Stratifications and Mackey Functors I: Functors for a Single Group

Peter Webb* School of Mathematics, University of Minnesota, Minneapolis MN 55455, USA email: webb@math.umn.edu

Abstract

In the context of Mackey functors we introduce a category which is analogous to the category of modules for a quasi-hereditary algebra which have a filtration by standard objects. Many of the constructions which work for quasi-hereditary algebras can be done in this new context. In particular we construct an analogue of the ‘Ringel dual’, which turns out here to be a standardly stratified algebra. The Mackey functors which play the role of the standard objects are constructed in the same way as functors which have been previously used in parametrizing the simple Mackey functors, but instead of using simple modules in their construction (as was done before) we use p-permutation modules. These Mackey functors are obtained as adjoints of the operations of forming the Brauer quotient and its dual. The filtrations which have these Mackey functors as their factors are closely related to the filtrations whose terms are the sum of induction maps from specified subgroups, or are the common kernel of restriction maps to these subgroups. These latter filtrations appear in Conlon’s decomposition theorems for the Green ring, as well as in other places, where they arise quite naturally.

* Partially supported by the NSF.

1. Introduction

This work is a study of several fundamental constructions in representation theory and their interactions in the context of Mackey functors. There are really four main ingredients and we will now describe them.

The first ingredient is a pair of filtrations of each Mackey functor M for a finite group G. These are constructed by considering for each family of subgroups X, closed under conjugation and taking subgroups, at each subgroup H the sum of the images in M(H) of the inductions from subgroups in X, and the intersection of the kernels of the restrictions to subgroups in X. The filtrations are constructed by enlarging X by adjoining one conjugacy class of subgroups at a time. We call the two filtrations the ascending and descending filtrations of M associated to the order in which the conjugacy classes of subgroups of G are taken. Such constructions have appeared widely before; for example, they underlie Conlon’s theorems on the Green ring [5, Section 81], and also one of the proofs of semisimplicity of Mackey functors when |G| is invertible [13]. This in turn makes use of a theorem of Dress [6, Theorems 2 and 4] which really has to do with these filtrations.

The second ingredient in our work is the notion of the Brauer quotient, and its dual, the restriction kernel. Brauer quotients have played a fundamental role in recent representation theory, as described in [12]. The restriction kernel has appeared less overtly, in spite of the fact that over a field it is the dual of the Brauer quotient applied to the dual Mackey functor. It appears explicitly in a conjecture of Feshbach (see [1]) where it is called the essential elements. The Brauer quotient and restriction kernel turn out to be certain values of the factors in ascending and descending filtrations of the Mackey functor.

The next ingredient is a pair of Mackey functors Δ_{H,U} and ∇_{H,U} associated to each subgroup H and N_{G}(H)/H-representation U. The Mackey functors ∇_{H,U} were first defined by Lewis [8], and they also appeared without special terminology in [13] where it was shown that when U is a simple module, ∇_{H,U} has a unique simple subfunctor. Arising from this a complete parametrization of the simple Mackey functors was obtained. It is because of this property that we use the ∇ notation here. Similarly, when U is simple, Δ_{H,U} has a unique simple quotient Mackey functor. However, we should point out that the notation is flawed in that not all of our functors ∇_{H,U} have a simple socle, and neither do all of the functors Δ_{H,U} have a simple top. An indication of the fundamental nature of these Mackey functors is given by the fact that the functor U ↦ ∇_{H,U} is the right adjoint of the functor which assigns to a Mackey functor M its Brauer quotient at H; and the functor U ↦ Δ_{H,U} is the left adjoint of the functor which assigns to M its restriction kernel at H. Indeed, these properties may be used to define these Mackey functors.

The fourth main ingredient in our work is the class of p-permutation modules, by which we mean modules which are direct summands of permutation modules. Their appearance here is not accidental, and really derives from the fact that if M is a projective Mackey functor, then all of its Brauer quotients are p-permutation modules (or zero). We will deal with the functors Δ_{H,U} and ∇_{H,U} where U is a p-permutation module for N_{G}(H)/H.

One of the main objects of our study will be the full subcategory D of the category of all Mackey functors for G whose objects are the Mackey functors which have a filtration whose factors are of the form Δ_{H,U} with U a p-permutation module. Such a filtration we will call a Δ-filtration. There is also an analogous category whose objects have a ∇-filtration, and the results we prove for D have dual statements for this other category. We will show that the projective Mackey functors are objects in D (Theorem 7.1). The ascending filtrations have an importance for the objects of D in that any ascending filtration of an object in D can be refined to a Δ-filtration (Proposition 6.1). We follow the direction of the work of Ringel [10] and we prove a theorem on the vanishing of Ext groups between the Δ_{H,U} (Theorem 5.1). This allows us to invoke Ringel’s work and deduce that D is functorially finite in the full category of Mackey functors and has relative almost split sequences. We go on to investigate the relative Auslander-Reiten quiver for D, and make several characterizations of particular Mackey functors in D (Section 11); but most importantly we characterize the Ext-projective and Ext-injective objects in D. This is done through an axiomatic notion of Ext-projective covers and Ext-injective hulls developed in Section 8. The work in Section 8 applies in greater generality than the context of Mackey functors, and may be of separate interest. The indecomposable Ext-projective objects and Ext-injective objects in D are in fact precisely the Ext-projective covers Π_{H,U} and the Ext-injective hulls T_{H,U} of the Δ_{H,U}. We now do what Ringel did and form the algebra E = End(⨁_{H,U} T_{H,U}). Since we are not working with a quasi hereditary algebra here (in contrast to Ringel’s situation) we do not expect E to be quasi-hereditary. It is, however, standardly stratified in the sense of [4] with respect to an obvious pre-order on the set of pairs (H, U) which parametrize the simple E-modules. The algebra E stands with respect to the group algebra of G in the same way as a Schur algebra stands with respect to the group algebra of a symmetric group, in that the group algebra is Morita equivalent to eEe for a certain idempotent e.

We will use the notation and basic constructions which appear in [14]. Indeed, many of the main ideas of [14] are used throughout this work and a good acqaintance with these ideas is really a prerequisite to understanding some of our arguments. Let us summarize the most basic notation. Unless otherwise stated we will be working with Mackey functors defined on a finite group G and its subgroups. Thus when H is a subgroup of G and M is a Mackey functor we have operations I_{H}^{G} : M(H) → M(G) and R_{H}^{G} : M(G) → M(H) which behave like induction and restriction in representation theory, and for each g ∈ G we have a conjugation morphism c_{g} : M(H) → M(^{g}H). We will always work with Mackey functors over a commutative ring R with a 1, and in the later parts of this paper we will require that R be a complete p-local ring, by which we mean that R is either a field of characteristic p or a complete discrete valuation ring with residue field of characteristic p. The category of Mackey functors for G over R is denoted Mack_{R}(G).

I wish to thank the referee for his or her careful reading of the manuscript and helpful comments.

2. Filtrations, Brauer quotients and restriction kernels

We start by describing the subfunctors of an arbitrary Mackey functor M which will form the terms of its ascending and descending filtrations. For this we will need the notion of the subfunctor of M generated by a collection of subsets A_{H} ⊆ M(H), where H ranges over (not necessarily all) subgroups of G. By this we mean the intersection of all subfunctors N of M for which N(H) ⊇ A_{H} for all H, and it is the unique minimal subfunctor of M containing all these subsets. A useful description of this subfunctor is stated in [14, 2.4].

Suppose that X is a set of subgroups of G. We define

I_{X} M = ⟨M(J) | J ∈ X ⟩

to be the subfunctor of M generated by its values on groups in X. We also define R_{X} M to be the largest subfunctor of M satisfying the condition

R_{J}^{K} : R_{X} M(K) → M(J) is the zero map, for every group J ∈ X and K ⊇ J.

Most of the time we will assume that X is closed under taking subgroups and conjugation, and in this context these functors were considered in [13, Sect. 3] where they were denoted Im t_{X} and Ker r_{X}.

We list some of the properties of these Mackey functors. In part (e) of our first result we use the term ‘cogenerated’ in a sense dual to that of the word ‘generated’. We will say that a Mackey functor M is cogenerated by subsets A_{H} ⊆ M(H) if every morphism of Mackey functors M → M′ which is injective on all the subsets A_{H} is in fact a monomorphism. In part (g) we refer to a construction introduced by Dress [6] in the study of relative projectivity, and at this point we use an alternative notation for Mackey functors in which they are defined on G-sets, rather than on subgroups of G (see [14]). Whenever X is a G-set and M a Mackey functor we may define a Mackey functor M_{X} whose value at a G-set Y is M(Y × X). There is a natural transformation θ_{X} : M_{X} → M which is determined at the G-set Y by applying the covariant part of M to projection onto the first coordinate, giving a homomorphism M(Y × X) → M(Y). Similarly there is a natural transformation θ^{X} : M → M_{X} determined by applying the contravariant part of M to projection onto the first coordinate. Part (j) of the result is a special case of the induction theorem of Dress [6]. We do not give the complete formulation so as to save space, and we will not use it, but merely take the opportunity to point out this important result.

(2.1) PROPOSITION. Let X, X_{1} and X_{2} be sets of subgroups of G.

(a) I_{∅}M = R_{all subgroups}M = 0 and R_{∅}M = I_{all subgroups}M = M.

(b) If X_{1} ⊆ X_{2} then I_{X_{1}}M ⊆ I_{X_{2}}M and R_{X_{1}}M ⊇ R_{X_{2}}M.

(c) Let X^{+} be the closure of X under adjoining conjugates. Then I_{X} M = I_{X^{+}}M and R_{X} M = R_{X^{+}}M.

(d) If X is closed under taking subgroups and conjugation then

I_{X} M(K) = ∑_{K⊇L∈X} I_{L}^{K}(M(L))  and  R_{X} M(K) = ⋂_{K⊇L∈X} Ker R_{L}^{K}.

(e) If X_{1} ⊆ X_{2} then I_{X_{2}}M/I_{X_{1}}M is generated by its values on the subgroups in X_{2}−X_{1} and R_{X_{1}}M/R_{X_{2}}M is cogenerated by its values on the subgroups in X_{2} −X_{1}. If additionally X_{1} and X_{2} are closed under taking subgroups then I_{X_{2}}M/I_{X_{1}}M and R_{X_{1}}M/R_{X_{2}}M are zero on subgroups which do not contain a conjugate of a subgroup in X_{2} − X_{1}.

(f) I_{X} (M_{1} ⊕ M_{2}) = I_{X} M_{1} ⊕ I_{X} M_{2} and R_{X} (M_{1} ⊕ M_{2}) = R_{X} M_{1} ⊕ R_{X} M_{2}.

(g) Suppose X is closed under taking subgroups. Put X = ⨆_{K∈X} G/K. Then I_{X} M equals the image of θ_{X} : M_{X} → M and R_{X} M equals the kernel of θ^{X} : M → M_{X}.

(h) Suppose that X is closed under taking subgroups, and if H is a subgroup of G put

X_{H} = {H ∩ ^{g}K | K ∈ X , g ∈ G},

a set of subgroups of H also closed under taking subgroups. Then when M is a Mackey functor for H we have

\begin{matrix}I_{X} (M ↑_{H}^{G}) = & I_{X_{H}} (M ↑_{H}^{G}) = (I_{X_{H}} M) ↑_{H}^{G}, \\ R_{X} (M ↑_{H}^{G}) = & R_{X_{H}} (M ↑_{H}^{G}) = (R_{X_{H}} M) ↑_{H}^{G}, \end{matrix}

and when M is a Mackey functor for G we have

\begin{matrix}(I_{X} M) ↓_{H}^{G}= & (I_{X_{H}} M) ↓_{H}^{G}= I_{X_{H}} (M ↓_{H}^{G}), \\ (R_{X} M) ↓_{H}^{G}= & (R_{X_{H}} M) ↓_{H}^{G}= R_{X_{H}} (M ↓_{H}^{G}). \end{matrix}

(i) ([15, 3.4.2]) If X_{1} ⊆ X_{2} and M is projective relative to a set of subgroups Y then I_{X_{2}}M/I_{X_{1}}M and R_{X_{1}}M/R_{X_{2}}M are both projective relative to Y.

(j) (Dress [6]) |G| · M ⊆ I_{X} M + R_{X} M and |G| · (I_{X} M ∩ R_{X} M) = 0.

Proof. (a) and (b) are self-evident. For (c) we have in one direction that I_{X} M ⊆ I_{X^{+}}M and R_{X} M ⊇ R_{X^{+}}M by (b). On

the other hand if L ∈ X^{+} then L = ^{g}L_{1} for some L_{1} ∈ X and now M(L) = c_{g}M(L_{1}) is the image under conjugation by g. Thus M(L) ⊆ I_{X} M(L) and so I_{X^{+}}M ⊆ I_{X} M. A similar argument shows that R_{X^{+}}M ⊇ R_{X} M.

Assertion (d) follows from [13, 2.1] and the comment at the start of Section 3 in [13].

In (e), we have from the definitions that I_{X_{2}}M is generated by I_{X_{1}}M and the values of M on the subgroups in X_{2} −X_{1}. Hence I_{X_{2}}M/I_{X_{1}}M is simply generated by its values on the subgroups in X_{2}−X_{1}. We leave the statement about cogeneration as an exercise. If now X_{1} and X_{2} are closed under taking subgroups then by (c) it suffices to assume that X_{1} and X_{2} are closed under conjugacy, and by (d) we see that I_{X_{2}}M is zero on subgroups which do not contain a conjugate of a subgroup in X_{2} and hence the same is true of I_{X_{2}}M/I_{X_{1}}M. If a subgroup contains a conjugate of a subgroup in X_{2}, but not a conjugate of a subgroup in X_{2} −X_{1}, then I_{X_{2}}M and I_{X_{1}}M take the same values on this subgroup by (d), and so the quotient functor is zero there. Hence the quotient functor is only non-zero on subgroups which contain a conjugate of a subgroup in X_{2} − X_{1}. Again we leave the statement about R_{X_{1}}M/R_{X_{2}}M as an exercise.

(f) is immediate.

(g) From the definitions of θ_{X} and θ^{X} we see that if X′ is another G-set which has the same isomorphism types of G=orbits as X, but with different multiplicities, then θ_{X} and θ_{X′} have the same image, and θ^{X} and θ^{X′} have the same kernel. From this we see that Im θ_{X} = Im θ_{X^{+}} and Ker θ^{X} = Ker θ^{X^{+}} where X^{+} = ⨆_{K∈X^{+}} G/K, since enlarging X to include conjugate subgroups does not change the isomorphism types of orbits in the G-sets.

We have already seen that I_{X} M = I_{X^{+}}M and R_{X} M = R_{X^{+}}M, and so it suffices to prove the result when X is closed under taking subgroups and conjugation. With this hypothesis the result was proved in [15, 3.4.3] for I_{X} M. We leave the rest of the details for R_{X} M to the reader.

(h) Let us put X = ⨆_{K∈X} G/K as a G-set, and X_{H} = ⨆_{K∈X_{H}} H/K as an H-set. Then X_{H} ↑_{H}^{G} may be identified as the G-set ⨆_{K∈X_{H}} G/K. Since

G/K ↓_{H}^{G}= ⨆_{g∈[H\G/K]} H/H ∩ ^{g}K

and H ∩ ^{g}K ∈ X_{H} when K ∈ X we see that X ↓_{H}^{G}, X_{H} ↑_{H}^{G}↓_{H}^{G} and X_{H} all contain the same isomorphism types of H-orbits, but perhaps with different multiplicities. We have that I_{X} (M ↑_{H}^{G}) is the image of θ_{X} : (M ↑_{H}^{G})_{X} → M ↑_{H}^{G}. At a G-set G/L we have

(M ↑_{H}^{G})_{X}(G/L) = M ↑_{H}^{G} (G/L × X)
= M((G/L × X) ↓_{H}^{G})
= M(G/L ↓_{H}^{G} ×X ↓_{H}^{G})
= M_{X↓_{H}^{G}} ↑_{H}^{G} (G/L)

so that (M ↑_{H}^{G})_{X} ≅ M_{X↓_{H}^{G}} ↑_{H}^{G}, and with this identification θ_{X} becomes

(θ_{X↓_{H}^{G}}) ↑_{H}^{G}: M_{X↓_{H}^{G}} ↑_{H}^{G}→ M ↑_{H}^{G} .

Since ↑_{H}^{G} is an exact functor we see that I_{X} (M ↑_{H}^{G}) = (I_{X_{H}} M) ↑_{H}^{G} since X ↓_{H}^{G} has the same orbits as X_{H}. We may do the same calculation with X_{H} ↑_{H}^{G} instead of X, to prove that I_{X_{H}} (M ↑_{H}^{G}) = (I_{X_{H}} M) ↑_{H}^{G}. This establishes the first line of equalities for I_{X}, and the second line for R_{X} is proved similarly.

It is immediate from (d) that if K ⊇ H then

(I_{X} M)(K) = (I_{X_{H}} M)(K) = (I_{X_{H}} M ↓_{H}^{G})(K)

which shows that

(I_{X} M) ↓_{H}^{G}= (I_{X_{H}} M) ↓_{H}^{G}= I_{X_{H}} (M ↓_{H}^{G}),

and similarly with R_{X} M.

For (i) we quote Proposition 3.4.2 from [15] (which uses different notation). A first application of that result shows that I_{X_{2}}M is projective relative to Y. Then a second application shows that I_{X_{2}}M/I_{X_{1}}M is projective relative to Y. We may prove the result for R_{X_{1}}M/R_{X_{2}}M by an analogous argument; or when R is a field we may deduce it from what we have already done by duality using the result in the next proposition.

Over a field, the subfunctors I_{X} M and R_{X} M are in fact dual to each other in the sense that doing one of these to M is equivalent to doing the other to the dual of M. When R is a field, recall from [14, Sect. 4] that for each Mackey functor M we define the dual Mackey functor M^{∗} by M^{∗}(K) = Hom_{R}(M(K), R) with operations R_{J}^{K} = (I_{J}^{K})^{∗} : M^{∗}(K) → M^{∗}(J), I_{J}^{K} = (R_{J}^{K})^{∗} : M^{∗}(J) → M^{∗}(K) and conjugations c_{g} = (c_{g^{−1}})^{∗} : M^{∗}(J) → M^{∗}(^{g}J). If N ⊆ M is a subfunctor, let us put N^{⊥}(K) = {f ∈ M^{∗}(K) | f|_{N(K)} = 0}. Then by a standard piece of linear algebra if N_{1} ⊆ N_{2} ⊆ M we have (N_{2}/N_{1})^{∗} ≅ N_{1}^{⊥}/N_{2}^{⊥}.

(2.2) PROPOSITION. Let M be a Mackey functor defined over a field and let X, X_{1} and X_{2} be sets of subgroups of G closed under conjugation and taking subgroups. Then (I_{X} M)^{⊥} = R_{X} (M^{∗}) and (R_{X} M)^{⊥} = I_{X} (M^{∗}). Thus if X_{1} ⊆ X_{2} we have

(I_{X_{2}}M/I_{X_{1}}M)^{∗} ≅ (I_{X_{1}}M)^{⊥}/(I_{X_{2}}M)^{⊥} = R_{X_{1}}(M^{∗})/R_{X_{2}}(M^{∗})

and

(R_{X_{1}}M/R_{X_{2}}M)^{∗} ≅ (R_{X_{2}}M)^{⊥}/(R_{X_{1}}M)^{⊥} = I_{X_{2}}(M^{∗})/I_{X_{1}}(M^{∗}).

Identifying M with its double dual we have (I_{X} (M^{∗}))^{∗} = R_{X} M and (R_{X} (M^{∗}))^{∗} = I_{X} M.

Proof. Since I_{X} M(K) is the sum of the images of all the maps I_{H}^{K} with H ∈ X and H ≤ K we have

(I_{X} M)^{⊥}(K) = {f ∈ M^{∗}(K) | f(I_{H}^{K}(x)) = 0 for all H ∈ X , H ≤ K and x ∈ M(H)}.

Now f(I_{H}^{K}(x)) = (R_{H}^{K}f)(x) from the definition of the restriction maps in M^{∗}, and so the condition is that R_{H}^{K}f is the zero map for all H ∈ X, H ≤ K. This is exactly the condition that f should belong to R_{X} (M^{∗})(K), and so (I_{X} M)^{⊥} = R_{X} (M^{∗}). The argument which shows that (R_{X} M)^{⊥} = I_{X} (M^{∗}) is similar, and the remaining assertions are standard deductions.

Proposition 2.1(b) says that the mappings X ↦ I_{X} M and X ↦ R_{X} M are order preserving and order reversing maps from the lattice of subsets of subgroups of G to the lattice of subfunctors of M. Using this we may construct filtrations of M by ordering the conjugacy classes of subgroups of G as 1 = H_{1}, H_{2}, …, H_{n} = G in such a way that if H_{i} is conjugate to a subgroup of H_{j} then i ≤ j. Let us write M_{i} = I_{{H_{1},...,H_{i}}}M and M^{i} = R_{{H_{1},...,H_{i}}}M with M_{0} = 0 and M^{0} = M. We will call

0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{n} = M  and  M = M^{0} ⊇ M^{1} ⊇ ⋯ ⊇ M^{n} = 0

the ascending filtration and the descending filtration of M associated to the list H_{1}, …, H_{n}. The last results immediately apply to the terms of these filtrations, and we have for all i,

• (M ⊕ N)_{i} = M_{i} ⊕ N_{i},

• M_{i}/M_{i−1} is generated by its value at H_{i},

• if M is projective relative to Y then so are all the factors M_{i+r}/M_{i},

• M_{i} + M^{i} ⊇ |G| · M and |G| · (M_{i} ∩ M^{i}) = 0,

• M_{i} = M^{∗i∗} when R is a field.

We are interested in the factors M_{i}/M_{i−1} and M^{i−1}/M^{i} of these filtrations. Evidently these are associated in some way to the subgroup H_{i}, but they depend also on the ordering of the rest of the list of conjugacy classes of subgroups and need not be uniquely determined by H_{i}. We will see from the next result that as the position of a particular subgroup H varies between different lists, there is a largest such factor associated to H, which occurs when H appears as early as possible in a list, and a smallest such factor which occurs when H appears as late as possible in a list. For lists in which H occurs between the extreme positions, these factors are related by homomorphisms. We will also see in the next results that the values of the factors at H_{i} are in fact determined independently of the position of H_{i} in the list of subgroups, and they are the Brauer quotient and restriction kernel of M at H_{i}.

The Brauer quotient of M at H is defined to be

\overline{M}(H) = M(H)/ ∑_{J<H} I_{J}^{H}(M(J).

We also define the restriction kernel of M at H, namely

\underline{M}(H) = ⋂_{J<H} Ker R_{J}^{H}.

These are constructions which appear throughout representation theory. When R is a field the restriction kernel is in fact dual to the Brauer quotient, in that it is the dual of the Brauer quotient of the dual Mackey functor: \underline{M}(H) = (\overline{M^{∗}})^{∗}. This is a particular consequence of Proposition 2.2 and the next result, Proposition 2.3. Thus in the context of Mackey functors over a field we may work equally well with either the Brauer quotient or the restriction kernel, although for some reason the Brauer quotient appears more frequently than the restriction kernel. In group cohomology the restriction kernel has been termed the essential elements, as studied for example in [1]. A construction with similar formalism (not to do with Mackey functors) underlies the ‘kernel intersection theorems’ of James in the context of representations of the symmetric and general linear groups, and our descending filtration of M is obtained by an analogous procedure to that which may be used to obtain Specht filtrations of the Young modules as described in [4].

(2.3) PROPOSITION.

(a) Suppose that X and Y are sets of subgroups of G with X ⊆ Y. There is an epimor-

phism

I_{X ∪{H}}M/I_{X} M → I_{Y∪{H}}M/I_{Y}M

and a monomorphism

R_{Y}M/R_{Y∪{H}}M → R_{X} M/R_{X ∪{H}}M.

(b) If X is a set of subgroups of G which contains a conjugate of every subgroup of H, but no conjugate of H itself, then

(I_{X ∪\{H\}}M/I_{X} M)(H) ≅ \overline{M}(H)

and

(R_{X} M/R_{X ∪\{H\}}M)(H) ≅ \underline{M}(H).

Proof. (a) Since X ⊆ Y we have that I_{X ∪{H}}M is a subfunctor of I_{Y∪{H}}M and I_{X} M is a subfunctor of I_{Y}M. Thus the composite of inclusion and quotient maps

I_{X ∪{H}}M → I_{Y∪{H}}M → I_{Y∪{H}}M/I_{Y}M

contains I_{X} M in the kernel and induces a morphism

I_{X ∪{H}}M/I_{X} M → I_{Y∪{H}}M/I_{Y}M.

It is an epimorphism since I_{Y∪{H}}M is generated by I_{Y}M and M(H), and M(H) = I_{X ∪{H}}M(H) is in the image of the morphism from I_{X ∪{H}}M.

The construction and properties of the morphism

R_{Y}M/R_{Y∪{H}}M → R_{X} M/R_{X ∪{H}}M

are established in a dual manner.

(b) Let X^{+} be the conjugacy closure of X. We have from Proposition 2.1

I_{X} M(H) = I_{X^{+}}M(H) = ∑_{H⊇L∈X^{+}} I_{L}^{H}(M(L))

and this equals ∑_{L<H} I_{L}^{H}M(L) since only proper subgroups of H appear in X^{+}. Since I_{X ∪\{H\}}M(H) = M(H), the quotient is \overline{M}(H). This establishes the first formula, and the proof of the second is very similar.

(2.4) COROLLARY. In any ascending or descending filtration of M corresponding to a list of subgroups 1 = H_{1}, H_{2}, …, H_{n} = G we have (M_{i}/M_{i−1})(H_{i}) ≅ \overline{M}(H_{i}) and (M^{i−1}/M^{i})(H_{i}) ≅ \underline{M}(H_{i}).

The structure we are observing is connected with the following well-known result.

(2.5) PROPOSITION. Let f : N → M be a morphism of Mackey functors.

(a) If the composite morphism N(H) → M(H) → \overline{M}(H) is surjective for every subgroup H then f is an epimorphism.

(b) If the composite morphism \underline{N}(H) → N(H) → M(H) is injective for every subgroup H then f is an monomorphism.

Proof. for (a), let 0 = M_{0} ⊆ M_{1} ⊆ ⋯ be an ascending filtration of M. We show by induction that M_{i} ⊆ Im f, which is certainly true when i = 0. Assuming that M_{i−1} ⊇ Im f we also know that (M_{i} ∩Im f)/M_{i−1} = M_{i}/M_{i−1} since this is generated by (M_{i}/M_{i−1})(H_{i}) by 2.1(e), which equals \overline{M}(H_{i}) by 2.3(b) and is covered by f. Hence M_{i} ⊆ Im f. The proof of (b) is dual.

3. The Δ and ∇ functors

Whenever K is a group, U is an RK-module and J is a subgroup of K we will use the notation F P_{U}(J) = U^{J} = H^{0}(J, U) for the fixed points of J on U, and F Q_{U}(J) = U_{J} = H_{0}(J, U) for the fixed quotient (or cofixed points) of J on U. Thus F P_{U} and F Q_{U} are the fixed point and fixed quotient Mackey functors.

Let H be a subgroup of G and U an R[N_{G}(H)/H]-module. The functors defined by

Δ_{H,U} = (Inf_{N(H)/H}^{N(H)} F Q_{U}) ↑_{N(H)}^{G}

and

∇_{H,U} = (Inf_{N(H)/H}^{N(H)} F P_{U}) ↑_{N(H)}^{G}

in the notation of [13] and [14] will be very important to us, and we now summarize the basic facts about them. The ∇_{H,U} were first defined in different notation by Lewis [8] and several of the statements we make in this section were familiar to him. They also play an important role in the work of Luca [9], who describes a number of their properties. When H and K are subgroups of G we will write

T (H, K) = {g ∈ G | ^{g}H ⊆ K}

for the transporter of H into K. By a straightforward modification of the argument in [14, 8.8] we have:

(3.1) PROPOSITION.

Δ_{H,U}(K) = ⨁_{g∈[K\T (H,K)/N_{G}(H)]} U_{N_{K^{g}} (H)}

and

∇_{H,U}(K) = ⨁_{g∈[K\T (H,K)/N_{G}(H)]} U^{N_{K^{g}} (H)}.

In particular Δ_{H,U}(H) = U = ∇_{H,U}(H), and both Δ_{H,U}(K) and ∇_{H,U}(K) vanish unless K contains a conjugate of H.

In this statement we are using the notation U_{N_{K^{g}} (H)} and U^{N_{K^{g}} (H)} for the fixed quotient and fixed points of N_{K^{g}}(H) on U.

Proof. By the formula for an induced functor [13, 4.3] we have

Δ_{H,U}(K) = ⨁_{g∈[K\G/N_{G}(H)]}(Inf_{N(H)/H}^{N(H)} F Q_{U})(K^{g} ∩ N_{G}(H)).

The terms are zero unless K^{g} ⊇ H, i.e. g ∈ T (H, K). We may write K^{g} ∩ N_{G}(H) = N_{K^{g}}(H), and now we have all the ingredients in the formula.

The formula for ∇_{H,U}(K) is the same, except that we take fixed points instead of fixed quotients.

The main significance of the Δ and ∇ functors is that they satisfy an adjoint property with respect to the Brauer quotients and restriction kernels.

(3.2) PROPOSITION. For each subgroup H the functor

Mack_{R}(G) → R[N_{G}(H)/H]-mod

specified by M ↦ \underline{M}(H) has left adjoint U ↦ Δ_{H,U}. Similarly the functor specified by M ↦ \overline{M}(H) has right adjoint U ↦ ∇_{H,U}. The natural isomorphisms Hom(Δ_{H,U}, M) → Hom(U, \underline{M}(H)) and Hom(M, ∇_{H,U}) → Hom(\overline{M}(H), U) are given by evaluating morphisms at H.

Proof. This appears, for example, in the proof of [13, 8.1]. We have (in the notation of [13])

Hom(Δ_{H,U}, M) = Hom((Inf_{N(H)/H}^{N(H)} F Q_{U}) ↑_{N(H)}^{G}, M)
≅ Hom(Inf_{N(H)/H}^{N(H)} F Q_{U}, M ↓_{N(H)}^{G})
≅ Hom(F Q_{U}, (M ↓_{N(H)}^{G})^{−})
≅ Hom(U, (M ↓_{N(H)}^{G})^{−}(H/H))

passing through a chain of adjunctions. Here

(M ↓_{N(H)}^{G})^{−}(K/H) = ⋂_{{J⊆K \atop J⊉H}} Ker R_{J}^{K}

and so its value on the identity subgroup H/H is exactly \underline{M}(H). The precise description of the adjunction isomorphism comes by tracing these isomorphisms through the chain of adjunctions.

The proof for ∇_{H,U} is similar. When R is a field we may also exploit duality to deduce the properties of ∇_{H,U} from those of Δ_{H,U^{∗}}.

The next corollary may also be expressed as an adjoint property in which Δ_{H,U} and ∇_{H,U} are regarded as functors to the full subcategory of Mackey functors which vanish on subgroups which do not contain a conjugate of H.

(3.3) COROLLARY. Let H be a subgroup of G and let M be a Mackey functor for which M(K) = 0 unless K contains a conjugate of H. Then every morphism of R[N_{G}(H)/H]-modules U → M(H) extends uniquely to a morphism of Mackey functors Δ_{H,U} → M, and every morphism of R[N_{G}(H)/H]-modules M(H) → U extends uniquely to a morphism of Mackey functors M → ∇_{H,U}.

Proof. For such a Mackey functor M we have \overline{M}(H) = \underline{M}(H) = M(H) and so the adjunction isomorphisms become Hom(Δ_{H,U}, M) ≅ Hom(U, M(H)) and Hom(M, ∇_{H,U}) ≅ Hom(M(H), U). The statement of the corollary is now another way of saying that these isomorphisms are given by evaluating morphisms at H.

From this we obtain the following conclusion:

(3.4) COROLLARY. Δ_{H,U} is generated by its value at H, and ∇_{H,U} is cogenerated by its value at H. If U is an indecomposable module then the Mackey functors Δ_{H,U} and ∇_{H,U} are indecomposable.

Proof. Suppose f : M → Δ_{H,U} is a morphism for which M(H) → Δ_{H,U}(H) = U is surjective, and consider the quotient morphism Δ_{H,U} → Δ_{H,U}/f(M). This morphism is zero on U and by 3.3 it is the unique extension of its value on U. Since zero is such an extension we obtain that the quotient map is zero, and hence f(M) = Δ_{H,U}. The proof for ∇_{H,U} is dual.

Now if U is indecomposable and Δ_{H,U} = M_{1} ⊕ M_{2} for some Mackey functors M_{1} and M_{2} then Δ_{H,U}(H) = U = M_{1}(H) ⊕ M_{2}(H) so one of the terms on the right — say M_{2}(H)

— is zero. Since Δ_{H,U} is generated by its value at H, so is M_{2}, and hence M_{2} = 0. Again the proof for ∇_{H,U} is similar.

We have seen in Proposition 3.2 that the functor Δ_{H,U} is associated by means of an adjoint property with the restriction kernel, and ∇_{H,U} is associated with the Brauer quotient. In the next result it is a little surprising that these associations are reversed, and Δ is used in combination with a Brauer quotient while ∇ is used in combination with a restriction kernel. We comment further on this after Corollary 3.6.

(3.5) COROLLARY. Consider the ascending and descending filtrations of a Mackey functor M associated to a list 1 = H_{1}, H_{2}, …, H_{n} = G of subgroups of G taken up to conjugacy, in non-decreasing order. For each i there are epimorphisms

Δ_{H_{i},\overline{M}(H_{i})} → M_{i}/M_{i−1}

and there are monomorphisms

M^{i−1}/M^{i} → ∇_{H_{i},\underline{M}(H_{i})}.

Proof. By 2.4 we have that M_{i}/M_{i−1}(H_{i}) = \overline{M}(H_{i}) is the Brauer quotient at H_{i} and M^{i−1}/M^{i}(H_{i}) = \underline{M}(H_{i}) is the restriction kernel. Thus the identity morphisms on these modules correspond via the adjunctions implicit in 3.3 to the claimed morphisms of Mackey functors in the present statement, since M_{i}/M_{i−1} and M^{i−1}/M^{i} vanish on groups which do not contain a conjugate of H_{i} as a subgroup. The first morphism is an epimorphism because M_{i}/M_{i−1} is generated by its value at H_{i}, and the second is a monomorphism because M^{i−1}/M^{i} is cogenerated by its value at H_{i}.

(3.6) COROLLARY. Let M be a Mackey functor defined over a field R. For each subgroup K we have

dim M(K) ≤ ∑_{H≤G up to conjugacy} dim Δ_{H,\overline{M}(H)}(K)

and

dim M(K) ≤ ∑_{H≤G up to conjugacy} dim ∇_{H,\underline{M}(H)}(K).

Proof. We take a list 1 = H_{1}, H_{2}, …, H_{n} = G of subgroups of G taken up to conjugacy, in non-decreasing order. Now dim M(K) = ∑_{i=1}^{n} dim(M_{i}/M_{i−1})(K) and each of these dimensions in the sum is bounded by dim Δ_{H_{i},\overline{M}(H_{i})}(K), by 3.5. The second inequality is proved similarly.

These inequalities bound the dimension of M(K) solely in terms of the Brauer quotients of M, or the restriction kernels of M, since the dimensions of Δ and ∇ may be deduced from the explicit formulas for them given in Proposition 3.1. The bounds are best possible in general, and indeed the rest of this paper is devoted to a study of Mackey functors for which equality holds. As a particular example, it will be proved that the first inequality is an equality whenever M is projective, from which we can deduce by duality that the second inequality is an equality whenever M is injective.

We point out a difference between Corollary 3.6 and a confusingly similar construction which underlies the twin functor of Thévenaz [11]. By Proposition 3.2 the identity morphism \overline{M}(H) → \overline{M}(H) = U corresponds to a morphism of Mackey functors M → ∇_{H,\overline{M}(H)}, and also the identity morphism U = \underline{M}(H) → \underline{M}(H) corresponds to a morphism Δ_{H,\underline{M}(H)} → M. We may take the first of these morphisms to be the components in a morphism

M → ∏_{H⊆G up to conjugacy} ∇_{H,\overline{M}(H)}

and at first sight it appears that we may be able to deduce something like the inequalities of Corollary 3.6 from this. The target is in fact the twin functor of M as studied by Thévenaz [11] (see also [13, 9.3]). It is shown there that when |G| is invertible in R this morphism is an isomorphism. When |G| is not invertible it need not be even an monomorphism, and so an inequality does not immediately follow. On closer examination the ∇ inequality of 3.6 involves the restriction kernel, whereas the twin functor is expressed in terms of the Brauer quotient, so the ingredients in these constructions are rather different.

4. Induction, restriction and inflation of Δ_{H,U} and ∇_{H,U}

In this section we consider the effect of some standard operations on the Mackey functors Δ_{H,U}. In order to specify the group for which these are Mackey functors we will quite often write this group as a superscript, so that for example Δ_{H,U}^{K} indicates that this is an object of Mack_{R}(K), where H ⊆ K. The results in this section are rather formal and at times uninspiring, but they do provide crucial simplifications in our arguments later on. As always, similar results hold for both the Δ_{H,U} and ∇_{H,U}. We will work mainly with the Δ_{H,U} and leave it to the reader to formulate and prove the results for ∇_{H,U} when necessary.

(4.1) LEMMA. Suppose that H ⊆ J ⊆ G and that U is an R[N_{J}(H)/H]-module. Then Δ_{H,U}^{J} ↑_{J}^{G}≅ Δ_{H,U↑_{N_{J} (H)}^{N_{G}(H)}}^{G}.

Proof. We are abusing notation by writing U ↑_{N_{J} (H)}^{N_{G}(H)} instead of U ↑_{N_{J} (H)/H}^{N_{G}(H)/H}, and we will repeatedly do this. We have

Δ_{H,U}^{J} ↑_{J}^{G} ≅ (Inf_{H}^{N_{J} (H)} F Q_{U}) ↑_{N_{J} (H)}^{J}↑_{J}^{G}
≅ (Inf_{H}^{N_{J} (H)} F Q_{U}) ↑_{N_{J} (H)}^{N_{G}(H)}↑_{N_{G}(H)}^{G}
≅ (Inf_{H}^{N_{G}(H)}((F Q_{U}) ↑_{N_{J} (H)/H}^{N_{G}(H)/H})) ↑_{N_{G}(H)}^{G}
≅ (Inf_{H}^{N_{G}(H)}(F Q_{U↑_{N_{J} (H)/H}^{N_{G}(H)/H}})) ↑_{N_{G}(H)}^{G} by [14, 5.2]
≅ Δ_{H,U↑_{N_{J} (H)}^{N_{G}(H)}}^{G}.

The third isomorphism comes from the definitions of inflation and induction.

(4.2) COROLLARY.

(a) If N_{G}(H) ⊆ J then Δ_{H,U}^{J} ↑_{J}^{G}≅ Δ_{H,U}^{G} and ∇_{H,U}^{J} ↑_{J}^{G}≅ ∇_{H,U}^{G}.

(b) Suppose that R is a complete p-local ring. Then every Mackey functor Δ_{H,U}^{G} with U an indecomposable p-permutation module arises as a summand of a functor Δ_{H,R}^{K} ↑_{K}^{G} where H ◁ K and H/K is a p-group.

(c) Suppose that R is a complete p-local ring and that H ⊆ J ⊆ G. If U is an indecomposable p-permutation R[N_{J}(H)/H]-module then Δ_{H,U}^{J} ↑_{J}^{G} is a direct sum of indecomposable Mackey functors Δ_{H,W} where W is a p-permutation R[N_{G}(H)/H]- module with vertex vx W ⊆_{N_{G}(H)/H} vx U.

Proof. (a) is a particular case of Lemma 4.1, but it is also immediate from the definitions by transitivity of induction.

(b) Since U is a p-permutation module for R[N_{G}(H)/J] it is a summand of a module R ↑_{K}^{N_{G}(H)} for some subgroup K with H ◁K and K/H a p-group. Thus Δ_{H,U}^{G} is a summand of Δ_{H,R↑_{K}^{N_{G}(H)}}^{G}, which is isomorphic to Δ_{H,R}^{K} ↑_{K}^{G} by Lemma 4.1.

(c) Here Δ_{H,U}^{J} ↑_{J}^{G}≅ Δ_{H,U↑_{N_{J} (H)}^{N_{G}(H)}}^{G} is the direct sum of Mackey functors Δ_{H,W} where W

ranges over the indecomposable summands of U ↑_{N_{J} (H)}^{N_{G}(H)}. These modules have the stated form, and the functors Δ_{H,W} are indecomposable by 3.4.

(4.3) LEMMA. Let L be a subgroup of G. Then

Δ_{H,U}^{G} ↓_{L}^{G}≅ ⨁_{x∈[L\T (H,L)/N_{G}(H)]} ^{x}(Δ_{H,U}^{L^{x}})

where on the right we interpret U as a module for N_{L^{x}}(H)/H by restriction from N_{G}(H)/H. It follows that when R is a complete p-local ring and U is an indecomposable p-permutation module then Δ_{H,U}^{G} ↓_{L}^{G} is a direct sum of indecomposable functors ^{x}(Δ_{H,W}^{L^{x}}) ≅ Δ_{^{x}H,^{x}W}^{L} where ^{x}W is a p-permutation R[N_{L}(^{x}H)/^{x}H]-module with vx ^{x}W ⊆_{N_{L}(^{x}H)/^{x}H} vx ^{x}U.

Proof. We can write Δ_{H,U}^{G} = Δ_{H,U}^{N_{G}(H)} ↑_{N_{G}(H)}^{G} and so

\begin{aligned}Δ_{H,U}^{G} ↓_{L}^{G} & = Δ_{H,U}^{N_{G}(H)} ↑_{N_{G}(H)}^{G}↓_{L}^{G} \\ & = ⨁_{x∈[L\G/N_{G}(H)]} ^{x}(Δ_{H,U}^{N_{G}(H)} ↓_{L^{x}∩N_{G}(H)}^{N_{G}(H)}↑_{L^{x}∩N_{G}(H)}^{L^{x}}). \end{aligned}

Now since Δ_{H,U}^{N_{G}(H)} = Inf_{N_{G}(H)/H}^{N_{G}(H)} F Q_{U} the restriction Δ_{H,U}^{N_{G}(H)} ↓_{L^{x}∩N_{G}(H)}^{N_{G}(H)} is zero unless

L^{x} ∩ N_{G}(H) ⊇ H, that is, x ∈ T (H, L) and then Δ_{H,U}^{N_{G}(H)} ↓_{L^{x}∩N_{G}(H)}^{N_{G}(H)}= Δ_{H,U}^{L^{x}∩N_{G}(H)}. Thus

\begin{aligned}Δ_{H,U}^{G} ↓_{L}^{G} & ≅ ⨁_{x∈[L\T (H,L)/N_{G}(H)]} ^{x}(Δ_{H,U}^{L^{x}∩N_{G}(H)} ↑_{L^{x}∩N_{G}(H)}^{L^{x}}) \\ & ≅ ⨁_{x∈[L\T (H,L)/N_{G}(H)]} ^{x}(Δ_{H,U}^{L^{x}}). \end{aligned}

The last statements are clear.

(4.4) COROLLARY. If N_{G}(H) ⊆ L then Δ_{H,U}^{L} is isomorphic to the subfunctor of Δ_{H,U}^{G} ↓_{L}^{G} generated by the value of this Mackey functor at H, and this is a direct summand of Δ_{H,U}^{G} ↓_{L}^{G}.

Proof. We have Δ_{H,U}^{G} ↓_{L}^{G}≅ ⨁_{x∈[L\T (H,L)/N_{G}(H)]} ^{x}(Δ_{H,U}^{L^{x}}) and the summand corresponding to x = 1 is isomorphic to Δ_{H,U}^{L}. Since Δ_{H,U}^{L}(H) ≅ Δ_{H,U}^{G}(H) ≅ U it follows that all the other summands in the direct sum are zero at H. Hence the subfunctor of Δ_{H,U}^{G} ↓_{L}^{G} generated by the value at H is precisely the summand Δ_{H,U}^{L}, since this summand is itself generated by its value at H (and furthermore we see that this summand is a uniquely determined subfunctor).

In the same spirit we note the following particular identities.

(4.5) COROLLARY. If H ◁ J ⊆ G and L ⊆ G then

(Δ_{H,R↑_{J}^{N_{G}(H)}}^{G}) ↓_{L}^{G} = (Δ_{H,R}^{J} ↑_{J}^{G}) ↓_{L}^{G}

= ⨁_{x∈[L\T (H,L)/J]} Δ_{^{x}H,R↑_{L∩^{x}J}^{N_{L}(^{x}H)}}^{L}

= ⨁_{x∈[L\T (H,L)/J]}(Δ_{^{x}H,R}^{L∩^{x}J}) ↑_{L∩^{x}J}^{L} .

The following result is really present in [14, 11.4], but it also follows in a straightforward way from our results about induction and restriction.

(4.6) COROLLARY. Suppose that R is a complete p-local ring and that U is an indecomposable R[N_{G}(H)/H]-module with vertex K/H where H ◁ K. Then K is a vertex of Δ_{H,U}.

Proof. If U is a summand of R ↑_{K}^{N_{G}(H)} then Δ_{H,U} is a summand of Δ_{H,R↑_{K}^{N_{G}(H)}} ≅ Δ_{H,R}^{K} ↑_{K}^{G}. This shows that K contains a vertex of Δ_{H,U}. On the other hand, suppose that Δ_{H,U} is a summand of Δ_{H,U} ↓_{L}^{G}↑_{L}^{G} where L ⊆ K. We can write this as

⨁_{x∈[L\T (H,L)/N_{G}(H)]}(Δ_{^{x}H,^{x}U}^{L}) ↑_{L}^{G}≅ ⨁_{x∈[L\T (H,L)/N_{G}(H)]} Δ_{^{x}H,^{x}U↑_{L}^{N_{G}(H)}}^{G}.

Now U = Δ_{H,U}(H) is a summand of the previous direct sum, evaluated at H, which is

⨁_{x∈[L\T (H,L)/N_{G}(H)]} ^{x}U ↑_{L}^{N_{G}(H)}

and since all summands are induced from L we deduce that L must contain a vertex of U. We deduce that L = K and so K is a vertex of Δ_{H,U}.

The following lemma is immediate from the definitions.

(4.7) LEMMA. Let N ◁ G, N ⊆ H ⊆ G, and let U be an N_{G}(H)/H module. Then Inf_{G/N}^{G}(Δ_{H/N,U}^{G/N}) ≅ Δ_{H,U} and Inf_{G/N}^{G}(∇_{H/N,U}^{G/N}) ≅ ∇_{H,U}.

As a consequence of what we have just done we see that the Mackey functors Δ_{H,U} where U is a p-permutation module are precisely what we obtain by starting with the Mackey functor F Q_{R} and repeatedly applying inflation and induction, and taking direct sums and direct summands.

To conclude this section we define one of the main objects of our study. This is the category D which is the full subcategory of the category Mack_{R}(G) of Mackey functors whose objects are the functors with a finite filtration

0 = F_{0}M ⊆ F_{1}M ⊆ ⋯ ⊆ F_{n}M = M

in which for every i the factor F_{i}M/F_{i−1}M is isomorphic to Δ_{H_{i},U_{i}} for some subgroup H_{i} and finitely-generated p-permutation R[N_{G}(H_{i})/H_{i}]-module U_{i}. We will refer to objects in D as ‘functors with a Δ-filtration’. As G varies, the categories D behave remarkably well with respect to induction and restriction.

(4.8) COROLLARY. Let H be a subgroup of G. Then restriction and induction between G and H preserve the categories D. More specifically, if we denote by D^{H} and D^{G} the categories D for each of these groups then if M ∈ D^{G} we have M ↓_{H}^{G}∈ D^{H} and if M ∈ D^{H} we have M ↑_{H}^{G}∈ D^{H}.

5. Vanishing of Ext-groups

From this point on we will assume all the time that R is either a field of characteristic p or a complete discrete valuation ring with residue field of characteristic p — what we have been calling a complete p-local ring. We have defined the functors Δ_{H,U} and ∇_{H,U} for arbitrary R[N_{G}(H)/H]-modules U, but now we will restrict these modules to be ppermutation modules, that is, summands of permutation R[N_{G}(H)/H]-modules. The first result in this section will be the key tool which makes our results on stratifications work. Extensions of Mackey functors are the same thing as extensions of modules for the Mackey algebra μ_{R}(G). To simplify the notation as much as possible in the proof we will sometimes write Ext_{μ(G)} or just Ext instead of Ext_{μ_{R}(G)}^{1}.

(5.1) THEOREM. Suppose that R is a complete p-local ring. Let H and K be subgroups of G and let U and W be p-permutation modules for R[N_{G}(H)/H] and R[N_{G}(K)/K] respectively. Then

(a) Ext_{μ_{R}(G)}^{1}(Δ_{H,U}, Δ_{K,W}) = 0 unless H >_{G} K.

(b) Ext_{μ_{R}(G)}^{1}(Δ_{H,U}, ∇_{K,W}) = 0 unless H =_{G} K.

(c) Ext_{μ_{R}(G)}^{1}(∇_{H,U}, ∇_{K,W}) = 0 unless H <_{G} K.

Proof. (a) There will be several reductions. The first is that we may assume that H and K are p-subgroups of G, which by [14, 9.14] is the same thing as requiring that Δ_{H,U} and Δ_{K,W} lie in Mack_{R}(G, 1). We recall from [14] that the expression 1 = ∑_{p−perfect J≤G} f_{J} as a sum of primitive orthogonal idempotents in the Burnside ring R ⊗_{ℤ} B(G) gives rise to a decomposition M = ⨁_{p−perfect J≤G} f_{J}M of every Mackey functor M where f_{J}M ∈ Mack_{R}(G, J). Summands of different f_{J}M lie in different blocks of Mackey functors, and so to have a non-zero Ext group between indecomposable Δ_{H,U} and Δ_{K,W} these Mackey functors must belong to the same category Mack_{R}(G, J). This is the same as requiring that O^{p}(H) and O^{p}(K) are both conjugate to J, and in our notation we will assume that they are equal to J. Now there is an equivalence of categories

(Inf_{N(J)/J}^{N(J)} −) ↑_{N(J)}^{G}: Mack_{R}(N(J)/J, 1) → Mack_{R}(G, J)

(see [14, 10.1]) and by Lemmas 4.7 and 4.1 the functors Δ_{H/J,U}^{N_{G}(J)/J} are sent to the functors Δ_{H,U}^{G} under this equivalence. In verifying this we use the fact that if J is p-perfect and H/J is a p-group then N_{G}(H) ⊆ N_{G}(J). Thus to determine whether or not Ext_{μ_{R}(G)}^{1}(Δ_{H,U}, Δ_{K,W}) is zero it suffices to determine whether or not Ext_{μ_{R}(N_{G}(J)/J)}^{1}(Δ_{H/J,U}^{N_{G}(J)/J}, Δ_{K/J,W}^{N_{G}(J)/J}) is zero, and now H/J and K/J are p-groups. In view of all this we may henceforth assume H and K are p-subgroups.

By 4.2 every Δ_{H,U} is a direct summand of a functor Δ_{H,R↑_{F}^{N_{G}(H)}}^{G} ≅ Δ_{H,R}^{F} ↑_{F}^{G} where

H ◁ F and F is a p-group, and similarly Δ_{K,W} is a summand of a functor Δ_{K,R↑_{J}^{N_{G}(K)}}^{G} ≅

Δ_{K,R}^{J} ↑_{J}^{G} where K◁J and J is a p-group. It suffices to show that Ext(Δ_{H,R}^{F} ↑_{F}^{G}, Δ_{K,R}^{J} ↑_{J}^{G}) = 0 in this situation, or in other words we may assume that U = R ↑_{F}^{N_{G}(H)} and W = R ↑_{J}^{N_{G}(K)} where F and J are p-groups.

It also suffices to prove the result when R is a field. This is because if R is a complete discrete valuation ring with maximal ideal (π) then Δ_{K,W} /πΔ_{K,W} ≅ Δ_{K,W/πW} since factoring out π commutes with the operations of induction, inflation and taking fixed quotients of the trivial module. Now the exact sequence

Δ_{K,W} →^{π}Δ_{K,W} → Δ_{K,W/πW}

gives an exact sequence

Ext(Δ_{H,U}, Δ_{K,W})→^{π} Ext(Δ_{H,U}, Δ_{K,W}) → Ext(Δ_{H,U}, Δ_{K,W/πW})
≅ Ext(Δ_{H,U/πU}, Δ_{K,W/πW})

which embeds

Ext(Δ_{H,U}, Δ_{K,W})/π Ext(Δ_{H,U}, Δ_{K,W})

as a subgroup of Ext(Δ_{H,U/πU}, Δ_{K,W/πW}). If we can prove that the latter group is zero then it follows that Ext(Δ_{H,U}, Δ_{K,W}) = 0 by Nakayama’s lemma. This justifies our claim that it suffices to assume R is a field.

In what follows we will use various change of rings isomorphisms of the type

Ext_{μ(G)}(M ↑_{F}^{G}, N) ≅ Ext_{μ(F)}(M, N ↓_{F}^{G}).

These are valid because the induction functor is both the left and the right adjoint of restriction [13] so that the conditions in e.g. [7, IV.12] are satisfied. We have

Ext_{μ(G)}(Δ_{H,R}^{F} ↑_{F}^{G}, Δ_{K,R}^{J} ↑_{J}^{G}) ≅ Ext_{μ(F)}(Δ_{H,R}^{F}, Δ_{K,R}^{J} ↑_{J}^{G}↓_{F}^{G})
≅ ⨁_{g∈[F \T (K,F)/J]} Ext_{μ(F)}(Δ_{H,R}^{F}, Δ_{^{g}K,R}^{F ∩^{g}J} ↑_{F ∩^{g}J}^{F})  by 4.5

≅ ⨁_{g∈[F \T (K,F)/J]} Ext_{μ(F ∩^{g}J)}(Δ_{H,R}^{F} ↓_{F ∩^{g}J}^{F}, Δ_{^{g}K,R}^{F ∩^{g}J})

≅ ⨁_{g∈[F \T (K,F)∩T (H,J)^{−1}/J]} Ext_{μ(F ∩^{g}J)}(Δ_{H,R}^{F ∩^{g}J}, Δ_{^{g}K,R}^{F ∩^{g}J})

using 4.3 in the last isomorphism and the fact that H ◁F. Writing L = F ∩^{g}J, the various summands all have the form Ext_{μ(L)}(Inf_{L/H}^{L} F Q_{R}, Inf_{L/^{g}K}^{L} F Q_{R}) where L is a p-group in which both H and ^{g}K are normal. We will show that all Ext groups of this kind are zero unless H properly contains ^{g}K.

As a functor on L/^{g}K, F Q_{R} has simple socle S_{L/^{g}K,R} (by the dual result to [14, 15.3], or by [14, 12.4] from which it may also be deduced). Hence Inf_{L/^{g}K}^{L} F Q_{R} has simple socle S_{L,R} and we have an injective presentation

0 → Inf_{L/^{g}K}^{L} F Q_{R} → B^{L∗} → X → 0  (∗)

where B^{L∗} is the dual of the Burnside ring functor for the group L and X is defined as the cokernel in this sequence. The fact that B^{L∗} is the injective hull of S_{L,R} as a Mackey functor for L follows from [14, 8.9(ii)], and we use this to compute Ext via the exact sequence

Hom(Inf_{L/H}^{L} F Q_{R}, B^{L∗}) → Hom(Inf_{L/H}^{L} F Q_{R}, X)
→ Ext(Inf_{L/H}^{L} F Q_{R}, Inf_{L/^{g}K}^{L} F Q_{R}) → 0.

By adjointness (Proposition 3.2) this is isomorphic to an exact sequence

Hom(R, \underline{B^{L∗}}(H)) → Hom(R, \underline{X}(H)) → Ext(Inf_{L/H}^{L} F Q_{R}, Inf_{L/^{g}K}^{L} F Q_{R}) → 0.

If H ⊉ ^{g}K then (Inf_{L/^{g}K}^{L} F Q_{R})(A) = 0 for all subgroups A of H so from (∗) we obtain that \underline{B^{L∗}}(H) → \underline{X}(H) is an isomorphism and homomorphisms to \underline{X}(H) factor through \underline{B^{L∗}}(H). If H = ^{g}K then consider (∗) again evaluated at H:

0 → Inf_{L/H}^{L} F Q_{R}(H) = R⟶^{α}B^{L∗}(H)⟶^{β}X(H) → 0

We claim that \underline{B^{L∗}}(H) = R. This is equivalent to the statement that \overline{B^{L}}(H) = R, by 2.2 and 2.3 as pointed out there. In B^{L}(H) all H-sets except the one-point set are induced from proper subgroups, so that the Brauer quotient has dimension 1. Also R = Im(α) ⊆ \underline{B^{L∗}}(H) since the domain of α restricts to 0 on all proper subgroups of H so Im(α) = \underline{B^{L∗}}(H) since R is currently assumed to be a field. We show that \underline{X}(H) = 0. If 0 ≠ x ∈ X(H) we can write x = β(y) with y ∉ Im(α). Thus there exists H_{1} ⊂ H with H_{1} ≠ H and R_{H_{1}}^{H}(y) ≠ 0. Now since Inf_{L/H}^{L} F Q_{R}(H_{1}) = 0 we have R_{H_{1}}^{H}(x) ≠ 0. We conclude that \underline{X}(H) = 0. By examining the sequence which computes Ext we deduce that this Ext group is zero when H = ^{g}K, and so in all cases Ext(Inf_{L/H}^{L} F Q_{R}, Inf_{L/^{g}K}^{L} F Q_{R}) = 0 if H ≯ ^{g}K.

(b) We proceed as in part (a), but with F P_{R} instead of F Q_{R} in the second variable, and reduce to considering Ext_{μ(L)}(Inf_{L/H}^{L} F Q_{R}, Inf_{L/^{g}K}^{L} F P_{R}). Now as a functor on L/^{g}K, F P_{R} has simple socle S_{^{g}K/^{g}K,R}, so Inf_{L/^{g}K}^{L} F P_{R} has simple socle S_{^{g}K,R}. We take an injective presentation

0 → Inf_{L/^{g}K}^{L} F P_{R} → I_{^{g}K,R} → Y → 0  (∗∗)

giving an exact sequence

Hom(Inf_{L/H}^{L} F Q_{R}, I_{^{g}K,R}) → Hom(Inf_{L/H}^{L} F Q_{R}, Y)
→ Ext(Inf_{L/H}^{L} F Q_{R}, Inf_{L/^{g}K}^{L} F P_{R}) → 0

which by adjointness is isomorphic to

Hom(R, \underline{I_{^{g}K,R}}(H)) → Hom(R, \underline{Y} (H)) → Ext(Inf_{L/H}^{L} F Q_{R}, Inf_{L/^{g}K}^{L} F P_{R}) → 0.

If H ⊉ ^{g}K then (Inf_{L/^{g}K}^{L} F P_{R})(A) = 0 for all subgroups A of H so from (∗∗) we obtain that \underline{I_{^{g}K,R}}(H) → \underline{Y} (H) is an isomorphism and homomorphisms to \underline{Y} (H) factor through \underline{I_{^{g}K,R}}(H). This shows that if H ⊉ ^{g}K then

Ext(Inf_{L/H}^{L} F Q_{R}, Inf_{L/^{g}K}^{L} F P_{R}) = 0.

We may proceed further and deduce immediately by duality that when H ⊉ ^{g}K then

Ext((Inf_{L/^{g}K}^{L} F P_{R})^{∗}, (Inf_{L/H}^{L} F Q_{R})^{∗}) = 0,

or in other words Ext(Inf_{L/^{g}K}^{L} F Q_{R}, Inf_{L/H}^{L} F P_{R}) = 0 since F P_{R} and F Q_{R} are dual. Interchanging the roles of H and ^{g}K we obtain that when ^{g}K ⊉ H then

Ext(Inf_{L/H}^{L} F Q_{R}, Inf_{L/^{g}K}^{L} F P_{R}) = 0.

Putting everything together we have that Ext(Inf_{L/H}^{L} F Q_{R}, Inf_{L/^{g}K}^{L} F P_{R}) = 0 unless H ⊇ ^{g}K and also ^{g}K ⊇ H, or in other words H = ^{g}K.

Finally (c) is proved dually to (a). We may also deduce it from (a) since when R is a field ∇_{H,U} ≅ (Δ_{H,U^{∗}})^{∗} and Ext(∇_{H,U}, ∇_{K,W}) ≅ Ext(Δ_{K,W^{∗}}, Δ_{H,U^{∗}}) which equals 0 unless H <_{G} K by (a).

Improvements on 5.1 seem hard to come by. For example, we might hope that Ext_{μ_{R}(G)}^{1}(Δ_{H,U}, ∇_{K,W}) = 0 always, in the same manner as for a quasi-hereditary algebra [10], but this is not so. When G = C_{2} and R is the field of two elements we calculate directly that Ext_{μ_{R}(G)}^{1}(Δ_{1,R}, ∇_{1,R}) ≅ R.

In the same spirit as 5.1 we present a similar result about homomorphisms.

(5.2) PROPOSITION. Let H and K be subgroups of G and let U and W be ppermutation modules for N_{G}(H)/H and N_{G}(K)/K respectively. Then

(a) Hom_{μ_{R}(G)}(Δ_{H,U}, Δ_{K,W}) = 0 unless H ≥_{G} K.

(b) Hom_{μ_{R}(G)}(Δ_{H,U}, ∇_{K,W}) = 0 unless H =_{G} K.

(c) Hom_{μ_{R}(G)}(∇_{H,U}, ∇_{K,W}) = 0 unless H ≤_{G} K.

Proof. If M is any Mackey functor for which M(H) = 0 then Hom_{μ_{R}(G)}(Δ_{H,U}, M) = 0 since Δ_{H,U} is generated by its value at H, and if f : Δ_{H,U} → M is non-zero then f(Δ_{H,U})(H) must be non-zero also. Since Δ_{K,W} (H) = ∇_{K,W} (H) = 0 unless H ≥_{G} K this proves (a) and establishes half of (b). Dually, if M is any Mackey functor for which M(K) = 0 then Hom_{μ_{R}(G)}(M, ∇_{K,W}) = 0 since ∇_{K,W} is cogenerated at K. Since Δ_{H,U}(K) = ∇_{H,U}(K) = 0 unless H ≤_{G} K this proves (c) and establishes the other half of (b).

6. D has relative almost split sequences

As a step towards proving that D has relative almost split sequences we need to know that D is closed under taking direct summands, and we prove this in the next results. In the first, we show that any Δ-filtration is compatible, in a certain sense, with any ascending filtration. We continue to suppose that R is a complete p-local ring.

(6.1) PROPOSITION. Let 1 = H_{1}, H_{2}, …, H_{n} = G be a list of representatives of the conjugacy classes of subgroups of G with the property that whenever H_{i} is conjugate to a subgroup of H_{j} then i ≤ j. Let M be a Mackey functor in D. Let M_{i} = I_{{H_{1},...,H_{i}}}M be the terms of the corresponding ascending filtration.

(a) We have M_{i}/M_{i−1} ≅ Δ_{H_{i},\overline{M}(H_{i})} where \overline{M}(H_{i}) is a p-permutation module for N_{G}(H_{i})/H_{i}. Thus the ascending filtration of M is a Δ-filtration.

(b) If ⋯ ⊆ F_{k} ⊆ F_{k+1} ⊆ ⋯ ⊆ M is any filtration of M with indecomposable Δ factors then the terms of the filtration may be changed one by one in the fashion F_{k} → F′_{k} where

\begin{aligned}F_{k+1}/F_{k−1} & ≅ F_{k+1}/F_{k} ⊕ F_{k}/F_{k−1} \\ & ≅ F′_{k}/F_{k−1} ⊕ F_{k+1}/F′_{k} \end{aligned}

such that we start with the given filtration and finish with a refinement of the filtration whose terms are the M_{i}.

(c) The multiplicity of each indecomposable Δ_{H_{i},U_{ij}} in any filtration of M with Δ factors is determined independently of the choice of filtration, and equals the multiplicity of U_{ij} as a summand of \overline{M}(H_{i}).

Proof. We first prove (b). Because Ext(Δ_{H,U}, Δ_{K,W}) = 0 unless H >_{G} K, starting with any filtration ⋯ ⊆ F_{k} ⊆ F_{k+1} ⊆ ⋯ we may perform a sequence of changes in an inductive fashion starting with all the factors of the form F_{k+1}/F_{k} ≅ Δ_{H_{1},U_{1j}} indexed by H_{1} and moving them to the bottom of the filtration, then moving all the factors F_{k+1}/F_{k} ≅ Δ_{H_{2},U_{2j}} immediately above them, and so on until all Δ_{H_{i},U_{ij}} factors occur together for a given i, after the ones for i − 1 and before i + 1. If F_{k} is the term that has beneath it all the Δ_{H_{r},U_{rj}} factors with r ≤ i then F_{k} is generated by its values at the H_{r} with r ≤ i and so F_{k} ⊆ M_{i}. On the other hand, since Δ_{H_{s},U_{sj}}(H_{r}) = 0 if r < s we deduce that M/F_{k} is zero on H_{r} with r ≤ i and hence M_{i}/F_{k} is zero on these H_{r}. But M_{i} and also M_{i}/F_{k} are generated by their values on such H_{r}, and so M_{i}/F_{k} = 0 and hence M_{i} = F_{k}. This establishes (b).

(a) From what we just did we see that M_{i}/M_{i−1} has a filtration by the Δ_{H_{i},U_{ij}} indexed by H_{i}. These functors do not extend themselves, so the factor is a direct sum. If this sum is ⨁_{j} Δ_{H_{i},U_{ij}} then it may also be written Δ_{H_{i},⨁_{j} U_{ij}} and the value of this Mackey functor

at H_{i} is (M_{i}/M_{i−1})(H_{i}) = ⨁_{j} U_{ij}, which by 2.4 is \overline{M}(H_{i}). This establishes (a).

(c) The multiplicity of Δ_{H_{i},U_{ij}} in the original filtration by the F_{k} is unchanged by the operations used in the proof of (b) and so equals the multiplicity of Δ_{H_{i},U_{ij}} as a summand of M_{i}/M_{i−1}. We have already seen that this is the multiplicity of U_{ij} as a summand of \overline{M}(H_{i}). Since any Δ-filtration can be transformed into a refinement of this ascending filtration without changing multiplicities, these multiplicities are independently determined.

The following is a characterization of the Mackey functors in D when R is a field.

(6.2) COROLLARY. Let R be a field. A Mackey functor M is an object of D if and only if the following two conditions are satisfied:

(a) for all subgroups H the Brauer quotient \overline{M}(H) is a p-permutation N_{G}(H)/H-module, and

(b) for all subgroups K of G

dim M(K) = ∑_{H≤G up to conjugacy} dim Δ_{H,\overline{M}(H)}(K).

Proof. Let 1 = H_{1}, H_{2}, …, H_{n} = G be a list of representatives of the conjugacy classes of subgroups of G with the property that whenever H_{i} is conjugate to a subgroup of H_{j} then i ≤ j and let M_{i} = I_{{H_{1},...,H_{i}}}M. Then we know in general from 3.6 that

dim M(K) ≤ ∑_{i=1}^{n} dim Δ_{H_{i},\overline{M}(H_{i})}(K)

and that we have equality if and only if every factor M_{i}/M_{i−1} is isomorphic to Δ_{H_{i},\overline{M}(H_{i})}. The only remaining condition for M to lie in D is supplied by (a).

The following result is also clear from our analysis.

(6.3) COROLLARY. Let M be a Mackey functor in D and suppose that a particular subgroup H appears as H_{i} in a list 1 = H_{1}, H_{2}, …, H_{n} = G and as K_{j} in a list 1 = K_{1}, K_{2}, …, K_{n} = G of conjugacy classes of subgroups, where both lists satisfy the condition that the conjugates of subgroups of each term appear earlier in the list. Let M_{i} and M′_{j} denote the terms of the corresponding ascending filtrations of M. Then M_{i}/M_{i−1} ≅ M′_{j}/M′_{j−1}. In other words, the factor associated to H does not depend on the position of H in the list.

It is the following corollary which will in fact be used to deduce the main result of this section.

(6.4) COROLLARY. Let M = A ⊕ B be a functor in D. Then A and B also lie in D.

Proof. We have M_{i} = A_{i} ⊕ B_{i} and so M_{i}/M_{i−1} = ⨁ Δ_{H_{i},U_{ij}}^{m_{ij}} = A_{i}/A_{i−1} ⊕ B_{i}/B_{i−1}. Since the Δ_{H_{i},U_{ij}} are indecomposable, by the Krull-Schmidt theorem we have that each of A^{i}/A^{i−1} and B^{i}/B^{i−1} is a direct sum of functors Δ_{H_{i},U_{ij}} and so A and B are objects in D.

We come to one of the principal results of this paper. It relies on the work of [10] which deals with representations of artin algebras. Because of this we require R to be a field.

(6.5) THEOREM. Let R be a field. The category D is functorially finite in Mack_{R}(G), and has relative almost split sequences.

Proof. This is a consequence of Theorem 1 and Corollary 1 of [10]. That theorem has to do with a subcategory of modules with filtrations with factors in a list Θ(1), …, Θ(n) where Ext(Θ(j), Θ(i)) = 0 for j ≥ i. If we arrange our Mackey functors Δ_{H,U} with U a p-permutation module in a list so that whenever K is conjugate to a subgroup of H then Δ_{H,U} precedes Δ_{K,W}, this list will satisfy Ringel’s condition by Theorem 5.1. Ringel’s theorem now shows that D is functorially finite in Mack_{R}(G). Because D is closed under taking direct summands, Corollary 1 of [10] says that D has relative almost split sequences.

7. Projective Mackey functors lie in D

We now show that projective Mackey functors are objects in D. The proof uses the formal machinery we have developed, and for this reason appears late in this exposition. We have also been able to find a more direct argument which could have been presented earlier, but it is longer and requires quite elaborate combinatorial manipulation. It seems less satisfactory conceptually than the one we are about to give.

(7.1) THEOREM. Let M be a finitely-generated projective Mackey functor over a complete p-local ring. Then M is an object in D.

Proof. We start with the same reduction that was made at the start of the proof of 5.1, namely we may assume that our projective Mackey functor M lies in Mack_{R}(G, 1). For suppose that M happens to lie in Mack_{R}(G, J) where J is a p-perfect subgroup of G. The reverse functor in the equivalence

(Inf_{N(J)/J}^{N(J)} −) ↑_{N(J)}^{G}: Mack_{R}(N(J)/J, 1) → Mack_{R}(G, J)

is restriction followed by deflation. If we regard M as a functor in Mack_{R}(N_{G}(J)/J, 1) by this means and produce a Δ-filtration of it in this category, then on applying (Inf_{N(J)/J}^{N(J)} −) ↑_{N(J)}^{G} we will obtain a Δ-filtration of M in Mack_{R}(G, J), by Lemmas 4.7 and 4.1. In view of this we will now assume that M ∈ Mack_{R}(G, 1).

We next claim that it suffices to assume M = B^{Q} ↑_{Q}^{G} for some p-subgroup Q, where B^{Q} is the Burnside ring Mackey functor for the subgroup Q. This is because every projective functor in Mack_{R}(G, 1) is a direct summand of a direct sum of functors of the form B^{Q} ↑_{Q}^{G} with Q a p-group [14, 8.7], and direct summands of objects in D again lie in D by 6.4.

In order to show that B^{Q} ↑_{Q}^{G} has a Δ-filtration it suffices to show that B^{Q} has a Δfiltration, as a Mackey functor for the p-group Q. This is because the property of having a Δ-filtration is preserved under induction, by 4.8.

Let 1 = H_{1}, H_{2}, …, H_{n} = Q be representatives of the conjugacy classes of subgroups of Q, listed so that whenever H_{i} is conjugate to a subgroup of H_{j} then i ≤ j. We will show that the corresponding ascending filtration of B^{Q} is a Δ-filtration by showing that each of the epimorphisms

Δ_{H_{i},\overline{B^{Q}}(H_{i})} → (B^{Q})_{i}/(B^{Q})_{i−1}

of Corollary 3.5 is an isomorphism and also that \overline{B^{Q}}(H_{i}) is a p-permutation R[N_{Q}(H_{i})/H_{i}]- module for all subgroups H_{i} of Q. In fact \overline{B^{Q}}(H_{i}) = R since all H_{i}-sets except the one-point set are induced from proper subgroups; and R is indeed a p-permutation module. Thus it remains to show that the epimorphisms are isomorphisms, and we claim that it is sufficient to show that

∑_{i=1}^{n} rank_{R} Δ_{H_{i},\overline{B^{Q}}(H_{i})}(K) = rank_{R} B^{Q}(K) for every subgroup K of Q.

To see why this is so, we note from Proposition 3.1 that in this particular situation

Δ_{H_{i},\overline{B^{Q}}(H_{i})}(K) = ⨁_{g∈[K\T (H_{i},K)/N_{Q}(H_{i})]} R

for each subgroup K, and this is a free R-module. Also (B^{Q})_{i}(K) is the span in B(K) of the K-sets K/H_{j} where j ≤ i and H_{j} ⊆ K, so that (B^{Q})_{i}/(B^{Q})_{i−1}(K) is also a free R-module. Thus to show that the epimorphisms are isomorphisms it suffices to show that

rank_{R} Δ_{H_{i},\overline{B^{Q}}(H_{i})}(K) = rank_{R}((B^{Q})_{i}/(B^{Q})_{i−1}(K))

for all H_{i} and for all subgroups K of Q. Since the rank on the left is always at least the rank on the right, it suffices to show that

∑_{i=1}^{n} rank Δ_{H_{i},\overline{B^{Q}}(H_{i})}(K) = ∑_{i=1}^{n} rank_{R}((B^{Q})_{i}/(B^{Q})_{i−1}(K)) for every subgroup K of Q

and the quantity on the right is rank B^{Q}(K). This establishes the most recent claim.

Now to verify that the claimed equality is indeed so we use again the formula from Proposition 3.1 to obtain

rank_{R} Δ_{H_{i},R}(K) = |K\T (H_{i}, K)/N_{Q}(H_{i})|.

This quantity is well known to be the number of K-conjugacy classes of subgroups of K which are Q-conjugate to H_{i}. The sum of these numbers over all subgroups H_{i} (taken up to Q-conjugacy) equals the number of K-conjugacy classes of subgroups of K since these are partitioned by the relation of being conjugate in Q. This in turn is equal to rank B^{Q}(K), which completes the proof.

We point out that the factors in a Δ-filtration of a projective Mackey functor M may be determined by combining results from [14] and [3] with Proposition 6.1. For by 6.1(a) M has a filtration with factors Δ_{H,\overline{M}(H)}, and assuming that M ∈ Mack_{R}(G, 1) the modules \overline{M}(H) are determined in [3, 5.10] as Brauer quotients

M(1)^{H}/ ∑_{J<H} tr_{J}^{H} M(1)^{J}.

Finally if M is indecomposable and has simple quotient S_{K,W}, then M(1) is the Green correspondent of the projective cover (as a R[N_{G}(K)/K]-module) of V, by [14, 12.7(ii)].

8. Ext-projective and Ext-injective objects in D

In this section and the next we go further with the methods used by Ringel [10] and extend them to a situation where weaker hypotheses are in force than the ones he considered. Our general set-up will be the following situation:

(8.1) HYPOTHESIS. D is the full subcategory of the finite length modules for an artin algebra μ whose objects are the modules with a filtration whose factors lie in a set {Θ(λ) | λ ∈ Λ}, where Λ is a finite pre-ordered set.

A pre-ordered set (also called a quasi-poset) is a set with a reflexive, transitive binary relation ≤. We write ρ < λ if ρ ≤ λ and λ ≰ ρ. In our application to Mackey functors we will take Λ to be the set of pairs (H, U) where H is a subgroup of G taken up to conjugacy and U is an indecomposable p-permutation R[N_{G}(H)/H]-module taken up to isomorphism. The pre-order on this set is specified by (H, U) ≤ (K, V) if and only if H is conjugate to a subgroup of K. Then (H, U) < (K, V) will mean that H is conjugate to a proper subgroup of K.

In Ringel’s work [10] the modules Θ(λ) were linearly ordered, rather than being related by a pre-order. We may still apply his results here by putting the elements of Λ in order Λ = {λ_{1}, …, λ_{n}} in such a way that λ_{i} > λ_{j} implies i < j (we deliberately choose to place the λ’s in non-ascending order so as to be consistent with Ringel). For example, we will work with the condition that Ext(Θ(λ), Θ(ρ)) ≠ 0 implies λ > ρ. If we place the elements of Λ in a total order in this way, this condition implies Ext(Θ(λ_{i}), Θ(λ_{j})) = 0 unless i < j, which is the condition which Ringel required in [10].

We begin with some generalities about Ext-projectivity and Ext-injectivity. We say that an object M of D is Ext-injective if Ext(A, M) = 0 for all objects A in D, and it is Ext-projective if Ext(M, A) = 0 for all objects A in D. Although we only really work explicitly with Ext-injective objects in this section, all of our results have dual versions which relate to Ext-projective objects, since there is nothing in our hypotheses to favour injectivity rather than projectivity.

We start by showing that Ext-injective objects have familiar lifting and splitting properties.

(8.2) LEMMA. In the situation of Hypothesis 8.1, let I be an Ext-injective object in D.

(a) Given mappings

0 ⟶ A ⟶^{α} B ⟶ C ⟶ 0
θ↓
I

in which the row is a short exact sequence and all objects lie in D, there exists a mapping β : B → I so that θ = βα.

(b) Every short exact sequence 0 → I → B → C → 0 in which all objects belong to D is split.

Proof. (a) Consider the long exact sequence

0 → Hom(C, I) → Hom(B, I) → Hom(A, I) → Ext(C, I) = 0

This shows that the mapping θ is the image of a mapping β : B → I.

(b) We apply (a) with A = I, taking θ to be the identity map on I.

We will say that a morphism A⟶^{α}I of objects in D is an Ext-injective hull of A if the following four conditions are satisfied:

1. α is a monomorphism.

2. I is Ext-injective.

3. Coker α is an object in D.

4. Whenever β : I → J is a morphism where J is an object of D, βα is a monomorphism and Coker βα is an object of D then β is a monomorphism.

We define an Ext-projective cover by the similar dual property.

(8.3) PROPOSITION. In the situation of Hypothesis 8.1, if an object of D has an Ext-injective hull it is unique up to isomorphism. Similarly, if an object of D has an Ext-projective cover it is unique up to isomorphism.

Proof. Let α : A → I and α′ : A → I′ be two Ext-injective hulls of A. We show that they are isomorphic. By Ext-injectivity we can find β : I → I′ with α′ = βα and γ : I′ → I with α = γα′. The second equation shows that γ is a monomorphism by condition 4. Putting the two equations together gives α = γβα which shows that γβ : I → I is a monomorphism, again by condition 4. Since I has finite length, it follows that γβ is an isomorphism, and hence γ is an epimorphism. This shows that γ is an isomorphism, and so the two Ext-injective hulls are isomorphic.

The uniqueness of Ext-projective covers is proved dually.

The argument we have just seen will also prove the following statement, whose dual formulation for Ext-projectivity we leave to the reader.

(8.4) PROPOSITION. Let α : A → I be an Ext-injective hull of A in D, and let α′ : A → I′ be any monomorphism to an Ext-injective object I′ such that Coker α′ is an object in D. Then there is a morphism β : I → I′ so that α′ = βα, and furthermore any such β is a split monomorphism.

Proof. As in the last result we construct β : I → I′ with α′ = βα and γ : I′ → I with γα′ = α. Thus α = γβα and by condition 4 we deduce that γβ : I → I is a monomorphism. Hence γβ is an isomorphism since I has finite length, and so β is a split monomorphism.

We now come to the existence of Ext-injective hulls and Ext-projective covers, and for this we impose further hypotheses on top of 8.1.

(8.5) PROPOSITION. In the situation of Hypothesis 8.1, suppose further that direct summands of modules in D again lie in D and that for all λ, ρ ∈ Λ we have

Ext(Θ(λ), Θ(ρ)) = 0

unless λ > ρ. Then each object A in D has an Ext-injective hull and an Ext-projective cover in D.

Proof. Let us start by writing Λ = {λ_{1}, …, λ_{n}} in such a way that λ_{i} > λ_{j} implies i < j. Then Ext(Θ(λ_{i}), Θ(λ_{j})) = 0 unless i < j and we may apply the results of [10]. From Lemma 4′ of [10] we see that it is possible to find a mapping α : A → I to an object of D satisfying conditions 1, 2 and 3 of the definition of an Ext-injective hull. Among such mappings, choose one with I of minimal length. We show that condition 4 is satisfied. Suppose β : I → J is a morphism as in condition 4. Then by the Ext-injective property there exists γ : J → I with α = γβα and hence α = (γβ)^{n}α for every n ≥ 1. By Fitting’s lemma there exists n so that I = (γβ)^{n}I ⊕ Ker(γβ)^{n} and now α factorizes as A⟶^{α_{1}}(γβ)^{n}I ↪ I. We show that A⟶^{α_{1}}(γβ)^{n}I satisfies conditions 1, 2 and 3 of the definition of an Ext-injective hull. First, (γβ)^{n}I is an object of D, since it is a direct summand of I, and for the same reason it is Ext-injective. Also, α_{1} is a monomorphism and Coker α_{1} appears in the exact sequence 0 → Ker(γβ)^{n} → Coker α_{1} → Coker α → 0, which shows that Coker α_{1} is an object in D since Ker(γβ)^{n} is a summand of I and so lies in D. It follows by minimality of I that I = (γβ)^{n}I and hence β is a monomorphism. This shows that condition 4 is satisfied.

It is clear that Ext-projective covers may be produced by a dual construction.

We will denote by T (λ) the Ext-injective hull in D of Θ(λ), and by Π(λ) the Extprojective cover in D of Θ(λ).

(8.6) THEOREM. In the situation of Hypothesis 8.1, suppose further that direct summands of modules in D again lie in D and that for all λ, ρ ∈ Λ we have

Ext(Θ(λ), Θ(ρ)) = 0

unless λ > ρ. Then the objects T (λ) are precisely the indecomposable Ext-injective objects in D. Each T (λ) has the property that in any filtration with Θ factors, Θ(λ) always occurs as the bottom factor, and the other Θ(ρ) factors all have ρ < λ. The objects Π(λ) are precisely the indecomposable Ext-projective objects in D. Each Π(λ) has the property that in any filtration with Θ factors, Θ(λ) always occurs as the top factor, and the other Θ(ρ) factors all have ρ > λ.

Proof. We first show that the T (λ) are the only indecomposable Ext-injectives in D. Suppose that J is an indecomposable Ext-injective in D. Then there is a short exact sequence 0 → Θ(λ) → J → C → 0 with C in D, for some λ. Let Θ(λ) → T (λ) be the Ext-injective hull of Θ(λ). By 8.4 there exists a morphism β : T (λ) → J which is a split monomorphism. Since J is indecomposable we deduce that J ≅ T (λ). This shows that the list of indecomposable Ext-injective objects in D is complete. The T (λ) have the claimed properties by the construction in Lemma 4′ of [10], and by the vanishing of Ext groups between the Θ(ρ).

The proof of the statement about Ext-projectives is dual.

In the specific case of the category D of Mackey functors whose objects are the Mackey functors with Δ-filtrations, we denote by T_{H,U} the Ext-injective hull of Δ_{H,U} and Π_{H,U} the Ext-projective cover of Δ_{H,U}. By 8.6 we get a complete list of indecomposable Extinjectives and Ext-projectives in D as (H, U) ranges through all pairs with H taken up to conjugacy and the indecomposable p-permutation R[N_{G}(H)/H]-module U taken up to isomorphism.

9. A standardly stratified algebra

The similarity between the properties of Mackey functors in D and the modules with a good filtration of Ringel [10] allows us to imitate one of his main constructions, that of the ‘Ringel dual’. In the case of a quasi-hereditary algebra the modules with a Δfiltration which also have a ∇-filtration coincide with the Ext-injective modules with a Δfiltration. The Ringel dual is the endomorphism ring of the direct sum of these canonical indecomposable modules. In the generality of the last section having to do with modules for a more general artin algebra μ, we replace the condition that modules have both a Δand a ∇-filtration by the condition that they are Ext-injective, which comes to the same thing for a quasi-hereditary algebra. Using these modules which we classified in Theorem 8.6, we show with one further hypothesis that the endomorphism ring of their direct sum is standardly stratified in the sense of [4].

(9.1) THEOREM. In the situation of Hypothesis 8.1, suppose that D is closed under taking direct summands, that Ext(Θ(λ), Θ(ρ)) = 0 unless λ > ρ, and also that Hom(Θ(λ), Θ(ρ)) = 0 unless λ > ρ. Let T = ⨁_{λ∈Λ} T (λ) be the direct sum of all the isomorphism types of indecomposable Ext-injective objects in D. Then the algebra E = End_{μ}(T) is standardly stratified in the sense of [4] with pre-ordered set Λ.

Proof. We verify the conditions of Definition 2.2.1 of [4]. The indecomposable projective modules for E are Q(λ) = Hom_{μ}(T (λ), T) and we define Δ(λ) = Hom_{μ}(Θ(λ), T).

The homomorphism Q(λ) → Δ(λ) which comes by functoriality from the inclusion Θ(λ) → T (λ) is epi since T is Ext-injective. Let us denote the simple E-module which is the unique simple quotient of Q(λ) by L(λ).

We will use the following result, which is similar to Yoneda’s lemma.

(9.2) LEMMA. Hom_{E}(Q(ρ), Δ(λ)) ≅ Hom_{μ}(Θ(λ), T (ρ)).

Proof. We show that the homomorphism

Hom_{μ}(Θ(λ), T (ρ)) → Hom_{E}(Hom(T (ρ), T), Hom(Θ(λ), T))
α ↦ (θ ↦ θ ∘ α)

is an isomorphism. It is a monomorphism because we may take θ to be the inclusion ι : T (ρ) ↪ T and then θ ∘α = 0 implies α = 0. It is an epimorphism: let ϕ : Hom(T (ρ), T) → Hom(Θ(λ), T) and consider ϕ(ι) : Θ(λ) → T. We show that ϕ(ι) factors through ι. Let π : T → T be the identity on T (ρ) and 0 on the other summands. Then πι = ι. Also ϕ(πι) = πϕ(ι) since ϕ is an E-map. Thus ϕ(ι) = πϕ(ι) so ϕ(ι) has image contained in ι(T (ρ)). Write ϕ(ι) = ιϕ̄ where ϕ̄ : Θ(λ) → T (ρ). We show that ϕ is the image of ϕ̄. To see this, let β : T (ρ) → T and let β̃ : T → T be β on T (ρ), and 0 on the other summands. Then ϕ(β) = ϕ(β̃ι) = β̃ϕ(ι) = β̃ιϕ̄ = βϕ̄ shows that ϕ is the image of ϕ̄. This completes the proof.

We verify condition (1) of Definition 2.2.1 of [4], which says the following: if the composition factor multiplicity [Δ(λ) : L(ρ)] is non-zero then ρ ≤ λ. This composition factor multiplicity is non-zero precisely if Hom_{μ}(Q(ρ), Δ(λ)) is non-zero, which by Lemma 9.2 is non-zero precisely if Hom_{μ}(Θ(λ), T (ρ)) ≠ 0. All Θ-factors of T (ρ) are Θ(σ) with σ ≥ ρ, and since Hom_{μ}(Θ(λ), Θ(σ)) = 0 unless λ ≥ σ we have Hom_{μ}(Θ(λ), T (ρ)) = 0 unless ρ ≤ λ. This shows that condition (1) is satisfied.

We now verify condition (2) of Definition 2.2.1 of [4], which says the following in our situation: for each λ ∈ Λ there is a short exact sequence 0 → R(λ) → Q(λ) → Δ(λ) → 0 where R(λ) has a filtration with factors Θ(ρ) with ρ > λ. By construction of the T (λ), each one has a filtration 0 = F_{0}T (λ) ⊆ F_{1}T (λ) ⊆ ⋯ ⊆ F_{n}T (λ) = T (λ) with F_{1}T (λ) ≅ Θ(λ) and F_{i}T (λ)/F_{i−1}T (λ) ≅ Θ_{ρ_{i}} with ρ_{i} > λ. Now we get a filtration 0 = F_{n}Q(λ) ⊆ F_{n−1}Q(λ) ⊆ ⋯ ⊆ F_{0}Q(λ) = Q(λ) where F_{i}Q(λ) = Ker(Hom(T (λ), T) → Hom(F_{i}T (λ), T)). Consider the following commutative diagram, in which the rows are exact at the right since T is Ext-injective.

0  0
↓  ↓
0 ⟶ F_{i}Q(λ) ⟶ Hom(T (λ), T) ⟶ Hom(F_{i}T (λ), T) ⟶ 0
↓  ∥  ↓
0 ⟶ F_{i−1}Q(λ) ⟶ Hom(T (λ), T) ⟶ Hom(F_{i−1}T (λ), T) ⟶ 0

↓

0

By the snake lemma we have

F_{i−1}Q(λ)/F_{i}Q(λ) ≅ Ker(Hom(F_{i}T (λ), T) → Hom(F_{i−1}T (λ), T)).

Now 0 → Hom(F_{i}T (λ)/F_{i−1}T (λ), T) → Hom(F_{i}T (λ), T) → Hom(F_{i−1}T (λ), T) is exact, so

F_{i−1}Q(λ)/F_{i}Q(λ) ≅ Hom(F_{i}T (λ)/F_{i−1}T (λ), T) ≅ Δ(ρ_{i}).

This gives a filtration of Q(λ) by standard objects with Δ(λ) at the top and other factors Δ(ρ_{i}) with ρ_{i} > λ, thus completing the verification of condition (2).

In the case of our application of this theory to Mackey functors, the standardly stratified algebra E stands in relation to the group algebra RG in the same way as the Schur algebra stands in relation to the corresponding symmetric group algebra, as shown in the next result.

(9.3) PROPOSITION. The algebra End_{μ_{R}(G)}(⨁_{U projective} T_{1,U}), where the sum is taken over indecomposable RG-projective modules U, has the form eEe for a certain idempotent e ∈ E. This subalgebra is Morita equivalent to RG.

Proof. We have Δ_{1,U} = F Q_{U} from the definition, and if U is projective this is both projective and injective as a Mackey functor by [14, 13.1]. Thus Δ_{1,U} is Ext-injective and so Δ_{1,U} = T_{1,U}. From [13, 6.1] we know that Hom_{μ_{R}(G)}(Δ_{1,U_{1}}, Δ_{1,U_{2}}) ≅ Hom_{RG}(U_{1}, U_{2}) and so

End_{μ_{R}(G)}( ⨁_{U projective} T_{1,U}) ≅ End_{RG}( ⨁_{U projective} U),

which is Morita equivalent to RG. Letting e : T → T be the idempotent which is the identity on the summands T_{1,U} with U projective and zero on the other summands we have eEe ≅ End_{μ_{R}(G)}(⨁_{U projective} T_{1,U}).

10. Two examples

We describe the categories D when G is the cyclic group of order 2 and the cyclic group of order 4, and when R is the field with 2 elements. Detailed arguments justifying our description will be given after we have presented all the tables. According to [13] there are two simple functors S_{1,R} and S_{C_{2},R} for C_{2} and three simple functors S_{1,R}, S_{C_{2},R} and S_{C_{4},R} for C_{4} given explicitly by

S_{H,R}(J) = \{ \begin{matrix}R & if H = J \\ 0 & otherwise. \end{matrix}

To shorten the notation we will simply write 1 for S_{1,R}, 2 for S_{C_{2},R} and 4 for S_{C_{4},R}. The projective covers of these functors may be described diagrammatically as follows.

G = C_{2}
(H, U)  1, R C_{2}, R
Simple functor  1  2
Projective ; cover ; P_{H,U}  1 2 1  2 1 2

G = C_{4}
(H, U)  1, R C_{2}, R C_{4}, R
Simple functor  1  2  4
Projective ; cover ; P_{H,U} _{1 {1 \atop 1} 2 4 2 1}  _{2 4 2 1 1 2 4 2}  4 2 1 2 4 4

In the diagrams for these functors we indicate a non-split extension by positioning composition factors underneath each other, but in the case of C_{4} not all non-split extensions are indicated in this way. The quotient of the radical by the socle of the functor P_{C_{2},R} for C_{4} has the structure 4 2 1 1 2 4 . In the middle of P_{1,R} for C_{4} there are non-split extension {1 \atop 2} and {2 \atop 1} in such a way that the functor Δ_{1, {R \atop R}} (to be described in the next table) is a homomorphic image of P_{1,R}, and the dual ∇_{1, {R \atop R}} = Δ_{1, {R \atop R}}^{∗} is a subfunctor.

We now present a description of the Mackey functors Δ_{H,U} where U is a p-permutation module for N_{G}(H)/H, their Ext-injective hulls T_{H,U} and Ext-projective covers Π_{H,U} and some modules for the algebra E = End_{μ_{R}(G)}(⨁ T_{H,U}). We will use the notation RC_{4} for the induced module R ↑_{1}^{C_{4}} and {R \atop R} for the induced module R ↑_{C_{2}}^{C_{4}}, both as a representation of C_{4} and also of C_{4}/C_{2}. We have the following tables of functor values:

(H, U)  1, R 1, {R \atop R} C_{2}, R
Mackey ; functors ; Δ_{H,U}  {1 \atop 2}  1 2 1  2
Ext- ; injectives ; T_{H,U}  2 1 2  1 2 1  2
Ext- ; projectives ; Π_{H,U}  {1 \atop 2}  1 2 1  2 1 2
Simple ; E-modules α β  γ
standard ; modules ; Δ for E  {α \atop β}  β  {γ \atop α}
Projectives ; for E α ; {β γ \atop α}  β ; α β  {γ \atop α}

Table of Mackey functors for C_{2} and modules for the stratified algebra E We may check from the table for C_{2} that E is not only standardly stratified but in fact quasi-hereditary; however this behaviour does not seem to be typical.

(H, U)  1, R 1, {R \atop R} 1, RC_{4} C_{2}, R C_{2}, {R \atop R} C_{4}, R
Mackey ; functors ; Δ_{H,U}  1 2 4  1 {1 2 4 \atop 2}  1 {1 \atop 1} 2 4 2 1  {2 \atop 4}  2 4 2  4
Ext- ; injectives ; T_{H,U}  4 2 4 1 4 2 4  2 4 2 1 1 2 4 2  1 {1 \atop 1} 2 4 2 1  4 2 4  2 4 2  4
Ext- ; projectives ; Π_{H,U}  1 2 4  1 {1 2 4 \atop 2}  _{1 {1 \atop 1} 2 4 2 1}  2 1 2 4 4  _{2 4 2 1 1 2 4 2}  4 2 1 2 4 4
Simple ; E- modules  α  β  γ  δ  ϵ  ζ
standard ; modules ; Δ for E  α β ; γ  β ; {γ α β \atop γ}  _{γ ; \begin{aligned}{γ \atop γ} & β ; α β \\ & γ \end{aligned}}  δ ; {α ϵ \atop β}  _{ϵ ; β δ ; {α ϵ \atop β}}  ζ ; {α δ \atop α}
Projectives ; for E Δ(α) ; {Δ(ζ) Δ(δ) \atop Δ(ζ)} {Δ(β) \atop Δ(ϵ)} Δ(γ)  {Δ(δ) \atop Δ(ζ)} Δ(ϵ) Δ(ζ)

Table of Mackey functors for C_{4} and modules for the stratified algebra E

We next present the relative Auslander-Reiten quivers for the categories D. In the quiver for C_{4} the Mackey functors 4 2 4 at the corners should be identified with the third copy of this Mackey functor in the middle of the opposite side, and similarly with the copies of the Mackey functor 1 2 4 . There are also some further identifications which should be made in cases when a single functor appears more than once in the diagram. We show only the sink and source maps which form part of relative almost split sequences, not the other sink and source maps involving Ext-projectives and Ext-injectives. Where there is no almost split sequence at a particular mesh in the diagram this is indicated with the symbol ♢. This happens because the Mackey functor on the left is Ext-injective and the one on the right is Ext-projective. In the case of the Mackey functors which are the projective covers of 1 and 2 we have omitted ♢. These functors are both projective and injective.

P_{2} = 2 1_{2}

↗ ↘
♢ {1 \atop 2}  2 ♢
P_{1} = 1 2 1

The relative Auslander-Reiten quiver of functors with Δ-filtration for C_{2} at p = 2.

4 2 4  ♢  1 {1 2 4 \atop 2}  2 4 2  ♢  1 2 4
↗  ↘  ↗  ↘  ↗  ↘  ↗
{2 \atop 4}  {1 4 \atop 2} 4  1 2 2 4 1 4 2 2  {2 \atop 4}
↘  ↗  ↘  ↗  ↘  ↗  ↘
♢  1 2 4  _{2 4 1 2 4 4 2}  1 2 2 4 1 4 2  4 2 4
↗  ↘  ↗  ↘  ↗  ↘  ↗
1 4 2 2 4 1 4 2 →_{2 ; \begin{aligned}4 ; {2 \atop P_{2}= 1} & & 1 2 4 \\ & 2 \end{aligned}} → 2 1 2 4 4 2  _{2 4 1 4 2 4}  1 4 2 2 4 1 4 2 → _{2 4 2 1 1 2 4 2}
↘  ↗  ↘  ↗  ↘  ↗  ↘
4 2 4  ♢  2 1 2 4 4  4 2 4 1 4 2 4  ♢  1 2 4
↘  ↗  ↘
♢ P_{4} = 4 2 1 2 4 4  4  ♢
P_{1} = _{1 {1 \atop 1} 2 4 2 1}

The relative Auslander-Reiten quiver of functors with Δ-filtration for C_{4} at p = 2.

To explain the calculations which produce these diagrams we will concentrate on Mackey functors for the group C_{4}. The case of C_{2} is much easier, and we leave it to the reader. We start with the functors Δ_{H,U} where U is a p-permutation module. The possibilities for H, U are listed at the top of the first table. The structures of Δ_{1,R} = F Q_{R}, Δ_{C_{2},R} = Inf_{C_{4}/C_{2}}^{C_{4}} F Q_{R} and Δ_{C_{4},R} = Inf_{C_{4}/C_{4}}^{C_{4}} F Q_{R} may be deduced by inflating the result of [14, 15.3]. Next Δ_{C_{2}, {R \atop R}} = Inf_{C_{4}/C_{2}}^{C_{4}} F Q_{{R \atop R}} is the inflation of a projective functor for C_{4}/C_{2} by [14, 13.3], and we know its structure from the tables at the end of [14]. The most difficult functors Δ_{H,U} to determine are Δ_{1, {R \atop R}} and Δ_{1,RC_{4}}. By [14, 13.3] we know that Δ_{1,RC_{4}} = P_{1,R} is the projective cover of 1. Since U → F Q_{U} is a right exact functor, the module epimorphism RC_{4} → {R \atop R} gives an epimorphism Δ_{1,RC_{4}} → Δ_{1, {R \atop R}}. We deduce that Δ_{1, {R \atop R}} has 1 as its unique simple quotient. By computing dimensions of fixed quotients we see that the composition factors of Δ_{1, {R \atop R}} are 1, 1, 2, 2 and 4. The exact sequence of modules 0 → R → {R \atop R} → R → 0 gives an exact sequence Δ_{1,R} → Δ_{1, {R \atop R}} → Δ_{1,R} → 0 and since we know the composition factors of each term we see that there is a short exact sequence of functors 0 → {1 \atop 2} → Δ_{1, {R \atop R}} → 1 2 4 → 0. By [14, 12.4], Δ_{1, {R \atop R}}^{∗} ≅ F P_{{R \atop R}} is an image of the projective P_{C_{2},R} so has 2 as its unique simple quotient. Hence the socle of Δ_{1, {R \atop R}} is 2. The subfunctor of Δ_{1, {R \atop R}} generated by its value at 2 is zero on the identity group since the operation R_{1}^{C_{2}} is zero (because C_{2} acts trivially on {R \atop R}). Hence there is a short exact sequence 0 → 2 4 2 → Δ_{1, {R \atop R}} → {1 \atop 1} → 0. From all this we deduce that Δ_{1, {R \atop R}} has the stated form.

After this, the structure of P_{1,R} = Δ_{1,RC_{4}} (also denoted P_{1} in the tables) is relatively easy to determine. The dimensions of this functor at the various subgroups give the composition factor multiplicities, and they are as presented. The functor is self-dual, with top and socle isomorphic to 1 [14, 13.3]. Its value at the identity group is RC_{4}, which is uniserial with four 1 factors. Thus no sections of the form 1 ⊕ 1 appear. There is an epimorphism Δ_{1,RC_{4}} → Δ_{1, {R \atop R}} whose codomain we have already determined. By looking at the composition factors we have a short exact sequence 0 → {1 \atop 1} → Δ_{1,RC_{4}} → Δ_{1, {R \atop R}} → 0. Combining these ingredients gives the form of the functor.

As a next step we compute the structure of the indecomposable projective Mackey functors. We have just done P_{1,R}. We know the Cartan invariants of these functors by the method of [14], and also by [14, 19.1] we know that P_{C_{2},R} is self-dual with socle 2. Its value at the identity group is P_{C_{2},R}(1) = {R \atop R} by [14, 12.7] and so F Q_{{R \atop R}} = Δ_{1, {R \atop R}} is a subfunctor by [14, 12.4]. In fact this is the subfunctor generated by the value at the identity group, so it is the start of a Δ-filtration by 6.1. Thus the quotient by this subfunctor has a filtration by Δ’s (whose general form we have already determined) and has composition factors 2, 2, and 4. It follows that we have a short exact sequence 0 → Δ_{1, {R \atop R}} → P_{C_{2},R} → Δ_{C_{2}, {R \atop R}} → 0 since this is the only possibility with these composition factors. Putting this together gives the structure presented for P_{C_{2},R}.

We may follow a similar line of argument for P_{C_{4},R}. We know its composition factors, that P_{C_{4},R}(1) = R and so Δ_{1,R} is the start of a Δ-filtration of P_{C_{4},R}. The quotient has composition factors 2, 4 and 4 so the Δ-filtration must have factors Δ_{C_{2},R} = {2 \atop 4} and Δ_{C_{4},R} = 4. Since P_{C_{4},R} has 4 as its unique simple quotient we deduce that there is a short exact sequence

0 → Δ_{1,R} → P_{C_{4},R} → {4 \atop 2} 4 → 0

where the right-hand functor is uniserial. Also by [14, 12.4], F P_{R} = 4 2 1 is an image of P_{C_{4},R}. From all this the structure follows.

The next step is to compute the groups Ext(Δ_{H,U}, Δ_{K,W}) where U and W are ppermutation modules. It speeds up the work to use Theorem 5.1, which implies that we can only get a non-zero Ext group when (K, W) = (1, R), (1, {R \atop R}) or (C_{2}, R). By direct calculation with projective presentations we find that Ext(Δ_{H,U}, Δ_{1,R}) = R when (H, U) = (C_{2}, R), (C_{2}, {R \atop R}) or (C_{4}, R), Ext(Δ_{C_{2}, {R \atop R}} , Δ_{1, {R \atop R}}) = R and Ext(Δ_{C_{4},R}, Δ_{C_{2},R}) = R. All other Ext groups between Δ’s are zero.

At this point we may directly construct the Ext-injective hull T_{H,U} of each Δ_{H,U} by the method of [10, Lemma 4]. The method is to start with a Mackey functor M = M_{0} (we do not mean an ascending filtration here) and work along the list of Δ_{H,U} in such a way that all Δ_{K,W} with K conjugate to a subgroup of H are considered prior to Δ_{H,U}. At each stage we have a functor M_{r} and we form a maximal extension

0 → M_{r} → M_{r+1} → Δ_{H,U}^{n} → 0

where n = dim Ext(Δ_{H,U}, M_{r}) and the extension corresponds to a basis for this Ext group. We now repeat the process with M_{r+1} instead of M_{r}, using the next Δ_{H′,U′} in the list. We finish this process with a monomorphism from the original functor M to an Ext-injective functor. Selecting a minimal direct summand of this in which M embeds gives the Extinjective hull of M.

For example, if we wish to find the Ext-injective hull of Δ_{1,R} = 1 2 4 the first non-zero

group Ext(Δ_{H,U}, 1 2 4 ) occurs with (H, U) = (C_{2}, R), in which case Δ_{C_{2},R} = {2 \atop 4}. We compute

Ext({2 \atop 4}, 1 2 4 ) = R with non-split extension M_{1} = 2 1 2 4 4 . Now Ext(Δ_{C_{2}, {R \atop R}} , M_{1}) = 0, and we

move on to Ext(Δ_{C_{4},R}, M_{1}) = R^{2}. The maximal extension here is

0 → _{2 1 2 4 4} → {4 \atop 2} 4 1 4 2 4 → 4 ⊕ 4 → 0

and the middle term is the Ext-injective hull of Δ_{1,R}. By means of the dual construction we also determine the Ext-projective covers of the Δ_{H,U}.

We now construct all the remaining objects in D. In the first instance this was done by trial and error, knowing that the ascending filtration of each object refines to a Δfiltration so that arbitrary objects may be constructed by successively extending by direct sums Δ_{H,U}^{n}. Given the above display of objects in D by means of the relative Auslander-Reiten quiver, we see immediately that these objects are indeed Mackey functors with a Δ-filtration. By computing all possible dimensions of homomorphisms between the Mackey functors listed (an 18 × 18 matrix in the case of C_{4}) we verify that all of the short exact sequences presented do satisfy the lifting property of an almost split sequence with respect to homomorphisms between the Mackey functors that we have found. A standard argument now shows that we have constructed a union of connected components of the quiver. To see that there are no more objects in D it remains to show that there are no more connected components of the quiver. We have failed to find an argument which justifies this final step, except to say by inspection that there are no more objects in D.

As regards our description of the standard modules and indecomposable projectives for E, we have first labeled the simple E-modules α, β, γ, δ, ϵ, ζ arbitrarily. These modules biject with the indecomposable summands T_{H,U} of T. The standard modules have the form Hom_{μ_{R}(G)}(Δ_{H,U}, T) and their structure is determined by explicit calculation of these homomorphisms, and an examination of how they factor through homomorphisms between the summands of T. Finally, the filtrations of the projective E-modules are obtained from the construction of these filtrations at the end of the proof of Theorem 9.1 11. Characterizations of objects in D and structure of the Auslander-Reiten quiver

Part of our motivation for this study has been the tantalizing possibility that the framework we have constructed may shed light on Alperin’s weight conjecture [2]. We recall that when R is a field of positive characteristic p, Alperin defines a weight to be a pair (H, U) where H is a p-subgroup of G and U is a block of defect zero for N_{G}(H)/H, that is, a simple projective R[N_{G}(H)/H]-module. Such a module is in particular a ppermutation module, and so the weights form a subset of the indexing set Λ which we have been considering.

There is a categorical property which distinguishes within D the functors Δ_{H,U} where (H, U) is a weight, namely, they are the objects of D which are simple as objects of the larger category Mack_{R}(G). Furthermore they are Ext-injective in D, so that they occur on the right-hand edge of the relative Auslander-Reiten quiver of D. We obtain also in this section a characterization of all the functors Δ_{H,U} which are Ext-injective as the ones for which U is a projective R[N_{G}(H)/H]-module. These also coincide with the functors satisfying Δ_{H,U} ≅ ∇_{H,U}. We show finally in this section that the relative Auslander-Reiten quiver of D is compatible with ascending filtrations, in a sense described in Proposition 11.3. We assume throughout this section that R is a field of characteristic p.

(11.1) PROPOSITION. Suppose that R is a field of characteristic p and let U be an R[N_{G}(H)/H]-module. Then

(a) U is a projective R[N_{G}(H)/H]-module if and only if Δ_{H,U} ≅ ∇_{H,U}, and

(b) (H, U) is a weight if and only if H is a p-group and Δ_{H,U} is a simple Mackey functor.

Proof. (a) By [14, 13.1] we have that U is projective if and only if F P_{U} ≅ F Q_{U}, which immediately implies Δ_{H,U} ≅ ∇_{H,U}. On the other hand, if Δ_{H,U}^{G} ≅ ∇_{H,U}^{G} then Δ_{H,U}^{G} ↓_{N_{G}(H)}≅ ∇_{H,U}^{G} ↓_{N_{G}(H)} and so the subfunctors of these two functors generated by their values at H are isomorphic. By Corollary 4.4 the subfunctor on the left is isomorphic to Δ_{H,U}^{N_{G}(H)}. On the right by a similar argument to the proof of 4.4 we have

\begin{aligned}∇_{H,U}^{G} ↓_{N_{G}(H)}^{G} & = ∇_{H,U}^{N_{G}(H)} ↑_{N_{G}(H)}^{G}↓_{N_{G}(H)}^{G} \\ & ≅ ⨁_{x∈[N_{G}(H)\G/N_{G}(H)]} ^{x}(∇_{H,U}^{N_{G}(H)} ↓_{N_{G}(H)^{x}∩N_{G}(H)}) ↑_{N_{G}(H)∩^{x}N_{G}(H)}^{N_{G}(H)} \end{aligned}

and the summand corresponding to x = 1 is isomorphic to ∇_{H,U}^{N_{G}(H)}. Since ∇_{H,U}^{N_{G}(H)}(H) ≅ ∇_{H,U}^{G}(H) ≅ U it follows that all the other summands in the direct sum are zero at H. Hence the subfunctor of ∇_{H,U}^{G} ↓_{N_{G}(H)}^{G} generated by the value at H is a subfunctor of the summand ∇_{H,U}^{N_{G}(H)}. This shows that Δ_{H,U}^{N_{G}(H)} is isomorphic to a subfunctor of ∇_{H,U}^{N_{G}(H)}.

From this it follows that F Q_{U} is isomorphic to a subfunctor of F P_{U}, since the previous two functors are inflated from these. We deduce that all the restriction maps R_{J}^{L} are injective for F Q_{U}, since the same is true for F P_{U}. It is well known that this condition is equivalent to the projectivity of U.

(b) If (H, U) is a weight then H is a p-group and by [14, 13.1(vii)] and [13, 8.1(ii)] we have that Δ_{H,U} = S_{H,U} is simple. On the other hand if Δ_{H,U} is simple then its value U at its minimum subgroup H must be a simple module by the parametrization of simple functors in [13], and by [13, 8.1(ii)] and exactness of induction and inflation we have that F Q_{U} must be a simple Mackey functor for N_{G}(H)/H. It follows from [14, 13.1] (using the condition dual to (vii) of that proposition) that U must be a projective R[N_{G}(H)/H]-module.

We now characterize those Δ_{H,U} which are Ext-injective, and those which are Extprojective.

(11.2) PROPOSITION. Suppose that R is a field of characteristic p.

(a) Let U be a projective R[N_{G}(H)/H]-module. Then

Ext(Δ_{K,W} , Δ_{H,U}) = 0  and  Ext(Δ_{H,U}, ∇_{K,W}) = 0

for all pairs (K, W) where W is a p-permutation R[N_{G}(K)/K]-module. In particular, Δ_{H,U} is Ext-injective in D.

(b) Conversely, if U is a p-permutation R[N_{G}(H)/H]-module and Ext(Δ_{K,W} , Δ_{H,U}) = 0 for all pairs (K, W) where W is a p-permutation R[N_{G}(K)/K]-module then U is projective.

Proof. (a) We know from Theorem 5.1 that Ext(Δ_{K,W} , Δ_{H,U}) can only be non-zero when K >_{G} H. On the other hand if U is a projective module then Δ_{H,U} ≅ ∇_{H,U} by 11.1 and Ext(Δ_{K,W} , ∇_{H,U}) can only be non-zero when K =_{G} H. We deduce that the Ext group is always zero.

Similarly, Ext(Δ_{H,U}, ∇_{K,W}) is only non-zero when H =_{G} K. But again Δ_{H,U} ≅ ∇_{H,U} and Ext(∇_{H,U}, ∇_{K,W}) is only non-zero when H <_{G} K, so the Ext group is always zero.

(b) We suppose without loss of generality that U is indecomposable. Let us write N = N_{G}(H) and suppose that K is a subgroup of G with H ≤ K ≤ N. Let V be a ppermutation N_{N}(K)/K-module. Then by 4.1 Δ_{K,V}^{N} ↑_{N}^{G} is a direct sum of functors Δ_{K,W} with W a p-permutation R[N_{G}(K)/K]-module and so

Ext(Δ_{K,V}^{N} ↑_{N}^{G}, Δ_{H,U}^{G}) ≅ Ext(Δ_{K,V}^{N} , Δ_{H,U}^{G} ↓_{N}^{G}) = 0.

Since Δ_{H,U}^{N} is a summand of Δ_{H,U}^{G} ↓_{N}^{G} by 4.4 we deduce that Ext(Δ_{K,V}^{N} , Δ_{H,U}^{N}) = 0 for all p-permutation N_{N}(K)/K-modules V.

Now consider the short exact sequence of Mackey functors for N_{G}(H)/H

0 → F Q_{U} → P → X → 0

where P is the indecomposable projective Mackey functor on N_{G}(H)/H with P (H/H) = U and the morphism from F Q_{U} to P extends the identity map on the evaluation at H/H. This morphism is considered in [14, 12.4], where it is shown that it is a monomorphism. We define X to be the cokernel of this morphism. In fact F Q_{U} = Δ_{H/H,U}^{N/H} and the image of F Q_{U} in P is I_{{H/H}}P since it is generated by the value of P at the identity subgroup H/H. It follows from 7.1 and 6.1 that this is the start of a filtration of P by Δ functors, which will have the form Δ_{K/H,W}^{N/H} with H < K ≤ N. Inflating this sequence to N gives an extension 0 → Δ_{H,U}^{N} → Inf_{N/H}^{N} P → Inf_{N/H}^{N} X → 0 where now by 4.7 Inf_{N/H}^{N} X has a filtration by functors Δ_{K,W}^{N} with H < K ≤ N and it follows that Ext(Inf_{N/H}^{N} X, Δ_{H,U}^{N}) = 0. Thus the sequence splits. Since P and Inf_{N/H}^{N} P are indecomposable it follows that F Q_{U} ≅ P. Now by [14, 13.1] it follows that U is projective.

(11.3) PROPOSITION. Suppose that R is a field of characteristic p. Then Δ_{H,U} is Ext-projective in D if and only if H = O^{p}(H).

Proof. By the equivalence of categories in [14, 10.1] and the reduction process indicated at the start of Theorems 5.1 and 7.1 it suffices to assume H is a p-group, and we will now do this.

Suppose that Δ_{H,U} is Ext-projective with 1 ≠ H a p-subgroup of G, and let K ⊂ H be a subgroup of index p in H. Then using Lemmas 4.1 and 4.3 we have

0 = Ext(Δ_{H,U}^{G}, Δ_{K,R}^{H} ↑_{H}^{G})
≅ Ext(Δ_{H,U}^{G} ↓_{H}^{G}, Δ_{K,R}^{H})
= Ext(Δ_{H,U}^{H}, Δ_{K,R}^{H}).

It follows that Ext(Δ_{H,R}^{H}, Δ_{K,R}^{H}) = 0 since U is now considered as a module for the trivial group H/H and so is a direct sum of copies of R. However, Δ_{H,R}^{H} = Inf_{H/K}^{H} S_{H/K,R}^{H/K} where S_{H/K,R}^{H/K} is a simple functor, and Δ_{K,R}^{H} = Inf_{H/K}^{H} F Q_{R}. From the methods of [14] applied to the cyclic group H/K of order p we see that there is a short exact sequence

0 → F Q_{R} → P_{H/K,R} → S_{H/K,R} → 0

where P_{H/K,R} is the projective cover of S_{H/K,R}, and this sequence is not split. Inflating it to H gives a non-split extension which contradicts our previous calculation that the Ext group was 0. Hence if Δ_{H,U} is Ext-projective it cannot happen that H ≠ 1.

Conversely if H = 1 then by 5.1 we know Ext(Δ_{H,U}, Δ_{K,W}) = 0 for all (K, W), and so Δ_{H,U} is Ext-projective.

We show now that the relative almost split sequences in D are compatible with ascending filtrations in that the terms of an ascending filtration of the Mackey functors in an almost split sequence also form an exact sequence, which is split most of the time.

(11.4) PROPOSITION. Suppose that R is a field of characteristic p. Let 1 = H_{1}, H_{2}, …, H_{n} = G be a list of representatives of the conjugacy classes of subgroups of G with the property that whenever H_{i} is conjugate to a subgroup of H_{j} then i ≤ j. For each Mackey functor M consider the terms M_{i} = I_{{H_{1},...,H_{i}}}M of the corresponding ascending filtration. Let 0 → A → B → C → 0 be a relative almost split sequence of Mackey functors in D.

(a) For every i the sequence 0 → A_{i}/A_{i−1} → B_{i}/B_{i−1} → C_{i}/C_{i−1} → 0 is split exact.

(b) For every i the sequence 0 → A_{i} → B_{i} → C_{i} → 0 is exact, and is split if C_{i} ≠ C.

(c) For every i the sequence 0 → A/A_{i} → B/B_{i} → C/C_{i} → 0 is exact, and is split if A_{i} ≠ 0.

Proof. We first note that by 6.1(a) if M is in D then so is every M_{i} and M/M_{i}. The exactness of the sequences in (b) is evidently equivalent to the exactness of the sequences in (c). From the definitions we see that whenever η : M → N is a morphism of Mackey functors then for every i the restriction η_{i} : M_{i} → N_{i} is a morphism. If η is a monomorphism then so is η_{i} and if η is an epimorphism then so is η_{i}.

Consider now the almost split sequence 0 → A → B → C → 0. Suppose for some i that C_{i} ≠ C. Then the inclusion C_{i} → C is not split epi, it factors C_{i} → B → C, and hence we obtain a factorization of the identity C_{i} → B_{i} → C_{i}. Thus B_{i} = C_{i} ⊕ K where K = Ker(B_{i} → C_{i}). We have A_{i} ⊆ K and from the last direct sum we deduce that K = K_{i}. Hence A_{i} = K, and B_{i} = C_{i} ⊕A_{i}. We deduce simultaneously that 0 → A_{i} → B_{i} → C_{i} → 0 is exact and split.

By the dual argument if A_{i} ≠ 0 then the morphism A → A/A_{i} is not split mono and factors as A → B → A/A_{i}, from which we obtain a factorization of the identity A/A_{i} → B/B_{i} → A/A_{i}. Thus B/B_{i} = A/A_{i} ⊕ B/L where B/L = Coker(A/A_{i} → B/B_{i}), i.e. L = A + B_{i}. Now B/L has C/C_{i} as an image and is an image of C. We deduce from the direct sum that (B/L)_{i} = 0. Since C/C_{i} is the largest quotient of C with (C/C_{i})_{i} = 0 we deduce that B/L ≅ C/C_{i}, and hence B/B_{i} = A/A_{i} ⊕ C/C_{i}. Again we deduce that 0 → A/A_{i} → B/B_{i} → C/C_{i} → 0 is exact and split.

We now claim that the conditions A_{i} = 0 and C_{i} = C cannot simultaneously happen, from which we will deduce that 0 → A_{i} → B_{i} → C_{i} → 0 is always exact. For if A_{i} = 0 then by 6.1(a) every Δ-factor of A has the form Δ_{H_{j},U_{j,l}} with j ≥ i, and if C_{i} = C then every Δ-factor of C has the form Δ_{H_{k},U_{k,l}} with k ≤ i. Since Ext(Δ_{H_{k},U_{k,l}}, Δ_{H_{j},U_{j,l}}) = 0 if j ≥ k we would deduce that 0 → A → B → C → 0 is split. As this is not the case we have established the claim.

All of the sequences 0 → A_{i} → B_{i} → C_{i} → 0 are now known to be exact, and it follows that the sequences 0 → A_{i}/A_{i−1} → B_{i}/B_{i−1} → C_{i}/C_{i−1} → 0 are all exact. Since all terms are direct sums of functors Δ_{H_{i},U_{i,l}}, these latter sequences split by 5.1.

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