{σ_{n−1}}(v, θ)]

= (v_{0}, ηι(θ_{0}))→^{σ_{0}} ⋯ → (v_{n−1}, ηι(θ_{n−1})) →^{σ_{n−1}}(v, ηι(θ))
= (w_{0}, α_{0})→^{γ_{0}} ⋯ → (w_{n−1}, α_{n−1}) →^{γ_{n−1}}(w, α),

it must be true that v = w, ηι(θ) = α and v_{i} = w_{i}, γ_{i} = σ_{i}, ηι(θ_{i}) = α_{i} for all i. Since θ_{i} = θσ_{n−1} ⋯ σ_{i} and α_{i} = αι(γ_{n−1}) ⋯ ι(γ_{i}), these imply θ(c) = c′ and μ_{z}(c′) = μ_{z}(θ̄(c)) = μ_{w}(c) in the commutative diagram, where

c = (w_{0}, α_{0}/α)→^{γ_{0}} ⋯ → (w_{n−1}, α_{n−1}/α) →^{γ_{n−1}}(w, 1_{w}).

Note that α_{i}/α = γ_{n−1} ⋯ γ_{i}. Hence the map τ_{M} is well defined. Using once again the universal property and similar arguments as above we can extend isomorphism of the form \underrightarrow{lim}_{Id_{D}↓_{y}} C_{n}(Id_{D} ↓_{?}) ∘ π = K(C_{n}(Id_{D} ↓_{?}))(y) ≅ C_{n}(ι ↓_{y}) to an isomorphism of functors K(C_{n}(Id_{D} ↓_{?})) ≅ C_{n}(ι ↓_{?}).

In the end, let ι : D → C be injective on Ob D. Then RC ⊗_{RD} − is well-defined and hence K ≅ RC ⊗_{RD} −. So we have all isomorphisms of (1).

Statement (2) follows directly from the definition of Kan extension.

Note that if all the overcategories ι ↓_{y}, y ∈ Ob C, are contractible, then |D| ≃ |C| by Quillen’s Theorem A. As a consequence, we get H^{∗}(C, \underline{R}) ≅ H^{∗}(D, \underline{R}). This is a special case of the following reduction theorem on higher limits, which also generalizes Corollary 3.2.3.

Corollary 3.3.4. (Jackowski-Słomińska [25]) Let ι : D → C satisfy the condition that every ι ↓_{y}, y ∈ Ob C, is R-acyclic. Then \underleftarrow{lim}_{C}^{∗} M ≅ \underleftarrow{lim}_{D}^{∗} Res_{ι} M for any RC-module M. In particular, when RC is a right projective RD-module and \underline{R} ↑_{D}^{C} ≅ \underline{R} these isomorphisms exist.

Proof. We take the bar resolution P of \underline{R}. By the previous lemma, its left Kan extension K(P) is a projective resolution of the RC-module \underline{R}. So we have for an arbitrary RC-module M

Hom_{RC}(K(P), M) ≅ Hom_{RD}(P, Res_{ι}M),

by the adjunction of these two functors K and Res_{ι}. Since the cochain complex \{Hom_{RC}(K(P), M)\} computes \underleftarrow{lim}_{C}^{∗} M and the other cochain complex \{Hom_{D}(P, M)\} calculates \underleftarrow{lim}_{D}^{∗} Res_{ι} M, the isomorphisms between the higher limits over C and D are established.

Since the isomorphism in Corollary 3.3.4 can be written in terms of Ext groups Ext_{RC}^{∗}(\underline{R}, N) ≅ Ext_{RD}^{∗}(\underline{R}, Res_{ι} N), it reminds us of the Eckmann-Shapiro Lemma in the cohomology theory of algebras, which compares Ext groups after a base ring change (Benson [5]). There are two different types of isomorphisms given by the Eckmann-Shapiro Lemma, one established using the induction and the other using the co-induction. We record them below as Lemmas 3.3.5 and 3.3.6.

Lemma 3.3.5. (Eckmann-Shapiro) Let M be an arbitrary RC-module. If RC is a right flat RD-module then for any RC-module N, we have

Ext_{RC}^{∗}(M ↑_{D}^{C}, N) ≅ Ext_{RD}^{∗}(M, N ↓_{D}^{C}).

Proof. Suppose P → M → 0 is a projective RD-resolution. Then P ↑_{D}^{C} → M ↑_{D}^{C} → 0 is also a projective resolution since RC is a right flat RD-module and ↑_{D}^{C} preserves projectives. The Ext groups Ext_{RD}^{∗}(M, N ↓_{D}^{C}) are the homology groups of Hom_{RD}(P, N ↓_{D}^{C}), which is isomorphic to Hom_{RC}(P ↑_{D}^{C}, N). Since the latter gives Ext_{RC}^{∗}(M ↑_{D}^{C}, N), it follows that

Ext_{RC}^{∗}(M ↑_{D}^{C}, N) ≅ Ext_{RD}^{∗}(M, N ↓_{D}^{C}).

If we replace M by \underline{R} in the Eckmann-Shapiro Lemma, we get Ext_{RC}^{∗}(\underline{R} ↑_{D}^{C}, N) ≅ Ext_{RD}^{∗}(\underline{R}, N ↓_{D}^{C}), something slightly different from the isomorphism of Jackowski and Słomińska: Ext_{RC}^{∗}(\underline{R}, N) ≅ Ext_{RD}^{∗}(\underline{R}, N ↓_{D}^{C}). The isomorphism Ext_{RC}^{∗}(\underline{R} ↑_{D}^{C}, N) ≅ Ext_{RD}^{∗}(\underline{R}, N ↓_{D}^{C}) doesn’t give a reduction for computing higher limits, unless \underline{R} ≅ \underline{R} ↑_{D}^{C} where the first \underline{R} is an RC-module while the second is an RD-module. We may readily ask if it’s possible to obtain \underline{R} ≅ \underline{R} ↑_{D}^{C} for some D ⊂ C, and this is indeed the reason why we look into the theory of relative projectivity and is our motivation for developing the theory of vertices and sources in Chapter 4.

When RC is projective as a left RD-module, using the right adjoint ⇑_{D}^{C} of ↓_{D}^{C} we can get a different formula, which is less useful than Lemma 3.3.6 in our situation. Recall that the co-induction of M is defined by M ⇑_{D}^{C} = Hom_{RD}(RC, M). The proof of the following lemma is similar to the previous one.

Lemma 3.3.6. (Eckmann-Shapiro) Suppose RC is a projective RD-module. Then for any RC-module M and RD-module N we have Ext_{RD}^{∗}(M ↓_{D}^{C}, N) ≅ Ext_{RC}^{∗}(M, N ⇑_{D}^{C}).

In transition to the main chapter of this thesis, we would like to make a comment on the proof of Corollary 3.3.4, which complements the comment after Lemma 3.3.5 and gives us some idea about what we do next. The essence of that proof is to use the isomorphism of complexes Hom_{RC}(K(P), M) ≅ Hom_{RD}(P, Res_{ι} M). In order to pass from this isomorphism to those between higher limits, we need to have the left Kan extension of P, K(P)(≅ RC ⊗_{RD} P) to be exact. Plus we want K(\underline{R})(≅ RC ⊗_{RD} \underline{R}) ≅ \underline{R} so we do have a projective resolution for \underline{R} (see comment after Lemma 3.3.6). In the original proof of Jackowski and Słomińska, the R-acyclicity of ι ↓_{y}, y ∈ Ob C resolves both issues we just mentioned. However, in practice it’s hard to detect the R-acyclicity of the overcategories ι ↓_{y}, y ∈ Ob C because of their complexity. Hence we turn to consider the induction ↑_{D}^{C} = RC⊗_{RD}−, which is isomorphic to K. The reasons are that firstly the flatness of RC will result in the exactness of K (hence the exactness of K(C_{∗}(Id_{D} ↓_{?}))), and secondly it’s relatively easier to formulate conditions on D to the conditions that RC becomes flat (see Lemma 1.3 of [6] for an early example). This naturally amounts to studying the flatness (in practice, the projectivity) of RC as an RD-module. In the next chapter, Section 4.2 deals with the projectivity of RC, and Section 4.4 tells us the existence of \underline{R} ≅ \underline{R} ↑_{D}^{C}.

In the end we give an example where all overcategories are R-acyclic, while RC is not a right flat RD-module.

Example 3.3.7. Let C be a poset and D be a subposet as follows.

\xymatrix{ & z \\ x \ar[ur]^{γ} &  & y \ar[ul]_{μ} & , & x &  & y \\  & w \ar[ul]^{α} \ar[ur]_{β} &  &  &  & w \ar[ul]^{α} \ar[ur]_{β}}

It’s easy to check all the overcategories are R-acyclic. In fact, ι ↓_{x}, ι ↓_{y} and ι ↓_{w} are contractible because each of them has a terminal object. The other one, ι ↓_{z}, has the reduced simplicial complex

0 → R{α, β}→^{δ}R{1_{x}, 1_{y}, 1_{w}}→^{ϵ}R → 0,

whose homology groups are zeros. Here ϵ is the augmentation map and δ sends α and β to 1_{x} − 1_{w} and 1_{y} − 1_{w}, respectively.

On the other hand, as a right RD-module, RC is a direct sum

RC = R{1_{w}} ⊕ R{1_{x}, α} ⊕ R{1_{y}, β} ⊕ R{1_{z}} ⊕ R{γ, μ, γα = μβ}.

The first 3 summands are right projective modules (see Section 4.1) which add up to RD, the fourth is a null module on which RD acts as zero and the last is isomorphic to the constant module \underline{R} which isn’t flat as a right RD-module.

However if we take E = {w} ⊂ C, then RC becomes a right projective RE-module, and meanwhile every overcategory associated with ι : E → C is R-acyclic.

EI-categories, relative projectivity,

vertices and sources

In this chapter, we investigate the representation theory of EI-categories and its applications to cohomology theory, especially to the computation of higher limits. We assume the base ring R is a field or a complete discrete valuation ring, in order to have the unique decomposition property for every RC-module. When R is a field of characteristic p > 0, we denote it by 𝔽_{p} (instead of 𝔽_{q} for q = p^{n}), and require it to be large enough (e.g. algebraically closed etc.) if necessary.

Definition An EI-category is a small category C in which all endomorphisms are isomorphisms.

Posets and groups are EI-categories. Recently some very interesting and important EI-categories have appeared in the representation theory of groups and the homotopy theory of classifying spaces of groups. We have in mind the Frobenius, Brauer and Puig categories described in the book by Thévenaz ([38] Chap.7), the orbit categories and other categories considered by Dwyer [10], and the fusion systems of Puig, used by Broto, Levi and Oliver [6].

Some of the general theory of EI-categories is given by tom Dieck in [9] (I.11), much of which was due to Lück, see also [26]. One of the important features of EIcategories is described as follows. Given an EI-category C, there is a preorder defined on Ob C, that is x ≤ y if and only if Hom_{C}(x, y) ≠ ∅. Let [x] be the isomorphism class of an object x ∈ Ob C. This preorder induces a partial order on the set Is C of isomorphism classes of Ob C (specified by [x] ≤ [y] if and only if Hom_{C}(x, y) ≠ ∅), which plays an important role in studying representations and cohomology of EIcategories. Because of the existence of an order for the isomorphism classes of objects in any EI-category, EI-categories are sometimes referred as “ordered categories” by some authors, see Oliver [31] and Jackowski and Słomińska [25]. The preorder makes it possible to construct the following useful full subcategories. For an EI-category C and an object x ∈ Ob C, we define a full subcategory D_{≤x} ⊂ D, consisting of all y ∈ Ob D such that Hom_{C}(y, x) ≠ ∅. Similarly we can define several other full subcategories of D: D_{<x}, D_{≥x} and D_{>x}.

Convention In the rest of this thesis, we’re going to assume that C is finite (i.e. Mor(C) is finite). Then RC becomes a finite dimensional algebra and 1_{RC} has a primitive decomposition.

When we talk about a full subcategory D of an EI-category C, we suppose D has the following property: if x ∈ Ob D, then [x] ⊂ Ob D, where [x] is the isomorphism class of x in C.

Let C be an EI-category and D ⊂ C a full subcategory. The above condition on D is a natural requirement, which won’t change the nature of any questions to be considered here and does save us from some unnecessary technical concerns. The thing is, if we are to investigate an RC-module M, then (as a functor) M has to take an isomorphic value on every object of an isomorphism class of objects in C. If D doesn’t meet our requirement, then we can construct a full subcategory E ⊂ C, which contains D as a full subcategory and is equivalent to D. In fact, we let Ob E consist of all objects in C which are isomorphic to some object(s) in D, and then the structure of E becomes clear. Since the representation and cohomology theories of E is the same as those of D and N ↑_{D}^{C} ≅ (N ↑_{D}^{E}) ↑_{E}^{C} for any N ∈ RD-mod, we can study the connections between representation and cohomology theories of RC-mod and those of RE-mod, in order to understand the connections between RC-mod and RD-mod.

In Section 4.1, we describe the structures of the simple and indecomposable projective RC-modules. Using the structures of indecomposable projective modules, we can generalize a result of Symonds [37] on the projectivity of the trivial module \underline{R}. Let D ⊂ C be a full subcategory. We show in Section 4.2 when a projective RC-module, regarded as an RD-module, remains projective. We briefly go over the general theory of relative projectivity of RC-modules in Section 4.3, where we also give an example to explain how it works with the Eckmann-Shapiro Lemma to calculate the Ext groups over C. Our main results accumulate in Section 4.4, in which we develop the theory of vertices and sources. In this section we associate to each indecomposable RC-module M a vertex, which is the smallest full convex subcategory of C, relative to which M is projective. This implies every indecomposable RC-module is an induced module from some subcategory (e.g. its vertex). Since we are particularly interested in the calculation of higher limits, we will describe in Section 4.5 the structure of the vertex of the trivial RC-module \underline{R}. We end this chapter with discussions of finite categories with subobjects. We show that some important topological properties of such a category are determined by those of its underlying poset.

4.1 Projective modules and simple modules

Most of the results in this section was obtained by Lück [26]. An account of his work can also be found in tom Dieck [9]. Remember we assume the base ring R is a field or a complete discrete valuation ring.

Let C be a small category and x ∈ Ob C an object. Suppose P_{x} is a projective R Aut_{C}(x)-module. Then the induced module RC ⊗_{R Aut(x)} P_{x} is a projective RCmodule, because RC ⊗_{R Aut(x)} R Aut(x) ≅ RC is projective and P_{x} is isomorphic to a direct summand of R Aut(x)^{n} for some positive integer n. If C is a finite EI-category, we show the list of projective modules of the form: RC ⊗_{R Aut_{C}(x)} P for some x ∈ Ob C and an indecomposable projective R Aut_{C}(x)-module P, exhausts all indecomposable projective RC-modules.

In order to achieve the goal, we analyze the structures of the regular RC-module and its indecomposable summands. Suppose C is a finite EI-category. It’s easy to see that the regular module of the category algebra RC decomposes into a direct sum ⊕_{x∈Ob C}RC · 1_{x}, where 1_{x} is the identity of Aut_{C}(x). If x ≅ x′ are two isomorphic objects, and f ∈ Hom_{C}(x, x′) is an isomorphism, then the assignment α ↦ α · f for each α ∈ RC · e defines an isomorphism of RC-modules RC · 1_{x′} → RC · 1_{x}. Hence RC ≅ ⊕_{x∈Ob C}[RC · 1_{x}]^{|[x]|}, where |[x]| denotes the size of the isomorphism class [x] of each x ∈ Ob C. These direct summands are projective RC-modules, but in general they are not indecomposable.

Lemma 4.1.1. Let C be a finite EI-category. One can write 1_{RC} = ∑_{x∈Ob C} ∑_{j=1}^{n_{x}} e_{xj}, where n_{x} is a positive integer for each x ∈ Ob C and 1_{x} = ∑_{j=1}^{n_{x}} e_{xj}. As a consequence, RC = ⊕_{x∈Ob C} ⊕_{j=1}^{n_{x}} RC · e_{xj} for some primitive orthogonal idempotents e_{xj} ∈ R Aut_{C}(x), x ∈ Ob C.

Proof. We know the identity of RC is 1 = ∑_{x∈Ob C} 1_{x}, where 1_{x} is the identity of R Aut(x). Let’s fix an object x. Without loss of generality, we suppose 1_{x} = ∑_{i=1}^{n_{x}} e_{i}, for some primitive orthogonal idempotents e_{i} of R Aut(x). We show these e_{i} are primitive idempotents of RC. Suppose some e_{j} ∈ R Aut(x) is not primitive. Then it’s of the form a + b, a sum of two orthogonal idempotents in RC, where both a and b are combinations of morphisms in C. If we write a = a_{1} + a_{2} + a_{3} and b = b_{1} + b_{2} + b_{3} where a_{1}, b_{1} ∈ R Aut(x), a_{2}, b_{2} ∈ ∑_{y≠x} R Aut(y), and a_{3}, b_{3} are combinations of nonisomorphisms. Using equalities like a^{2} = a, b^{2} = b and ab = ba = 0, we can show that a_{1} and b_{1} are orthogonal idempotents in R Aut_{C}(x), a_{2} and b_{2} are orthogonal idempotents in ∑_{y≠x} R Aut_{C}(y), while a_{1} + b_{1} = e_{j}, a_{2} + b_{2} = 0 and a_{3} + b_{3} = 0. So we must have either a_{1} = 0 or b_{1} = 0, and a_{2} = b_{2} = 0 by simple calculations. Let’s assume a_{1} = 0. Then a = a_{3} is an idempotent. From a_{3} + b_{3} = 0 and a_{3}^{2} = a_{3}, we can see b_{3}^{2} = −b_{3}. Now using ab = 0 and ba = 0 we can deduce b_{3}b_{1} = b_{3} and b_{1}b_{3} = b_{3}. But this is impossible since b_{1} = e_{j} ∈ R Aut_{C}(x) while b_{3} is a linear combination of non-isomorphisms. Therefore e_{j} has to be primitive in RC.

The above lemma gives us some indecomposable projective RC-modules of the form RC · e, where e ∈ R Aut_{C}(x) for some x ∈ Ob C. An indecomposable projective RC-module like that evaluated at x equals R Aut_{C}(x) · e, which is an indecomposable projective R Aut_{C}(x)-module. If y ≅ x, then it’s not hard to see that RC·e(y) = R Hom_{C}(x, y)⊗_{R Aut(x)}R Aut(x)e is isomorphic to R Aut(x)e as R-modules. These indecomposable projectives are induced modules because for each e the natural surjection RC ⊗_{R Aut(x)} R Aut(x) · e → RC · e has to be an isomorphism.

Corollary 4.1.2. (Lück [26]) Any projective RC-module is isomorphic to a direct sum of indecomposable projective modules of the form RC · e, where e ∈ R Aut_{C}(x) is a primitive idempotent, for some x ∈ Ob C.

Proof. Since any projective RC-module is a direct summand of some free RC-module, we can use the preceding lemma and the Krull-Schmidt Theorem to prove the statement.

Using the above corollary, one can deduce the structures of simple RC-modules, since with the assumption on the base ring, every simple module is a quotient of some indecomposable projective module by its radical. Because each indecomposable projective module is a direct summand of some RC · 1_{x}, x ∈ Ob C, we only have to consider the quotient of RC · 1_{x} by its radical in order to obtain a complete list of simple RC-modules. It’s not hard to see all the non-isomorphisms in Hom_{C}(x, −) span a submodule that is contained in the radical of RC · 1_{x}. This implies RC · 1_{x} is the projective cover of a semi-simple module, which is atomic, concentrated on the isomorphism class [x]. With further examination, we can conclude that every simple R Aut_{C}(x)-module gives rise to a simple RC-module, and all simple RC-modules can be obtained in this way.

Theorem 4.1.3. (Lück [26]) Let C be an EI-category. For each object x ∈ Ob C and simple R Aut_{C}(x)-module V there is a simple RC-module M such that [x] ∈ Is C is exactly the set of objects on which M is non-zero, and M(x) = V. On the other hand, if M is a simple RC-module, then there exists a unique isomorphism class of objects [x] ∈ Is C on which M is non-zero, and furthermore each M(x) is a simple R Aut_{C}(x)- module. These two processes are inverse to each other. Thus the isomorphism classes of the simple RC-modules biject with the pairs ([x], V), where x ∈ Ob C and V is a simple R Aut_{C}(x) module, taken up to isomorphism.

Proof. Let M be a simple RC-module and choose x ∈ Ob C for which M(x) ≠ 0. The submodule of M generated by M(x) is non-zero only on objects y with [x] ≤ [y], and equals M since M is simple. Therefore M(y) = 0 unless [x] ≤ [y]. The submodule of M generated by the M(y) with [x] < [y] is zero at x, hence is not the whole of M, so must be zero since M is simple. Thus M(y) = 0 unless [x] = [y]. Finally, if M(x) were to have a proper submodule as an R Aut_{C}(x)-module, it would generate a proper submodule of M, so M(x) must be simple R Aut_{C}(x)-module.

Conversely, given an object x ∈ Ob C and a simple R Aut(x)-module V. Then we can use it to construct an RC-module M as follows.

M(y) = \{ \begin{matrix}R Hom(x, y) ⊗_{R Aut(x)} M(x), & if y ≅ x \\ 0, & otherwise \end{matrix}

Thus M has the property that M(x) = V and hence we have shown that there is a bijection between simple RC-modules and pairs ([x], V) as claimed.

We denote a simple RC-module by S_{x,V}, if it comes from a simple R Aut(x)-module V, for some x ∈ Ob C. For consistency, we use P_{x,V} for the projective cover of S_{x,V}, whose structure is determined by its value at the object x. If R Aut(x) · e is the projective cover of the simple R Aut(x)-module V, then RC · e is the projective cover of S_{x,V}.

Example 4.1.4. Let C be a poset and R a field. Then both [x] and Aut(x) have just one element. All simple modules are of the form S_{x,1} = R · 1_{x}, with projective cover P_{x,1} = RC · 1_{x}. When we deal with simply a poset, we usually write S_{x}, P_{x} for simples and projectives, instead of S_{x,1}, P_{x,1}, because there is only one simple RCmodule associated to each object of C.

With the description of indecomposable projectives, we can also show when the trivial module \underline{R} is projective. The following proposition generalizes a result of Symonds [37].

Proposition 4.1.5. Let C be a finite EI-category. Then \underline{R} is projective if and only if each connected component of C has a unique isomorphism class of minimal objects [x], and furthermore such an x has the property that for all y in the same connected component as x, Aut(x) has a single orbit on Hom(x, y), and | Aut(x)| is invertible in R.

Proof. If \underline{R} is projective then \underline{R} ≅ ⨁ P_{x,V} for certain indecomposable projective modules P_{x,V}. The only V which can arise are V = R, and R must be projective as an R Aut(x)-module, forcing | Aut(x)| to be invertible in R for the x which appear in the direct sum, as in the first proof.

Since P_{y,R}(z) = 0 unless y ≤ z, P_{x,R} must appear as a summand for each isomorphism class of minimal x. Now P_{x,R}(z) = Hom(x, z) ⊗_{Aut(x)} R ≅ R^{n}, where n is the number of orbits of Aut(x) on Hom(x, z). For P_{x,R}(z) to be a summand of R we must have n = 1.

Finally, ⨁_{{minimal x}} P_{x,R} at an object z is R^{t} where t = number of isomorphism classes of minimal [x] with x ≤ z, so each component has a unique minimal x.

The other direction is easy. The conditions imply that \underline{R} = ⨁_{\{minimal x\}} P_{x,R} and this is projective.

Remark 4.1.6. Since we know the structures of indecomposable projective RC-modules, we can describe the indecomposable injective RC-modules via standard duality, when R is a field. Let P be a projective RC-module. Then P^{∗} = Hom_{R}(P, R) is a right injective RC-module. All right injective RC-modules can be obtained in this way, and furthermore if P is indecomposable, so is P^{∗}. We will describe the right RC-module structure of (RC)^{∗} which has every indecomposable injective module as a direct summand, up to isomorphism. It’s easy to see (RC)^{∗} is the R-span of the linear functions on RC: {f_{α}|α ∈ Mor(C)}, where f_{α}(β) = 0 for all β ≠ α and f_{α}(α) = 1. Routine calculations show that the right RC-module structure on (RC)^{∗} is given by

f_{α} · β = ∑_{α=βγ}f_{γ}.

The right-hand-side is assumed to be zero if there’s no γ satisfying α = βγ. We note that in general the set of right injective RC-modules cannot be identified with the set of right projective RC-modules.

4.2 Restrictions of projective modules

Let M and N be two RC-modules. For computing Ext_{RC}^{∗}(M, N), we plan to develop some machinery which reduces the size of C, by which we mean finding a (full) subcategory D ⊂ C such that Ext_{RC}^{∗}(M, N) ≅ Ext_{RD}^{∗}(Res_{ι} M, Res_{ι} N). This isomorphism will be established using the Eckmann-Shapiro Lemma and the theory of vertices and sources which we shall develop in the next two sections. In order to apply the Eckmann-Shapiro Lemma we have to require, for the given inclusion ι : D ↪ C, the functor ↑_{D}^{C} = RC ⊗_{RD} − to preserve the projective resolutions of RD-modules. The functor ↑_{D}^{C} always sends projective modules to projective modules so we just need to find out when it’s exact.

If RC is a right flat RD-module then ↑_{D}^{C} is exact. In our context, a feasible way to sort out the flatness of RC is to figure out when it is a right projective RD-module. However in order to be consistent with our general set-up, we will describe how to make RC into a left projective RD-module. We note that the argument for RC being a right projective RD-module follows immediately in a dual fashion. This means after present section we will know for which full subcategories ↓_{D}^{C} preserves left or right projective RC-modules.

We find it’s convenient to study the action of ↓_{D}^{C} : (R-mod)^{C} → (R-mod)^{D} on the representable functors in order to formulate the conditions on D for which ↓_{D}^{C} preserves (left and right) projective modules. We will concent ourselves with the full subcategories of C, for some reasons we will see soon in the course of developing our theory of vertices and sources.

The main theorems here are due to tom Dieck [9]. In order to use the characterization of projective modules, we assume R is a field or a complete discrete valuation ring.

Definition 4.2.1. Let D be a subcategory of C. Then we say a non-isomorphism f ∈ Mor(C) is irreducible (resp. co-irreducible) with respect to D if there’s no way to write f = f_{1} · f_{2} with f_{1}, f_{2} non-isomorphisms and f_{1} (resp. f_{2}) ∈ Mor(D).

Given x ∈ Ob C\ Ob D and y ∈ Ob D the subset of Hom_{C}(x, y) of all nonisomorphisms which are irreducible with respect to D is denoted by Irr_{D}(x, y). Similarly given x ∈ Ob D and y ∈ Ob C\ Ob D we define coIrr_{D}(x, y) to be the subset of Hom_{C}(x, y) consisting of non-isomorphisms that are co-irreducible with respect to D.

If a non-isomorphism f ∈ Mor(C) is irreducible with respect to C, then we simply say f is irreducible. Note that our irreducible morphisms are different from those in the representation theory of Artin algebras, see [2].

Lemma 4.2.2. Let D be a full subcategory of an EI-category C. Suppose x ∈ Ob C, y ∈ Ob D and Hom_{C}(x, y) is not empty. Then as an R Aut(y)-module,

RHom_{C}(x, y) = RIrr_{D}(x, y) ⊕ R r(x, y),

where Irr_{D}(x, y) is the set of all non-isomorphisms which are irreducible with respect to D, and r(x, y) is the complementary set of Irr_{D}(x, y) ⊂ Hom_{C}(x, y). Of course either of them could be empty.

Proof. Obvious.

Recall that we have made the following assumption: if D ⊂ C is a full subcategory then it satisfies the condition that if y ∈ Ob D then [y] ⊂ Ob D, where [y] is the isomorphism class of y ∈ Ob C. Let R Hom_{C}(x, −) be a representable functor for some x ∈ Ob C. We consider the restriction of it: R Hom_{C}(x, −) ↓_{D}^{C}. When [x] ⊂ Ob D, we simply have R Hom_{C}(x, −) ↓_{D}^{C} = R Hom_{D}(x, −), which is a projective RD-module. When [x] ∩ Ob D = ∅, we have

RHom_{C}(x, −) ↓_{D}^{C}= ∑_{y∈Ψ_{x}}RHom_{D}(y, −) · RIrr_{D}(x, y), (∗)

where Ψ_{x} is the set of all y ∈ Ob D_{≥x} = Ob D_{>x} such that Irr_{D}(x, y) is non-empty. It is so because, for each z ∈ Ob D_{≥x}, every morphism f ∈ Hom_{C}(x, z) has a suitable factorization f = f_{1} · f_{2} for some f_{2} ∈ Irr_{D}(x, y), y ∈ Ob D_{≥x} (y doesn’t have to be minimal in Ob D_{≥x}!). One can see that if y, y′ ∈ Ψ_{x} and y ≅ y′, then R Irr_{D}(x, y) ≅ R Irr_{D}(x, y′) and R Hom(y, −)R Irr_{D}(x, y) = R Hom(y′, −)R Irr_{D}(x, y′), and thus

RHom_{C}(x, −) ↓_{D}^{C}= ∑_{[y]⊂Ψ_{x}}RHom_{D}(y, −) · RIrr_{D}(x, y). (∗)

Here we regard RHom_{D}(y, −) · RIrr_{D}(x, y) as a covariant functor on D such that

[RHom_{D}(y, −) · RIrr_{D}(x, y)](z) = RHom_{D}(y, z) · RIrr_{D}(x, y).

We will keep using this kind of expressions even though it looks like a “product” of a functor and a module. If R Hom_{C}(x, −) ↓_{D}^{C} is a projective RD-module, then we show the right-hand-side of the above equality (∗) must be a direct sum.

Lemma 4.2.3. Let D be a full subcategory of a finite EI-category C. Suppose x ∈ Ob C and [x] ∩ Ob D = ∅. If R Hom_{C}(x, −) ↓_{D}^{C} is projective, then we must have

RHom_{C}(x, −) ↓_{D}^{C}= ⨁_{[y]⊂Ψ_{x}}RHom_{D}(y, −) · RIrr_{D}(x, y),

and thus R Hom_{D}(y, −)R Irr_{D}(x, y) is a projective RD-module for every y ∈ Ψ_{x}.

Proof. We can write R Hom_{C}(x, −) ↓_{D}^{C} = ⨁_{z,V} M_{z,V} for some z ∈ Ob D and simple R Aut(z)-modules V, with M_{z,V} ≅ P_{z,V}. Given w ∈ Ob D, M_{z,V} (w) consists of elements which are linear combinations of morphisms in Hom_{C}(x, w). Note that from the structure of indecomposable projective modules (Corollary 4.1.2), every M_{z,V} is determined by M_{z,V} (z) (an indecomposable projective R Aut(z)-module), and that M_{z,V} (z′) = 0 if z′ ≱ z. It implies that M_{z,V} (w) ⊂ R Hom_{C}(x, w) consists of linear combinations of morphisms which factor through z.

First of all, we show that whenever Irr_{D}(x, y) ≠ ∅ there exists some M_{y,V} as a direct summand of R Hom_{C}(x, −) ↓_{D}^{C}. The thing is

RHom_{C}(x, −)(y) ↓_{D}^{C}= ⨁_{z,V} M_{z,V} (y) = {⨁_{z≅y} M_{z,V} (y)} ⨁{⨁_{z<y} M_{z,V} (y)}.

Since the second summand is a submodule of R r(x, y), we must have M_{z,V} (y) ≠ 0 for some z ≅ y.

Second, we’ll prove, for every y ∈ Ob D_{>x}, ⨁_{z≅y} M_{z,V} (y) ≅ R Irr_{D}(x, y). Suppose w ∈ Ob D_{>x} is a minimal object. Then it’s easy to see R Irr_{D}(x, w) = R Hom_{C}(x, w) ≠ 0 and ⨁_{z≅w} M_{z,V} (w) = R Hom_{C}(x, w) = R Irr_{D}(x, w). Let M_{0} be the set of minimal objects in Ob D_{>x}. We consider the minimal objects of Ob D_{>x}\M_{0}. For every u ∈ Ob D_{>x}\M_{0}, we’ll show ⨁_{z≅u} M_{z,V} (u) ≅ R Irr_{D}(x, u) too. Then inductively we can prove, for every y ∈ Ob D, ⨁_{z≅y} M_{z,V} (y) ≅ R Irr_{D}(x, y), since Ob D is finite. As usual we consider the equality R Hom_{C}(x, u) = {⨁_{z≅u} M_{z,V} (u)} ⨁{⨁_{z<u} M_{z,V} (u)}. We have the second summand ⨁_{z<u} M_{z,V} (u) = ⨁_{z∈M_{0}} M_{z,V} equals R r(x, u), and therefore from

RIrr_{D}(x, u) ⨁ R r(x, u) = {⨁_{z≅u} M_{z,V} (u)} ⨁{ ⨁_{z∈M_{0}} M_{z,V} (u)},

we obtain R Irr_{D}(x, u) ≅ ⨁_{z≅u} M_{z,V} (u).

In the end from ⨁_{z≅y} M_{z,V} (y) ≅ R Irr_{D}(x, y) we get

RHom_{D}(y, −) · ⨁_{z≅y} M_{z,V} (y) = ⨁_{z≅y} M_{z,V} ≅ RHom_{D}(y, −) · RIrr_{D}(x, y).

Hence we have an isomorphism R Hom_{C}(x, −) ↓_{D}^{C} ≅ ⨁_{[y]⊂Ψ_{x}} RHom_{D}(y, −)·RIrr_{D}(x, y). Along with the equality R Hom_{C}(x, −) ↓_{D}^{C} = ∑_{[y]⊂Ψ_{x}} R Hom_{D}(y, −) · R Irr_{D}(x, y), it implies

RHom_{C}(x, −) ↓_{D}^{C}= ⨁_{[y]⊂Ψ_{x}} RHom_{D}(y, −) · RIrr_{D}(x, y),

because the R-ranks of ∑_{[y]⊂Ψ_{x}} R Hom_{D}(y, −)·R Irr_{D}(x, y) and ⊕_{[y]⊂Ψ_{x}}RHom_{D}(y, −)· RIrr_{D}(x, y) are equal.

Remark 4.2.4. In fact we can continue to show every R Irr_{D}(x, y) in Lemma 4.2.3 is a projective R Aut(y)-module (this is recorded as Lemma 4.2.8). Meanwhile, it’s not hard to see that

⨁_{z≅y,V} M_{z,V} = ⨁_{z≅y}{⨁_{V} M_{z,V} } ≅ |[y]| · ⨁_{V} M_{y,V} .

Here |[y]| denotes the size of the isomorphism class [y]. The reason is that M_{y,V} is determined by M_{y,V} (y) and every g ∈ Is(y, z) induces a permutation of direct summands of the leftmost term evaluated at y, because gR Irr_{D}(x, y) = R Irr_{D}(x, z) = R Irr_{D}(x, y). More explicitly, the action of g sends a direct summand, on evaluation at y, ⨁_{V} M_{y,V} (y) = ⨁_{V} R Aut(y) · b_{y,V} isomorphically to ⨁_{V} gR Aut(y)g^{−1} · gb_{y,V} = ⨁_{V} R Aut(z) · (gb_{y,V}), which equals ⨁_{V′} M_{z,V′}(z). Here b_{y,V} ∈ R Hom_{C}(x, y) is an element which satisfies M_{y,V} (y) = R Aut(y) · b_{y,V} for each V.

Before we go further, we present two simple examples where ↓_{D}^{C} preserves projectives.

Example 4.2.5. First, we consider the following category C and its full subcategory D

[figure: x; 1_{x}; f; α; y; 1_{y}; g; ;; y; 1_{y}; g] ,

where gα = α, αf = α, f^{2} = 1_{x} and g^{2} = 1_{y}. Suppose R = ℂ. We can get ℂ Hom_{C}(x, −) ↓_{D}^{C} = ℂ Hom_{D}(y, −)·ℂ Irr_{D}(x, y) = ℂ Hom_{D}(y, −)·α ≅ ℂ Hom_{D}(y, −)· \frac{(1_{y}+g)}{2} and Res_{D}^{C} ℂ Hom_{C}(y, −) = ℂ Hom_{D}(y, −), which are both projective ℂD-modules. (Note if R is a field of characteristic 2, then R Hom_{C}(x, −) ↓_{D}^{C} is not projective.)

Second, we consider the category C′ and its subcategory D′

\xymatrix{ & z &  &  &  & z \\ x \ar[ur]^{γ} &  & y \ar[ul]_{μ} & , & x \ar[ur]^{γ} &  & y, \ar[ul]_{μ} \\  & w \ar[ul]^{α} \ar[ur]_{β}}

where γα ≠ μβ. We have R Hom(w, −) ↓_{D′}^{C′} ≅ R Hom_{D′}(x, −) ⊕ R Hom_{D′}(y, −) due to the fact γα ≠ μβ. We need to emphasize that without the assumption γα ≠ μβ the isomorphism does not exist.

From the first example we know R Hom_{C}(x, y) = R Irr_{D}(x, y) is projective as an R Aut(y)-module, which verifies the assertion of the first sentence in Remark 4.2.4. We shall show R Irr_{D}(x, y) being projective, when x isn’t isomorphic to an object of D, is necessary for ↓_{D}^{C} to preserve projectives. From the second example we see a certain factorization property of the non-isomorphisms is required for ↓_{D}^{C} to preserve projectives. The factorization property actually guarantees the sum (∗) which appears before Lemma 4.2.3 to be direct. Thus the examples tell us both the base ring and the intrinsic structure of C will affect the action of ↓_{D}^{C} on representable functors.

Definition 4.2.6. Let x ∉ Ob D and z ∈ Ob D. Suppose f ∈ Hom_{C}(x, z) and f admits two factorizations f = f_{1}·f_{2} = f′_{1}·f′_{2} with f_{1}, f′_{1} ∈ Mor(D) and f_{2}, f′_{2} irreducible with respect to D. Then we say f is good if for every such pair of factorizations f_{2}, f′_{2} have isomorphic co-domains. We say D is good with respect to x ∈ Ob C\ Ob D if for all z ∈ Ob D with Hom_{C}(x, z) ≠ ∅, all f ∈ Hom_{C}(x, z) are good. We say D is good if D is good with respect to all x ∈ Ob C\ Ob D.

Lemma 4.2.7. Let [x] ∩ Ob D = ∅. The full subcategory D is good with respect to x if and only if

RHom_{C}(x, −) ↓_{D}^{C}= ⨁_{[y]⊂Ψ_{x}}RHom_{D}(y, −) · RIrr_{D}(x, y),

where Ψ_{x} is the union of all isomorphism classes [y] ⊂ Ob D_{≥x} such that Irr_{D}(x, y) is non-empty.

Proof. Given the goodness we show the intersection

(RHom_{D}(y, −) · RIrr_{D}(x, y)) ⋂(∑_{y′≅y̸} RHom_{D}(y′, −) · RIrr_{D}(x, y′)) = {0}.

For proving this, it suffices to show that the intersection is zero on evaluation at each z ∈ Ob D. Suppose

u ∈ [(RHom_{D}(y, −) · RIrr_{D}(x, y)) ⋂(∑_{y′≠y} RHom_{D}(y′, −) · RIrr_{D}(x, y′))](z).

Then we can write u = ∑_{i} r_{i}u_{i}^{1} · u_{i}^{2} with u_{i}^{1} ∈ Hom_{D}(y, z) and u_{i}^{2} ∈ Irr_{D}(x, y) for some y ∈ Ob D_{≥x}. Since u ∈ ∑_{y′≅y̸} R Hom_{D}(y′, −) · R Irr_{D}(x, y′), we can also put u = ∑ r′_{j}u_{j}^{′1} · u_{j}^{′2}, with u_{j}^{′1} ∈ Hom_{D}(y′, z) and u_{j}^{′2} ∈ Irr_{D}(x, y′) for y′ ≅ y̸ and y′ ∈ Ob D_{≥x}. Therefore

∑_{i} r_{i}u_{i}^{1} · u_{i}^{2} = ∑_{j} r′_{j}u′_{j}^{1} · u′_{j}^{2}.

But the equality implies that r_{i}, r′_{j} are all zero for our condition guarantees u_{i}^{1} · u_{i}^{2} ≠ u′_{j}^{1} · u′_{j}^{2} for any i, j since u_{i}^{2}, u_{j}^{2} have non-isomorphic targets, and morphisms are independent in R Hom_{C}(x, z), by definition. Hence u = 0, and we have the direct sum.

On the other hand, if

RHom_{C}(x, −) ↓_{D}^{C}= ⨁_{[y]⊂Ψ_{x}}RHom_{D}(y, −) · RIrr_{D}(x, y),

and f = f_{1} · f_{2} = f′_{1} · f′_{2} ∈ Hom_{C}(x, z) with f_{1} ∈ Hom_{D}(y, z), f′_{1} ∈ Hom_{D}(y′, z), f_{2} ∈ Irr_{D}(x, y) and f′_{2} ∈ Irr_{D}(x, y′), then we show y ≅ y′. For otherwise from f_{2} ∈ R Hom(y, −) · R Irr_{D}(x, y) and f′_{2} ∈ R Hom(y′, −) · R Irr_{D}(x, y′), we get

f ∈ R Hom(y, z) · RIrr_{D}(x, y) ∩ R Hom(y′, z) · RIrr_{D}(x, y′),

which is a contradiction.

Next we examine when the RD-module R Hom(y, −) · R Irr_{D}(x, y) is projective.

Lemma 4.2.8. Let [x]∩Ob D = ∅. Then the RD-module R Hom_{D}(y, −)·R Irr_{D}(x, y) being projective implies R Irr_{D}(x, y) is projective as an R Aut(y)-module.

Proof. If R Hom_{D}(y, −) · R Irr_{D}(x, y) is a projective RD-module, then from the structure theorem of the projective modules

RHom_{D}(y, −) · RIrr_{D}(x, y) ≅ ⨁_{i} RHom_{D}(y, −) · e_{i}

for some primitive idempotents e_{i} of R Aut(y). Evaluating the two functors on both sides at y, we get [R Hom_{D}(y, −) · R Irr_{D}(x, y)](y) ≅ [⨁_{i} R Hom_{D}(y, −) · e_{i}](y), or R Irr_{D}(x, y) ≅ ⨁_{i} R Aut(y)e_{i}.

The converse of the above lemma is not true.

Example 4.2.9. Let C be the following category

\xymatrix{x \ar[r]^{α} \ar[r]_{β} & y \ar[r]^{γ} & z} ,

with γα = γβ. Consider the full subcategory D

\xymatrix{y \ar[r]^{γ} & z} .

We have R Hom_{C}(x, −) ↓_{D}^{C} = R Hom_{D}(y, −) · R Irr_{D}(x, y) with Hom(y, −) = {1_{y}, γ} and Irr_{D}(x, y) = {α, β}. Obviously, R Irr_{D}(x, y) = Rα ⊕ Rβ is a free R Aut(y)- module. But R Hom(y, −) · R Irr_{D}(x, y) is not a projective RD-module since it equals R Hom_{D}(y, −)\{\frac{α+β}{2} \} ⊕ R Hom_{D}(y, −)\{\frac{α−β}{2} \} (suppose charR ≠ 2), while the first summand R Hom_{D}(y, −)\{\frac{α+β}{2} \} ≅ R Hom_{D}(y, −) is projective and second summand R Hom_{D}(y, −)\{\frac{α−β}{2} \} ≅ S_{y} is not.

Suppose R Irr_{D}(x, y) ≅ ⨁_{i} R Aut(y)e_{i}. In order to maintain a direct sum on the right-hand-side after multiplying both sides with R Hom_{D}(y, −), we need an extra condition, which we borrow from tom Dieck [9].

Lemma 4.2.10. Suppose [x] ∩ Ob D = ∅ and R Irr_{D}(x, y) is a projective R Aut(y)- module. Then the RD-module R Hom_{D}(y, −) · R Irr_{D}(x, y) is projective if and only if

RHom_{D}(y, −) · RIrr_{D}(x, y) ≅ RHom_{D}(y, −) ⊗_{R Aut(y)} RIrr_{D}(x, y).

Proof. The “if” part is easy since the tensor product R Hom_{D}(y, −) ⊗_{R Aut(y)} − preserves projectives. The “only if” part can be obtained by considering the natural transformation

π : RHom_{D}(y, −) ⊗_{R Aut(y)} RIrr_{D}(x, y) → RHom_{D}(y, −) · RIrr_{D}(x, y),

given on each object z ∈ Ob D via multiplication

π_{z} : RHom_{D}(y, z) ⊗_{R Aut(y)} RIrr_{D}(x, y) → RHom_{D}(y, z) · RIrr_{D}(x, y).

Every π_{z} is a splitting epimorphism because both ends are projective modules. Furthermore π_{y} is an isomorphism. Since both functors are determined by their evaluations at y, the natural transformation π is indeed an isomorphism.

We also have a combinatorial condition on the existence of the isomorphism in the previous lemma (compare with Lemma 4.2.6). It was formulated by tom Dieck ([9], 11.38(iii)), and it helps us to improve our written statement of the main result in this section. The condition (TD) is stronger than our definition of goodness.

(TD) Let x ∈ Ob C\ Ob D, y ∈ Ob D and f ∈ Hom_{C}(x, y). If f admits two factorizations f_{1} · f_{2} = f′_{1} · f′_{2} with f_{2}, f′_{2} irreducible with respect to D, then there is an isomorphism g ∈ Mor(D) such that f_{2} = gf′_{2} and gf_{1} = f′_{1}.

The following lemma is actually extracted from tom Dieck’s Proposition 11.39 [9].

Lemma 4.2.11. Let x ∈ Ob C\ Ob D, y ∈ Ob D. Then

RHom_{D}(y, −) ⊗_{R Aut(y)} RIrr_{D}(x, y) ≅ RHom_{D}(y, −) · RIrr_{D}(x, y),

if and only if every morphism f ∈ Hom_{C}(y, z) Irr_{D}(x, y), z ∈ Ob D satisfies condition (TD).

Proof. Again we consider the natural transformation π = {π_{z}}_{z∈Ob D}. It’s always an epimorphism, and it’s an isomorphism if and only if (TD) is satisfied.

The previous two lemmas imply that, given the condition (TD), R Hom(y, −) · R Irr_{D}(x, y) is projective if R Irr_{D}(x, y) is. We write this out as a corollary.

Corollary 4.2.12. Suppose D is a full subcategory of C and x ∈ Ob C\ Ob D with [x] ∩ Ob D = ∅. If all morphisms in Hom_{C}(x, z), z ∈ Ob D satisfy condition (TD), then R Hom(y, −) · R Irr_{D}(x, y) is projective if and only if R Irr_{D}(x, y) is projective.

Proof. Just combine Lemmas 4.2.8, 4.2.10 and 4.2.11.

At this point, we can see, just assuming condition (TD), we have

RHom_{C}(x, −) ↓_{D}^{C} = ⨁_{[y]⊂Ψ_{x}} RHom_{D}(y, −) · RIrr_{D}(x, y)
≅ ⨁_{[y]⊂Ψ_{x}} RHom_{D}(y, −) ⊗_{R Aut(y)} RIrr_{D}(x, y).

To summarize the results, we state the following theorem.

Theorem 4.2.13. Suppose R is a field or a complete discrete valuation ring. Let D ⊂ C be a full subcategory and [x] ∩ Ob D = ∅. Then R Hom_{C}(x, −) ↓_{D}^{C} is projective if and only if the following conditions are satisfied:

1. for each y ∈ Ob D, every f ∈ Hom_{C}(x, y) satisfies the condition (TD); and

2. for each y ∈ Ob D, R Irr_{D}(x, y) is a projective R Aut(y)-module.

Proof. When RHom_{C}(x, −) ↓_{D}^{C} is projective, we have

RHom_{C}(x, −) ↓_{D}^{C}= ⨁_{[y]⊂Ψ_{x}} RHom_{D}(y, −) · RIrr_{D}(x, y)

by Lemma 4.2.3. Hence both R Hom_{D}(y, −)·RIrr_{D}(x, y) and RIrr_{D}(x, y) are projective by Lemmas 4.2.4 and 4.2.8, respectively. In the end from Lemmas 4.2.10 and 4.2.11, we know all morphisms satisfy the condition (TD).

On the other hand, with condition (TD) we have

RHom_{C}(x, −) ↓_{D}^{C} = ⨁_{[y]⊂Ψ_{x}} RHom_{D}(y, −) · RIrr_{D}(x, y)
≅ ⨁_{[y]⊂Ψ_{x}} RHom_{D}(y, −) ⊗_{R Aut(y)} RIrr_{D}(x, y).

This along with condition (2) gives the projectivity of RHom_{C}(x, −) ↓_{D}^{C}.

We borrow the notion of “tautness” (with slight modifications) from Symonds [37].

Definition 4.2.14. We say a full subcategory D is R-taut in C if for every x ∈ Ob C with [x] ∩ D = ∅, D meets the conditions (1) and (2) in Theorem 4.2.13.

When D ⊂ C is R-taut, ↓_{D}^{C} preserves projectives by Theorem 4.2.13 because any projective RC-module is isomorphic to a direct summand of some RC^{n} and RC is a direct sum of representable functors. Next we mention a sufficient condition for ↓_{D}^{C} to preserve projectives.

Proposition 4.2.15. Let D ⊂ C be a full subcategory satisfying the following conditions:

1. For any two morphisms f_{1}, f_{2} ∈ Irr_{D}(x, y), if g_{1}f_{1} = g_{2}f_{2} for some g_{1}, g_{2} ∈ Mor(D) then f_{1} = gf_{2}, gg_{1} = g_{2} for some g ∈ Is(D); and

2. |Stab_{Aut(y)}(f)|^{−1} ∈ R for any f ∈ Irr_{D}(x, y).

Then D ⊂ C is R-taut.

Proof. In fact (1) implies (TD), and (2) says that R Irr_{D}(x, y) is a projective R Aut(y)- module due to a standard result on permutation modules of finite groups, and that the condition (2) of the previous theorem is satisfied.

For future reference, we give a dual version of Theorem 4.2.13. It’s easily to see that the right indecomposable projective RC-modules are of the form e·R Hom_{C}(−, x) for a primitive idempotent e ∈ R Aut(x), x ∈ Ob C. Let D ⊂ C be a full subcategory satisfying the condition that if y ∈ Ob D then [y] ⊂ Ob D. As before, we consider the restriction on the representable functor R Hom_{C}(−, x). If x ∈ Ob D then we simply have R Hom_{C}(−, x) ↓_{D}^{C} = R Hom_{D}(−, x). When x ∈ Ob C\ Ob D we have

RHom_{C}(−, x) ↓_{D}^{C}= ∑_{[y]⊂Ψ′_{x}} RcoIrr_{D}(y, x) · RHom_{D}(−, y),

where Ψ′_{x} is the set of all y ∈ Ob D such that coIrr_{D}(y, x) ≠ ∅.

In order to describe ↓_{D}^{C}, we need introduce the notion of co-goodness. Let x ∈ Ob C\ Ob D. Suppose f ∈ Hom_{C}(z, x), z ∈ Ob D and f admits two different factorizations f = f_{1} ·f_{2} = f′_{1} ·f′_{2} with f_{2}, f′_{2} ∈ Mor(D) and f_{1}, f′_{1} irreducible with respect to D. Then we say f is co-good if f_{1}, f′_{1} have isomorphic domains. We say D is co-good with respect to x ∈ Ob C\ Ob D if for all z ∈ Ob D, all f ∈ Hom_{C}(z, x) are good, provided Hom_{C}(z, x) ≠ ∅. We say D is co-good if D is co-good with respect to all x ∈ Ob C\ Ob D. There is also a dual condition to (TD), and for consistency we call it (co-TD), while its formulation is left for the readers.

Theorem 4.2.16. Suppose R is a field or a complete discrete valuation ring. Let D ⊂ C be a full subcategory and x ∈ Ob C\ Ob D. Then R Hom_{C}(−, x) ↓_{D}^{C} is a right projective RD-module if and only if the following conditions are satisfied:

1. for each y ∈ Ob D, every f ∈ Hom_{C}(y, x) satisfies the condition (co-TD); and

2. for each y ∈ Ob D, R coIrr_{D}(y, x) is a right projective R Aut(y)-module.

Definition 4.2.17. We say D is R-co-taut in C if for every x ∈ Ob C with [x] ∩ Ob D = ∅, D meets the conditions (1) and (2) in Theorem 4.2.16.

We show the existence of such R-taut and R-co-taut subcategories by constructing two classes of subcategories which we will work with later on.

Definition 4.2.18. Suppose D ⊂ C is a (full) subcategory. We say D is an ideal in C if, for any x ∈ Ob D, C_{≤x} ⊂ D. Similarly, we say D is a co-ideal in C if, for any x ∈ Ob D, we have C_{≥x} ⊂ D.

Let D ⊂ C be a full subcategory. Then we can form a full subcategory of C, named C\D, which consists of all objects not belonging to D. From the definitions it’s easy to verify that if D is an ideal (resp. a co-ideal) then C\D is a co-ideal (resp. an ideal). Note that if D ⊂ C is an ideal (resp. a co-ideal) then RD becomes a right ideal (resp. a left ideal) in RC.

If a full subcategory D forms an ideal (resp. a co-ideal) in C, then ↓_{D}^{C} preserves projectives (resp. right projectives). These can be regarded as trivial cases of Theorems 4.2.13 and 4.2.16.

Lemma 4.2.19. If D is an ideal in Ob C, then ↓_{D}^{C} preserves left projective modules. If D is a co-ideal in Ob C, then ↓_{D}^{C} preserves right projective modules.

Proof. We only prove the first assertion by computing R Hom_{C}(x, −) ↓_{D}^{C} explicitly. If x ∈ Ob D, then R Hom_{C}(x, −) ↓_{D}^{C} = R Hom_{D}(x, −). If x ∉ Ob D, then by definition R Hom_{C}(x, −) ↓_{D}^{C} = 0. Hence R Hom_{C}(x, −), and consequently RC, are projective RD-modules.

We record the essence of the argument used to prove Lemma 4.2.19 for future reference.

Corollary 4.2.20. Let D be a full subcategory of C which is an ideal (resp. a co-ideal) in Ob C. We have RC ↓_{D}^{C} ≅ RD as an RD-module (resp. a right RD-module).

Remark 4.2.21. As for the injective modules, from

Hom_{R}(RC, R) ↓_{D}^{C}= Hom_{R}(RC ↓_{D}^{C}, R)

as right RD-modules, we can see that ↓_{D}^{C} preserves right injective RC-modules if and only if ↓_{D}^{C} preserves left projective RC-modules, i.e., D is R-taut. (Here we assume R is a field.) In fact, when D is an ideal in C, we can see this by direct computation.

The thing is, the right RD-module Res_{D}^{C}(RC)^{∗} decomposes into a direct sum as follows (see Remark 4.1.6 for the structure of (RC)^{∗})

(RC)^{∗} ↓_{D}^{C} = R{f_{α} | α has both target and source in D}
⊕R{f_{α} | α only has its target in D}
⊕R{f_{α} | other α}.

The first summand is exactly (RD)^{∗}. The second summand is gone when D is an ideal in C, and RD acts on the third summand as zero.

Similarly, ↓_{D}^{C} preserves left injective RC-modules if and only if D is R-co-taut.

4.3 Relative projectivity

Results in last section tell us when we can use the Eckmann-Shapiro Lemma. However, in order to obtain desired isomorphisms (i.e. reduction formulas) we need to know if and how we can express the involved modules as induced modules. In this section, we briefly go over the theory of relative projectivity, which will provide us a general framework to study the induced modules. Fixing a subcategory D ⊂ C, this theory relates certain RC-modules with RD-modules. When D is required to be full in the next section, we can get a very clear picture about this relationship. But in present section we just deal with arbitrary subcategories of C in order to give some results of the most generality. After establishing necessary results in the theory of relative projectivity, we will recall the Eckmann-Shapiro Lemma and give our first example of calculating higher limits, using the knowledge we obtain in and before this section.

Suppose A is an R-subalgebra of the R-algebra B. Let M be a B-module. Then there is a natural epimorphism ϵ : B⊗_{A}M → M given by the multiplication ϵ(f⊗m) = fm. We shall only consider the case of a category algebra RC with a subalgebra RD, for some subcategory D of C.

Definition 4.3.1. Let M be an RC-module. If the RC-module epimorphism

ϵ = ϵ_{M} : M ↓_{D}^{C} ↑_{D}^{C} = RC ⊗_{RD} M → M

is split, then we say M is projective relative to D, or simply D-projective.

We have some equivalent descriptions of the relative projectivity of an RC-module M.

Proposition 4.3.2. Let D ⊂ C be a subcategory and M an RC-module. Then the following statements are equivalent:

1. The canonical surjective map ϵ : M ↓_{D}^{C} ↑_{D}^{C} → M splits;

2. M is a direct summand of M ↓_{D}^{C} ↑_{D}^{C};

3. M is a direct summand of N ↑_{D}^{C}, where N is an RD-module.

4. if 0 → M′′ → M′ → M → 0 is an exact sequence of RC-modules which splits as an exact sequence of RD-modules, then it splits as an exact sequence of RCmodules.

Proof. The statements 1, 2 and 4 are proved to be equivalent in a much more general context, see for instance [2], Chapter VI Proposition 3.6. When C is a finite group, statement 3 is well-known to be equivalent to the others, and the proof for category algebras is not seen in print even though the proof similar to that for group algebras.

We’ll prove in this direction 1 ⇒ 2 ⇒ 3 ⇒ 4 ⇒ 1. The proofs of 1 ⇒ 2 ⇒ 3 are straightforward. Now we show 3 ⇒ 4. Let M be a direct summand of N ↑_{D}^{C} for some RD-module N. Then we have two RC-maps α : M ↪ N ↑_{D}^{C} and β : N ↑_{D}^{C} → M satisfying the equality βα = 1_{M}. Suppose 0 → M′′ → M′→^{π}M → 0 is an exact sequence of RC-modules, which splits as an RD-sequence. Then there is an RD-map ι : M ↓_{D}^{C} → M′ ↓_{D}^{C} such that π|_{RD} ∘ ι = 1_{M ↓_{D}^{C}}. We need to extend ι to an RC-map ι′, with the property that π ∘ ι′ = 1_{M}. But first we want to construct an RC-map from N ↑_{D}^{C} to M′. In fact, since we have a split epimorphism β|_{RD} : N ↑_{D}^{C} ↓_{D}^{C} → M ↓_{D}^{C} and a split monomorphism ι : M ↓_{D}^{C} → M′ ↓_{D}^{C}, we can compose them to get an RD-map ι ∘ β|_{RD} : N ↑_{D}^{C} ↓_{D}^{C} → M′ ↓_{D}^{C}. By the canonical adjunction between ↑_{D}^{C} and ↓_{D}^{C}, we obtain an RC-map

ϵ_{M′} ∘ (RC ⊗_{RD} (ι ∘ β_{RD})) : (N ↑_{D}^{C}) ↓_{D}^{C} ↑_{D}^{C} → M′,

corresponding to ι∘β|_{RD}. Similarly let ϵ_{(N ↑_{D}^{C}) ↓_{D}^{C} ↑_{D}^{C}}∘(RC⊗_{RD}γ) : N ↑_{D}^{C} → (N ↑_{D}^{C}) ↓_{D}^{C} ↑_{D}^{C} be the RC-map corresponding to the natural RD-map γ : N → (N ↑_{D}^{C}) ↓_{D}^{C} ↑_{D}^{C} ↓_{D}^{C}, where γ(g) = 1 ⊗ 1 ⊗ g for any g ∈ N. Then we get an RC-map

ϵ_{M′} ∘ (RC ⊗_{RD} (ι ∘ β|_{RD})) ∘ ϵ_{(N ↑_{D}^{C}) ↓_{D}^{C} ↑_{D}^{C}} ∘ (RC ⊗_{RD} γ) : N ↑_{D}^{C} → M′,

from which we construct the RC-map ι′ : M ↪ N ↑_{D}^{C} → M′ as

ϵ_{M′} ∘ (RC ⊗_{RD} (ι ∘ β|_{RD})) ∘ ϵ_{(N ↑_{D}^{C}) ↓_{D}^{C} ↑_{D}^{C}} ∘ (RC ⊗_{RD} γ) ∘ α.

It’s easy, though tedious, to check that π ∘ ι′ = 1_{M}.

In the end we show 4 ⇒ 1. Since ϵ_{M} is surjective, we can form an exact RCsequence 0 → M′′ → M ↓_{D}^{C} ↑_{D}^{C} → M → 0, where M′′ is the kernel of ϵ_{M}. This sequence splits after restriction to RD, since there exists an RD-map ι : M ↓_{D}^{C} → (M ↓_{D}^{C} ↑_{D}^{C}) ↓_{D}^{C} sending M ↓_{D}^{C} isomorphically onto 1⊗_{RD} M ↓_{D}^{C}, and ϵ_{M}|_{RD} ∘ι = 1_{M ↓_{D}^{C}} as RD-maps. Thus the exact RC-sequence splits by hypothesis, that is, ϵ_{M} : M ↓_{D}^{C} ↑_{D}^{C} → M splits. So we’re done.

Suppose M and N are two RC-modules. Then we write M|N if M is isomorphic to a direct summand of N. We collect some useful results below.

Proposition 4.3.3. Let C be a finite EI-category. Then

1. if E ⊂ D are subcategories of C and M is E-projective then M is D-projective;

2. if E ⊂ D are subcategories of C, N is an RD-module which is E-projective, and M is a direct summand of N ↑_{D}^{C}, then M is E-projective;

3. if an RC-module M is D-projective, and it’s projective as RD-module, then M is a projective RC-module.

Proof. Since M is a direct summand of M ↓_{E}^{C}↑_{E}^{C} which can be written as (M ↓_{E}^{C}↑_{E}^{D}) ↑_{D}^{C}, we have M|N ↑_{D}^{C} for an RD-module. So M is D-projective.

Next we prove 2. From N|N ↓_{E}^{D}↑_{E}^{D}, we get

M | N ↑_{D}^{C}| (N ↓_{E}^{D}↑_{E}^{D}) ↑_{D}^{C}= (N ↓_{E}^{D}) ↑_{E}^{C} .

It means M is E-projective.

As for 3, the reason is that M ↓_{D}^{C} must be a direct summand of RD^{n} for some positive cardinal n. Then M ↓_{D}^{C} ↑_{D}^{C} is a direct summand of RD^{n} ↑_{D}^{C} ≅ RC^{n}. Thus M is a direct summand of a free RC-module RC^{n}.

We need the following terminology before introducing our next two results.

Definition 4.3.4. For each RC-module M, we define the M-minimal objects to be those x ∈ Ob C which satisfy the condition that M(y) = 0 if y ≅ x̸ and Hom(y, x) is non-empty. Similarly we can define M-maximal objects.

For example, the \underline{R}-minimal objects are the minimal objects of C, and the \underline{R}- maximal objects are the maximal objects of C. The S_{x,V} -minimal and S_{x,V} -maximal objects are the same: all y ∈ [x].

Lemma 4.3.5. If M is D-projective: M ↓_{D}^{C} ↑_{D}^{C} ≅ M ⊕ M′, then Ob D contains all M-minimal objects, and if furthermore D is full then M′(x) = 0 for all x ∈ Ob D.

Proof. If z is M-minimal and z ∉ Ob D, then M ↓_{D}^{C} ↑_{D}^{C}(z) = ∑_{y∈Ob D} R Hom(y, z) ⊗ M(y) = 0, since z is M-minimal. Therefore M cannot be a direct summand of M ↓_{D}^{C} ↑_{D}^{C} which is a contradiction. Hence Ob D contains all M-minimal objects.

If x ∈ Ob D, then M ↓_{D}^{C} ↑_{D}^{C}(x) ≅ M(x) when D is a full subcategory. It forces M′(x) = 0.

We explain what’s special about these M-minimal objects.

Lemma 4.3.6. Suppose D ⊂ C is a subcategory and M is an RC-module which is D-projective. Then M(x) is Aut_{D}(x)-projective as an R Aut_{C}(x)-module, if x is M-minimal.

Proof. We just evaluate the relation M | M ↓_{D}^{C}↑_{D}^{C} at x, and then the result follows.

If \underline{R} is D-projective, then for any minimal object x ∈ Ob C, \underline{R}(x) = R is Aut_{D}(x)- projective as an R Aut_{C}(x)-module. When R = 𝔽_{p} for some prime p dividing the order of Aut_{C}(x), then Aut_{D}(x) has to contain a Sylow p-subgroup of Aut_{C}(x), by a standard result from the theory of vertices and sources for group algebras, see for instance Alperin [1] and Webb [41].

Note that M(x) is not necessarily an indecomposable R Aut_{C}(x)-module even if M is indecomposable. The following is a simple example.

Example 4.3.7. Let R = 𝔽_{2} and C the poset x → y. Suppose M is the RC-module given by M(x) = R ⊕ R and M(y) = R, where the unique non-isomorphism M(x) → M(y) is defined via (a, b) ↦ a+b. Then it’s easy to verify that M is an indecomposable module. Obviously M(x) = R ⊕ R is decomposable.

It’s time to give some examples for the theory of relative projectivity. The first example is taken from group representation theory.

Example 4.3.8. Let C = Ĝ where G is a finite group. Suppose 𝔽_{p} is a field of characteristic p, a prime which divides the order of G. If M is an indecomposable RC-module with vertex P, then M is projective relative to the subcategory D = P̂.

The second example shows that an RC-module can be projective relative to two distinct proper full subcategories of C.

Example 4.3.9. Let C be the following EI-category

[figure: x; 1_{x}; f; α; y; 1_{y}; g; β; z; 1_{z}; h] ,

along with two full subcategories D and E:

[figure: x; 1_{x}; f; α; y; 1_{y}; g; ;; x; 1_{x}; f] ,

with αf = α, gα = α, βg = β and hβ = β. Let R be a commutative ring with an identity. We can diagrammatically describe the following three modules \underline{R} ↓_{E}↑^{C}

\xymatrix{R \ar[r]^{α} & (α ⊗ R) \ar[r]^{β} & βα ⊗ R} ,

(\underline{R} ↓_{E}↑^{C}) ↓_{D}

\xymatrix{R \ar[r]^{α} & (α ⊗ R)} ,

and (\underline{R} ↓_{E}↑^{C}) ↓_{D}↑^{C}

\xymatrix{R \ar[r]^{α} & (α ⊗ R) \ar[r]^{β} & (βα ⊗ R)} .

So we have \underline{R} ↓_{E}↑^{C}= \underline{R} and \underline{R} ↓_{D}↑^{C}≅ \underline{R}.

The main tool we will use for computing Ext groups Ext_{RC}^{∗}(M, N) is the Eckmann-Shapiro Lemma (Lemma 3.3.6). Here we content ourselves with the special form for a pair of algebras RD ⊂ RC. For convenience, we recall the lemma we stated in Section 3.3. Let M be RC-module. If RC is a right flat RD-module then for any RC-module N, we have

Ext_{RC}^{∗}(M ↑_{D}^{C}, N) ≅ Ext_{RD}^{∗}(M, N ↓_{D}^{C}).

Here is a simple example showing us how the Eckmann-Shapiro Lemma works.

Example 4.3.10. Suppose C is a finite poset with a unique minimal object m. We choose a subposet D containing only the minimal object m. Then it’s easy to verify that D is R-co-taut since RD = R · 1_{m}, and furthermore \underline{R} is D-projective with \underline{R} ↑_{D}^{C}≅ \underline{R}. The Eckmann-Shapiro Lemma implies

Ext_{RC}^{∗}(\underline{R}, N) ≅ Ext_{RD}^{∗}(R, N(m)).

Since RD ≅ R, we have Ext_{RD}^{0}(R, N(m)) = N(m) and Ext_{RD}^{n}(R, N(m)) = 0 for all n > 0. It’s not a surprise because \underline{R} ≅ P_{x,1} is projective.

In the next section, we will show every indecomposable RC-module M is an induced module, i.e., of the form N ↑_{D}^{C} for some subcategory D and an RD-module N. Hence if D is R-co-taut, we can directly apply the Eckmann-Shapiro Lemma to get an isomorphism of Ext groups. In Section 4.6 we shall see that a certain class of EI-categories have some important homological properties determined by subcategories which are indeed posets. When such a category has a unique minimal object, the calculations of higher limits in the preceding example apply to it without any difficulties.

4.4 Vertices and sources

We continue to explore the ideas we presented in last section. From now on, we restrict our attention to the set of full subcategories of a fixed EI-category C. We develop a theory of vertices and sources for category algebras, parallel to that for group algebras, which we describe now. For each indecomposable RC-module M, we associate to it a full subcategory V_{M}, which is unique and will be named the vertex of M. This subcategory V_{M} has the following important properties: M ↓_{V_{M}}, which will be called the source for M, is an indecomposable RV_{M}-module and satisfies M ≅ M ↓_{V_{M}} ↑^{C}.

Since we can also prove N ↑_{D}^{C} is indecomposable if N is an indecomposable RDmodule for a full subcategory D. The properties of V_{M} allow us to parameterize all indecomposable RC-module according to their vertices. Such a parametrization of indecomposable RC-modules will be expressed in terms of a certain equivalence between two module categories. Since every indecomposable RC-module is an induced module, we can get a variation of the Eckmann-Shapiro Lemma : Ext_{RC}^{∗}(M, N) ≅ Ext_{RD}^{∗}(M ↓_{D}^{C}, N ↓_{D}^{C}), which in particular gives a formula for computing higher limits.

Let M be an indecomposable RD-module that is D-projective for some full subcategory D. If D is a disjoint union of several connected components {D_{i}}_{i∈I}, then from M ↓_{D}^{C} ↑_{D}^{C} = ⊕_{i∈I}M ↓_{D_{i}}^{C}↑_{D_{i}}^{C} we know M is really projective relative to some D_{i}. Indeed such a D_{i} has to be unique because it contains all the M-minimal objects. Thus for the rest of this chapter, when we talk about the D-projectivity of some indecomposable M ∈ RC-mod we usually mean that D is connected.

Proposition 4.4.1. If M is D-projective for a connected full subcategory D ⊂ C, then M is generated by its values on D, that is, M ↓_{D}^{C} ↑_{D}^{C} ≅ M.

Proof. Suppose M ↓_{D}^{C} ↑_{D}^{C} = M′ ⊕ M′′ for some RC-module M′, M′′ with M′ ≅ M and M′′(x) = 0 for all x ∈ Ob D. Let’s take y ∉ Ob D and consider M ↓_{D}^{C} ↑_{D}^{C}(y) = M′(y) ⊕ M′′(y). We claim M ↓_{D}^{C} ↑_{D}^{C}(y) = ∑_{y>x∈Ob D} R Hom_{RC}(x, y) ⊗_{RD} M(x) equals M′(y). In fact M′(x) = 1_{D} ⊗ M(x) for all x ∈ Ob D, and given any α ∈ Hom(x, y), α · M′(x) ⊂ M′(y), which means α ⊗ M(x) ⊂ M′(y). When x and α run over all possible choices, we get exactly ∑_{y>x∈Ob D} R Hom_{RC}(x, y) ⊗_{RD} M(x) ⊂ M′(y) which is indeed an equality since the converse direction inclusion is certainly true. Thus the statement is correct.

If H ⊂ G are finite groups and M is an RG-module which is H-projective, then we usually have M ↓_{H}↑^{G}≅ M ⊕ M′, with M′ ≠ 0 in general. We can explain the difference between this and our result by the fact that, regarded as categories, Ĥ isn’t a full subcategories of Ĝ (i.e. Ĥ ≠ Ĝ) in general.

If M is D-projective (D full), then the natural surjection ϵ : M ↓_{D}^{C} ↑_{D}^{C} → M is an isomorphism. Let y ∈ Ob C. From ϵ_{y} : M ↓_{D}^{C} ↑_{D}^{C}(y)→^{≅}M(y) we get

ϵ_{y}(M ↓_{D}^{C} ↑_{D}^{C}(y)) = ϵ_{y}(∑_{x∈Ob D_{≤y}} R Hom_{C}(x, y) ⊗_{RD} M(x))
= ∑_{x∈Ob D_{≤y}} R Hom_{C}(x, y) · M(x)
= M(y).

This implies the RC-module M is generated by its values on objects in D. However, ∑_{x∈Ob D_{≤y}} R Hom_{C}(x, y)·M(x) = M(y) for any y ∈ Ob C\ Ob D doesn’t guarantee M is D-projective. We can consider the category x ⇉ y with two non-isomorphisms and two trivial isomorphisms. Then the trivial module \underline{R} is projective relative to the whole category, not \{x\} although both non-isomorphisms send \underline{R}(x) = R isomorphically to \underline{R}(y) = R.

Proposition 4.4.1 gives us some hints about the existence of the next two important results.

Theorem 4.4.2. Let D ⊂ C be a (connected) full subcategory and N an indecomposable RD-module. Then the RC-module N ↑_{D}^{C} is indecomposable and moreover is D-projective.

Proof. Suppose N ↑_{D}^{C}= N_{1} ⊕ N_{2}, where N_{1}, N_{2} are both non-zero. Then N = N ↑_{D}^{C}↓_{D}^{C}= N_{1} ↓_{D}^{C} ⊕N_{2} ↓_{D}^{C}, and since N is indecomposable we must have N_{1} ↓_{D}^{C}= N and N_{2} ↓_{D}^{C}= 0 (or the other way around). Now that N ↑_{D}^{C} is generated by its values on D implies N_{2} = 0. Hence inductively we can show N ↑_{D}^{C} is indecomposable, and its D-projectivity follows from Definition 4.3.1.

One can compare the above theorem with the Green’s indecomposability theorem in group representation theory (see for instance Alperin [1] or Benson [4]).

Definition 4.4.3. Let x be an object of an EI-category C and H ⊂ Aut_{C}(x). Then we use {x}_{H} to denote the subcategory of C whose object is x and morphisms are elements of H. If H = Aut_{C}(x) then we will use {x}, instead of {x}_{Aut_{C}(x)}, to denote the full subcategory of C with a single object x. We use {[x]} to denote the full subcategory of C consisting of all objects which are isomorphic to x.

Given an x, one can choose the full subcategory {[x]} and use an indecomposable R{[x]}-module N to generate an RC-module N ↑_{{[x]}}^{C}. Then Theorem 4.4.2 asserts that such an induced module is indecomposable. This implies that RC is not of finite representation type if, for some x ∈ Ob C, R Aut_{C}(x) is not. There is a theorem of Higman says that if R = 𝔽_{p} and G is a finite group then 𝔽_{p}G has finite representation type if and only if the Sylow p-subgroups of G are cyclic, see Webb [41].

Proposition 4.4.4. Let M be an indecomposable RC-module which is D-projective for a connected full subcategory D ⊂ C. Then M ↓_{D}^{C} is indecomposable.

Proof. Suppose M ↓_{D}^{C}= M_{1}⊕⋯⊕M_{n} is a direct sum of indecomposable RD-modules. Then M ≅ M ↓_{D}^{C}↑_{D}^{C}= M_{1} ↑_{D}^{C} ⊕ ⋯ ⊕ M_{n} ↑_{D}^{C}. Since M is indecomposable, we have M|M_{i} ↑_{D}^{C} for some index i. This implies M(x)|M_{i} ↑_{D}^{C} (x) = M_{i}(x) for all x ∈ Ob D, hence M ↓_{D}^{C}= M_{i} is indecomposable.

The preceding two results give us an equivalence of two module categories.

Definition 4.4.5. Let D ⊂ C be a connected full subcategory. We define RC_{D}-mod to be the full subcategory of RC-mod consisting of all D-projective RC-modules.

For the sake of simplicity, we write Hom_{RC_{D}}(M, N) for the set of morphisms between two modules M, N ∈ RC_{D}-mod.

Proposition 4.4.6. The functor ↓_{D}^{C} : RC_{D}-mod → RD-mod is an equivalence with ↑_{D}^{C} as its inverse.

Proof. From the previous results we see the two functors are well-defined on objects, while ↓_{D}^{C} ↑_{D}^{C} ≅ Id_{C_{D}} and ↑_{D}^{C} ↓_{D}^{C} ≅ Id_{D}. The actions of induction and restriction on the morphisms are very clear. Furthermore on morphisms we have the following isomorphisms

Hom_{RD}(M ↓_{D}^{C}, N ↓_{D}^{C}) ≅ Hom_{RC_{D}}(M ↓_{D}^{C} ↑_{D}^{C}, N) ≅ Hom_{RC_{D}}(M, N),

and

Hom_{RC_{D}}(M ↑_{D}^{C}, N ↑_{D}^{C}) ≅ Hom_{RD}(M, N ↑_{D}^{C} ↓_{D}^{C}) ≅ Hom_{RD}(M, N).

So ↓_{D}^{C} ↑_{D}^{C} and ↑_{D}^{C} ↓_{D}^{C} are also identities on morphisms, because both M and N are generated by their values on D.

Note that the second isomorphism doesn’t depend on the relative projectivity of the modules, since we always have Hom_{RC}(M ↑_{D}^{C}, N ↑_{D}^{C}) ≅ Hom_{RD}(M, N), as long as D ⊂ C is a full subcategory.

Now we’re ready to develop a theory of vertices and sources for category algebras. The next proposition and its corollary suggest that we should restrict our attention to the subcategories which are ideals in C, in order to give a proper definition of the vertex of an indecomposable module.

Proposition 4.4.7. Suppose D and E are full subcategories of C whose objects form ideals in Ob C. Suppose M is an indecomposable RC-module. Then M ↓_{D}↑^{C}↓_{E}↑^{C}≅ M ↓_{D∩E}↑^{C}.

Proof. We need to consider the structure of _{RE}RC ⨂_{RD} M. Since Ob E forms an ideal in Ob C, we get RC ≅ RE as an RE-module. The only terms in this direct sum on which D is non-zero in the action from the right are the ones where x is in Ob D. Regarded as a right RD-module, RC can be identified with RD = ⨁_{x∈Ob D} R Hom_{C}(x, −). So as an RE-RD-bimodule, RC ≅ ⨁_{x∈Ob(D∩E)} R Hom_{C}(x, −). Thus

RC⨂_{RE}RC⨂_{RD}M ≅ RC⨂_{RE}{⨁_{x∈Ob(D∩E)}RHom_{C}(x, −)}⨂_{RD}M
= RC⨂_{RE}{⨁_{x∈Ob(D∩E)}RHom_{C}(x, −)}⨂_{R(D∩E)}M
≅ RC⨂_{RE}RE⨂_{R(D∩E)}M
≅ RC⨂_{R(D∩E)}M.

We note that this argument does not work for an arbitrary pair of full (R-taut, R-co-taut etc) subcategories relative to which M is projective.

Corollary 4.4.8. Suppose D and E are full subcategories of C which are ideals. Let M be an indecomposable RC-module, which is both D-projective and E-projective. Then M is also D ∩ E-projective. Thus for any indecomposable RC-module M, there exists a unique smallest full subcategory Ṽ_{M}, among all ideals in C, relative to which M is projective.

Proof. We just need to check that D ∩ E forms an ideal in C, and then the results follow from the above lemma.

Obviously, Ṽ_{M} has to be connected, because if Ṽ_{M} = D_{1}∪D_{2}, then M ↓C ; Ṽ_{M}= M ↓_{D_{1}}^{C} ⊕M ↓_{D_{2}}^{C}, and M must be projective relative to one of its connected components, which contradicts with the minimality of Ṽ_{M}.

At this point, one who is familiar with the definition of the vertex for an indecomposable group module might want to define the category Ṽ_{M} to be the “vertex” of M. However, there is a possibility to throw away some objects x ∈ Ṽ_{M} for which M(x) = 0, and subsequently obtain a category smaller than Ṽ_{M} for M, relative to which M is still projective.

Definition 4.4.9. For any RC-module M we define the full subcategory of C, C_{M} to be a category consisting of all y ∈ Ob C with Hom_{C}(x, y) ≠ ∅ for some M-minimal object x. Similarly we define C^{M} to be the full subcategory containing all z ∈ Ob C with Hom(z, x) ≠ ∅ for some M-maximal object x. In particular, we have C_{S_{x,V}} = C_{≥x} and C^{S_{x,V}} = C_{≤x}.

In fact, C_{M} is a co-ideal in C generated by the M-minimal objects, and C^{M} is an ideal in C generated by the M-maximal objects. In particular, we have C\underline{R} = C\underline{R} = C.

Definition 4.4.10. The full subcategory V_{M} = Ṽ_{M} ∩ C_{M} is called the vertex of M.

Note that V_{M} is connected but is not necessarily an ideal in C, though it’s an ideal in C_{M}. Also note that by Proposition 4.4.1, every indecomposable RC-module is an induced module. Next we want to get an alternative description of the vertex of M, free of the use of C_{M}.

Definition 4.4.11. Let D ⊂ C be a subcategory. Then D is said to be convex if whenever there is a sequence of morphisms x→^{α}y→^{β}z in C with x, z ∈ Ob D, then both α and β are in Mor(D).

Ideals and co-ideals in C are full convex subcategories. Let M be an indecomposable RC-module. Then its vertex V_{M} is convex. Note that in general a convex subcategory D doesn’t have to be full. Since the intersection of two convex subcategories is still convex, if not empty, it’s natural to define the convex hull of a subcategory D of C as the smallest convex subcategory containing D.

Let’s make a detour to get an interesting property of convex subcategories before we return to the developing of our own theory. When we have a short exact sequence of groups 1 → N → G → H → 1. Then any RH-module can be lifted to an RGmodule via the projection. But in general an RN-module cannot be regarded as an RG-module. From next result, it seems the reason is that N̂ is not convex in Ĝ (actually the only convex subcategory of Ĝ is itself).

Proposition 4.4.12. Let D ⊂ C be a convex subcategory. Then there exists a functor τ_{D} from RD-mod to RC-mod.

Proof. Suppose M ∈ RD-mod. Let τ_{D}(M) = M as R-modules. We shall define an RC-module structure on τ_{D}(M). For any α ∈ Mor(C), we require α·τ_{D}(M) = 0 unless α ∈ Mor(D). This gives a well-defined RC-module structure on τ_{D}(M) because for any α, β ∈ Mor(C) we must have (βα) · τ_{D}(M) = β · (α · τ_{D}(M)) since D is convex. The action of τ_{D} on homomorphisms is the same as the identity.

The above result give us a different parametrization of RC-modules through the convex hull of the support of each RC-module. Note that the condition “convex” cannot be dropped. Let’s take C to be a poset of three objects x ⟶ y ⟶ z and choose the subposet D : x ⟶ z. Then the trivial RD-module \underline{R} is not an RCmodule. We should warn readers that the vertex of τ_{V_{M}} (M ↓_{V_{M}}^{C}) is usually larger than V_{M}, which means τ_{V_{M}} (M ↓_{V_{M}}^{C}) is not V_{M}-projective in general. One can jump to Propositions 4.4.18 and 4.4.19 for some examples (where we give the vertices of indecomposable projective and atomic modules).

Proposition 4.4.13. Let M be an indecomposable RC-module and D a full subcategory of C. Then the following statements are equivalent:

1. D is the vertex of M;

2. D is the smallest ideal in C_{M}, relative to which M is projective;

3. D is the smallest full convex subcategory of C, relative to which M is projective.

Proof. 1 ⇒ 2 : If D = V_{M}, then by definition D is full convex and M is D-projective. Suppose E is an ideal in C_{M}, relative to which M is projective. We claim D ⊂ E. In fact, D is an ideal in C_{M}, and so is D ∩ E. We can naturally extend D ∩ E to an ideal { \atop D ∩ E} ⊂ Ṽ_{M} in C, relative to which M is projective. But then by definition we have

Ṽ_{M} = { \atop D ∩ E}, which implies D = Ṽ_{M} ∩ C_{M} = { \atop D ∩ E} ∩ C_{M} = D ∩ E.

2 ⇒ 3 : Let E be a full convex subcategory for which M is E-projective. Then E contains all M-minimal objects, and thus E ∩ C_{M} must be an ideal in C_{M}. Since as an RC_{M}-module M is projective relative to E ∩ C_{M}, we have D ⊂ E.

3 ⇒ 1 : Let E be an ideal in C, relative to which M is projective. Then E ∩ C_{M} is a full convex subcategory in C, which means D ⊂ E ∩ C_{M}. We can take E to be Ṽ_{M}, and this results in an inclusion D ⊂ V_{M}, which can be shown to be an equality by extend D to an ideal in Ṽ_{M}.

The term “convex” in part 3 of above proposition can not be removed, see Remark 4.4.15 below.

Proposition 4.4.14. Let D be a connected full subcategory of C and N an indecomposable RD-module with the vertex V_{N} ⊂ D. Then the indecomposable RC-module M = N ↑_{D}^{C} is V_{N}-projective. If V_{N} is a (connected and full) convex subcategory of C, then V_{M} = V_{N}.

If M is an indecomposable RC-module whose vertex is V_{M}, and D is a connected full subcategory containing V_{M}, then M ↓_{D}^{C} is an indecomposable RD-module whose vertex is V_{M}.

Proof. The first statement makes sense because of Theorem 4.4.2, and it is by Proposition 4.3.3 (2) we know V_{M} ⊂ V_{N}. After we prove the second part, we can show V_{M} is exactly V_{N}, if V_{N} is a convex subcategory of C.

The second statement makes sense because of Proposition 4.4.4. Suppose M ↓_{D}^{C} has vertex E′ ⊂ D. Since M ≅ (M ↓_{D}^{C}) ↓_{V_{M}}^{D} ↑_{V_{M}}^{C} ≅ (M ↓_{V_{M}}^{C} ↑_{V_{M}}^{D}) ↑_{D}^{C}, we obtain M ↓_{D}^{C} ≅ M ↓_{V_{M}}^{C} ↑_{V_{M}}^{D} = (M ↓_{D}^{C}) ↓_{V_{M}}^{D} ↑_{V_{M}}^{D}. Hence M ↓_{D}^{C} is V_{M}-projective and E′ ⊂ V_{M}. But by the first part, M ↓_{D}^{C} ↑_{D}^{C} has a vertex that is contained in E′. Since M ↓_{D}^{C} ↑_{D}^{C} ≅ M, we must have V_{M} ⊂ E′.

Now we go back to finish proving part 1. Let V_{M} be the vertex of M = N ↑_{D}^{C}, from part 2 and M ↓_{D}^{C} = N ↑_{D}^{C} ↓_{D}^{C} we get desired equality V_{M} = V_{N} because M ↓_{D}^{C} still has vertex V_{M} and N ↑_{D}^{C} ↓_{D}^{C} = N has vertex V_{N}.

Remark 4.4.15. In the first part of Proposition 4.4.14, if V_{N} is not a convex subcategory in C, then it’s not necessarily true that V_{M} = V_{N}, where M = N ↑_{D}^{C}. One can check Example 4.5.8 in the next section, where we have a pair of categories D ⊂ C. If we choose the RD-module N = \underline{R}, then V_{N} = D, which is not convex in C. The induced module M = N ↑_{D}^{C} = \underline{R} ↑_{D}^{C} is isomorphic to the trivial RC-module \underline{R}, which is V_{N}-projective and whose vertex is shown to be V_{M} = C.

Now we define the source. Since M ↓_{V_{M}}^{C} ↑_{V_{M}}^{C} ≅ M and M ↓_{V_{M}}^{C} is indecomposable, M is determined up to isomorphism by the indecomposable RV_{M}-module M ↓_{V_{M}}^{C}.

Definition 4.4.16. Suppose M is an indecomposable RC-module with the vertex V_{M}. Then M ↓_{V_{M}} is called the source for M.

It’s time to give some examples, which can be helpful for understanding the established results.

Example 4.4.17. Recall in Example 4.3.9 we had the category C

[figure: x; 1_{x}; f; α; y; 1_{y}; g; β; z; 1_{z}; h] ,

and two full subcategories D and E:

[figure: x; 1_{x}; f; α; y; 1_{y}; g; ;; x; 1_{x}; f] ,

with αf = α, gα = α, βg = β and hβ = β. Both D and E are ideals in C and we showed \underline{R} ↓_{E}↑^{C}= (\underline{R} ↓_{E}↑^{C}) ↓_{D}↑^{C}≅ \underline{R}. Obviously the vertex of \underline{R} is V\underline{R} = E, and the source for it is the RE-module \underline{R}.

Next we provide two more examples, which give us the vertices and sources for two kinds of important RC-modules: indecomposable projectives and indecomposable atomic modules (e.g. simple modules). Recall that we denote by {[x]} the full subcategory of C (equivalent to { \atop Aut_{C}(x)}) consisting of objects in the isomorphism class [x].

Proposition 4.4.18. The vertex of the indecomposable projective module P_{x,V} is {[x]}. The source for P_{x,V} is P_{V} = P_{x,V} ↓_{[x]}^{C}, the projective cover of V as an R{[x]}- module.

Proof. For simplicity, we can assume C is skeletal. Then we can easily check that P_{x,V} ↓_{C_{≤x}}^{C}↑_{C_{≤x}}^{C}= P_{V} ↑_{C_{≤x}}^{C}= RC ⊗_{RC_{≤x}} R Aut(x) · e_{x,V} = RC ⊗_{RC_{≤x}} e_{x,V} ≅ P_{x,V}. The isomorphism is defined via multiplication α ⊗ e_{x,V} ↦ α · e_{x,V}, which is obviously surjective and is indeed injective since e_{x,V} is an idempotent. Now C_{P_{x,V}} = C_{≥x}. So by definition the vertex of P_{x,V} is C_{≤x} ∩ C_{≥x} = {x}.

Recall that a module M is called atomic if M(y)(= 1_{y} · M) ≠ 0 if and only if y ≅ x for an object x ∈ Ob C. For such a module M, it is indecomposable if and only if M(x) is indecomposable as an R Aut(x)-module.

Proposition 4.4.19. Let M be an indecomposable atomic module concentrated on [x] ⊂ Ob C. Let D be the full subcategory of C_{≥x} whose object set consists of [x] and those y ≅ x̸ which satisfy the condition that Irr_{C}(x, y) ≠ ∅. Then M is D-projective, and V_{M} is the smallest convex full subcategory that contains D. The source for M is itself (but regarded as an RV_{M}-module).

Proof. It’s easy to verify that M ↓_{D}^{C} ↑_{D}^{C} ≅ M and there is no proper full subcategory of D having the same property, because if y ∈ Ob C_{≥x}, Irr_{C}(x, y) ≠ ∅ and y ∉ Ob D then 0 ≠ R Irr_{C}(x, y) ⊗_{RD} M(x) ⊂ R Hom(x, y) ⊗_{RD} M(x) ⊂ M ↓_{D}^{C}↑_{D}^{C} (y), which contradicts with the fact M ↓_{D}^{C}↑_{D}^{C} (y) ≅ M(y) = 0.

From above, we see D ⊂ V_{M}. By 4.4.13 (3), V_{M} is exactly the smallest full convex subcategory containing D.

The last example first discusses the vertex and source for an indecomposable module other than indecomposable projective or atomic modules, and then describes the representation type of RC.

Example 4.4.20. Given a category C

[figure: x; 1_{x}; f; i_{1}; i_{2}; y; 1_{y}; g] ,

with i_{1}f = i_{1}, i_{2}f = i_{2}, gi_{1} = i_{2} and gi_{2} = i_{1}. Let R = 𝔽_{2} be a field of characteristic 2. We consider the indecomposable module M such that M(x) = 𝔽_{2} and M(y) = 𝔽_{2}⊕𝔽_{2}. The maps i_{1}, i_{2} send M(x) = 𝔽_{2} to the first and the second component, respectively, of M(y) = 𝔽_{2} ⊕ 𝔽_{2}, and g interchanges the two entries of 𝔽_{2} ⊕ 𝔽_{2} (it’s easy to verify these define a functor M : C → R-mod, and it gives rise to a 𝔽_{2}C-module which isn’t projective or simple).

We can see M is an induced module, which is not projective or simple. It has D = \{x\} ≅ { \atop Aut(x)} as its vertex with source M(x) = 𝔽_{2}, since 𝔽_{2} ↑_{D}^{C}(y) = (i_{1}⊗𝔽_{2})⊕(i_{2}⊗𝔽_{2}) as 𝔽_{2} Aut(y)-module.

This category C has exactly three full subcategories: {x}, {y} and C itself, all of which are connected and convex. By Proposition 4.4.6, the indecomposable RCmodules can be put into three distinct classes, each of them is determined by one of the three subcategories. If we consider {x}, then any indecomposable RC-module, which is {x}-projective, is generated by its value on x and corresponds uniquely to an indecomposable 𝔽_{2} Aut_{C}(x)(≅ 𝔽_{2}C_{2})-module. There are finitely many of them, up to isomorphism, because any indecomposable 𝔽_{2}C_{2}-modules is isomorphic to either 𝔽_{2}C_{2} or 𝔽_{2}, and RC ⊗_{R Aut_{C}(x)} − is exact. Similar statement can be posed for indecomposable RC-modules which are {y}-projective. There exist modules which are not projective relative to either of the two proper subcategories of C. The typical ones are the atomic modules concentrated on x, e.g., S_{x,1}. However, there are modules other than the atomic ones. For instance, \underline{𝔽_{2}} has vertex V\underline{𝔽_{2}} = C. In general if M has vertex C then M(x) ≠ {0} because otherwise it’ll be atomic, concentrated on y with vertex {y}. We show 𝔽_{2}C has infinite representation type by constructing infinitely many non-isomorphic modules whose vertices are C. Define for any n ∈ N an (indecomposable) 𝔽_{2}C-module M_{n} such that M_{n}(x) = [𝔽_{2} Aut_{C}(x)]^{n} and M(y) = 𝔽_{2}i_{1} + 𝔽_{2}i_{2}. It’s certainly not {y}-projective, and is not {x}-projective when n > 1 because [RC ⊗_{R Aut_{C}(x)} M(x)](y) has dimension 2n. If n ≠ m are both bigger than 1 then M_{n} ≅ M̸_{m} since M_{n}(x) ≅ M̸_{m}(x).

Note that the construction at the end of last paragraph can be generalized to a class of EI-categories which contains a convex subcategory consisting of two non-isomorphic objects x and y such that Aut_{C}(x) acts freely and transitively on Hom_{C}(x, y). This kind of categories are of infinitely representation type.

As we mentioned in last section, the Eckmann-Shapiro type lemma has a better form when D ⊂ C is full.

Lemma 4.4.21. Let M be an RC-module which is D-projective for a full subcategory of C. If D is R-co-taut in C, then

Ext_{RC}^{∗}(M, N) ≅ Ext_{RD}^{∗}(M ↓_{D}^{C}, N ↓_{D}^{C}).

In particular we get Ext_{RC}^{∗}(M, N) ≅ Ext_{RC_{M}}^{∗} (M ↓_{C_{M}}^{C} , N ↓_{C_{M}}^{C}). If M = \underline{R} is Dprojective and D is full and R-co-taut in C then for any RC-module N, we have

\underleftarrow{lim}_{C}^{∗}N ≅ \underleftarrow{lim}_{D}^{∗}N ↓_{D}^{C} .

Hence if V\underline{R} is R-co-taut in C then \underleftarrow{lim}_{C}^{∗} N ≅ \underleftarrow{lim}_{V\underline{R}}^{∗} N ↓_{V\underline{R}} for any RC-module N.

Proof. We can replace the M in Lemma 3.3.6 by M ↓_{D}^{C}, and then use the fact that M ↓_{D}^{C} ↑_{D}^{C} ≅ M. The special cases for higher limits come from the fact that \underline{R} ↓_{D}^{C} = \underline{R}.

Note that an R-co-taut V\underline{R} is in general a rather big subcategory in C, because it contains all minimal objects, and if furthermore it’s a co-ideal then we must have V\underline{R} = C. But in some special cases we may have V\underline{R} really small. See our further investigations of V\underline{R} in the next section.

Corollary 4.4.22. Suppose C has a unique isomorphism class of minimal objects [x] which forms an R-co-taut full subcategory \{[x]\} in C. If \{[x]\} equals V\underline{R}, then we have \underleftarrow{lim}_{C}^{n} M ≅ Ext_{R Aut(x)}^{n}(R, M(x)) for any RC-module M.

When we go to the section on categories with subobjects, we will see that any such category with a unique minimal object x will have \{x\} = V\underline{R}, which is indeed R-co-taut in C, and hence we can apply the above result. This can be thought as a generalization of the result for a poset with an initial object (Example 4.3.11). We conclude the applications of the theory of vertices and sources with an isomorphism of cohomology rings.

Proposition 4.4.23. Let D ⊂ C be a full subcategory, relative to which M is projective. Suppose R is a field or a complete discrete valuation ring and RC is a right flat RD-module. Then there is a ring isomorphism between Ext_{RC}^{∗}(M, M) and Ext_{RD}^{∗}(M ↓_{D}^{C}, M ↓_{D}^{C}).

Proof. Let 0 → M ↓_{D}^{C} → N_{n−1} → ⋯ → N_{0} → M ↓_{D}^{C} → 0 represent an element of Ext_{RD}^{n}(M ↓_{D}^{C}, M ↓_{D}^{C}) for some positive integer n. Then it gives rise to an element of Ext_{RC}^{n}(M, M) vis induction 0 → M → N_{n−1} ↑_{D}^{C} → ⋯ → N_{0} ↑_{D}^{C} → M → 0, since M ↓_{D}^{C} ↑_{D}^{C} ≅ M and RC is a right flat RD-module. But this element of Ext_{RC}^{n}(M, M) restricts back to the given element of Ext_{RD}^{n}(M ↓_{D}^{C}, M ↓_{D}^{C}). Hence the composite of these two maps Ext_{RD}^{n}(M ↓_{D}^{C}, M ↓_{D}^{C}) → Ext_{RC}^{n}(M, M) → Ext_{RD}^{n}(M ↓_{D}^{C}, M ↓_{D}^{C}) is the identity, which implies the first map Ext_{RD}^{n}(M ↓_{D}^{C}, M ↓_{D}^{C}) → Ext_{RC}^{n}(M, M) is injective. Since we know these two Ext groups are isomorphic (Lemma 4.4.21), this map has to be bijective. Now it’s easy to check that this map also respects the Yoneda splice, and it defines a ring isomorphism.

4.5 Structure of the vertex of the trivial module

Because of the special interests in the trivial module \underline{R}, we try to obtain a precise description of D, relative to which \underline{R} is projective. Suppose \underline{R} is D-projective and D is full. Based on Lemma 3.3.3 (2) and the fact \underline{R} ≅ \underline{R} ↓_{D}^{C} ↑_{D}^{C}, we’re able to give a simple topological characterization of D (Proposition 4.5.1). However, this characterization is not easy to use as a criterion to separate such categories D from others, because it requires understanding of the structures of the overcategories associated to the inclusion ι : D ↪ C. So we turn to characterize these D by the “weakly essential objects” which have been introduced and considered in the case of G-posets by authors such as Quillen [34], Puig [32] and Bouc [5]. These weakly essential objects are easy to find and we prove D must contain them if \underline{R} is D-projective.

Recall that Lemma 3.3.3 (2) says if all overcategories associated to the functor ι : D ↪ C are connected, then \underline{R} is D-projective. Our first proposition will ensure us the converse of that is also true if D is a full subcategory.

Proposition 4.5.1. Let ι : D ↪ C be the inclusion of a full subcategory. Then \underline{R} is D-projective if and only if ι ↓_{y} is connected for any y ∈ Ob C.

Proof. By Lemma 3.3.3 (2), we know if ι ↓_{y} is connected for any y ∈ Ob C then \underline{R} ↓_{D}^{C} ↑_{D}^{C} ≅ K(\underline{R}) ≅ \underline{R}.

On the other hand, if \underline{R} is D-projective we must have \underline{R} ≅ \underline{R} ↓_{D}^{C} ↑_{D}^{C} because D is a full subcategory. From K(\underline{R}) ≅ \underline{R} ↓_{D}^{C} ↑_{D}^{C} we get K(\underline{R})(y) = \underrightarrow{lim}_{ι↓_{y}} \underline{R} ≅ \underline{R}(y) = R. Hence ι ↓_{y} has to be connected.

The proposition is not true for subcategories which are not full in C. A simple example will be a group G with a proper Sylow-p subgroup P, both of which are regraded as categories with a single object ∗. When R = 𝔽_{p}, \underline{R} is P̂-projective, while P̂ ↓_{∗} has [G : P] connected components hence is not connected. When C is a poset, the overcategory ι ↓_{y} can be identified with D_{≤y} for any y ∈ Ob C. When y ∈ Ob D, D_{≤y} has a terminal object and thus is contractible. When y ∈ Ob C\ Ob D, we have D_{≤y} = D_{<y}, and the connectedness of the latter has been studied using the so-called “weakly essential objects” by Puig [32], Quillen [34] and Bouc [5]. In fact, the weakly essential objects are quite useful in analyzing the topological structures of subposets of C which contain them. Because of this and that every EI-category has an underlying poset, we would like to see if they can be used in our case. It turns out the answer is positive.

Recall that for an EI-category C and an object x ∈ Ob C, we denote by C_{<x} the full subcategory whose objects are {y ∈ Ob C| Hom_{C}(y, x) ≠ ∅, y ≅ x̸}.

Definition 4.5.2. An object x ∈ Ob C is weakly essential if the full subcategory C_{<x} is empty or has more than one component. The full subcategory of C, consisting of all weakly essential objects, is named Wess_{0}(C). There is a larger full subcategory Wess(C) consists of objects x ∈ Ob C which satisfy the condition that the full subcategory C_{<x} are not contractible.

Obviously the minimal objects of C are weakly essential, contained in Wess_{0}(C). The notations Wess_{0}(C) and Wess(C) belong to Symonds [37], though the first was studied by Puig [32] and the second by Quillen [34] and Bouc (see [4] Definition 6.6.2). We’ve seen in last section that all minimal objects are contained in V\underline{R}. This will be strengthened shortly by showing Wess_{0}(C) ⊂ V\underline{R}. Furthermore when C is a poset, V\underline{R} is totally determined by Wess_{0}(C).

Let \underline{R} ≅ \underline{R} ↓_{D}^{C} ↑_{D}^{C}. Since \underline{R} is D-projective, it is generated by its values on D. If we take an object y ∈ Ob C\ Ob D, then \underline{R}(y) is generated by \underline{R}’s values on D_{<y}. This forces D_{<y} to be connected, because otherwise \underline{R}(y) would have an R-rank bigger than one which is impossible.

Lemma 4.5.3. Let D be a connected full subcategory of C. Suppose \underline{R} is D-projective. Then for any y ∉ Ob D, D_{<y} is non-empty and connected.

Proof. Obviously D_{<y} is non-empty, because otherwise it’ll contradict with the definition of D. If D_{<y} were disconnected, then we prove

\underline{R} ↓_{D}^{C}↑_{D}^{C} (y) = ∑_{x∈Ob D} R Hom(x, y) ⊗ \underline{R}(x)

is a direct sum of at least two non-zero summands. Hence a contradiction since \underline{R} ↓_{D}^{C}↑_{D}^{C} (y) = R.

If D_{<y} were disconnected, then \underline{R} ↓_{D}^{C}↑_{D}^{C} (y) = ∑_{x∈Ob D} R Hom(x, y) ⊗ \underline{R}(x) contains two elements α ⊗ 1 and β ⊗ 1, where α ∈ Hom(x_{1}, y) and β ∈ Hom(x_{2}, y) for x_{1}, x_{2} from different components of D_{<y}. Let’s assume x_{1}, x_{2} minimal. Now, since \underline{R} ↓_{D}^{C}↑_{D}^{C} (y) = R has rank 1, we have rα⊗_{RD}1 = β⊗_{RD}1 for some r ∈ R. But it means that rαγ⊗_{RD}1_{x_{2}} = β⊗_{RD}1_{x_{2}} (or rα⊗_{RD}1_{x_{1}} = βγ⊗_{RD}1_{x_{1}}) for some γ ∈ R Hom(x_{2}, x_{1}) (or in R Hom(x_{1}, x_{2})), which implies Hom(x_{2}, x_{1}) (or Hom(x_{1}, x_{2})) is non-empty. So x_{1} and x_{2} belong to the same connected component which is a contradiction.

The above result leads us to a corollary which is more convenient to use than Proposition 4.5.1 as a criterion for the D-projectivity of \underline{R}.

Corollary 4.5.4. Let D be a full subcategory of C, relative to which \underline{R} is projective. Then Wess_{0}(C) ⊂ D. In particular Wess_{0}(C) ⊂ V\underline{R}.

Proof. We show if y ∉ Ob D, then y ∉ Wess_{0}(C). First of all since D contains all minimal objects in C, C_{<y} can not be empty because y ∉ Ob D. We claim C_{<y} is connected. Assume the opposite. Since D_{<y} ⊂ C_{<y} and C_{<y} is disconnected, by Lemma 4.5.2 D_{<y} must lie in only one of the components of C_{<y}. But then C_{<y} contains at least one minimal object x which doesn’t belong to Ob D_{<y}. Actually, x isn’t in Ob D either, because otherwise D_{<y} is disconnected. Now x ∉ Ob D contradicts with the fact that D contains all minimal objects.

The following result extends a theorem of Bouc (Proposition 4.6.7) on posets, saying that Wess_{0}(C) and C have the same numbers of connected components. Recall that there is a poset P (C) associated to every EI-category C. When C is finite we call the maximal length of chains of non-isomorphisms in P (C) the length of C, denoted by l(C).

Lemma 4.5.5. Let D ⊂ C be a full subcategory. If Wess_{0}(C) ⊂ D, then Wess_{0}(C) ⊂ D ⊂ C induce bijections on connected components.

Proof. Firstly we assume C is connected and prove D has to be connected too. We do induction on the length of C. Assume D = D_{1} ∪ D_{2}. Then for any y ∈ Ob C, from Wess_{0}(C)_{<y} = Wess_{0}(C_{<y}) and Wess_{0}(C_{<y}) ⊂ D_{<y} ⊂ C_{<y} we deduce that D_{<y} = D_{1}_{<y} ∪ D_{2}_{<y} is connected since l(C_{<y}) < l(C) and C_{<y} is connected. Thus for every non-minimal y ∈ Ob C, one of the D_{i}_{<y}, i = 1, 2 must be empty. Let Y_{1} be the set of maximal objects in C such that D_{2}_{<y} = ∅ and Y_{2} be the set of maximal objects such that D_{1}_{<y} = ∅. If both Y_{1} and Y_{2} were not empty, then for y_{1} ∈ Y_{1} and y_{2} ∈ Y_{2} we must have C_{<y_{1}}∩C_{<y_{2}} = ∅ because otherwise it would contain a minimal object z which belongs to Wess_{0}(C) ⊂ D_{1} ∪ D_{2}. Hence z ∈ D_{1}_{<y_{2}} (or z ∈ D_{2}_{<y_{1}}), a contradiction to the definition of Y_{1} and Y_{2}. But Y_{1} ≠ ∅ and Y_{2} ≠ ∅ leads to an equality C = C_{1} ∪ C_{2}, where C_{i} = {x ∈ Ob C | x ∈ Ob C_{<y}, y ∈ Y_{i}}, i = 1, 2. This is impossible. Thus either Y_{1} or Y_{2} is empty, which means either D_{1} or D_{2} is empty. Hence D is connected.

Secondly suppose C = C_{1} ∪ C_{2} ∪ ⋯ ∪ C_{n}, where C_{i}, i = 1, ⋯, n are connected components. We show D has the same number of components. Let D = D_{1} ∪ D_{2} ∪ ⋯ ∪ D_{m} as a union of connected components. Then every D_{j} is contained in some C_{i}, while no two distinct D_{j}’s are contained in the same C_{i} because of the above arguments and the fact that Wess_{0}(C) = Wess_{0}(C_{1}) ∪ ⋯ ∪ Wess_{0}(C_{n}). This forces m = n and induces a natural bijection on the connected components of D and C. We’re done.

If C is connected and \underline{R} is D-projective, then D is connected because we always have Wess_{0}(C) ⊂ D.

Corollary 4.5.6. Let C be an EI-category and D a connected full subcategory, relative to which \underline{R} is projective. For every y ∉ Wess_{0}(C), Wess_{0}(C)_{<y}, Wess(C)_{<y} and D_{<y} are all connected.

Proof. By definition of Wess_{0}(C), C_{<y} is always connected. The results follow from the inclusions Wess_{0}(C)_{<y} ⊂ D_{<y} ⊂ C_{<y} and Wess_{0}(C)_{<y} ⊂ Wess(C)_{<y} ⊂ C_{<y}, combined with Lemma 4.5.5 and the fact that Wess_{0}(C) ⊂ D.

When C is a poset and D is a subposet containing Wess_{0}(C), by ι ↓_{y}≅ D_{≤y} and Proposition 4.5.1, we see \underline{R} is D-projective, because if y ∈ Ob D then ι ↓_{y} is contractible and if y ∈ Ob C\ Ob D, Wess_{0}(C_{<y}) ⊂ D_{<y} ⊂ C_{<y} implies D_{<y} hence ι ↓_{y} is connected. Thus as a special case \underline{R} is Wess_{0}(C)-projective and Wess_{0}(C) becomes the smallest full category, relative to which \underline{R} is projective. This means V\underline{R} is the “convex hull” of Wess_{0}(C), which is uniquely determined by Wess_{0}(C). We give a sufficient condition on the connectedness of overcategories and the relative projectivity of \underline{R}.

Proposition 4.5.7. Let C be an EI-category and D a full subcategory containing Wess_{0}(C). Then every ι ↓_{y}, y ∈ Ob C, is connected if for any pair of objects x ∈ Ob D and y ∈ Ob C\ Ob D, Aut_{C}(x) acts transitively on Hom_{C}(x, y). Hence \underline{R} is D-projective.

Proof. According to Lemma 3.3.3 (2) and Proposition 4.5.1 we only need to show the overcategory ι ↓_{y} is connected for each y ∈ Ob C. When y ∈ Ob D, ι ↓_{y} is connected because there is only one isomorphism class of maximal objects, of the form (y, g) where g ∈ Aut(y). Now we assume y ∈ Ob C\ Ob D. The objects of ι ↓_{y} are of the form (x, α), where x ∈ Ob D_{<y} and α ∈ Hom_{C}(x, y). If for every pair of objects x ∈ Ob D and y ∈ Ob C\ Ob D, Aut_{C}(x) acts transitively on Hom_{C}(x, y), then ι ↓_{y} has the same underlying poset as D_{<y} because (x, α) ≅ (x, β) for any two morphisms α, β ∈ Hom_{C}(x, y). Now D_{<y} is connected if and only if its underlying poset is connected. Since Wess_{0}(C) ⊂ D, we know D_{<y} is connected for any y ∈ Ob C by Corollary 4.5.6.

In some sense, the above lemma tells us what objects in C are needed to be kept in D in order to make \underline{R} D-projective, which could be used to find V\underline{R}. For instance in Example 4.5.8 below, we have a category with three objects: x, y and z. Since the automorphism groups Aut_{C}(x) and Aut_{C}(y) do not act transitively on the corresponding sets of non-isomorphisms ending at z, z is contained in V\underline{R}. Furthermore since Wess_{0}(C) = \{x\}, V\underline{R} must contain x and z. In fact the smallest full subcategory D ⊂ C, relative to which \underline{R} is projective, consisting of exactly the two objects x and z, while V\underline{R} = C because it needs to be convex. Note that in the example, we also show that in general there is no containment between Wess(C) and D, even if \underline{R} is D-projective.

Example 4.5.8. Suppose C is the following category

[figure: x; 1_{x}; f; τ; y; 1_{y}; α; β; z; 1_{z}; h] ,

with R arbitrary, Aut_{C}(x) acting trivially on Hom_{C}(x, y) = {τ}, Aut_{C}(z) interchanging α and β, and βτ ≠ ατ. From direct calculations, we can see that V\underline{R} = C. The following category is Wess(C)

[figure: x; 1_{x}; f; τ; y; 1_{y}] .

We can check \underline{R} is projective relative to the full subcategory D

[figure: x; 1_{x}; f; ατ; βτ; z; 1_{z}; h] ,

which is not convex and is the smallest full subcategory among all those relative to which \underline{R} is projective. Comparing these categories, we get Wess(C) ⊂ V\underline{R} = C, D ⊄ Wess(C) and Wess(C) ⊄ D. Note that Wess_{0}(C) = \{x\} is contained in V\underline{R}, Wess(C) and D.

We end this section with a little bit further discussion of the structure of the overcategories. The first result describes the maximal objects in the overcategories when D ⊂ C is full. It somehow reveals certain connections between representationtheoretic and topological aspects of the overcategories, because we will see the maximal objects are related to the irreducible morphisms. The second and the third results continue the discussion for D ⊂ C that is not full.

Lemma 4.5.9. Let D be a full subcategory of C along with the inclusion ι : D ↪ C. Then

1. if y ∈ Ob D, an object (x, α) ∈ Ob ι ↑^{y} is maximal if and only if x ≅ y and α ∈ Is_{C}(x, y); and

2. if y ∈ Ob C\ Ob D, an object (x, α) ∈ Ob ι ↑^{y} is maximal if and only if α ∈ coIrr_{D}(x, y).

Proof. Just use the definitions of ι ↓_{y} and coIrr_{D}(x, y).

This result can be used to establish Proposition 4.5.1. When y ∈ Ob D, we know that ι ↓_{y} is always contractible, since its skeleton has a terminal object, i.e. the isomorphism class of (y, 1_{y}). If \underline{R} is D-projective, then for every y ∈ Ob C\ Ob D we have (RC ⊗_{RD} \underline{R})(y) = ∑_{y>x∈Ob D} R Hom(x, y) ⊗_{RD} 1_{x} = R. Furthermore we can write ∑_{y>x∈Ob D} R coIrr_{D}(x, y) ⊗_{RD} 1_{x} = R. Now by Lemma 4.5.9, (x, α) ∈ ι ↓_{y}, for α ∈ coIrr_{D}(x, y), is a maximal object, and indeed all maximal objects are of this form. If ι ↓_{y} were disconnected, then there are two maximal objects, say (x, α) and (z, β), belonging to two different components of ι ↓_{y}, which implies that α ⊗_{RD} 1_{x} and β ⊗_{RD} 1_{z} are linearly independent. Hence ∑_{y>x∈Ob D} R coIrr_{D}(x, y) ⊗_{RD} 1_{x} has rank > 1, a contradiction.

If D ⊂ C is a full subcategory, ι is the inclusion and y ∈ Ob D, then the skeleton of ι ↓_{y} has a terminal object given by (y, 1_{y}), hence ι ↓_{y} is connected (contractible indeed). When D is not full in C, ι ↓_{y} is not necessarily connected, and there usually exists more than one maximal object in each connected component. We count the number of isomorphism classes of maximal objects in ι ↓_{y} if y ∈ Ob D.

Lemma 4.5.10. Let D be a subcategory of C along with the inclusion ι : D ↪ C. Suppose y ∈ Ob D. Then the number of isomorphism classes of maximal objects of ι ↓_{y} is exactly | Aut_{C}(y) : Aut_{D}(y)|. Furthermore the isomorphism class of any (y, g), g ∈ Aut_{C}(y), contains exactly | Aut_{D}(y)| maximal objects, and each maximal object has a trivial automorphism group.

Proof. Without loss of generality, let’s assume C is skeletal. Then (y, g) ≅ (y, g′) if and only if there are in the same (left) coset of [Aut_{C}(y)/ Aut_{D}(y)].

These maximal objects may lie in different connected components of ι ↓_{y}. When D ⊂ C isn’t full, it’s not easy to count the number of the connected components, even if y ∈ Ob D. A connected component may contain two non-isomorphic maximal objects (y, g) and (y, g′) as long as there exists a (x, α) and β, β′ ∈ Hom(x, y) such that α = gβ and α = g′β′. From gβ = g′β′ we get β = g^{−1}g′β′ where g^{−1}g′ ∈ Aut_{C}(y)\ Aut_{D}(y). This proves the following.

Lemma 4.5.11. Let ι : D ↪ C be an inclusion and y ∈ Ob D. Then some connected component of ι ↓_{y} contains more than one isomorphism class of maximal objects if and only if there exists an x ∈ Ob D_{<y} and a β ∈ Hom(x, y), so that | Aut_{C}(y) · β| is strictly bigger than | Aut_{D}(y) · β|.

It’s easy to see that Aut_{C}(y) acts on the set of connected components of ι ↓_{y} via g : (x, α) ↦ (x, gα), because it permutes the set of maximal objects. When y ∈ Ob D, Aut_{C}(y) acts transitively and the stabilizer of this action contains Aut_{D}(y). When y ∉ Ob D, the action of Aut_{C}(y) is not transitively in general, since if there are two components then Aut_{C}(y) cannot send a maximal object in one component to one in another. This is so because the maximal objects in ι ↓_{y} are those objects (x, α) with α ∈ coIrr_{D}(x, y) and Aut_{C}(y) coIrr_{D}(x, y) = coIrr_{D}(x, y). It’s not clear how to tell whether or not ι ↓_{y} is connected, when y ∉ Ob D. But obviously when x ≅ x̸′, (x, α) ≅ (x′, α′) , and when x ≅ x′, (x, α) ≅ (x′, α′) if and only if there is an f ∈ Is_{D}(x, x′) such that fα′ = α.

4.6 Categories with subobjects

This part of the work grows out of our observation that if C is a finite poset then V\underline{R} is determined by Wess_{0}(C), see comments after Corollary 4.5.6. Let ι : D ↪ C be the inclusion of a connected full subcategory. Proposition 4.5.1 tells us that the RC-module \underline{R} is D-projective if and only if every overcategory ι ↓_{y}, y ∈ Ob C, is connected. We mentioned before that when C is a poset, the overcategory ι ↓_{y} can be identified with a subposet D_{≤y} for each y ∈ Ob C. Since the connectedness and the contractibility of such subposets determine useful topological properties of the poset C through Wess_{0}(C) and Wess(C), we cannot help wondering if there are other categories such that the overcategories associated to the inclusion of full subcategories are isomorphic to posets, and to what extent these posets are related to the weakly essential objects? In this section, our focus goes to the categories with subobjects, which were introduced and studied by Oliver [31] and include posets as special cases. A category with subobjects is usually dubbed as (C, I), where C is a small category and I is a subcategory (see definition below). We show any finite category with subobjects (C, I) is EI, and I is indeed a poset. One of the key properties of an EIcategory with subobjects is that all its morphisms are monomorphic. Given ι : D ↪ C we deduce that the overcategories in question can be identified with subposets of I.

This means some homological properties of EI-categories with subobjects are governed by the subposets of I so that we can extend a Bouc’s result to these categories. Investigation in this direction goes beyond what prompts it, and we’ve recorded some properties of EI-categories with subobjects which are not necessarily related to our studies in the preceding sections. But as an explicit and generalized proposition of our opening statement, we can say, given a finite category with subobjects (C, I), Wess_{0}(I) uniquely determines the smallest full subcategory, relative to which \underline{R} is projective. In fact V\underline{R} can be described as the “full convex hull” of Wess_{0}(I).

Following Oliver’s original definition, a category with subobjects is a pair of categories I ⊂ C such that Ob I = Ob C, and such that the following two conditions are satisfied:

1. | Hom_{I}(x, y)| ≤ 1 for any pair of objects x, y; and

2. each morphism α ∈ Hom_{C}(x, y) can be written in a unique way as a composite α = α_{0} · f, where f ∈ Is_{C}(x, x′) for some x′, and α_{0} ∈ Hom_{I}(x′, y).

When C is EI, the morphisms in Mor(I) are necessarily monomorphic. From a category-theoretic viewpoint, if there is a monomorphism from x to y, two objects in some category C, then x is called a subobject of y. If furthermore the category C is concrete, i.e. every object has an underlying set structure, then the meaning of the term “subobject” is obvious.

Example 4.6.1. Let mSets be the category of finite sets, whose morphisms are all monomorphic functions. It’s natural to choose I_{mSets} to be the (connected) subcategory whose morphisms are inclusions, which is indeed a connected poset since mSets is an EI-category. But it’s easy to see there are other choices of I. In (mSets, I_{mSets}), we fix two sets A and B, of sizes i and j, respectively, where i, j are two positive integers with i ≤ j. Then any α ∈ Hom_{mSets}(A, B) can be uniquely factored as α_{0}f, where f ∈ Is_{mSets}(A, α(A)) and α_{0} ∈ Hom_{mSets}(α(A), B) is the inclusion (in I_{mSets}). Note that | Hom_{mSets}(A, B)| = C_{j}^{i}·i! (here C_{j}^{i} means j choose i), and | End_{mSets}(A)| = | Aut_{mSets}(A)| = |S_{i}| = i!. A surprising fact is that Hom_{I_{mSets}}(x, y) can be empty even if Hom_{mSets}(x, y) is not. Consider two sets A and B, and suppose Hom_{mSets}(A, B) is non-empty. Then as objects in I_{mSets}, Hom_{I_{mSets}}(A, B) ≠ ∅ if and only if A is a subset of B. Particularly two objects do not have to be isomorphic in I_{mSets}, even if they are isomorphic in mSets, since two finite sets of the same cardinality are isomorphic in mSets, but not in I_{mSets} unless they are equal. Because mSets is an EI-category, there is no non-isomorphism between two isomorphic objects. Hence the picture of I_{mSets} is very clear.

In this example I_{mSets} is connected, which is not necessarily true as a general rule. For instance we can consider the (infinite) full subcategory mSets_{≤B}. There is a (finite) full subcategory mSets_{⊆B} ⊂ mSets_{≤B}, whose objects are the subsets of B. These two categories are categories with subobjects with distinct I’s. If we name the subcategories as I_{⊆B} and I_{≤B}, then I_{≤B} is the disjoint union of I_{⊂B} and infinitely many discrete objects, which are isomorphic, but not equal, to an object in mSets_{⊆B}.

We say (C, I) is a skeletal category with subobjects, if C is skeletal (this implies I is skeletal as well). It is not true that if (C, I) is a category with subobjects, then the skeleton of C can be made into a category with subobjects. A convenient example will be the category mSets we just mentioned. The skeleton of mSets, \overline{mSets}, can be constructed by taking one set from each isomorphism class and then form a full subcategory with these sets. Hence its object set consists of finite sets of pairwise distinct sizes. This category cannot be made into a category with subobjects since End_{mSets}(A) acts freely, but not transitively, on Hom\overline{mSets}(A, B) = Hom_{mSets}(A, B) whenever |A| < |B| (hence there is no way to find a suitable I\overline{mSets}).

If we assume C is an EI-category, then we naturally have a definition of EIcategories with subobjects. When (C, I) is an EI-category with subobjects, the subcategory I becomes a poset (Lemma 4.6.2) whose structure is usually different from the underlying poset P(C) of C (one can check this for the category mSets). We note that Jackowski and Słomińska introduced EI-categories with quotients in their paper [25], which is the dual concept to EI-categories with subobjects. All results in this section have their counterparts for EI-categories with quotients, because the opposite of an EI-category with quotients is an EI-category with subobjects, and vice versa. Lemma 4.6.2. Let (C, I) be a category with subobjects. Then the endomorphism group of each x ∈ Ob I is precisely {1_{x}}, and the only isomorphisms in Mor(I) are these 1_{x}, for all x ∈ Ob I. If furthermore C is EI, I is a poset.

Proof. It’s easy to see End_{I}(x) = Aut_{I}(x) = {1_{x}} since | End_{I}(x)| ≤ 1 and 1_{x} ∈ End_{I}(x). Now we show Is_{I}(x, y) = ∅ if x ≅ y (in C) and x ≠ y. If there were an α_{0} ∈ Is_{I}(x, y), then we would have two distinct factorizations α_{0} = 1_{y}α_{0} = α_{0}1_{x}, a contradiction to the definition of a category with subobjects.

If C is EI and Hom_{I}(x, y) ≠ ∅ for some x ≠ y, we show Hom_{I}(y, x) = ∅. This implies I is a poset. Indeed if there exists α ∈ Hom_{I}(x, y) and β ∈ Hom_{I}(y, x), then αβ = 1_{y} and βα = 1_{x} in I (and C). Hence α and β are isomorphisms, which is impossible because from above we know Is_{I}(x, y) = ∅ if x ≠ y.

The above result implies that if (C, I) is a (not necessarily EI) category with subobjects, a non-empty set Hom_{I}(x, y) (if there is any) will consist of a non-isomorphism, given that x ≅ y and x ≠ y in C. This is the situation we want to avoid. Our next result says, if C satisfies a particular condition, then we don’t have to worry about it.

Proposition 4.6.3. Let (C, I) be a category with subobjects. If every isomorphism class of objects in C is finite, then C is an EI-category.

Proof. Suppose there exists an α ∈ End_{C}(x)\ Aut_{C}(x) for some x ∈ Ob C. We show this assumption leads to a contradiction. By definition of a category with subobjects, α = α_{0}f for some α_{0} ∈ Hom_{I}(x′, x) and f ∈ Is_{C}(x, x′), where x′ ≅ x in C. We claim x′ ≠ x. If x = x′, then α_{0} ∈ Hom_{I}(x, x) = End_{I}(x) = {1_{x}} by preceding lemma. But then α = f is an isomorphism, a contradiction to our assumption. From x ≠ x′, we know α_{0} ∈ Hom_{I}(x′, x) is not an isomorphism.

Now since x′ ≅ x, there exists a β ∈ End_{C}(x′)\ Aut_{C}(x′), and β = β_{0}g for some β_{0} ∈ Hom_{I}(x′′, x′) and g ∈ Is_{C}(x′, x′′). As is shown in last paragraph, x′′ ≠ x′ and β_{0} is not an isomorphism. In fact x′′ cannot be x either, since if they were equal, we would have two morphisms in I: β_{0} : x → x′ and α_{0} : x′ → x, which implies α_{0}β_{0} = 1_{x} and β_{0}α_{0} = 1_{x′}. The two equalities assert that α_{0} and β_{0} are isomorphisms, inverse to each other, hence a contradiction to our assumptions on α_{0} and β_{0}. Thus any two of x, x′ and x′′ are not equal, and we can find a third non-isomorphism γ ∈ End_{C}(x′′)\ Aut_{C}(x′′) so that we can repeat what we’ve done for β ∈ End_{C}(x′)\ Aut_{C}(x′). Gradually, we’re going to produce an infinite list of isomorphic objects in C, x, x′, x′′, ⋯, while any two of them are not equal. This leads to a contradiction since we assume [x] is finite. Thus there is no such α ∈ End_{C}(x)\ Aut_{C}(x) for any x ∈ Ob C, and then End_{C}(x) = Aut_{C}(x) for all x ∈ Ob C, or C is EI.

Categories with subobjects constructed from subgroups of (finite) groups have the property that each isomorphism class of objects is finite. Hence they are always EI. More generally, any finite category with subobjects (C, I) is EI, and I is a poset. When (C, I) is skeletal, C is automatically an EI-category as a corollary of last result. However, we can prove it directly using the definition of a category with subobjects.

Corollary 4.6.4. Let (C, I) be a skeletal category with subobjects. Then C is an EI-category, and (C, I) is an EI-category with subobjects. In fact I ≅ P (C), the underlying poset of C.

Proof. Let γ ∈ End_{C}(x) for a fixed object x. We show γ is an isomorphism. In fact, by the second condition in the definition of a category with subobjects, γ admits a factorization as γ_{0}f, where γ_{0} ∈ Hom_{I}(x′, x) and f ∈ Is_{C}(x, x′). Since C is skeletal, we have x = x′, hence γ_{0} ∈ Hom_{I}(x, x) = End_{I}(x) = {1_{x}}, by the first condition of the definition of a category with subobjects. Then γ = f is an isomorphism of x, and hence C is EI.

Also by the definition of a category with subobjects, when C is skeletal the following statement has to be true: if x, y ∈ Ob C and Hom_{C}(x, y) ≠ ∅, then | Hom_{I}(x, y)| = 1. This implies I ≅ P (C).

We give some useful characterizations of EI-categories with subobjects.

Lemma 4.6.5. Let (C, I) be an EI-category with subobjects. Then

1. all morphisms in C are monomorphisms;

2. if C is a finite category with subobjects, then the number of objects z ∈ [x] for which Hom_{I}(z, y) ≠ ∅ equals | Hom_{C}(x, y)|/| Aut_{C}(x)|, for any y ≅ x̸ such that Hom_{C}(x, y) ≠ ∅;

3. if C is skeletal Aut_{C}(x) acts freely and transitively on Hom_{C}(x, y), for any pair of objects x, y ∈ Ob C.

Proof. To prove 1, let α ∈ Hom_{C}(y, z) and γ, γ′ ∈ Hom_{C}(x, y) satisfying αγ = αγ′. We show γ = γ′, and then α is monomorphic.

Since α = α_{0}f for some α_{0} ∈ Hom_{I}(y′, z) and f ∈ Is_{C}(y, y′) (where y′ ≅ y in C), we can write αγ = α_{0}(fγ) = α_{0}(γ_{0}g) and αγ′ = α_{0}(fγ′) = α_{0}(γ′_{0}g′) where g ∈ Is_{C}(x, x′), γ_{0} ∈ Hom_{I}(x′, y′), g′ ∈ Is_{C}(x, x′′), γ′_{0} ∈ Hom_{I}(x′′, y′) for some x′ ≅ x and x′′ ≅ x, which satisfy the equalities fγ = γ_{0}g and fγ′ = γ′_{0}g′. By condition (2) in the definition of a category with subobjects and (α_{0}γ_{0})g = (α_{0}γ′_{0})g′, we get g = g′ hence γ_{0} = γ′_{0} because we must have x′ = x′′ and as a consequence Hom_{I}(x′, y′) = Hom_{I}(x′′, y′). So γ = f^{−1}(γ_{0}g) = f^{−1}(γ′_{0}g′) = γ′, which means α is a monomorphism.

Now suppose C is finite. Then statement 2 makes sense because every [x] is finite and the morphism sets, such as Hom_{C}(x, y), are always finite. Since C is EI by Proposition 4.6.3, Aut_{C}(x) acts freely on Hom_{C}(x, y) due to part 1. There are n = | Hom_{C}(x, y)|/| Aut_{C}(x)| orbits of morphisms. Let’s write the orbits as \overline{α_{1}}, ⋯, \overline{α_{n}}, given by the representatives α_{1}, ⋯, α_{n} ∈ Hom_{C}(x, y). Note that if α_{i} = α_{i0}f for some α_{i0} ∈ Hom_{I}(x′, y) and f ∈ Is_{C}(x, x′), then any other representative α′_{i}, satisfying \overline{α_{i}} = \overline{α′_{i}}, can be written as α′_{i} = α_{i0}(fg) where g ∈ Aut_{C}(x) and α_{i}g = α′_{i}. We claim if i ≠ j, then α_{i} and α_{j} must factor through distinct x′ and x′′ which are isomorphic to x. Let’s say α_{i} = α_{i0}f and α_{j} = α_{j0}f′. If x′ = x′′, then α_{i0} = α_{j0}. Hence α_{i} = α_{i0}f = α_{j0}f = α_{j0}f′(f^{′−1}f) = α_{j}(f^{′−1}f), a contradiction. Thus the set A_{xy} = {z ∈ [x]| Hom_{I}(z, y) ≠ ∅} has size at least n = | Hom_{C}(x, y)|/| Aut_{C}(x)|, and we show |A_{xy}| cannot be larger than this number. Suppose x′, x′′ ∈ A_{xy}. Let f ∈ Is_{C}(x, x′), f′ ∈ Is_{C}(x, x′′), α_{i0} ∈ Hom_{I}(x′, y) and α_{j0} ∈ Hom_{I}(x′′, y). Then the two morphism sets α_{i0}f Aut_{C}(x) and α_{j0}f′ Aut_{C}(x) are mutually exclusive, because if α_{i0}fg = α_{j0}f′g′ for g, g′ ∈ Aut_{C}(x) we would obtain α_{i0} = α_{j0}f′g′g^{−1}f^{−1}, which is impossible. If we choose for every x_{i} ∈ A_{xy} an isomorphism f_{i} ∈ Is_{C}(x, x_{i}), and assume Hom_{I}(x_{i}, y) = {α_{i0}}. Then Hom_{C}(x, y) is a disjoint union of sets ∐_{A_{xy}} α_{i0}f_{i} Aut_{C}(x), each of which has the same size | Aut_{C}(x)|. So we must have |A_{xy}| = n.

Part 3 is true by definition.

Let (C, I) be a category with subobjects. Suppose D ⊂ C is a full subcategory satisfying the condition that if x ∈ Ob D then [x] ⊂ Ob D. We can naturally make D into a category with subobjects (D, I ∩D). Then we can talk about full subcategories of a category with subobjects. In the rest of this section, we will consider the subposets Wess(I) and Wess_{0}(I) of I. We show finite categories with subobjects have some similar homological properties to certain subposets of I, and thus Wess_{0}(I) and Wess(I) may be used in a similar way as Bouc has done for posets. It’s well-known that when C is a poset and ι : D ↪ C is an inclusion, the overcategory ι ↓_{y} can be identified with the subposet D_{≤y}, and this is the key to prove a Bouc’s result by Quillen’s Theorem A. The next lemma generalizes this result to skeletal categories with subobjects. In Section 3.3 we’ve seen that for a pair of categories ι : D ↪ C, if D is a full subcategory and y ∈ Ob D, ι ↓_{y} is contractible because it has a terminal object (y, 1_{y}). When C is an EI-category with subobjects and y ∈ Ob C\ Ob D, we can also say something about the overcategory ι ↓_{y}.

Lemma 4.6.6. Let (C, I) be an EI-category with subobjects, and ι : D ↪ C a full subcategory. Then for any y ∈ Ob C\ Ob D, the skeleton of ι ↓_{y} is isomorphic to a poset, which can be identified with the poset I_{<y} ∩ D. If D has a unique minimal object, then I_{<y} ∩ D, if not empty, has an initial object and is contractible. If C is skeletal, we have I_{<y} ∩ D ≅ P (D_{<y}).

Proof. If we fix an x ∈ Ob D_{<y} then every object (x′, α′) ∈ ι ↓_{y} with x′ ≅ x is isomorphic to some (x_{i}, α_{i}), where x_{i} ≅ x and α_{i} ∈ Hom_{I}(x_{i}, y). Since (x_{i}, α_{i}) ≅ (x_{j}, α_{j}) if and only if x_{i} = x_{j}, the skeleton of ι ↓_{y} is isomorphic to the full subcategory consisting of objects {(x, α)|x ∈ Ob D_{<y}, α ∈ Hom_{I}(x, y)}. Using the definition of a category with subobjects, it’s easy to see the full subcategory is a poset, and is isomorphic to I_{<y} ∩ D by our assumption.

When D has a unique minimal object, so does I_{<y} ∩ D. Hence it’s contractible because in the poset the unique minimal object is indeed an initial object. If C is skeletal, every isomorphism class of objects contains only one object. So the identification I_{<y} ∩ D ≅ P (D_{<y}) follows.

Definition 4.6.7. Let (C, I) be a finite category with subobjects. Then we denote the two full subcategories of C which share the same object sets with Wess_{0}(I) and Wess(I), respectively, by D_{Wess_{0}} and D_{Wess}. Obviously D_{Wess_{0}} ⊂ D_{Wess}.

We comment here that in general Wess_{0}(C) ⊊ D_{Wess_{0}} and Wess(C) ⊊ D_{Wess}. But when (C, I) is skeletal, we do have Wess_{0}(C) = D_{Wess_{0}} and Wess(C) = D_{Wess}. Our next two propositions show the importance of these two new full subcategories of C.

Proposition 4.6.8. Let C be a finite category with subobjects. Then D_{Wess_{0}} is the smallest full subcategory among all full subcategories of C, relative to which \underline{R} is projective. Consequently, \underline{R} is projective relative to D_{Wess}, and V\underline{R} is the smallest convex subcategory (or ideal) of C that contains D_{Wess_{0}}.

Proof. By Proposition 4.5.1 and Lemma 4.6.4, we know \underline{R} is D_{Wess_{0}}-projective. If D contains D_{Wess} then \underline{R} is D-projective. On the other hand if D_{Wess} ⊄ D, then by Lemma 4.6.4 again, there exists an overcategory associated with ι : D ↪ C which is disconnected. Thus \underline{R} won’t be D-projective.

Our next result is a generalization of Bouc’s result on finite posets (see [5] Proposition 6.6.5, or [37] Proposition 3.10). Let I be a finite poset and I′ a subposet. Bouc proved if Wess(I) ⊂ I′ ⊂ I then the inclusions induce homotopy equivalences of the classifying spaces of these posets.

Proposition 4.6.9. Let (C, I) be a finite category with subobjects. Given a full subcategory D with D_{Wess} ⊂ D, the inclusions |D_{Wess}| ⊂ |D| ⊂ |C| induce homotopy equivalences.

Proof. Following Bouc’s idea, we’re going to use Quillen’s Theorem A to prove D_{Wess} ⊂ D induces an equivalence. Let’s take any object y ∈ Ob D. We want to show the overcategory category ι ↓_{y} is always contractible, where ι : D_{Wess} ↪ D is the inclusion. If y ∈ Ob D_{Wess}, then this category has a terminal object (y, 1_{y}), and hence is contractible. If y ∉ Ob D_{Wess}, we claim ι ↓_{y} is still contractible. By our assumption and Lemma 4.6.4, we know the skeleton of ι ↓_{y} is isomorphic to the poset (I ∩ D)_{<y} ∩ Wess(I), because (D_{Wess}, Wess(I)) ⊂ (D, I ∩ D) is a full subcategory. Since (I ∩ D)_{<y} ∩ Wess(I) = Wess(I_{<y}) ⊂ I_{<y} and I_{<y} is contractible by definition, from Bouc’s original result we have Wess(I)_{<y} contractible. Hence so is ι ↓_{y} and we’re done.

The above two propositions result in the following reduction of higher limits.

Corollary 4.6.10. Let (C, I) be a category with subobjects and D a full subcategory satisfying either one of the following conditions:

1. D_{Wess} ⊂ D; or

2. D_{Wess_{0}} ⊂ D and RC is a right flat RD-module,

then we have \underleftarrow{lim}_{C}^{∗} N ≅ \underleftarrow{lim}_{D}^{∗} N ↓_{D}^{C}. Note that if N = \underline{R}, this isomorphism gives rise to a ring isomorphism.

Proof. If the condition in (1) is satisfied, then every overcategory associated to the inclusion ι : D ↪ C is contractible as is shown in Proposition 4.6.6. So we can use Corollary 3.2.6 to establish the isomorphisms.

If the conditions in (2) are satisfied, then we can use Proposition 4.6.5 and Lemma 4.4.19 to get the isomorphism.

Example 4.6.11. Let R = 𝔽_{2} and C be the following EI-category

[figure: x; 1_{x}; g; α; β; y; 1_{y}; h] ,

where αg = β and βg = α. We’re not going to use other morphisms since we can let y have an arbitrary automorphism group, which don’t affect the higher limits. First of all, it’s easy to see that C is a category with subobjects, where I can be chosen to be the poset x→^{α}y. Since Wess_{0}(I) = Wess(I) consists a single object x, D_{Wess_{0}} = D_{Wess} are equal to following full subcategory.

[figure: x; 1_{x}; g]

Since V\underline{𝔽_{2}} is the smallest convex subcategory containing \{x\}, V\underline{𝔽_{2}} = \{x\}, which can be verified to be 𝔽_{2}-co-taut in C. Then using the above corollary we deduce that \underleftarrow{lim}_{\{x\}}^{n} ↓_{\{x\}}^{C} − ≅ \underleftarrow{lim}_{C}^{n} −, and as a special case we have \underleftarrow{lim}_{\{x\}}^{n} \underline{𝔽_{2}} ↓_{\{x\}}^{C}≅ \underleftarrow{lim}_{C}^{n} \underline{𝔽_{2}} = 𝔽_{2}. It’s interesting to note that Wess_{0}(C) ⊂ Wess(C) = C.

Resolutions and their applications

Using the results in last chapter we can establish an Eckmann-Shapiro type lemma

Ext_{RC}^{∗}(M, N) ≅ Ext_{RC_{M}}^{∗} (M, N)

for any RC-modules M and N, because C_{M} is a co-ideal in C relative to which M is projective. This can be shown in a more straightforward way: if P → M → 0 is the minimal projective resolution, then every P_{n} should have C_{P_{n}} ⊂ C_{M} which immediately results in the above isomorphism. In this chapter we study (minimal) projective resolutions of RC-modules. We will see that not only the support of M but the support of N affects the computation of Ext_{RC}^{∗}(M, N). As an example, we show how to use projective resolutions to give interesting interpretations of groups Ext_{RC}^{∗}(S_{x,V} , S_{y,W}). Finally, the characterization of the minimal projective resolutions of RC-modules gives us some ideas about computing various homological dimensions.

5.1 The minimal projective resolution of an RC-

module

Fix an M ∈ RC-mod, we describe its minimal projective resolution, which will help us to remove the objects beyond M-minimal and N-maximal objects which are “irrelevant with respect to computing Ext_{RC}^{∗}(M, N)”. One can check Definition 4.4.9 for the meanings of C_{M} and C^{M} for any M ∈ RC-mod. As in last chapter, the base ring R is assumed to be a field or a complete discrete valuation ring.

Lemma 5.1.1. Suppose M is an RC-module. If P_{M} ≅ ⊕_{y,U}P_{y,U} is the projective cover of M, then every such y belongs to C_{M}, which means that P_{M} is supported on C_{M}. Thus M/Rad M is a semi-simple module which is isomorphic to a direct sum of some simple modules supported on C_{M}.

If D is an ideal in C such that D ∩ C_{M} ≠ ∅, then P_{M} ↓_{D}^{C} is the projective cover of M ↓_{D}^{C}. Particularly, if x is an M-minimal object, then P_{M}(x) is the projective cover of M(x).

Proof. We only have to show that any indecomposable projective module P_{y′,U′}, with y′ ∉ Ob C_{M} and Hom_{C}(y′, x) ≠ ∅ for some M-minimal x, isn’t isomorphic to a direct summand of P_{M}. If it were a summand, then the restriction of the defining (essential) surjection π : P_{M} → M satisfying the following commutative diagram for all f ∈ Hom(y′, x)

\xymatrix{P_{y′,U′}(x) \ar[r]^{π_{x}} & M(x) \\ P_{y′,U′}(y′) \ar[u]^{P_{y′,U′}(f)} \ar[r]_{π_{y′}} & M(y′) = 0. \ar[u]_{M(f)}}

It forces π_{x} to be the zero map. Hence π restricted on P_{y′,U′} is trivial which is a contradiction to the definition of the projective cover of a module.

Now let D be an ideal in C with D ∩ C_{M} ≠ ∅. Then by our result in last chapter P_{M} ↓_{D}^{C} is a projective RD-module which admits a surjection onto M ↓_{D}^{C}. So we just have to show P_{M} ↓_{D}^{C} is the smallest among modules satisfying these properties. Let π : P_{M} → M be the natural (essential) surjection. Then after restriction we obtain a surjection π|_{D} : P_{M} ↓_{D}^{C} → M ↓_{D}^{C}. Suppose P_{M} ≅ [⊕_{x∈Ob D}P_{x,V}] ⊕ [⊕_{y∉Ob D}P_{y,U}]. Then by the structure theorem of projective modules P_{y,U} ↓_{D}^{C} = 0 and P_{x,V} ↓_{D}^{C} = P̃_{x,V}, where P̃_{x,V} is a projective RD-module whose values on z ∈ Ob D are the same as those of P_{x,V}, and zero otherwise. If P_{M} ↓_{D}^{C} were not the projective cover of M ↓_{D}^{C}, by the standard property of the projective cover of a module, Res_{D}^{C} P_{M} would be a direct summand P̃_{1} ⊕ P̃_{2}, where P_{1} and P_{2} are two direct summands of P_{M} such that P̃_{1} is the projective cover of Res_{D}^{C} M and P̃_{2} ≠ 0. Since π|_{D} sends P̃_{1} surjectively onto M ↓_{D}^{C} and P̃_{2} to 0, this implies π(P_{2}) = 0 because P_{2} is generated by its value on some object in D. Thus we obtain a contradiction to the minimality of P_{M}.

Now let x be an M-minimal object. Then C_{≤x} is an ideal in C and C_{≤x} ∩ C_{M} = {[x]} ≠ ∅. Hence P_{M} ↓_{C_{≤x}}^{C} is a projective cover of M ↓_{C_{≤x}}^{C}. But x is a maximal object in C_{≤x}. We get immediately P_{M}(x) is the projective cover of M(x).

With the above lemma we can go on to describe the minimal projective resolution of an arbitrary RC-module. Given an RC-module M and its projective cover P_{M}, from the previous lemma we know P_{M} ≅ ⊕_{y,U}P_{y,U} with y ∈ Ob C_{M}. When we look at the minimal resolution of M

P_{M} : ⋯ → P_{n} → ⋯ → P_{1} → P_{0} → M → 0,

we know P_{0} is simply P_{M} and P_{1} is the projective cover of K_{0}, the kernel of the map P_{0} = P_{M} → M. Since C_{K_{0}} ⊂ C_{P_{0}} ⊂ C_{M}, we have C_{P_{1}} ⊂ C_{M} too by the lemma. Hence we conclude the following result on the minimal projective resolution by repeating the same argument for every P_{n}.

Corollary 5.1.2. Let M be an RC-module and P_{M} its minimal projective resolution. Then P_{M} is supported on C_{M}. Suppose D is an ideal in C. Then P_{M} ↓_{D}^{C} is the minimal projective resolution of M ↓_{D}^{C}.

If M is an atomic module whose support is {[x]} for some x ∈ Ob C. Then in the minimal projective resolution of M, P_{1} ≅ ⊕_{y,U}P_{y,U} with y ≅ x or Irr_{C}(x, y) ≠ ∅ if y ≅ x̸.

Proof. The first statement is a direct consequence of the preceding lemma and the discussion after it, and so we prove the second part here. Indeed, it’s earlier to see this using a larger projective resolution of M. Suppose P̃_{M} is constructed in this way: P̃_{0} = (RC ·1_{x})^{n} for some positive integer n, and P̃_{1} is the projective cover of the kernel K̃_{0} of the map P̃_{0} → M → 0. Then K̃_{0} contains all non-isomorphisms in P̃_{0}, and thus Rad(RC) · K̃_{0} = Rad(K̃_{0}) contains all reducible morphisms in P̃_{0}. This implies P̃_{1}, as the projective cover of K̃_{0}, is isomorphic to a direct sum of the form ⊕_{y,U}P_{y,U}, where y ≅ x or y ≅ x̸ and Irr_{C}(x, y) ≠ ∅. Since P_{M} is the minimal projective resolution of M, P_{M} must be isomorphic to a direct summand of P̃_{M}. In particular, P_{1} is isomorphic to a direct summand of P̃_{1}. Hence the statement follows.

Using our results on relative projectivity, Lemma 5.1.1 can be strengthened.

Proposition 5.1.3. Let M be an RC-module which is D-projective for a full subcategory D. Then M/Rad(M) has a support contained in D. As a consequence, the projective cover P_{M} is D-projective. proof. Since M is D-projective, for any y ∈ Ob C\ Ob D, we have

M(y) = ∑_{x∈Ob D_{≤y}} RHom_{C}(x, y) · M(x) ⊂ RC · M = Rad(M).

Hence M/Rad(M) has a support contained in D. This implies P_{M} ≅ ⊕_{y∈Ob D}P_{y,U}, and then P_{M} is D-projective because D contains the vertex of every P_{y,U}.

Note that if M is indecomposable, then M and P_{M} usually have different vertices. One can consider P_{x,V} → S_{x,V} when they aren’t equal. From the same example we can see the above proposition cannot be strengthened: M being D-projective doesn’t imply M/Rad M is D-projective. Furthermore the kernel of the surjection P_{M} → M does not have to be D-projective if M is. We can construct a category C with two objects x and y, in which Aut_{C}(x) ≅ C_{2}, Aut_{C}(y) = {1_{y}} and Hom_{C}(x, y) = {α}. Suppose R = 𝔽_{2}. Then the trivial module \underline{𝔽_{2}} is \{x\}-projective and has projective cover P_{x,1} = 𝔽_{2}C · 1_{x}. The kernel of the map 𝔽_{2}C · 1_{x} → \underline{𝔽_{2}} is isomorphic to (𝔽_{2})_{x}, a dimension 1 atomic module concentrated on x, whose vertex is equal to C, not {x}. This example implies that, given a D-projective module M and its minimal resolution P_{M} → M → 0, we cannot expect all P_{n} to be D-projective, except that D = C_{M}.

Next we show that C_{M} and C^{N} have some impact on computing Ext_{RC}^{∗}(M, N).

Proposition 5.1.4. Given two RC-modules M and N, we have

Ext_{RC}^{∗}(M, N) = Ext_{RC_{M}^{N}}^{∗} (M ↓_{C_{M}^{N}}^{C} , N ↓_{C_{M}^{N}}^{C}),

where C_{M}^{N} = C_{M} ∩ C^{N}.

Proof. We take the minimal RC-resolution of M

P_{M} : ⋯ → P_{1} → P_{0} → M → 0.

It is supported on C_{M} hence an RC_{M}-resolution of M ↓_{C_{M}}^{C}. It’s obvious that

Hom_{RC}(P_{n}, N) = Hom_{RC_{M}^{N}} (P_{n} ↓_{C_{M}^{N}}^{C} , N ↓_{C_{M}^{N}}^{C}),

and it passes to an isomorphism of cochain complexes

{Hom_{RC}(P_{M}, N)} ≅ {Hom_{RC_{M}^{N}} (P_{M} ↓_{C_{M}^{N}}^{C} , N ↓_{C_{M}^{N}}^{C})}.

If we can show P_{M} ↓_{C_{M}^{N}}^{C} is still a projective resolution of M ↓_{C_{M}^{N}}^{C} as an RC_{M}^{N}-module then we’re done. But this comes from Lemma 5.1.1 since C_{M}^{N} is an ideal in C_{M}.

Corollary 5.1.5. Let C_{x}^{y} = C_{≥x} ∩ C_{≤y}. Then

Ext_{RC}^{∗}(S_{x,V} , S_{y,W}) ≅ Ext_{RC_{x}^{y}}^{∗}(S_{x,V} , S_{y,W}).

In particular we have Ext_{RC}^{∗}(S_{x,V} , S_{x,W}) ≅ Ext_{R Aut(x)}^{∗}(V, W).

Note that in the above corollary if Hom(x, y) = ∅ then C_{x}^{y} = ∅ hence Ext_{RC}^{∗}(S_{x,V} , S_{y,W}) vanish. Proposition 5.1.6. Let S_{x,V} and S_{y,W} be two simple RC-modules. If Irr(x, y) = ∅, then Ext_{RC}^{1}(S_{x,V} , S_{y,W}) = 0.

Proof. Suppose ⋯ → P_{1} → P_{0} → S_{x,V} → 0 is the minimal projective resolution of S_{x,V}. Then since S_{y,W} is simple, we have Ext_{RC}^{1}(S_{x,V} , S_{y,W}) ≅ Hom(P_{1}, S_{y,w}). But P_{1} ≅ ⊕P_{z,V} for z ≅ x or z ∈ Ob C_{≥x} such that Irr(x, z) ≠ ∅. By our assumption the degree one Ext group has to be zero because P_{y,W} cannot be a direct summand of P_{1}.

5.2 Computing Ext_{RC}^{n}(S_{x,R}, S_{y,R})

As an interesting example, we try to calculate the groups Ext_{RC}^{∗}(S_{x,V} , S_{y,W}). By Corollary 5.1.4, we only need to deal with the situation when Hom_{C}(x, y) ≠ ∅. When the two simples are S_{x,R} and S_{x,R}, we can give rather explicit interpretations of the Ext groups. For simplicity, in this section we assume C is skeletal and has x as the unique minimal object and y as the unique maximal object, i.e. C = C_{x}^{y}.

Consider the trivial RC-module \underline{R}. It has a unique maximal submodule, denoted by \underline{R}_{>x}, which takes value 0 at x, and R elsewhere. The quotient module is obviously isomorphic to the simple module S_{x,R}. So we have a short exact sequence of RCmodules

0 → \underline{R}_{>x} →^{i} \underline{R} → S_{x,R} → 0.

It gives rise to a long exact sequence (a)

0 → Hom(S_{x,R}, S_{y,R}) → Hom(\underline{R}, S_{y,R}) →^{i^{∗}} Hom(\underline{R}_{>x}, S_{y,R})
→ Ext^{1}(S_{x,R}, S_{y,R}) → Ext^{1}(\underline{R}, S_{y,R}) →^{i^{∗}} Ext^{1}(\underline{R}_{>x}, S_{y,R})
⋯ ⋯ ⋯
→ Ext^{n}(S_{x,R}, S_{y,R}) → Ext^{n}(\underline{R}, S_{y,R}) →^{i^{∗}} Ext^{n}(\underline{R}_{>x}, S_{y,R})
⋯  ⋯  · · · .

We’re going to use this long exact sequence to calculate the two groups Ext_{RC}^{n}(\underline{R}, M) and Ext_{RC}^{n}(\underline{R}_{>x}, M) over some special category C.

Proposition 5.2.1. Let C = C_{x}^{y} be a connected EI-category, where x ≅ y̸. Suppose \underline{R} is a projective RC-module. Then we have Ext_{RC}^{n}(S_{x,R}, M) ≅ \underleftarrow{lim}_{C_{>x}}^{n−1} M ↓_{C_{>x}}^{C} for n ≥ 2 and ι : C_{>x} ↪ C.

If \underline{R} is a projective RC^{op}-module then we have Ext_{RC}^{n}(\underline{R}, S_{y,R}) ≅ H̃^{n−1}(|C_{<y}|, R) for n ≥ 1, and Ext_{RC}^{0}(\underline{R}, S_{y,R}) = 0.

Proof. From the preceding long exact sequence we get

Ext_{RC}^{n−1}(\underline{R}_{>x}, M) ≅ Ext_{RC}^{n}(S_{x,R}, M)

when n ≥ 2. Then by Proposition 3.1.4 we have Ext_{RC}^{n−1}(\underline{R}_{>x}, M) ≅ \underleftarrow{lim}_{C_{>x}}^{n−1} Res_{ι} M.

Next we consider the long exact sequence derived from the sequence of RC-modules 0 → S_{y,R} → \underline{R} → \underline{R}_{C_{<y}} → 0 by applying Ext_{RC}^{∗}(\underline{R}, −), where \underline{R}_{<y} is the quotient \underline{R}/S_{y,R}. In the long exact sequence we’ll have Ext_{RC}^{n}(\underline{R}, \underline{R}) = 0 for n ≥ 1 because \underline{R} is a projective RC^{op}-module and Ext_{RC}^{n}(\underline{R}, \underline{R}) ≅ Ext_{RC^{op}}^{n}(\underline{R}, \underline{R}). Thus Ext_{RC}^{n}(\underline{R}, S_{y,R}) ≅ Ext_{RC}^{n−1}(\underline{R}, \underline{R}_{<y}) ≅ H^{n−1}(|C_{<y}|, R) for n ≥ 2. For n = 0, we look at the exact sequence

0 → Hom_{RC}(\underline{R}, S_{y,R}) → Hom_{RC}(\underline{R}, \underline{R}) → Hom_{RC}(\underline{R}, \underline{R}_{<y}) → Ext_{RC}^{1}(\underline{R}, S_{y,R}) → 0,

and it’s easy to see Ext_{RC}^{0}(\underline{R}, S_{y,R}) ≅ Hom_{RC}(\underline{R}, S_{y,R}) = 0 and Ext_{RC}^{1}(\underline{R}, S_{y,R}) = 0.

Suppose C = C_{x}^{y}. We use C_{(x,y)} to denote the full subcategory of C after removing [x] and [y].

Corollary 5.2.2. Suppose C = C_{x}^{y} and \underline{R} is projective as both an RC-module and an RC^{op}-module. Then

1. if C_{(x,y)} ≠ ∅ then for n ≥ 2, Ext_{RC}^{n}(S_{x,R}, S_{y,R}) ≅ H̃^{n−2}(|C_{(x,y)}|, R) and

Ext_{RC}^{0}(S_{x,R}, S_{y,R}) = Ext_{RC}^{1}(S_{x,R}, S_{y,R}) = 0;

2. if C_{(x,y)} = ∅ we have

Ext_{RC}^{n}(S_{x,R}, S_{y,R}) ≅ Ext_{R Aut_{C}(x)}^{n−1}(R, R) ≅ Ext_{R Aut_{C}(y)}^{n−1}(R, R)

when n ≥ 1, while Ext_{RC}^{0}(S_{x,R}, S_{y,R}) = 0.

Proof. By the preceding proposition, for n ≥ 2 we have

Ext_{RC}^{n}(S_{x,R}, S_{y,R}) ≅ \underleftarrow{lim}_{C_{>x}}^{n−1}S_{y,R},

and the latter turns out to be isomorphic to Ext_{C_{(x,y)}}^{n−1}(\underline{R}, \underline{R}), where C_{(x,y)} = (C_{>x})_{<y}. Now for n = 0 we have Ext_{RC}^{0}(S_{x,R}, S_{y,R}) = Hom_{RC}(S_{x,R}, S_{y,R}) = 0. For n = 1 from the exact sequence

0→Hom_{RC}(S_{x,R},S_{y,R})→Hom_{RC}(\underline{R},S_{y,R})→Hom_{RC}(\underline{R}_{>x},S_{y,R})→Ext_{RC}^{1}(S_{x,R},S_{y,R})→0,

we know Ext_{RC}^{1}(S_{x,R}, S_{y,R}) is the cokernel of the natural map Hom_{RC}(\underline{R}, S_{y,R}) → Hom_{RC}(\underline{R}_{>x}, S_{y,R}), which is 0.

The second statement can be proved in a similar way.

We comment that Corollary 5.2.2 does not apply to categories with a unique minimal object and a unique maximal object where the automorphism groups do not act transitively.

Example 5.2.3. We consider the category C

\xymatrix{x \ar[r]^{α} & z \ar[r]^{β} \ar[r]_{γ} & w \ar[r]^{μ} & y} .

If βα = γα then C has x as the initial object, and if μβ = μγ then y is the terminal object. When these two conditions are satisfied, \underline{R} is projective as both an RC- and an RC^{op}-module. Therefore we can apply Corollary 5.2.2 to compute Ext_{RC}^{n}(S_{x,R}, S_{y,R}).

However, if either x isn’t initial or y isn’t terminal, the statements of Corollary 5.2.2 are no longer correct. In fact we can easily find it’s always true that Ext_{RC}^{3}(S_{x,R}, S_{y,R}) = 0 and H̃^{1}(|C_{(x,y)}|, R) = R for any field R.

When C is a poset, every subposet C_{x}^{y} satisfies the conditions in Corollary 5.2.2, and this gives rise to our next corollary, which is the main theorem of Igusa and Zacharia [22] where they used the theorem of Gerstenhaber and Shack [16] (quoted as Theorem 3.2.5 here in this thesis) which identifies the Hochschild cohomology and simplicial cohomology of posets.

Corollary 5.2.4. Let C be a poset and x < y ∈ Ob C. Then Ext_{RC}^{n}(S_{x,R}, S_{y,R}) = H̃^{n−2}(|C_{(x,y)}|, R) for n ≥ 2 if C_{(x,y)} ≠ ∅, and Ext_{RC}^{0}(S_{x,R}, S_{y,R}) = Ext_{RC}^{1}(S_{x,R}, S_{y,R}) = 0. If C_{(x,y)} = ∅ then Ext_{RC}^{1}(S_{x,R}, S_{y,R}) = R and Ext_{RC}^{n}(S_{x,R}, S_{y,R}) = 0 for any n ≠ 1.

Note that in Corollary 5.2.4 when C_{(x,y)} is not empty then Irr_{C}(x, y) = ∅ because x is the initial and y is the terminal object in C_{x}^{y}. One can compare this with Proposition 5.1.6.

5.3 Homological dimensions

Our work in the previous sections can be used to study the homological dimension of RC-modules. With the description of the minimal resolution of any RC-module we’re able to give a theorem on the global dimension of RC.

Recall that there is a poset P (C) associated to every EI-category C. When C is finite we call the maximal length of chains of non-isomorphisms in P (C) the length of C, denoted by l(C).

Theorem 5.3.1. Let C be a finite EI-category. Then RC has finite global dimension if and only if for all x ∈ Ob C, | Aut(x)|^{−1} ∈ R. In fact, proj.dim(M) ≤ l(C_{M}), the length of C_{M}. Hence gl.dim(RC) ≤ l(C).

Proof. Since RC has finitely many simple modules, RC having finite global dimension is equivalent to the statement that every simple RC-module has a finite projective resolution.

Suppose | Aut(x)|^{−1} ∈ R for all x ∈ Ob C. We do an induction on l(C), the length of C. Fix a simple S_{y,W}. It has a minimal projective resolution P written as

⋯ → P_{n} → ⋯ → P_{1} → P_{0} → S_{y,W} → 0.

Since P(y) → S_{y,W} (y) = W → 0 is the minimal projective resolution of W and | Aut(y)|^{−1} ∈ R, we have P_{0}(y) ≅ W for R Aut(y) is semi-simple. Then it implies that all P_{n}, n > 0 are supported on C_{>y} for P_{n}(y) = 0 when n > 0. Therefore if we take K_{0} as the kernel of the map P_{0} → S_{y,W}, K_{0} must be an RC-module supported on C_{>y}, and

⋯ → P_{n} → ⋯ P_{1} → K_{0} → 0

becomes a minimal projective resolution of K_{0}. Because C_{>y} has smaller length than C, the resolution of K_{0} is finite. So is P, the resolution of S_{y,W}.

On the other hand if any S_{y,W} has a finite projective resolution P, then P(y) is a finite resolution of the simple R Aut(y)-module S_{y,W} (y) = W. Since projective R Aut(y)-modules are the same as injective R Aut(x)-modules, the finite exact sequence P(y) → W → 0 splits. Hence each and every W is projective which means R Aut(y) is semi-simple, or equivalently | Aut(y)|^{−1} ∈ R.

Under the circumstance if we consider an arbitrary RC-module M, we can show proj.dim(M) ≤ l(C_{M}). Let

P_{M} : 0 → P_{n} → ⋯ → P_{1} → P_{0} → M → 0

be the minimal projective resolution of M. Then P_{0} is supported on C_{M}. Just like what we’ve done above, it’s easy to see P_{1} is supported on C_{M}\{M-minimal objects} ⊂ C_{M}, thus the size of C_{P_{i}} decreases strictly when i grows bigger and bigger. This means proj.dim(M) ≤ l(C_{M}). Hence gl.dim(RC) ≤ l(C).

Abusing the notation, when R is understood we will say C has finite global dimension if RC has finite global dimension.

Corollary 5.3.2. Let D, C be two EI-categories equipped with a functor ι : D → C. Suppose for any x ∈ Ob D, y ∈ Ob C and α ∈ Hom_{C}(ι(x), y), |ι^{−1}(Stab_{Aut(ι(x))}(α))| is invertible in R. Then every overcategory ι ↓_{y} has finite global dimension.

Proof. Fixing a y ∈ Ob D. Then objects of ι ↓_{y} are of the form (x, α) for some x ∈ Ob D and α ∈ Hom_{C}(ι(x), y). The automorphism group of such an object is {g ∈ Aut_{D}(x) | αι(g) = α}, which equals ι^{−1}(Stab_{Aut(ι(x))}(α)). Hence the result follows from the previous theorem.

When ι : D ↪ C is an inclusion, the stabilizer group becomes Stab_{Aut_{D}(x)}(α). If each group has an order invertible in R, then ι ↓_{y} has finite global dimension. Recall in Section 3.3 we studied the exactness of RC ⊗_{RD} P_{D} through the contractibility of overcategories. The above result says that if every stabilizer group has an order invertible in R, then the homology of RC ⊗_{RD} P_{D} vanish after certain stage, though RC ⊗_{RD} P_{D} isn’t necessarily exact.

In the same spirit we can give the following statement. When D ⊂ C is a subcategory and Mor(C)\ Mor(D) contains only monomorphisms, all the stabilizers in the previous corollary are trivial, and hence the overcategories are of finite global dimension. As a special case we can take a finite category with subobjects (C, I). Then every morphism is monomorphic and hence the overcategories associated to the inclusion ι : D ↪ C, of any subcategory D, are equivalent to disjoint unions of finite posets, which have lengths less than or equal to that of C.

Cohomology rings of finite

EI-categories

We already know the cohomology ring of a finite EI-category is the same as the cohomology ring of the topological realization of the category. Inspired by the known results for finite posets and finite groups, whose cohomology rings are finitely generated (see Evens [12] and Venkov [39]), in this chapter we look for further structure of the cohomology ring of an EI-Category. We note here that the finite generation of the cohomology ring of C does not immediately appear to follow from the result of Friedlander and Suslin [13] that the cohomology ring of a cocommutative Hopf algebra is finitely generated, because a category algebra usually isn’t a cocommutative Hopf algebra.

6.1 The cohomology ring is not finitely-generated

The cohomology ring of an arbitrary finite EI-category is not finitely generated. Below is an example.

We consider the following category C over 𝔽_{2}:

[figure: x; 1_{x}; g; α; y; 1_{y}; h] ,

where α · g = α = h · α, g^{2} = 1_{x} and h^{2} = 1_{y}.

We’re going to use the simplicial complex associated to C for computations:

⋯ → C_{n} → ⋯ → C_{2} → C_{1} → C_{0} → 0,

where C_{0} is the 𝔽_{2}-vectorspace generated by the objects, i.e., x and y, and C_{n} is the 𝔽_{2}-vectorspace having a basis consisting of all chains made by n consecutive morphisms. For instance, C_{2} has base elements x→^{g}x→^{g}x, x→^{g}x→^{α}y and x→^{α}y→^{h}h. For brevity, we will simply denote them by (g, g), (g, α), (α, h) and (h, h). Note that we can and do skip all the degenerated chains like (1_{x}, g) and (α, 1_{y}), because they don’t contribute to our calculations.

The differential is given as follows

d(f_{1}, f_{2}, ⋯, f_{n}) = (f_{2}, ⋯, f_{n}) + ∑_{i≥1}^{n−1}(−1)^{i}(f_{1}, ⋯, m_{i} · f_{i+1}, ⋯, f_{n})

+(−1)^{n}(f_{1}, f_{2}, ⋯, f_{n−1}),

where (f_{1}, f_{2}, ⋯, f_{n}) is an element of C_{n}.

Now we pass to the dual complex in order to compute cohomology:

⋯ ← (C_{n})^{∗} ← ⋯ ← (C_{2})^{∗} ← (C_{1})^{∗} ← (C_{0})^{∗} ← 0,

where (C_{n})^{∗} is the 𝔽_{2}-vectorspace generated by the dual base of C_{n}, i.e. {δ_{(m_{1},m_{2},··· ,m_{n})}}, the set of functions indexed by {(m_{1}, m_{2}, ⋯, m_{n})} ⊂ C_{n} such that

δ_{(m_{1},m_{2},··· ,m_{n})}(m′_{1}, m′_{2}, ⋯, m′_{n}) = 0

unless (m′_{1}, m′_{2}, ⋯, m′_{n}) = (m_{1}, m_{2}, ⋯, m_{n}), whence the value of the function equals 1.

For n > 0, C_{n}^{∗} has base elements of the following types δ_{(g,··· ,g)}, δ_{(h,··· ,h)}, δ_{(g,··· ,g,α)}, δ_{(α,h,··· ,h)} and δ_{(g,··· ,g,α,h,··· ,h)}. Direct calculations show that the cocycles δ_{(g,··· ,g,α)}, δ_{(α,h,··· ,h)} ∈ C_{n+1}^{∗} are the images of δ_{(g,··· ,g)} and δ_{(h,··· ,h)} ∈ C_{n}^{∗} respectively. Since the differential restricted to all δ_{(g,··· ,g,α,h,··· ,h)} is the zero map, the cohomology groups for n > 0 is actually generated by all δ_{(g,··· ,g,α,h,··· ,h)} with at least one g and one h on each end. Therefore we get Ext_{𝔽_{2}C}^{n}(\underline{𝔽_{2}}, \underline{𝔽_{2}}) = 𝔽_{2}^{n−2} for n ≥ 3, and Ext_{𝔽_{2}C}^{1}(\underline{𝔽_{2}}, \underline{𝔽_{2}}) = Ext_{𝔽_{2}C}^{2}(\underline{𝔽_{2}}, \underline{𝔽_{2}}) = 0.

Since the cup product of arbitrary two elements in ⊕_{i≥1} Ext_{𝔽_{2}C}^{i}(\underline{𝔽_{2}}, \underline{𝔽_{2}}) is 0, the cohomology ring Ext_{𝔽_{2}C}^{∗}(\underline{𝔽_{2}}, \underline{𝔽_{2}}) is not finitely generated.

6.2 The cohomology ring modulo a nilpotent ideal

However, the above example is not the end of our story, because all elements in ⊕_{i≥1} Ext_{𝔽_{2}C}^{i}(\underline{𝔽_{2}}, \underline{𝔽_{2}}) are can be easily seen to be nilpotent, and the cohomology ring modulo these nilpotents is indeed finitely generated! Nevertheless, this property stays true in all other examples we’ve computed so far. Hence the observation leads to the following conjecture.

Conjecture For a finite EI-category, its cohomology ring modulo nilpotents is finitely generated.

At the first sight one might think of the short exact sequence of complexes

0 → 𝔸 → ℂ → ℂ/𝔸 → 0,

where 𝔸 is the subcomplex of ℂ generated by all automorphisms in C. The short exact sequence gives rise to a long exact sequence which contains all necessary ingredients

⋯ → H^{n−1}(𝔸) → H^{n}(ℂ/𝔸) → H^{n}(ℂ) → H^{n}(𝔸) → H^{n+1}(ℂ/𝔸) → · · · .

If the maps H^{n−1}(𝔸) → H^{n}(ℂ/𝔸) were all zero, then the long exact sequence would break up to some short exact sequences like the following

0 → H^{n}(ℂ/𝔸) → H^{n}(ℂ) → H^{n}(𝔸) → 0.

Then if the image of H^{n}(ℂ/𝔸) in H^{n}(ℂ) were always nilpotent, we could prove our conjecture is correct. However, the break-down is not possible in general. For instance, the long exact sequence of the following category doesn’t break into the desirable pieces.

[figure: x; 1_{x}; g; k; gk; α; y; 1_{y}; h] .

A segment of the associated long exact sequence is recorded below.

0 → 0 → 𝔽_{2} → 𝔽_{2}^{2}
→ 𝔽_{2} → 0 → 𝔽_{2}^{3}
→ 𝔽_{2}^{3} → 0 → 𝔽_{2}^{4}
→ 𝔽_{2}^{6} → 𝔽_{2}^{2} → 𝔽_{2}^{5}
→ 𝔽_{2}^{10} → 𝔽_{2}^{5} → 𝔽_{2}^{6}.

For n ≥ 1, we actually have some short exact sequences like

0 → H^{n}(𝔸) → H^{n+1}(ℂ/𝔸) → H^{n+1}(ℂ) → 0,

which do not agree with the proposed break-down of the long exact sequence, and are not very helpful in describing the structure of the cohomology ring. However, the use of 𝔸 helps us make a certain connection between the cohomology of C and cohomology of the automorphism groups insider it.

Proposition 6.2.1. The image of H^{∗}(ℂ/𝔸), denoted by I = ⊕_{n≥0}I_{n}, is an ideal of H^{∗}(ℂ) consisting of nilpotents. Furthermore, i^{∗}(H^{∗}(ℂ)) is a subring of H^{∗}(𝔸).

Proof. The first observation is from the long exact sequence induced by

0 → 𝔸 →^{i} ℂ →^{π} ℂ/𝔸 → 0.

The long exact sequence looks like

⋯ → H^{n−1}(𝔸) → H^{n}(ℂ/𝔸) →^{π^{∗}} H^{n}(ℂ) →^{i^{∗}} H^{n}(𝔸) → H^{n+1}(ℂ/𝔸) → · · · .

It’s a canonical result that i^{∗} and π^{∗} induce two ring homomorphisms, still denoted by i^{∗} and π^{∗}. We get the image I = π^{∗}(H^{∗}(ℂ/𝔸)) ⊂ ⊕_{n≥1} H^{n}(ℂ) because of H^{0}(ℂ/𝔸) = 0. Now we show I is an ideal. Consider an arbitrary cocycle f ∈ C_{n}^{∗}(ℂ/𝔸). For any cocycle g ∈ C_{m}^{∗}(ℂ), we get a cocycle (π^{∗}(f) ∪ g) ∈ C_{n+m}^{∗}(ℂ). Let’s compute (π^{∗}(f) ∪ g)(σ) for any base element σ ∈ C_{n+m}. By definition of cup product, we have

(π^{∗}(f)∪g)(σ) = π^{∗}(f)(σ|_{(x_{0},··· ,x_{n})})·g(σ|_{(x_{n},··· ,x_{n+m})}) = f(π(σ|_{(x_{0},··· ,x_{n})}))·g(σ|_{(x_{n},··· ,x_{n+m})}).

It’s zero when σ only consists of automorphisms. So π^{∗}(f) ∪ g is actually of the form ∑_{σ∈Γ} r_{σ}f_{σ}, where Γ is a set of chains of length n + m containing at least one non-isomorphism. This element h = ∑_{σ∈Γ} r_{σ}f_{σ} can be regarded as a cocycle in C_{n+m}^{∗}(ℂ/𝔸), and now π^{∗}(h) = π^{∗}(f) ∪ g. So I is an ideal of the ring H^{∗}(ℂ). Elements of I are nilpotent because the category C is finite. If k is the maximum length of chains of non-isomorphisms in C, then for any f ∈ I, we must have f^{k+1} = 0 by the definition of cup product.

As for the image i^{∗}(H^{∗}(ℂ)), it contains the identity ∑_{x∈Ob C} f_{x}. So it’s a subring of H^{∗}(𝔸). In fact, it’s isomorphic to H^{∗}(ℂ)/I because of the exactness of the long exact sequence.

6.3 The existence of a cohomology ring with non-

nilpotent elements

It’s necessary to show that there exist non-nilpotent elements in the cohomology ring of at least one EI-category, because otherwise there is no need to make the above conjecture. One of the examples is the following category C:

[figure: x; 1_{x}; g; k; gk; α; β; y; 1_{y}; h] ,

where αk = α, αg = α(gk) = β, βk = β, βg = β(gk) = α and hα = β, hβ = α.

There is a full subcategory D of C, which is equivalent to the cyclic group of order 2, on the left of the following diagram

[figure: y; 1_{y}; h; x; 1_{x}; g; k; gk; α; β; y; 1_{y}; h] .

Using Quillen’s theorem A we can show these two categories are homotopy equivalent. Therefore the cohomology ring of C is isomorphic to that of D, which is well-known to contain non-nilpotent elements.

An example

In this chapter, we plan to use an example to illustrate our main results throughout this thesis. From now on, C_{n} denotes a cyclic group of order n.

Let’s take the symmetric group on four letters S_{4}. There are three Sylow 2subgroups of S_{4}, which are isomorphic to D_{8} ( the dihedral group of order 8) and contain the same subgroup V of order 4, where V = {(), (12)(34), (13)(24), (14)(23)}. In fact, it’s not hard to see that S_{4} ≅ V ⋊ S_{3}. Below is a poset C which consists of the non-trivial 2-subgroups of a Sylow 2-subgroup, denoted as D_{8}, from which one can get the poset of all (non-trivial) 2-subgroups of S_{4}.

[figure: D_{8}; C_{2} × C_{2}; C_{4}; V; C_{2}; C_{2}; C_{2}; C_{2}; C_{2}]

In this particular poset, we have the following groups:

D_{8} = {(), (12), (34), (12)(34), (13)(24), (14)(23), (1324), (1423)},

V = {(), (12)(34), (13)(24), (14)(23)},

C_{4} = {(), (1324), (12)(34), (1423)}, C_{2} × C_{2} = {(), (12), (34), (12)(34)}, and the five C_{2}’s: {(), (12)}, {(), (34)}, {(), (12)(34)}, {(), (13)(24)} and {(), (14)(23)}. Note that the center of this D_{8} is {(), (12)(34)}.

Let O_{2}(S_{4}) be the 2-orbit category of S_{4}. This is a category whose objects are the non-trivial 2-subgroups of S_{4}. If H, K are two 2-subgroups of S_{4}, then the set of morphisms from H to K is defined as Hom(H, K) = K\N_{S_{4}}(H, K), where N_{S_{4}}(H, K) is called the transporter, consisting of elements g ∈ S_{4} such that ^{g}H ⊂ K. It’s easy to see that the 2-orbit category O_{2}(S_{4}) is equivalent to the the full subcategory whose objects are the 2-subgroups of a fixed Sylow 2-subgroup of S_{4}, because every 2subgroup is isomorphic in O_{2}(S_{4}) to a subgroup of a fixed Sylow 2-subgroup. The following category is such a full subcategory of O_{2}(S_{4}), and as an abuse of notations, we’ll still name it O_{2}(S_{4}).

[figure: D_{8}; 1; C_{2} × C_{2}; 2; 1; C_{4}; 2; 1; V; 6; 3; C_{2}; 2; 2; 2; C_{2}; 2; 2; 2; C_{2}; 4; 2; 2; 6; 4; C_{2}; 4; 6; 4; 4; C_{2}; 4; 6; 4]

Here the numbers denote the sizes of the sets of irreducible morphisms between two adjacent objects, or the sizes of endomorphisms groups. Note that there are some other non-isomorphisms from the C_{2}’s to D_{8}. For instance, there is one (reducible) morphism from {(), (12)} to D_{8} and three (reducible) morphisms from {(), (12)(34)} to D_{8}. A good way to calculate the morphisms in such an orbit category is to use the powerful computer software GAP. Let’s list some of the morphism sets (by representatives):

Hom(C_{2} × C_{2}, D_{8}) = {()}, Hom(⟨(12)⟩, C_{2} × C_{2}) = {(), (13)(24)},

Hom(⟨(12)(34)⟩, C_{2} × C_{2}) = {(), (13)(24)}, Hom(⟨(12)(34)⟩, C_{4}) = {(), (12)}, Hom(⟨(12)(34)⟩, V) = {(), (12), (13), (14), (123), (132)}, Hom(V, D_{8}) = {(), (123), (132)}. By direct computation, we can also completely understand the actions of the

automorphism groups Aut(x) and Aut(y) on Hom(x, y), for any x, y ∈ Ob O_{C}. For future reference, we describe some of the actions. Note that all morphisms in such an orbit category are epimorphisms, and this means that αβ = α′β implies α = α′. We can find, for instance, Aut(V) acts transitively and freely on Hom(⟨(12)(34)⟩, V) and Aut(C_{4}) acts transitively and freely on Hom(⟨(12)(34)⟩, C_{4}). Here Aut(V) ≅ S_{3} is presented by the set of representatives {(), (12), (13), (14), (123), (132)}.

In practice, people care about the contravariant functors from O_{2}(S_{4}) to R-mod. While in order to be consistent with our conventions, we consider the covariant functors from the opposite category O_{2}^{op}(S_{4}) to R-mod. For simplicity, we use the skeleton of O_{2}^{op}(S_{4}), instead of O_{2}^{op}(S_{4}) itself, to illustrate our calculations. Thus the category O to be studied is as follows

[figure: C_{2}; 2; C_{2}; 4; C_{2} × C_{2}; 2; 2; 2; C_{4}; 2; 2; V; 6; 6; D_{8}; 1; 1; 1; 3; .]

We’re going to give the following information for this category O (for R = ℂ or 𝔽_{2}):

1. The vertex V\underline{R} The vertex of \underline{R} is the following full subcategory:

[figure: V; 6; D_{8}; 3; 1]

This category V\underline{R} is R-co-taut in O. So we have \underleftarrow{lim}_{O}^{∗} M ≅ \underleftarrow{lim}_{V\underline{R}}^{∗} Res_{ι} M by Lemma 4.4.22. In order to produce the minimal projective resolution for the trivial RV\underline{R}-module, we first give the structure of RV\underline{R} as follows. First of all we have RV\underline{R} = RV\underline{R} ·1_{D_{8}} ⊕RV\underline{R} ·1_{V}. When R = ℂ, the two projective modules can be described by

\begin{aligned}& 1 \\ 1 & & 2 & and & 1 & ⊕ & −1 & ⊕ & 2 & ⊕ & 2, \end{aligned}

or ℂV\underline{ℂ} = P_{D_{8},1}⊕[P_{V,1}⊕P_{V,−1}⊕P_{V,2}^{2}]. When R = 𝔽_{2}, 𝔽_{2}V\underline{𝔽_{2}} = P_{D_{8},1}⊕[P_{V,1}⊕P_{V,2}^{2}], or we can describe the two projectives as

\begin{aligned}& 1 & & & 1 \\ 1 & & 2 & and & 1 & ⊕ & 2 & ⊕ & 2. \end{aligned}

Now for either choice of the base ring R, the trivial RV\underline{R}-module \underline{R} has a (minimal) projective resolution of length 2: 0 → P_{V,2} → P_{D_{8},1} → \underline{R} → 0, where P_{D_{8},1} = RV\underline{R} · 1_{D_{8}} is the projective cover of \underline{R}, and P_{V,2} = S_{V,2} is the projective simple of R Aut(V) ≅ RS_{3} of dimension 2, regarded as an RV\underline{R}-module. Thus we have \underleftarrow{lim}_{O}^{i} M ≅ \underleftarrow{lim}_{V\underline{R}}^{i} M = 0 for any M ∈ RC-mod if i > 1. One can compare these calculations with a formula of Jackowski-Słomińska [25] (quoted here as Corollary 3.3.4). Fixing a finite group G and a prime p, a subgroup H is pradical if the group N_{G}(H)/H has no normal p-subgroups. Their formula asserts that the p-orbit category of a group G, O_{p}(G), can always be replaced by its full subcategory consisting of just p-radical subgroups when one wants to compute higher limits over O_{p}^{op}(G). Note that in our example the two subgroups of S_{4}, D_{8} and V, are the only 2-radical subgroups of S_{4} which are contained in D_{8}. Furthermore Grodal [17] found that the higher limits over the full subcategory of O_{p}^{op}(G), which consists of all non-trivial p-radical subgroups, vanish if the degree is bigger than the length of the underlying poset of the subcategory. In our example, the 2-orbit category O_{2}^{op}(G) is replaced by its full subcategory V\underline{R}, which consists of the two 2-radical subgroups D_{8}, V. Since the underlying poset of V\underline{R} has length one, we must have \underleftarrow{lim}_{V\underline{R}}^{i} M = 0 if i > 1, and this verifies our previous calculations.

Remark 7.0.1. If we consider the 2-orbit category of D_{8}, instead of S_{4}, then the only 2-radical group is D_{8} itself. It’s easy to see that D_{8} becomes the initial object of O_{2}^{op}(D_{8}), and \{D_{8}\} is the vertex of \underline{R}.

Remark 7.0.2. We can check that the above category V\underline{R} is isomorphic to the opposite of the 2-orbit category of S_{3} which has objects C_{2} and {()}. Thus

H^{∗}(O_{2}^{op}(S_{4}), M) ≅ H^{∗}(O_{2}^{op}(S_{3}), M),

for any M ∈ RO_{2}^{op}(S_{4})-mod. If we recall Jackowski-McClure-Oliver’s notation [24]

Λ^{∗}(K, M) = H^{∗}(O_{p}^{op}(K), M({()})),

where M is an atomic functor concentrated on the trivial subgroup {()} of a group K, then we can compare our formula for computing \underleftarrow{lim}_{O_{2}^{op}(S_{4})}^{∗} M with theirs

H^{∗}(O_{p}^{op}(G), M) ≅ Λ^{∗}(N_{G}(P)/P, M({()})),

where M is an atomic functor concentrated on the conjugacy class of the subgroup P. Indeed, if we take N = S_{4}, P = V and p = 2 in their formula, then their formula implies H^{∗}(O_{2}^{op}(S_{4}), M) ≅ H^{∗}(O_{2}^{op}(S_{3}), M) for any atomic M that is concentrated on V, since N_{S_{4}}(V)/V ≅ S_{3}.

2. Vertices for other modules and Green correspondence According to Proposition 4.4.19, the only objects of the vertex of each indecomposable projective module P are the P -minimal objects. If M is an indecomposable atomic module (Definition 3.3.3), concentrated on [x], then V_{M} is the full subcategory consisting of its support [x] and the objects y such that Irr(x, y) ≠ ∅ (Proposition 4.4.20). Simple modules are atomic. Thus the simple module S_{D_{8},1} has as its vertex the full subcategory of O consists of objects D_{8}, C_{2} × C_{2}, C_{4} and V. The simple ℂO-modules S_{V,1}, S_{V,−1} (if R = ℂ) and S_{V,2} all have the full subcategory with two objects, V and C_{2}, as their vertex. The object C_{2} = {(), (12)(34)} has an automorphism of order 4, which is isomorphic to C_{2} × C_{2}. Since there are infinitely many indecomposable 𝔽_{2}O-modules whose vertex is {C_{2}}, 𝔽_{2}O is of infinite representation type because there exists a Green correspondence (Corollary 4.4.8). More generally, if an EI-category E has an object x whose automorphism group algebra R Aut(x) is of infinite representation type, then so is RE because, by Theorem 4.4.4, each indecomposable R Aut(x)-module gives rise to an indecomposable RE-module by tensor induction, and any two modules obtained in this way are not isomorphic to each other since their evaluations at x are not.

3. The groups Ext_{RO}^{∗}(S_{1}, S_{2}) and \underleftarrow{lim}_{O}^{∗} M etc.

In this example, if there exists only one reducible morphism from the support of S_{1} to that of S_{2}, then Ext_{RO}^{0}(S_{1}, S_{2}) = Ext_{RO}^{1}(S_{1}, S_{2}) = 0 (Proposition 5.1.5). For S_{1} and S_{2} that share the same support, the Ext groups are isomorphic to those over the automorphism group of their support while S_{1} and S_{2} are regraded as group modules (Corollary 5.1.4). We compute two examples for which there exists at least one irreducible nonisomorphism between the supports of the two simples. First of all, if we take the supports to be D_{8} and V, then there are three different kinds of groups that we need to consider: Ext_{RO}^{∗}(S_{D_{8},1}, S_{V,1}), Ext_{RO}^{∗}(S_{D_{8},1}, S_{V,2}) and Ext_{RO}^{∗}(S_{D_{8},1}, S_{V,−1}) when R = ℂ. Note that we can restrict our attention to the full subcategory consisting of only two objects D_{8} and V (Theorem 5.1.3). Within this full subcategory we can write out the minimal resolution of S_{D_{8},1} for each situation. Then the Ext groups are isomorphic to the Hom’s, because S_{V,?} and S_{V,2} are simple.

(a) When R = ℂ, the minimal resolution is

0 → P_{V,2} ⊕ P_{V,1} → P_{D_{8},1} → S_{D_{8},1} → 0,

since the ℂ-span of Hom_{O}(D_{8}, V) = {(), (123), (132)}, as a ℂ Aut(V)(≅ ℂS_{3})-module, can be decomposed into a direct sum of two projective simples

ℂ{() + (123) + (132)} ⊕ ℂ{(123) − (), (132) − ()}.

Thus we have

Ext_{ℂO}^{0}(S_{D_{8},1}, S_{V,1}) ≅ Hom(P_{D_{8},1}, S_{V,1}) = 0,
Ext_{ℂO}^{0}(S_{D_{8},1}, S_{V,−1}) ≅ Hom(P_{D_{8},1}, S_{V,−1}) = 0,
Ext_{ℂO}^{0}(S_{D_{8},1}, S_{V,2}) ≅ Hom(P_{D_{8},1}, S_{V,2}) = 0,
Ext_{ℂO}^{1}(S_{D_{8},1}, S_{V,1}) ≅ Hom(P_{V,1} ⊕ P_{V,2}, S_{V,1}) ≅ ℂ,
Ext_{ℂO}^{1}(S_{D_{8},1}, S_{V,−1}) ≅ Hom(P_{V,1} ⊕ P_{V,2}, S_{V,−1}) = 0,
Ext_{ℂO}^{1}(S_{D_{8},1}, S_{V,2}) ≅ Hom(P_{V,1} ⊕ P_{V,2}, S_{V,2}) ≅ ℂ

and Ext_{ℂO}^{i}(−, −) = 0 for i ≥ 2; (b) When R = 𝔽_{2}, the minimal resolution is

⋯ → P_{V,1} → P_{V,1} → P_{V,1} ⊕ P_{V,2} → P_{D_{8},1} → S_{D_{8},1} → 0,

which is infinite with all places filled in by P_{V,1} except at degrees 0 and 1. It is so because the 𝔽_{2}-span of Hom(D_{8}, V) = {(), (123), (132)} still decomposes into a direct sum of the trivial module and the dimension 2 projective simple of 𝔽_{2}S_{3}, but the trivial module is no longer projective which has a projective cover of dimension 2 whose socle (= radical) is isomorphic to 𝔽_{2}. Thus we use the property of minimal resolutions to get

Ext_{𝔽_{2}O}^{0}(S_{D_{8},1}, S_{V,1}) ≅ Hom(P_{D_{8},1}, S_{V,1}) = 0,
Ext_{𝔽_{2}O}^{0}(S_{D_{8},1}, S_{V,2}) ≅ Hom(P_{D_{8},1}, S_{V,2}) = 0,
Ext_{𝔽_{2}O}^{1}(S_{D_{8},1}, S_{V,1}) ≅ Hom(P_{V,1} ⊕ P_{V,2}, S_{V,1}) ≅ 𝔽_{2},
Ext_{𝔽_{2}O}^{1}(S_{D_{8},1}, S_{V,2}) ≅ Hom(P_{V,1} ⊕ P_{V,2}, S_{V,2}) ≅ 𝔽_{2},
Ext_{𝔽_{2}O}^{i}(S_{D_{8},1}, S_{V,1}) ≅ Hom(P_{V,1}, S_{V,1}) ≅ 𝔽_{2}

and Ext_{𝔽_{2}O}^{i}(S_{D_{8},1}, S_{V,2}) ≅ Hom(P_{V,1}, S_{V,2}) ≅ 0 for i ≥ 2.

If we want to know \underleftarrow{lim}_{O}^{∗} S_{D_{8},1} and \underleftarrow{lim}_{O}^{∗} S_{V,?}, then we can first apply the formula \underleftarrow{lim}_{O}^{∗} M ≅ \underleftarrow{lim}_{V\underline{R}}^{∗} Res_{ι} M so that these groups become \underleftarrow{lim}_{V\underline{R}}^{∗} S_{D_{8},1} and \underleftarrow{lim}_{V\underline{R}} S_{V,?}. Using the minimal projective resolution of \underline{R}, we can get that all these groups vanish except that \underleftarrow{lim}_{O} S_{D_{8},1} ≅ \underleftarrow{lim}_{V\underline{R}} S_{D_{8},1} ≅ R and \underleftarrow{lim}_{O}^{1} S_{V,2} ≅ \underleftarrow{lim}_{V\underline{R}}^{1} S_{V,2} ≅ R.

4. The cohomology ring In part 1, we obtained a projective resolution of the RV\underline{R}-module \underline{R}, and it can be used to calculate the cohomology ring Ext_{RO}^{∗}(\underline{R}, \underline{R}). By Proposition 4.4.25, this cohomology ring is isomorphic to Ext_{RV\underline{R}}^{∗}(\underline{R}, \underline{R}) for the full subcategory V\underline{R}. Now from the projective resolution of RV\underline{R}-module \underline{R},

0 → P_{V,2} → P_{D_{8},1} → \underline{R} → 0,

we get a cochain complex 0 → Hom(P_{D_{8},1}, \underline{R}) → Hom(P_{V,2}, \underline{R}) → 0. Since Hom_{V\underline{R}}(P_{V,2}, \underline{R}) ≅ Hom_{RS_{3}}(S_{V,2}, R) = 0, we have Ext_{RV\underline{R}}^{1}(\underline{R}, \underline{R}) ≅ Hom(P_{V,2}, \underline{R}) ≅ 0, and hence Ext_{RV\underline{R}}^{0}(\underline{R}, \underline{R}) ≅ Hom(P_{D_{8},1}, \underline{R}) which is isomorphic to R. Of course we can tell Ext_{RV\underline{R}}^{0}(\underline{R}, \underline{R}) ≅ R directly from the fact that V\underline{R} is connected. The cohomology ring of O, Ext_{RO}^{∗}(\underline{R}, \underline{R}), is isomorphic to R for R = ℂ or R = 𝔽_{2}.

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