The Low-Dimensional Cohomology of Categories Peter Webb

1. A Resolution of \underline{R}

The homology and cohomology of a small category C over a commutative ring R with 1 are defined as H_{∗}(C, B) = Tor_{∗}^{RC}(B, \underline{R}) and H^{∗}(C, A) = Ext_{RC}^{∗}(\underline{R}, A) where \underline{R} is the constant functor, B is a right RC-module and A is a left RC-module, RC being the category algebra. For these definitions we refer to [14] and a discussion of the use of representations of categories can be found there as well as in many of the other references we give, such as [1, 2, 3, 4, 6, 9, 10, 13]. The category algebra RC is the free R-module with the morphisms of C as a basis, the multiplication being composition of morphisms when defined, and otherwise zero.

Homology and cohomology of C may be computed by taking a projective resolution of \underline{R}, and a resolution analogous to the bar resolution in group cohomology was constructed in [12]. We observe here that a resolution may also be associated to any surjection from a free category (see [11] for the definition) to C in the same way that in group cohomology one associates the Gruenberg resolution to a presentation of a group. Let F → C be a functor where F is a free category with the same objects as C, and which is surjective on morphisms. This gives rise to an R-algebra homomorphism RF → RC whose kernel N is a 2-sided ideal in RF. There is a left RC-module homomorphism RC → \underline{R} specified on the basis morphisms of RC by α ↦ 1_{cod α} where cod α is the codomain of α and we define the left augmentation ideal of RC to be its kernel ^{L}IC. Similarly there is a morphism of right modules RC → \underline{R} whose kernel is the right augmentation ideal IC^{R}. By the same arguments as those which establish the Gruenberg resolution [5, p. 34] we have the following.

Theorem 1.1. With the above notation we have a free resolution of RC-modules

⋯ → N^{2}/N^{3} → (N·^{L}IF)/(N^{2}·^{L}IF) → N/N^{2} → ^{L}IF/(N·^{L}IF) → RC → \underline{R} → 0.

A crucial fact which makes this work is that as left RF-modules, N and ^{L}IF are both free.

We obtain formulas for homology

H_{2n}(C, \underline{R}) ≅ (N^{n} ∩ IF^{R} · N^{n−1} ·^{L}IF)/(IF^{R} · N^{n} + N^{n} ·^{L}IF)

and

H_{2n+1}(C, \underline{R}) ≅ (IF^{R} · N^{n} ∩ N^{n} ·^{L}IF)/(N^{n+1} + IF^{R} · N^{n} ·^{L}IF)

which fit into a picture exactly exactly like that on [5, p. 48]. In the case of first homology the formula simplifies to give

H_{1}(C, \underline{R}) ≅ (IC^{R} ∩^{L}IC)/(IC^{R} ·^{L}IC)

thereby extending the result for groups that the abelianization of the group is isomorphic to the augmentation ideal (over the integers) modulo its square.

2. Relation Modules and Projective Extensions

We will use the notion of an extension of a category which appears in Hoff [8], bearing in mind that there are also other approaches (see [1, 4, 6, 8, 14]).

Starting with any surjective functor F → C where F is a free category with the same objects as C we construct a relation module with similar properties to relation modules coming from presentations of groups. With categories we do not have a satisfactory notion of the kernel of the functor, and it is hard to see how to proceed by taking the abelianization of the kernel as in group theory. Instead we define the relation module to be the RC-module N/(N ·^{L}IF), and it appears in a short exact sequence of RC-modules 0 → N/(N ·^{L}IF) → RC^{d} → ^{L}IC → 0, which is part of the resolution we have constructed. It has the property that it occurs as the left hand term in an extension of the category C which is projective in a certain category of such extensions, in the manner of [5, p. 197]. To construct this extension observe that H^{2}(C, A) ≅ Ext_{RC}^{1}(^{L}IC, A) for all modules A, so that the short exact sequence of RC-modules just mentioned corresponds to a category extension of C by N/(N ·^{L}IF), and this has the projective property among extensions of C by an RC-module because the short exact sequence of RC-modules is projective among extensions of ^{L}IC.

3. Five term exact sequences

By following the proof of the analogous result for group cohomology on [7, p. 202] we obtain the following.

Theorem 3.1. Let K → E → C be an extension of categories, let B be a right RC-module and let A a left RC-module. There are exact sequences

H_{2}(E, B) → H_{2}(C, B) → B ⊗_{RC} H_{1}(K) → H_{1}(E, B) → H_{1}(C, B) → 0

and

H^{2}(E, A) ← H^{2}(C, A) ← Hom_{RC}(H_{1}(K), A) ← H^{1}(E, A) ← H^{1}(C, A) ← 0.

In this result H_{1}(K) = H_{1}(K, \underline{ℤ}) ≅ H_{1}(|K|) is the product of the abelianizations of the groups K(x) which are the automorphism groups of the objects x of K, and because we have an extension of categories it becomes a left RC-module.

We may define the Schur multiplier of C to be H_{2}(C, \underline{ℤ}), and it has a property which generalizes a corresponding result for groups.

Theorem 3.2. Let K → E → C be an extension of categories and suppose that the induced homomorphism H_{1}(E) → H_{1}(C) is an isomorphism. Then \underrightarrow{lim}H_{1}(K) = \underline{ℤ} ⊗_{ℤC} H_{1}(K) is a homomorphic image of H_{2}(C, ℤ).

For groups the corresponding theorem is sometimes stated for central extensions with the property that the normal subgroup is contained in the derived subgroup of the extension group, such extensions being called stem extensions. With categories we replace this condition by the stated hypothesis on H_{1}.

4. Lyndon-Hochschild-Serre Spectral Sequences

In group cohomology the five term sequences may be viewed as coming from the Lyndon-Hochschild-Serre spectral sequences, and we ask if the same is true for categories in general. This is not entirely clear, and we report on a result of Fei Xu [15].

Theorem 4.1. Let K → E → C be such that the corresponding functors between the opposite categories are an extension of the opposite category C^{op}. Let B be a right RE-module and A a left RE-module. There are spectral sequences whose second pages are

E_{p,q}^{2} = H_{p}(C, H_{q}(K, B)) ⇒ H_{p+q}(E, B)

and

E_{2}^{p,q} = H^{p}(C, H^{q}(K, A)) ⇒ H^{p+q}(E, A).

The fact that we have an extension of C^{op} is here used to make H_{q}(K, B) into a right RC-module, and H^{q}(K, A) into a left RC-module, and this is not the same dependence as the one which appeared in the five term exact sequences presented in the previous section.

References

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[15] F. Xu, Homological properties of category algebras, Ph.D. thesis, University of Minnesota 2006.