SOME SPECTRAL SEQUENCES IN FUNCTOR

COHOMOLOGY AND APPLICATIONS TO FUSION SYSTEMS

Markus Linckelmann

Abstract. We prove that the group H^{2}(C; k^{×}) for C a right ideal in a fusion system and an algebraically closed field k is finite. The main tools are spectral sequences approximating the cohomology of a functor on an EI-category C in terms of the cohomology of certain functors on the underlying partially ordered set of isomorphism classes of C, in conjunction with the contractibility results from [16].

Contents

§1 Introduction 2 §2 Background material on functor cohomology 5 §3 Ideals in categories 12 §4 On limits of bifunctors 15 §5 Subdivisions of EI-categories and regular functors 19 §6 The base change spectral sequence of the functor C → [C] 25 §7 Comparing contravariant functors over C to covariant functors over [C] 28 §8 Constant coefficients over regular EI-categories 31 §9 Invariance under passage to subdivisions and proof of Theorem 1.3 33 §10 Orbit spaces with vanishing low degree cohomology 35 §11 Further invariance properties of H^{∗}(C; k^{×}) for fusion systems 39 §12 On H^{1}(C; k^{×}) 42 §13 Proof of Theorem 1.8 44

Typeset by AMS-TEX

§1 Introduction

Fusion systems are finite EI-categories which formalise in a category theoretic language the concepts of local structure of finite groups and blocks at a prime p. Besides their relevance in representation theory, they give rise to a class of topological spaces called p-local finite groups, providing a notion of “classifying spaces” of fusion systems. In both the representation and homotopy theoretic side of the picture, some of the main open problems can be stated in terms of the cohomology of certain functors on fusion systems. In particular, the gluing problem [13, 4.2] suggests the need to investigate second cohomology groups of the form H^{2}(C, k^{×}) for C a finite EI-category and k an algebraically closed field. While H^{2}(G; k^{×}) is finite if G is a finite group, it is easy to construct finite EI-categories C such that H^{2}(C; k^{×}) is not finite. For fusion systems we have the following finiteness result:

Theorem 1.1. Let p be a prime, let P be a finite p-group, let F be a fusion system on P and let C be a right ideal in F. Let k be an algebraically closed field. The abelian group H^{2}(C; k^{×}) is finite.

Here k^{×} is understood as contravariant constant functor on C. See §10 below for a quick review of the basic terminology regarding fusion systems, as well as further references. Theorem 1.1 will be proved in §10 as a consequence of a more precise result, Theorem 1.2, in terms of the orbit space of a finite EI-category C; this is the partially ordered set [S(C)] of isomorphism classes [σ] of finite chains of non-isomorphisms σ = X_{0} →^{φ_{0}} X_{1} →^{φ_{1}} ⋯ →^{φ_{n−1}} X_{n} in C with partial order induced by taking subchains; see §5 below for more details.

Theorem 1.2. Let C be a finite EI-category and let k be an algebraically closed field. If the orbit space [S(C)] is acyclic then the abelian group H^{2}(C; k^{×}) is finite.

Theorem 1.1 follows from 1.2 in conjunction with the contractibility results in [16]. Theorem 1.2 in turn is a consequence of a more general result:

Theorem 1.3. Let C be a finite EI-category and let k be an algebraically closed field. There is a group homomorphism H^{2}([S(C)]^{op}; k^{×}) → H^{2}(C; k^{×}) whose kernel and cokernel are both finite abelian groups. In particular, if H^{2}([S(C)]; k^{×}) = {0} then H^{2}(C; k^{×}) is finite.

The proof of 1.3 is given at the end of §9. As follows from 1.3, in order to show that H^{2}(C; k^{×}) is finite it suffices to know that H^{2}([S(C)]; k^{×}) vanishes. We will see in §10 that the stronger hypothesis of having an acyclic orbit space can be used to obtain more precise results embedding H^{2}(C; k^{×}) into a 5-term exact sequence:

Theorem 1.4. Let C be a finite EI-category and let k be an algebraically closed field. Suppose that the orbit space [S(C)] is acyclic. There is an exact sequence of abelian groups

0 → H^{1}([S(C)]; A^{1}) → H^{2}(C; k^{×}) → H^{0}([S(C)]; A^{2}) → H^{2}([S(C)]; A^{1}) → H^{3}(C; k^{×})

where A^{i} is a covariant functor sending [σ] ∈ [S(C)] to H^{i}(Aut_{S(C)}(σ); k^{×}) for i = 1, 2.

This is proved in 10.1; it implies a criterion for the existence and uniqueness of a solution of the gluing problem [13, 4.2] for arbitrary fusion systems (cf. 10.6). We include further properties of H^{∗}(C; k^{×}) for right ideals C in fusion systems in §11. The following result on H^{1}(C; k^{×}) is proved in §12.

Theorem 1.5. Let p be a prime, let P be a finite p-group, let F be a fusion system on P and let C be a right ideal in F. Let k be an algebraically closed field of characteristic p. The group homomorphism H^{1}(C; k^{×}) ⟶ H^{1}(Aut_{F}(P); k^{×}) induced by restriction is injective.

The main tools to prove the above statements include spectral sequences in work of Gabriel-Zisman [5], Jackowski-Słomińska [10] and Słomińska [21, 22]. In particular, the standard base change spectral sequence [22, 1.4.(ii)] or [10, 5.3] (or [5, Appendix II, 3.6] for the homology version) of the canonical functor C → [C] from an EI-category C to the underlying partially ordered set of isomorphism classes [C], takes the following form:

Theorem 1.6. Let C be an EI-category and let k be a commutative ring. For any contravariant functor F : C → Mod(k) there is a cohomology spectral sequence

E_{2}^{p,q} = H^{p}([C]; F^{q}) ⇒ H^{p+q}(C; F) ,

where F^{q} : [C] → Mod(k) is a contravariant functor satisfying F^{q}([X]) = H^{q}(C_{≤X}; F|_{C_{≤X}}) for any object X in C and any integer q ≥ 0.

Here C_{≤X} denotes the full subcategory of C of all objects Y such that Hom_{C}(Y, X) ≠ ∅. We include a proof in §6 for the convenience of the reader. There is another spectral sequence which looks similar but does not seem to be the base change spectral sequence of some functor; it relates the cohomology of contravariant functors on C to covariant functors on [C]:

Theorem 1.7. Let C be an EI-category and let k be a commutative ring. For any contravariant functor F : C → Mod(k) there is a cohomology spectral sequence

E_{2}^{p,q} = H^{p}([C]^{op}; F^{q}) ⇒ H^{p+q}(C; F) ,

where F^{q} : [C] → Mod(k) is a covariant functor satisfying F^{q}([X]) = H^{q}(C_{≥X}; F|_{C_{≥X}}) for any object X in C and any integer q ≥ 0.

The proof of 1.7 is given in §7. For C a right ideal in a fusion system F on a finite p-group P, the methods in [16] reduce the computation of the cohomology of certain functors on the orbit space [S(C)] to the subcategory of normal chains [S_{◁}(C)], The following theorem approximates the cohomology of functors on [S_{◁}(C)] in terms of the cohomology of certain functors on subposets of the form [S(N_{F}(Q)/Q)] for Q fully Fnormalised in P. In particular, if C consists of F-centric subgroups of P then N_{F}(Q) is actually the fusion system of a finite group L_{Q} (as a consequence of [1, Proposition C]) and so the computation of the cohomology of functors on [S(C)] is played back to a problem involving only fusion systems of finite groups; see 10.6 for a more detailed motivation. The following Theorem is proved in §13:

Theorem 1.8. Let p be a prime, let P be a finite p-group, let F be a fusion system on P and let C be a right ideal in F. Let {R_{q} | 0 ≤ q ≤ r} be a set of representatives of the Fisomorphism classes of subgroups of P belonging to C such that R_{q} is fully F-normalised for 0 ≤ q ≤ r and such that |R_{q}| ≥ |R_{q+1}| for 1 ≤ q ≤ r. Denote by C_{q} the right ideal in N_{F}(R_{q})/R_{q} consisting of all non trivial subgroups of N_{P} (R_{q})/R_{q}, where 1 ≤ q ≤ r. Let k be a commutative ring and let A : [S_{◁}(C)] ⟶ Mod(k) be a covariant functor. Then, for 1 ≤ q ≤ r, the functor A induces a covariant functor A^{q} : [S_{◁}(C_{q})] → Mod(k) and there is a spectral sequence

E_{1}^{p,q} ⇒ H^{p+q}([S_{◁}(C)]; A)

with the following properties.

(i) We have E_{1}^{p,q} = H^{p+q−1}([S_{◁}(C_{q})]; A^{q}) for p + q ≥ 2 and 1 ≤ q ≤ r.

(ii) We have E_{1}^{p,q} = {0} if q < 0 or q > r or p + q < 0, we have E_{1}^{p,0} = {0} for p ≠ 0, and we have E_{1}^{0,0} = A([P]). (iii) If there is an integer m ≥ 1 such that H^{m}([S_{◁}(C_{q})]; A^{q}) = {0} for 1 ≤ q ≤ r then H^{m+1}([S_{◁}(C)]; A) = {0}. (iv) If A vanishes on all chains of length zero in [S_{◁}(C)] then also E_{1}^{p,q} = H^{0}([S_{◁}(C_{q})]; A^{q}) for p + q = 1 and 1 ≤ q ≤ r, and E_{1}^{p,q} = {0} if p + q = 0 or q = 0.

Notation. Motivated by some problems arising in modular representation theory we made one slightly non standard choice in that we privilege, from a notational point of view, contravariant functors over covariant functors when it comes to functor categories over small categories. We will use the notation H^{∗}(C; F) indiscriminately for the cohomology of a covariant or contravariant functor over a category C since it will always be clear from the context whether a functor is covariant or contravariant. In statements where both covariant and contravariant functors appear we will denote the cohomology of a functor F over a category C by H^{∗}(C; F) if F is contravariant and by H^{∗}(C^{op}; F) if F is covariant.

Remark. There is by now an extensive literature on cohomology of functors in general and of EI-categories in particular - see for instance [24] for a survey which includes further references. In order to keep this paper as self-contained as possible, we have chosen a rather expository style for the first six sections - given the technical nature of the subject we hope the reader will forgive redundancies for the sake of convenience. In the Sections 2 and 3 we collect some well-known technicalities around the general theme of functor cohomology. Section 4 relates well-known properties of certain bifunctors to the construction principle associating pairs of spectral sequences with bifunctors [4, Ch. XVI, §1]. Section 5 reviews general properties of subdivisions of EI-categories, regular categories and functors. In the Sections 6 and 7 we describe the spectral sequences 1.6 and 1.7 which relate the cohomology of a contravariant functor over an EI-category C to the cohomology of contravariant and covariant functors over the poset of isomorphism classes [C], respectively. In Section 8 we apply this material in order to prove a result on cohomology with constant coefficients over regular EI-categories. Section 9 contains some invariance properties of cohomology with constant coefficients over arbitrary EIcategories, and we prove Theorem 1.3. Theorem 1.1, Theorem 1.2 and Theorem 1.4 are proved in Section 10. Further invariance properties of cohomology with constant coefficients over fusion systems are described in Section 11. The proofs of Theorem 1.5 and Theorem 1.8 are given in Section 12 and 13, respectively.

§2 Backround material on functor cohomology

We review in this section well-known material on functor cohomology. Let k be a commutative ring and let A be an abelian category. For any small category C we denote by F(C, A) the category of contravariant functors from C to A, with natural transformations of functors as morphisms. If A = Mod(k) we write F(C, k) instead of F(C, Mod(k)). The category F(C, A) is again an abelian category. By Yoneda’s lemma, for any object X in C, the functor kHom_{C}(−, X) sending an object Y in C to the free k-module kHom_{C}(Y, X) with basis Hom_{C}(Y, X) is projective in F(C, k); in particular, F(C; k) has enough projectives. More precisely, for any F ∈ F(C, k) the map sending a natural transformation φ : kHom_{C}(−, X) → F to the element φ(X)(Id_{X}) ∈ F(X) is a k-linear isomorphism

2.1.

Hom_{F(C,k)}(kHom_{C}(−, X), F) ≅ F(X) .

By a result of Grothendieck, F(C; k) has also enough injectives. A limit of a contravariant functor F : C → A is an object lim_{⟵}_{C}(F) in A together with a family of morphisms

π_{X} : lim_{⟵}_{C}(F) → F(X) for any object X in C, such that for any morphism φ : X → X′ in C we have π_{X} = F(φ) ∘ π_{X′} and such that for any other object A in A and any other family of morphisms τ_{X} : A → F(X) for any object X in C satisfying τ_{X} = F(φ) ∘ τ_{X′} for any morphism φ : X → X′ in C, there is a unique morphism α : A → lim_{⟵}_{C} F such

that τ_{X} = π_{X} ∘ α for any object X in C. Dually, a colimit of F is an object lim_{⟶}_{C}(F)

in A with a family of morphisms ι_{X} : F(X) → lim_{⟶}_{C}(F) satisfying the dual universal

property. By replacing C by its opposite category C^{op} one gets the analogous definitions for covariant functors. If there is no ambiguity we write lim_{⟵} and lim_{⟶} instead of lim_{⟵}_{C} and

lim_{⟶}_{C}, respectively.

If A = Mod(k) then limits and colimits exist and can be constructed explicitly as follows: take for lim_{⟵}(F) the subset of all (f_{X}) ∈ ∏_{X∈Ob(C)} F(X) satisfying F(φ)(f_{X′}) =

f_{X} for any morphism φ : X → X′ in C with the maps π_{X} : lim_{⟵}(F) → F(X) induced by the canonical projections of the product onto its factors. Dually, take for lim_{⟶}(F) the quotient of ∑_{X∈Ob(C)} F(X) by the submodule generated by all elements of the form

f_{X} − F(φ)(f_{X′}) for any morphism φ : X → X′ in C with the maps ι_{X} : F(X) → lim_{⟶}(F) induced by the canonical injections of F(X) into the coproduct.

It follows from the universal properties that lim_{⟵} and lim_{⟶}, provided they exist, define actually covariant functors from F(C, A) to A. Here is another well-known interpretation of these functors:

Theorem 2.2. Let C be a small category and let A be an abelian category having small limits and colimits. Denote by Γ : A ⟶ F(C, A) the exact functor which maps any object A in A to the contravariant constant functor A in F(C, A) defined by A(X) = A for any object X in C and A(φ) = Id_{A} for any morphism φ in C. Then the functor lim_{⟵} is right adjoint to Γ and the functor lim_{⟶} is left adjoint to Γ. In particular, lim_{⟵} preserves injectives and lim_{⟶} preserves projectives.

Proof. One checks that the map sending a morphism α : A → lim_{⟵}(F) in A to the family of morphisms (π_{X} ∘ α)_{X∈Ob(C)} is a bijection Hom_{A}(A, lim_{⟵}(F)) ≅ Hom_{F(C,A)}(A, F), where the π_{X} are as above. The inverse of this map is constructed using the universal property. Thus lim_{⟵} is the right adjoint of an exact functor, and hence lim_{⟵} preserves injectives. The dual argument proves the statements for lim_{⟶}. □

One easily checks the well-known fact that if C has a terminal object Y then lim_{⟵}(F) ≅ F(Y), and if C has an initial object X then lim_{⟶}(F) ≅ F(X), for any functor F in F(C; A).

If A = Mod(k), the functor lim_{⟵} has another well-known characterisation:

Theorem 2.3. ([5, 3.3]) Let C be a small category and let k be a commutative ring. There is an isomorphism of functors from F(C, k) to Mod(k)

lim_{⟵} ≅ Hom_{F(C,k)}(k, −) .

Proof. By 2.2 we have Hom_{k}(k, lim_{⟵}(F)) ≅ Hom_{F(C,k)}(k, F) and the left side is canonically isomorphic to lim_{⟵}(F). □

If F, G are two contravariant functors in F(C, A) we denote as usual by Ext_{C}^{i}(F, G) the i-th right derived functor of Hom_{F(C,A)}(F, −) evaluated at G if A has enough injectives; this is well-known to coincide with the i-th left derived functor of Hom_{F(C,A)}(−, G) evaluated at F provided that A has enough projectives.

We denote by lim_{⟵}^{i} the i-th right derived functor of the functor lim_{⟵} : F(C; A) → A.

If A = Mod(k) we set H^{i}(C; F) = Ext_{C}^{i}(k, F); by 2.3 this is equal to lim_{⟵}^{i}(F), and by the previous remarks this is also the i-th left derived functor of Hom_{F(C,k)}(−, F) evaluated at k (which is sometimes easier to compute, using the standard projective resolution of the contravariant constant functor k). We will, as usual, say that a functor F in F(C; k) is acyclic if H^{i}(C; F) = 0 for any positive integer i. It is well-known that if C has a terminal object then every functor F in F(C; k) is acyclic. This can easily be seen using the above formalism: if T is a terminal object in C then k ≅ kHom_{C}(−, T) is a projective functor, and hence lim_{⟵} ≅ Hom_{F(C;k)}(k, −) is exact. If C has an initial object, not every functor on F(C; k) need to be acyclic, but constant functors are (we give a proof for the convenience of the reader in 2.8 below).

As before, let A be an abelian category having small limits and colimits. Given two small categories C, D and a covariant functor Φ : C → D we denote by Φ^{∗} : F(D; A) → F(C; A) the covariant functor obtained by restriction through Φ. This functor has a right adjoint denoted by Φ_{!} : F(C; A) → F(D; A) and a left adjoint denoted by Φ_{∗} : F(C; A) → F(D; A), the right and left Kan extensions.

Given an object Y in D we denote by Φ^{Y} the category whose objects are pairs (X, φ) with X an object in C, φ : Φ(X) → Y in D, and where a morphism (X, φ) → (X′, φ′) in Φ^{Y} is a morphism α : X → X′ in C such that φ′ ∘Φ(α) = φ. We have a forgetful functor Φ^{Y} → C mapping (X, φ) to X; for any F in F(C; A) we denote by F^{Y} the functor in F(Φ^{Y}; A) obtained from restricting F through this forgetful functor. Similarly, given an object Y in D we denote by Φ_{Y} the category whose objects are pairs (φ, X) with X an object in C, φ : Y → Φ(X) in D, and where a morphism (φ, X) → (φ′, X′) in Φ_{Y} is a morphism α : X → X′ in C such that Φ(α) ∘ φ = φ′. We have a forgetful functor Φ_{Y} → C mapping (φ, X) to X; for any F in F(C; A) we denote by F_{Y} the functor in F(Φ_{Y}; A) obtained from restricting F through this forgetful functor.

Theorem 2.4. ([7, Ch. IX, §5]) Let C, D be small categories, let Φ : C → D be a covariant functor and let A be an abelian category having small limits and colimits. The restriction functor Φ^{∗} : F(D; C) → F(C; A) has a right adjoint Φ_{!} : F(C; A) → F(D; A) such that

Φ_{!}(F)(Y) = lim_{⟵}_{Φ^{Y}}(F^{Y})

and a left adjoint Φ_{∗} : F(C; A) → F(D; A) such that

Φ_{∗}(F)(Y) = lim_{⟶}_{Φ_{Y}}(F_{Y})

for any functor F in F(C; A) and any object Y in D.

Since the restriction functor Φ^{∗} is exact, its left adjoint Φ_{∗} preserves projectives and its right adjoint Φ_{!} preserves injectives. One can use 2.4 to give a proof of the well-known fact that evaluation functors preserve projectives and injectives:

Corollary 2.5. Let C be a small category, let A be an abelian category having small products and coproducts, and let X be an object in C. The evaluation functor

\{ \begin{matrix}F(C; A) & → A \\ F & ↦ F(X) \end{matrix}

preserves projectives and injectives.

Proof. We apply 2.4 to the functor Φ from the category with one object ∗ and one morphism to C sending ∗ to X. Clearly F(∗, A) ≅ A, and through this isomorphism, the evaluation functor in the statement of 2.5 is the same as the restriction functor Φ^{∗}. Given an object Y in C, the category Φ^{Y} has as set of objects the morphism set Hom_{C}(X, Y) and no morphisms other than the identity morphisms; thus for any object A in A (viewed as functor from ∗ to A) we have Φ_{!}(A)(Y) = ∏_{φ∈Hom_{C}(X,Y)}A; in particular,

Φ_{!} is exact. A similar argument shows that Φ_{∗}(A)(Y) = ∐_{φ∈Hom_{C}(Y,X)}A, hence Φ_{∗} is

exact, too. Therefore Φ^{∗}, having exact left and right adjoints, preserves projectives and injectives. □

If D is a subcategory of a small category C and Φ : D → C the inclusion functor, we write F|_{D} instead of Φ^{∗}(F) for any functor F in F(C; A). Adjoint functors give rise to spectral sequences (cf. [4, Ch. XVI]), providing a useful tool to compute functor cohomology. The following spectral sequence from [22, 1.4.(ii)] or [10, 5.3] is referred to as the base change spectral sequence for functor cohomology; its homology version is in [5, Appendix II, Thm. 3.6].

Theorem 2.6. (cf. [22, 1.4.(ii), [10, 5.3], [5, Appendix II, 3.6]) Let C, D be small categories, let Φ : C → D be a covariant functor and let k be a commutative ring. For any functor F in F(C; k) there is a cohomology spectral sequence

H^{p}(D; R^{q}Φ_{!}(F)) ⇒ H^{p+q}(C; F) .

Proof. Since Φ_{!} is right adjoint to Φ^{∗} we have an isomorphism of functors Hom_{F(C;k)}(k, −) ≅ Hom_{F(D;k)}(k, −) ∘ Φ_{!}, where we denote abusively the constant contravariant functors with value k on C and D by the same letter k. Since Φ_{!} has an exact left adjoint it preserves injectives, and so the Grothendieck spectral sequence associated with this composition of functors yields the result. □

Corollary 2.7. ([8, 3.1]) If Φ_{!} is exact then H^{∗}(C; F) ≅ H^{∗}(D; Φ_{!}(F)) for any F in F(C; k).

Proof. The spectral sequence in 2.6 collapses to the isomorphism as stated. One can see this, of course, also without any spectral sequence argument: if Φ_{!} is exact then Φ^{∗} preserves projectives, hence Φ^{∗} maps a projective resolution of k (on C) to a projective resolution of k (on D). Thus H^{∗}(D; Φ_{!}(F)) = Ext_{D}^{∗}(k; Φ_{!}(F)) ≅ Ext_{C}^{∗}(Φ^{∗}(k); F) = H^{∗}(C; F). □

It is shown in [9, 5.1(ii)] that the situation in 2.7 arises in particular if Φ : C → D has a right adjoint Ψ : D → C; one checks that then Ψ^{∗} is right adjoint to Φ^{∗} and hence Φ_{!} ≅ Ψ^{∗} is exact.

Corollary 2.8. If C has an initial object then for any k-module A the constant contravariant functor A : C → Mod(k) sending every object in C to A is acyclic.

Proof. Let X be an initial object in C. Consider the inclusion functor Φ : {X} → C, where X is considered as subcategory of C having one object and one morphism. For any object Y in C, the category Φ^{Y} has then a unique object (X, φ), where φ : X → Y is the unique morphism in C. Thus, if we consider the k-module A as functor from {X} to Mod(k) we get that Φ_{!}(A) = A, the constant functor on C with value A. In particular, Φ_{!} is exact, and now 2.8 follows from 2.7. □

The following Theorem is a tool to compute the functors R^{q}Φ_{!} in the base change spectral sequence 2.6; when specialised in degree zero, this is 2.4. The homology version of this Theorem is again in [5]. We include a proof for the convenience of the reader.

Theorem 2.9. (cf. [5, Appendix II, §3]) Let C, D be small categories, let Φ : C → D be a covariant functor and let k be a commutative ring. Let Y be an object in D, let F be a functor in F(C; k) and let F^{Y} be the contravariant functor in F(Φ^{Y} , k) obtained from restricting F to Φ^{Y} through the forgetful functor. For any integer q ≥ 0 we have a natural isomorphism

R^{q}Φ_{!}(F)(Y) ≅ H^{q}(Φ^{Y}; F^{Y}) .

Proof. What we show is that restriction F(C; k) → F(Φ^{Y} , k) through the forgetfulfunctor Φ^{Y} → C preserves injectives. Before we do this, let us observe how this concludes the proof: if I is an injective resolution of F, then its restriction to Φ^{Y} through the forgetful functor Φ^{Y} → C is an injective resolution, denoted by I^{Y}, of the functor F^{Y}. Thus

R^{q}Φ_{!}(F)(Y) = H^{q}(Φ_{!}(I)(Y)) = H^{q}(lim_{⟵}_{Φ^{Y}}(I^{Y})) = H^{q}(Φ^{Y}; F^{Y}) .

Denote by Ψ : Φ^{Y} → C the forgetful functor. In order to show that Ψ^{∗} preserves injectives, it suffices to show that its left adjoint Ψ_{∗} is exact. For this in turn we have to show, by Kan’s Theorem 2.4, that for any object X in C the functor lim_{⟶}_{Ψ_{X}} is exact

whenever Ψ_{X} is non empty. We start by describing the categories Φ^{Y} and Ψ_{X}. The category Φ^{Y} has as objects pairs (X, φ) with X ∈ Ob(C) and φ ∈ Hom_{D}(Φ(X), Y). A morphism ν : (X, φ) → (X′φ′) in Φ^{Y} is a morphism ν : X → X′ in C such that the diagram in D

2.9.1.

\xymatrix{Φ(X) \ar[dr]_{φ} \ar[rr]^{Φ(ν)} &  & Φ(X′) \ar[dl]^{φ′} \\  & Y}

is commutative. The category Ψ_{X} has as objects pairs (α, (Z, τ)), where (Z, τ) ∈ Ob(Φ^{Y}) and α ∈ Hom_{C}(X, Z). A morphism μ : (α, (Z, τ)) → (α′, (Z′, τ′)) in Ψ_{X} is a morphism μ : (Z, τ) → (Z′, τ′) in Φ^{Y} making the diagram in C

2.9.2.

\xymatrix{ & X \ar[dl]_{α} \ar[dr]^{α′} \\ Z \ar[rr]_{μ} &  & Z′}

commutative, and since μ is a morphism in Φ^{Y}, it makes also the diagram in D 2.9.3.

\xymatrix{Φ(Z) \ar[dr]_{τ} \ar[rr]^{Φ(μ)} &  & Φ(Z′) \ar[dl]^{τ′} \\  & Y}

commutative. If we apply Φ to the diagram 2.9.2 and combine it with 2.9.3, we get a commutative diagram in D

\xymatrix{ & Φ(X) \ar[dl]_{Φ(α)} \ar[dr]^{Φ(α′)} \\ Φ(Z) \ar[dr]_{τ} \ar[rr]^{Φ(μ)} &  & Φ(Z′) \ar[dl]^{τ′} \\  & Y}

Thus if Hom_{D}(Φ(X), Y) = ∅ then Ψ_{X} = ∅. We therefore may assume that Hom_{D}(Φ(X), Y) ≠ ∅. For any morphism τ ∈ Hom_{D}(Φ(X), Y) we get an object in Ψ_{X} of the form (Id_{X}, (X, τ)). Given any object (α, (Z, κ)) in Ψ_{X} and a morphism μ : (Id_{X}, (X, τ)) → (α, (Z, κ)) then, by 2.9.1 and 2.9.2, the following two diagrams

\xymatrix{Φ(X) \ar[dr]_{τ} \ar[rr]^{Φ(μ)} &  & Φ(Z) \ar[dl]^{κ} \\  & Y}

and

\xymatrix{ & X \ar[dl]_{_{Id_{X}}} \ar[dr]^{α} \\ X \ar[rr]_{μ} &  & Z}

must be commutative. This forces μ = α and τ = κ∘Φ(μ). In other words, the connected component of (α, (Z, κ)) in Ψ_{X} has an initial object, namely (Id_{X}, (X, κ ∘ Φ(α))), and α is the unique morphism from this object to (α, (Z, κ)). Thus every connected component of Ψ_{X} has an initial object, and hence lim_{⟶}_{Ψ_{X}} is exact, hence so is Ψ_{∗}, which concludes the

proof. □

§3 Ideals in categories

When specialised to “ideals” in categories, the technology from Section 2 is wellknown to take a much simpler form. For future reference we describe this in some detail.

Definition 3.1. Let C be a category and let D be a subcategory of C. We say that D is a right ideal (resp. left ideal) in C if D is a full subcategory with the property that for any object X in D and any object Y in C such that Hom_{C}(X, Y) ≠ ∅ (resp. Hom_{C}(Y, X) ≠ ∅) the object Y belongs to D, too.

Note that a left or right ideal in a small category C is in particular closed under isomorphisms in C. The set of right ideals in C is closed under taking intersections, and this set is a partially ordered set with respect to the inclusion. The same holds for the set of left ideals.

Definition 3.2. Let C be a category, let D, E be full subcategories of C closed under isomorphisms. We denote by D ∪ E the full subcategory of C whose object class is the union of the object classes of D and E; we denote by D ∩ E the full subcategory of C whose object class is the intersection of the object classes of D and E, and we denote by C − D the full subcategory of C whose object class is the complement of the object class of D in C.

Proposition 3.3. Let C be a category and let D be a full subcategory of C closed under isomorphisms. Then D is a right (resp. left) ideal if and only if C − D is a left (resp. right) ideal.

Proof. Trivial. □

Lemma 3.4. Let C be a category, let D be a full subcategory of C closed under isomorphisms, let A be an abelian category and let F : C → A be a contravariant functor.

(i) If D is a left or right ideal there is a unique functor F_{D} : C → A such that the restrictions to D of F and F_{D} coincide and such that F_{D} is zero on C − D. (ii) If D is a right ideal there is a unique natural transformation F → F_{D} such that F(X) → F_{D}(X) is the identity map for all objects X in D.

(iii) If D is a left ideal there is a unique natural transformation F_{D} → F such that F_{D}(X) → F(X) is the identity for all objects X in D.

Proof. Let X → Y → Z be a sequence of two morphisms in C such that Y is not in D. If D is a right ideal then X is not in D, and if D is a left ideal then Z is not in D. In both cases, F_{D} is well-defined on morphisms, which proves (i), and the statements (ii), (iii) are trivial verifications. □

Proposition 3.5. Let C be a small category, let D be a full subcategory of C and denote by Φ : D → C the inclusion functor. Let A be an abelian category and let G : D → A be a contravariant functor.

(i) If D is a right ideal in C then Φ_{!}(G) is the unique contravariant functor on C whose restriction to D is G and which sends any object outside of D to zero. In particular, Φ_{!} is exact and Φ^{∗} preserves projective functors.

(ii) If D is a left ideal in C then Φ_{∗}(G) is the unique contravariant functor on C whose restriction to D is G and which sends any object outside of D to zero. In particular, Φ_{∗} is exact and Φ^{∗} preserves injective functors.

Proof. One can verify this either by using the descriptions in (i) and (ii) as definitions of Φ_{!}, Φ_{∗} and then verify that these descriptions yield adjoints to the restriction functor, or one can also just directly apply the descriptions of Kan extensions quoted in 2.4. □

Corollary 3.6. Let C be a small category, let D be a full subcategory of C and denote by Φ : D → C the inclusion functor. Let A be an abelian category having small limits and colimits, and let F : C → A be a contravariant functor.

(i) If D is a right ideal in C then F_{D} = Φ_{!}Φ^{∗}(F).

(ii) If D is a left ideal in C then F_{D} = Φ_{∗}Φ^{∗}(F).

In particular, in both cases the functor on F(C; A) sending F to F_{D} is exact.

Corollary 3.7. Let C be a small category, let D be a right ideal in C and let k be a commutative ring. For any functor F in F(C; k) the restriction from C to D induces an isomorphism

H^{∗}(C; F_{D}) ≅ H^{∗}(D; F|_{D}) .

Proof. This follows from 3.6(i) combined with 2.7. □

There is a Mayer-Vietoris type exact sequence:

Proposition 3.8. Let C be a small category and let D, E be right ideals in C such that C = D ∪ E. Let A be an abelian category and let F be a functor in F(C, A). There is a short exact sequence of functors in F(C, A)

0 ⟶ F ⟶ F_{D} ⊕ F_{E} ⟶ F_{D∩E} ⟶ 0 .

Proof. By Lemma 3.4.(ii) we have canonical natural transformations from F to F_{D}, F_{E}; taking their sum yields a natural transformation from F to F_{D} ⊕ F_{E} which is a monomorphism because D ∪ E = F. Similarly, 3.5.(ii) yields canonical natural transformations from F_{D}, F_{E} to F_{D∩E} which are both surjective; hence taking their difference yields a natural transformation from F_{D} ⊕ F_{E} to F_{D∩E} which makes the sequence in the statement exact. □

If A = Mod(k) for some commutative ring k, taking cohomology yields a long exact sequence which looks like this:

Corollary 3.9. Let C be a small category and let D, E be right ideals in C such that C = D ∪ E. Let k be a commutative ring and let F be a functor in F(C, k). There is a long exact cohomology sequence

→ H^{i−1}(D ∩E; F|_{D∩E}) → H^{i}(C; F) → H^{i}(D; F|_{D})⊕H^{i}(E; F|_{E}) → H^{i}(D ∩E; F|_{D∩E}) →

In particular, given δ ∈ H^{i}(D; F|_{D}) and ϵ ∈ H^{i}(E; F|_{E}) there is γ ∈ H^{i}(C; F) satisfying γ|_{D} = δ and γ|_{E} = ϵ if and only if δ|_{D∩E} = ϵ|_{D∩E}.

Proof. Apply the functor Hom_{F(C;k)}(k, −) to a short exact sequence of injective resolutions of the functors in 3.8. This yields a long exact sequence whose terms are as claimed, using the fact that H^{i}(C; F_{D}) ≅ H^{i}(D; F|_{D}) by 3.7 and the analogous isomorphisms for E, D ∩ E instead of D. □

Remark 3.10. With the above notation, if D, E are left ideals in C such that C = D ∪ E we also get a short exact sequence of functors

0 ⟶ F_{D∩E} ⟶ F_{D} ⊕ F_{E} ⟶ F ⟶ 0 ,

which yields a long exact sequence

→ H^{i−1}(C; F) → H^{i}(C; F_{D∩E}) → H^{i}(C; F_{D}) ⊕ H^{i}(C; F_{E}) → H^{i}(C; F) →

but H^{i}(C; F_{D}) is not isomorphic to H^{i}(D; F|_{D}) in general.

§4 On limits of bifunctors

The main result of this section, Theorem 4.6 below, is a functor cohomological version of a construction principle associating pairs of spectral sequences with bifunctors, as described in Cartan and Eilenberg [4, Ch. XVI, §1]. Given small categories C, P and an abelian category A containing small limits, the limit over P is a covariant functor F(P; F(C; A)) → F(C; A), and evaluation at an object X in C yields a covariant functor F(P; F(C; A)) → F(P; A). We are going to investigate connections between the two. Note that F(P; F(C; A)) can be identified with F(P ×C; A). The material of this section will be applied to the case where P is a poset of right ideals in a small category C, which motivates the notation. The starting point is a well-known result on commutation of limits (see e.g. [6, 25.4]):

Proposition 4.1. Let C, P be small categories and let A be an abelian category having small limits. We have a commutative diagram

F(P; F(C; A)) ⟶^{lim_{⟵}_{P}} F(C; A)
F(P;lim_{⟵}_{C})↓  ↓lim_{⟵}_{C}
F (P; A) ⟶_{lim_{⟵}_{P}} A ,

where F(P; lim_{⟵}_{C}) is the functor sending a functor G in F(P; F(C; A)) to the functor

D ↦ lim_{⟵}_{C}(G(D)) in F(P; A).

Proof. This is just a general formal property of limits: we can view G in F(P; F(C; A)) as bifunctor from P × C to A and then clearly lim_{⟵}_{P}(lim_{⟵}_{C}(G(D)(X))) ≅ lim_{⟵}_{C}(lim_{⟵}_{P}(G(D)(X))),

because both sides are isomorphic to the limit of G viewed as bifunctor on P × C. □

The functor lim_{⟵}_{P} preserves injectives by 2.2, and for similar reasons, the functor

F(P; lim_{⟵}_{C}) preserves injectives (we give a proof, for the convenience of the reader, in

4.2 below). Thus the commutative diagram in 4.1 gives rise to two spectral sequences involving the right derived functors of lim_{⟵}_{P} and F(P; lim_{⟵}_{C}).

Proposition 4.2. Let C, P be small categories and let A be an abelian category having small limits. The functor

F(P; lim_{⟵}_{C}) : \{ \begin{matrix}F(P; F(C; A)) & ⟶ F(P; A) \\ G & ↦ (D ↦ lim_{⟵}_{C}(G(D))) \end{matrix}

has an exact left adjoint, namely the functor

\{ \begin{matrix}F(P; A) & ⟶ F(P; F(C; A)) \\ H & ↦ (D ↦ \underline{H(D)}) \end{matrix}

where \underline{H(D)} is the constant contravariant functor in F(C; A) mapping every object X in C to H(D). In particular, the functor F(P; lim_{⟵}_{C}) preserves injectives.

Proof. We check first that this makes sense. If E → D is a morphism in P we get a natural transformation G(D) → G(E), hence upon taking limits a morphism lim_{⟵}_{C}(G(D)) → lim_{⟵}_{C}(G(E)), and so the assignment D → lim_{⟵}_{C}(G(D)) is in F(P; A). Clearly

the assignment G ↦ (D ↦ lim_{⟵}_{C}(G(D)) is covariant functorial. We need to construct

natural isomorphisms

Hom_{F(P;A)}(H, (D ↦ lim_{⟵}_{C}(G(D))) ≅ Hom_{F(P;F(C;A))}((D ↦ \underline{H(D)}), G) .

On the left side, a natural transformation H → (D ↦ lim_{⟵}_{C}(G(D))) corresponds to a

compatible family of morphisms H(D) → lim_{⟵}_{C}(G(D)), and on the right side, a natural

transformation corresponds to a compatible family of natural transformations \underline{H(D)} → G(D). Since, by 1.2, we have

Hom_{A}(H(D), lim_{⟵}_{C}(G(D))) ≅ Hom_{F(C;A)}(\underline{H(D)}, G(D))

the result follows. □

In the case A = Mod(k) for some commutative ring k the computation of the derived functors of F(P; lim_{⟵}_{C}) is straightforward:

Proposition 4.3. Let C be a small category, let P be a poset and let k be a commutative ring. Let i be a non negative integer. The i-th right derived functor of the functor

F(P; lim_{⟵}_{C}) : \{ \begin{matrix}F(P; F(C; k)) & ⟶ F(P; k) \\ G & ↦ (D ↦ lim_{⟵}_{C}(G(D))) \end{matrix}

is equal to the functor

\{ \begin{matrix}F(P; F(C; k)) & ⟶ F(P; k) \\ G & ↦ (D ↦ H^{i}(C; G(D))) \end{matrix}

Proof. Let I be an injective resolution of G in the category F(P; F(C; k)). Let D be an object in P. Then, by 2.5, I(D) is an injective resolution of the functor G(D) in F(C; k). Thus lim_{⟵}_{C}(I(D)) is a cochain complex of functors in F(C; k) whose i-th

cohomology is lim_{⟵}_{C}^{i}(G(D)) = H^{i}(C; G(D)). Thus the i-th cohomology of I is the functor

D ↦ H^{i}(C; G(D)) as claimed. □

The following well-known result (see [6, 25.5]) computes the functor lim_{⟵}_{P} :

F(P; F(C; A)) → F(C; A) in terms of an evaluation functor:

Proposition 4.4. Let C, P be small categories and let A be an abelian category having small limits. For any object X in C, evaluation at X defines an exact covariant functor

\{ \begin{matrix}F(P; F(C; A)) & ⟶ F(P; A) \\ G & ↦ G_{X} \end{matrix}

which preserves injectives, where G_{X} is the functor mapping an object D in P to G_{X}(D) = G(D)(X), and we have a natural isomorphism

(lim_{⟵}_{P}(G))(X) ≅ lim_{⟵}_{P}(G_{X}) .

Proof. We observe first that G_{X} as defined gives rise to a contravariant functor. If D, E are objects in P such that there is a morphism E → D in P, applying G to this morphism yields a morphism G(D) → G(E), that is, a natural transformation of contravariant functors which in turn is given by a family of morphisms G(D)(X) → G(E)(X) in A for all objects X in C, hence a family of morphisms G_{X}(D) → G_{X}(E), and so in particular G_{X} is a contravariant functor in F(P; A). The assignment G ↦ G_{X} is covariant functorial: if G → G′ is a natural transformation of functors G, G′ in F(P; F(C; A)), we have a family of natural transformations G(D) → G′(D) for any object D in P, any of which is in turn given by a family of morphisms G(D)(X) → G′(D)(X), with X running over the set of objects in C, and the naturality implies that this yields a natural transformation G_{X} → G′_{X} with the required functoriality properties. The exactness of this evaluation functor is trivial.

The evaluation functor \{ \begin{matrix}F(C; A) & → A \\ F & ↦ F(X) \end{matrix} has a canonical left adjoint, by 2.5.

Thus, for a fixed object D in P, the functor \{ \begin{matrix}F(P; F(C; A)) & → A \\ G & ↦ G_{X}(D) \end{matrix} has a canonical

exact left adjoint. Naturality implies that this yields an exact left adjoint of the functor

\{ \begin{matrix}F(P; F(C; A)) & ⟶ F(P; A) \\ G & ↦ G_{X} \end{matrix} , and hence this functor preserves injectives.

It remains to show that the functor X ↦ lim_{⟵}_{P}(G_{X}) in F(C; A) is a limit lim_{⟵}_{P}(G) of

the functor G in F(P; F(C; A)). For any object X in C and any object D in P we have a morphism (lim_{⟵}_{P}(G_{X}))(D) → G_{X}(D) = G(D)(X), which determines a family of

natural transformations from the functor (X ↦ lim_{⟵}_{P}(G_{X})) to G(D). We have to check

the universal property of this family of natural transformations. Let H be a functor in F(C, A) and suppose we have a compatible family of natural transformations H → G(D), where D runs over the set of objects in P. For fixed X this yields a compatible family of morphisms H(X) → G_{X}(D), and so the universal property of lim_{⟵} yields a unique morphism lim_{⟵}_{P}(G_{X}) → H(X), so a unique natural transformation from the functor X ↦

lim_{⟵}_{P}(G_{X}) to the functor H with the required commutativity property. □

Proposition 4.5. Let C, P be small categories and let k be a commutative ring. For any functor G in F(P; F(C; k)), any object X in C and any non negative integer i we have

(lim_{⟵}_{P}^{i}(G))(X) ≅ H^{i}(P; G_{X}) .

Proof. The evaluation functor G → G_{X} as in 4.4 is exact and preserves injectives, hence maps an injective resolution of G to an injective resolution of G_{X}. The result follows then from 4.4 applied to an injective resolution of G, together with the fact that lim_{⟵}_{P}^{i}(G_{X}) =

H^{i}(P; G_{X}) by the notation introduced after 2.3. □

The following two Grothendieck spectral sequences combine the results of this Section:

Theorem 4.6. Let C, P be small categories and let k be a commutative ring. For any functor G in F(P; F(C; k)) we have two cohomology spectral sequences starting at the E_{2}-page

H^{p}(C; X ↦ H^{q}(P; G_{X})) ⇒ (lim_{⟵}_{C} ∘ lim_{⟵}_{P})^{p+q}(G) ,

H^{p}(P; D ↦ H^{q}(C; G(D))) ⇒ (lim_{⟵}_{C} ∘ lim_{⟵}_{P})^{p+q}(G) .

Proof. The functors lim_{⟵}_{P} and F(P; lim_{⟵}_{C}) occurring in the commutative diagram in 4.1

both preserve injectives, by 2.2 and 4.4, respectively. Their right derived functors are computed in 4.5 and 4.3, respectively, from which follows that the two Grothendieck spectral sequences obtained from the commutative diagram in 4.1 have the form as claimed. □

§5 Subdivisions of EI-categories and regular functors

Following [18], an EI-category is a small category with the property that every endomorphism of an object is an isomorphism. If X, Y are non isomorphic objects in an EI-category C then at most one of the morphism sets Hom_{C}(X, Y) and Hom_{C}(Y, X) is non empty. Indeed, if there is a morphism φ : X → Y and a morphism ψ : Y → X in C, then ψ ∘ φ and φ ∘ ψ are automorphisms of X and Y, respectively. One easily deduces that both φ and ψ have to be isomorphisms. Thus the set of isomorphism classes of C has a partial order given by [X] ≤ [Y] if Hom_{C}(X, Y) ≠ ∅, where [X], [Y] are the isomorphism classes of the objects X, Y in C, respectively. We denote by [C] the poset of isomorphism classes of C, regarded itself as category in the usual way. There is an obvious functor C ⟶ [C] mapping any object X in C to its isomorphism class [X] and any morphism φ : X → Y in C to the unique morphism [X] → [Y] in [C]. In an EI-category, non-isomorphisms form an “ideal”; that is, any morphism in C composed with any non-isomorphism in C yields again a non-isomorphism. The particular role of non-isomorphisms in EI-categories motivates the definition of subdivisions [14. 1.1]:

Definition 5.1. (cf. [14, 1.1], [16, 2.1]) The subdivision of an EI-category C is the category S(C) defined as follows:

The objects of S(C) are the faithful functors σ : [m] → C, where m is any non negative integer and [m] = {0 < 1 < 2 < ⋯ m}, the totally ordered set viewed as category in the obvious way.

A morphism in S(C) from σ : [m] → C to τ : [n] → C is a pair (α, μ) consisting of a monotone increasing map α : [m] → [n] and a natural isomorphism of functors μ : σ ≅ τ ∘ α.

More explicitly, the objects of S(C) can be viewed as chains

σ = X_{0} ⟶^{φ_{0}} X_{1} ⟶^{φ_{1}} ⋯ ⟶^{φ_{n−1}} X_{m}

of objects X_{i}, where 0 ≤ i ≤ m, and of non isomorphisms φ_{i}, where 0 ≤ i ≤ m − 1, and where m is a non negative integer. Any such chain is called a chain of length m, where as usual chains of length 0 are just objects in C. The fact that chains of non isomorphisms are the same as faithful functors uses the property of EI-categories that the composition of any two non isomorphisms is again a non isomorphism.

With this point of view, a morphism in S(C) from such a chain of non-isomorphisms

σ = X_{0} ⟶^{φ_{0}} X_{1} ⟶^{φ_{1}} ⋯ ⟶^{φ_{m−1}} X_{m}

to a chain of non-isomorphisms

τ = Y_{0} ⟶^{ψ_{0}} Y_{1} ⟶^{ψ_{1}} ⋯ ⟶^{ψ_{n−1}} Y_{n}

is then a family μ = (μ_{i})_{0≤i≤m} where for each i there is α(i) ∈ {0, 1, .., n} such that μ_{i} : X_{i} → Y_{α(i)} is an isomorphism which makes the obvious diagrams commutative; that is,

μ_{i+1} ∘ φ_{i} = ψ_{α(i+1)−1} ∘ ⋯ ∘ ψ_{α(i)+1} ∘ ψ_{α(i)} ∘ μ_{i}

for any i ∈ {0, 1, .., m−1}. There is no need to specify here that α is monotone increasing - this follows automatically from C being an EI-category.

The subdivision construction is functorial with respect to covariant functors between EI-categories which preserve non-isomorphisms; more precisely, if C, C′ are EIcategories and if Φ : C ⟶ C′ is a covariant functor which maps any non-isomorphism in C to a non-isomorphism in C′, then Φ induces an obvious covariant functor S(C) ⟶ S(C′). If C is itself a poset, this subdivision construction yields just the usual barycentric subdivision of C as poset. In general, the five categories C, [C], S(C), [S(C)] and S([C]) are related by a commutative pentagon of canonical functors (cf. [14, 1.4])

5.2.

\xymatrix{ & S(C) \ar[dl] \ar[rr] &  & C \ar[dr] \\ [S(C)] \ar[drr] &  &  &  & [C] \\  &  & S([C]) \ar[urr]}

Definition 5.3. (cf. [14, 2.1]) An EI-category C is called regular if for any two objects X, Y in C the group Aut_{C}(X) acts regularly on the set Hom_{C}(X, Y) whenever this set is non-empty.

By passing from an EI-category to its subdivision one always ends up with a regular EI-category (which is one of the motivations for this definition):

Proposition 5.4. (cf. [14, 1.2, 1.3]) Let C be an EI-category. Then the subdivision S(C) is a regular EI-category and every morphism in S(C) is a monomorphism.

Definition 5.5. (cf. [16, 2.4]) The orbit space of an EI-category C is the partially ordered set [S(C)], viewed as topological space.

Proposition 5.6. Let C be an EI-category and let D be a full subcategory in C closed under isomorphisms. Then S(D) is a left ideal in S(C).

Proof. Let X → Y be a morphism in S(C). If all objects occurring in the chain Y belong to D the same is true for X because D is closed under isomorphism, and since D is a full subcategory of C clearly S(D) is a full subcategory of S(C). □

Proposition 5.7. Let C be an EI-category. We have an isomorphism of subdivision categories S(C) ≅ S(C^{op}).

Proof. We map a chain X_{0} → X_{1} → ⋯ → X_{m} in S(C) to its obvious image X_{0} ← X_{1} ← ⋯ ← X_{m}, and, with the notation of 5.1, we map a morphism (μ_{i})_{0≤i≤m} in S(C) to the morphism (μ_{i}^{−1})_{0≤i≤m} in S(C^{op}). □

Using 9.1 below twice and 5.7 yields H^{∗}(C; A) ≅ H^{∗}(S(C); A) ≅ H^{∗}(S(C^{op}); A) ≅ H^{∗}(C^{op}; A) for any EI-category C and any constant functor with value A, a particular case of the well-known fact that cohomology with constant coefficients is invariant under passage to opposite categories.

Corollary 5.8. Let C be an EI-category. There is a canonical covariant functor S(C) ⟶ C^{op} mapping a chain X_{0} → X_{1} → ⋯ → X_{m} in S(C)^{op} to X_{0}.

Proof. Precompose the canonical functor S(C^{op}) → C^{op} with the isomorphism S(C) ≅ S(C^{op}) in 5.7. □

The fact that every morphism in S(C) is a monomorphism is true in any regular EI-category:

Proposition 5.9. Let C be a regular EI-category. Every morphism in C is a monomorphism.

Proof. Let X, Y, Z be objects in C, let φ, ψ ∈ Hom_{C}(X, Y) and let τ ∈ Hom_{C}(Y, Z) such that τ ∘ φ = τ ∘ ψ. Since Aut_{C}(X) acts regularly on Hom_{C}(X, Y) there is a unique ζ ∈ Aut_{C}(X) such that ψ = φ ∘ ζ. Then τ ∘ φ = τ ∘ ψ = τ ∘ φ ∘ ζ. Since Aut_{C}(X) acts also regularly on Hom_{C}(X, Z) this implies ζ = Id_{X}, hence ψ = φ. □

Following [18], an EI-category C is free if Aut_{C}(Y) acts freely on Hom_{C}(X, Y) for any two objects X, Y in C for which the set Hom_{C}(X, Y) is non-empty. If every morphisms in C is an epimorphism then clearly C is free.

Proposition 5.10. Let C be an EI-category with the property that every morphism in C is an epimorphism. Then every morphism in S(C) is both a monomorphism and an epimorphism. In particular, S(C) is regular and free.

Proof. Clear by 5.4 and the above remarks. □

The following definition generalises regularity to functors between EI-categories; it generalises extensions of categories by a functor with values in abelian groups, and is, in part, motivated by the canonical functor of the transporter category onto the centric linking system of a finite group as described in [2, 1.3].

Definition 5.11. Let C, D be EI-categories and let Φ : C → D be a covariant functor. We say that Φ is regular if the following hold:

(a) Φ induces an isomorphism of posets [C] ≅ [D];

(b) for any two objects X, Y in C the map Hom_{C}(X, Y) ⟶^{Φ} Hom_{D}(Φ(X), Φ(Y)) induced by Φ is surjective;

(c) for any two objects X, Y in C such that Hom_{C}(X, Y) is non empty the group

K(X) = ker(Aut_{C}(X) ⟶^{Φ} Aut_{D}(Φ(X)))

acts freely on Hom_{C}(X, Y) through composition of morphisms and induces a bijection

Hom_{C}(X, Y)/K(X) ≅ Hom_{D}(Φ(X), Φ(Y)) .

The situation described in 5.11 arises in particular in the context of extensions of categories: if C is an extension of the category D by a contravariant functor Z : D ⟶ Ab then the structural functor C ⟶ D is regular. This definition is an obvious generalisation of the regularity of an EI-category as defined in 5.3:

Proposition 5.12. An EI-category C is regular if and only if the canonical functor C ⟶ [C] is regular.

Proof. Obvious. □

Regularity of functors passes on to subdivisions:

Proposition 5.13. Let C, D be EI-categories and let Φ : C → D be a regular functor. Then Φ induces a regular functor S(C) → S(D).

Proof. Since Φ induces an isomorphism of posets [C] ≅ [D], every non-isomorphism in C is mapped by Φ to a non-isomorphism in D. Thus Φ induces a functor S(C) → S(D), still denoted by the same letter Φ. Let

σ = X_{0} ⟶^{φ_{0}} X_{1} ⟶^{φ_{1}} ⋯ ⟶^{φ_{m−1}} X_{m}
τ = Y_{0} ⟶^{ψ_{0}} Y_{1} ⟶^{ψ_{1}} ⋯ ⟶^{ψ_{n−1}} Y_{n}

be two objects in S(C), and let Let μ = (μ_{i})_{0≤i≤m} be a morphism in S(D) from Φ(σ) to Φ(τ). That is, μ is a family μ = (μ_{i})_{0≤i≤m} where for each i there is α(i) ∈ {0, 1, .., n} such that μ_{i} : Φ(X_{i}) → Φ(Y_{α(i)}) is an isomorphism in D which makes the obvious diagrams commutative; that is,

μ_{i+1} ∘ Φ(φ_{i}) = Φ(ψ_{α(i+1)−1}) ∘ ⋯ ∘ Φ(ψ_{α(i)+1}) ∘ Φ(ψ_{α(i)}) ∘ μ_{i}

for any i ∈ {0, 1, .., m − 1}. We construct inductively a morphism ν = (ν_{i})_{0≤i≤m} from σ to τ in C as follows. Choose any ν_{m} : X_{m} → Y_{α(m)} such that Φ(ν_{m}) = μ_{m}; this is possible since Φ is surjective on morphisms. Since Φ induces an isomorphism of posets [C] ≅ [D] the morphism ν_{m} is automatically an isomorphism in C. Suppose we have already constructed isomorphisms ν_{i+1}, ν_{i+2}, .., ν_{m} in C satisfying Φ(ν_{j}) = μ_{j} for i + 1 ≤ j ≤ m and

ν_{j+1} ∘ φ_{j} = ψ_{α(j+1)−1} ∘ ⋯ ∘ ψ_{α(j)+1} ∘ ψ_{α(j)} ∘ ν_{j}

for i + 1 ≤ j < m, where i is an integer such that 0 ≤ i < m. Let ν′_{i} : X_{i} → Y_{α(i)} be any morphism in C such that Φ(ν′_{i}) = μ_{i}. Then, as before, ν′_{i} : X_{i} → Y_{α(i)} is an isomorphism in C, and the images under Φ of the two compositions ν_{i+1} ∘ φ_{i} and ψ_{α(i+1)−1} ∘ ⋯ ∘ ψ_{α(i)+1} ∘ ψ_{α(i)} ∘ ν′_{i} in D are equal. Thus, by condition (c) in 5.11, there is an automorphism α ∈ Aut_{C}(X_{i}) such that Φ(α) = Id_{Φ(X)} and such that

ν_{i+1} ∘ φ_{i} = ψ_{α(i+1)−1} ∘ ⋯ ∘ ψ_{α(i)+1} ∘ ψ_{α(i)} ∘ ν′_{i} ∘ α .

We set then ν_{i} = ν′_{i} ∘α. The family ν = (ν_{i})_{0≤i≤m} constructed in this way is a morphism in S(C) from σ to τ whose image in S(D) is the morphism μ. This shows that the map induced by Φ

Hom_{S(C)}(σ, τ) ⟶ Hom_{S(D)}(Φ(σ), Φ(τ))

is surjective. Moreover, μ is an isomorphism in S(D) if and only if ν is an isomorphism in S(C). Thus (a) and (b) in 5.11 hold. If ν = (ν_{i})_{0≤i≤m} and ν′ = (ν′_{i})_{0≤i≤m} are two morphisms in S(C) from σ to τ such that Φ(ν) = Φ(ν′) then Φ(ν_{i}) = Φ(ν′_{i}), hence by the regularity of Φ there are unique α_{i} ∈ Aut_{C}(X_{i}) such that Φ(α_{i}) = Id_{Φ(X_{i})} and such that ν′_{i} = ν_{i} ∘ α_{i}. Then α = (α_{i})_{0≤i≤m} is an automorphism in S(C) satisfying ν′ = ν ∘ α and Φ(α) = Id_{Φ(σ)}, and by the regularity of the category S(C), the morphism α is unique with these properties. This shows that also (c) in 5.11 holds. □

The kernels K(X) appearing in 5.11.(c) can be viewed as automorphism groups in the categories Φ^{Y} as follows:

Proposition 5.14. Let C, D be EI-categories and let Φ : C → D be a regular functor. Let X be an object in C and set Y = Φ(X).

(i) The poset [Φ^{Y}] has as unique maximal element the isomorphism class of the object (X, Id_{Y}) in Φ^{Y}.

(ii) We have Aut_{Φ^{Y}} (X, Id_{Y}) = ker(Aut_{C}(X) →^{Φ} Aut_{D}(Y)).

Proof. For (i), let (Z, ψ) be an object in Φ^{Y}; that is, Z is an object in C and ψ : Φ(Z) → Y is a morphism in D. Since Φ is regular, there is a morphism β : Z → X such that Φ(β) = ψ = Id_{Y} ∘ ψ. Thus β is a morphism in Φ^{Y} from (Z, ψ) to (X, Id_{Y}). In other words, [(Z, ψ)] ≤ [(X, Id_{Y})] for any object (Z, ψ) in Φ^{Y}. This proves (i), and (ii) is trivial. □

Proposition 5.15. Let C, D be EI-categories and let Φ : C → D be a regular functor. Suppose that every morphism in C is a monomorphism. Let X be an object in C and set Y = Φ(X).

(i) The category Φ^{Y} is regular.

(ii) For any object (Z, ψ) in Φ^{Y} we have Aut_{Φ^{Y}} (Z, ψ) = ker(Aut_{C}(Z) →^{Φ} Aut_{D}(Φ(Z)).

Proof. Let α, β : (Z, ψ) → (Z′, ψ′) be two morphisms in Φ^{Y}. That is, Z, Z′ are objects in C and ψ ∈ Hom_{D}(Φ(Z), Y), ψ′ ∈ Hom_{D}(Φ(Z′), Y) such that ψ′ ∘ Φ(α) = ψ = ψ′ ∘ Φ(β). Since Φ is regular there are morphisms φ ∈ Hom_{C}(Z, X), φ′ ∈ Hom_{C}(Z′, X) such that Φ(φ) = ψ and Φ(φ′) = ψ′. Then

Φ(φ′ ∘ α) = Φ(φ) = Φ(φ′ ∘ β)

The regularity of Φ implies that there is a unique automorphism γ ∈ Aut_{C}(X) such that φ′ ∘ α ∘ γ = φ′ ∘ β and such that Φ(γ) = Id_{Y}. Since φ′ is a monomorphism we get that α ∘ γ = β. It follows that γ is the unique automorphism of (Z, ψ) in Φ^{Y} satisfying α ∘ γ = β. This proves (i). When applied to the case (Z, ψ) = (Z′, ψ′) and β = Id_{Z} this argument proves (ii). □

Remark 5.16. If C is an EI-category then S(C) is an EI-category with subobjects in the sense of Oliver [20]. To see this, let N(C) be the subposet of S(C) such that Ob(N(C)) = Ob(S(C)) and such that σ ≤ τ in N(C) if there is a morphism (α, μ) : σ → τ with the property that the morphisms μ_{i} are identity morphisms; in other words, if σ = τ ∘ α, or equivalently, if σ is a subchain of τ. An arbitrary morphism σ → τ in S(C) can be written uniquely as composition of an isomorphism identifying σ to a subchain σ′ of τ followed by the morphism σ′ ≤ τ in N(C). Thus by 9.1 below, for the purpose of computing cohomology with constant coefficients we always may assume that C is an EI-category with subobjects.

Example 5.17. Let k be an algebraically closed field and let C be the partially ordered set defined by the graph

\xymatrix{ &  & • \\ • \ar[urr] \ar[rrr] \ar[dr] \ar[ddrr] &  &  & • \ar[ul] \ar[ddl] \\  & • \ar[dr] \ar[uur] &  &  & • \ar[uull] \ar[ul] \ar[lll] \ar[dll] \\  &  & •}

The nerve of C is homeomorphic to the sphere S^{2} and hence H^{2}(C; k^{×}) ≅ k^{×} is infinite.

§6 The base change spectral sequence of the canonical functor C → [C]

Theorem 1.6, which we restate here, is an immediate consequence of the base change spectral sequence 2.6 from [22, 1.4.(ii)] (or [5, Appendix II, 3.6] for the homology version) applied to the canonical functor C → [C], where C is an EI-category:

Theorem 6.1. Let C be an EI-category and let k be a commutative ring. For any contravariant functor F : C → Mod(k) there is a cohomology spectral sequence

E_{2}^{p,q} = H^{p}([C]; F^{q}) ⇒ H^{p+q}(C; F) ,

where F^{q} : [C] → Mod(k) is a contravariant functor satisfying F^{q}([X]) = H^{q}(C_{≤X}; F|_{C_{≤X}}) for any object X in C and any integer q ≥ 0.

Here C_{≤X} denotes the full subcategory of C of all objects Y such that Hom_{C}(Y, X) ≠ ∅. If [X] ≤ [Y] then C_{≤X} ⊆ C_{≤Y} and F^{q} maps the morphism [X] ≤ [Y] in [C] to the restriction map H^{q}(C_{≤Y}; F|_{C_{≤Y}}) → H^{q}(C_{≤X}; F|_{C_{≤X}}). This works in slightly greater generality for functors from C to a poset - which is the theme of [22] - we spell out the details for the sake of providing notation and references.

Definition 6.2. Let C be an EI-category, let P be a poset and let Φ : C → P be a covariant functor. For any object Y in P we denote by C_{≤Y} the left ideal in C consisting of all objects X in C such that Φ(X) ≤ Y, and we denote by C_{≥Y} the right ideal in C consisting of all objects X in C such that Y ≤ Φ(X). If P = [C] and Φ : C → [C] is the canonical functor, we write C_{≤X} instead of C_{≤[X]} and C_{≥X} instead of C_{≥[X]}, where X is any object in C.

The notation C_{≤Y} and C_{≥Y} does not keep track of the functor Φ, but since we will mainly be interested in the canonical functors occurring in 5.2, confusion should be impossible. Note that for any object X in C, the left ideal C_{≤X} consists of all objects X′ in C such that Hom_{C}(X′, X) ≠ ∅, and the right ideal C_{≥X} consists of all objects X′ in C such that Hom_{C}(X, X′) ≠ ∅. Any left or right ideal in an EI-category C is of the form C_{≤Y} or C_{≥Y} for a suitable poset P, a covariant functor Φ : C → P and an object Y in P. Indeed, let D be a left ideal in C, take for P the totally ordered set with two elements {1, 2}, and let Φ : C → P be the unique covariant functor sending every object in D to 1 and every object in C − D to 2. Then clearly D = C_{≤1}. The dual argument works for right ideals.

Lemma 6.3. Let C be an EI-category, let P be a poset and let Φ : C → P be a covariant functor. Let Y, Y′ be objects in P. The following are equivalent:

(i) Y ≤ Y′. (ii) C_{≤Y} ⊆ C_{≤Y′}. (iii) C_{≥Y′} ⊆ C_{≥Y}.

Proof. Trivial. □

Lemma 6.4. Let C be an EI-category, let P be a poset, let Φ : C → P be a covariant functor and let Y be an object in P. The forgetful functor Φ^{Y} → C induces an isomorphism of categories Φ^{Y} ≅ C_{≤Y}. and the forgetful functor Φ_{Y} → C induces an isomorphism of categories Φ_{Y} ≅ C_{≥Y}.

Proof. Any object in Φ^{Y} is of the form (X, φ), where X is an object in C and φ : Φ(X) → Y a morphism in P. Since P is a poset, the objects in Φ^{Y} can be identified with the objects X in C such that Φ(X) ≤ Y, thus with the objects of C_{≤Y}. Given two objects X, X′ in C_{≤Y}, if there is a morphism α : X → X′ in C, then Φ(X) ≤ Φ(X′) ≤ Y, hence the set of morphisms from X to X′ as objects of Φ^{Y} is equal to Hom_{C}(X, X′). A dual argument proves the second statement. □

Theorem 6.5. Let C be an EI-category, let P be a poset and let Φ : C → P be a covariant functor. Let k be a commutative ring and let F be a functor in F(C; k). For any integer q ≥ 0 there is a unique functor F^{q} in F(P; k) which sends an object Y in P to H^{q}(C_{≤Y}; F|_{C_{≤Y}}) and which sends a morphism Y → Y′ to the unique map H^{q}(C_{≤Y′}; F|_{C_{≤Y ′}}) → H^{q}(C_{≤Y}; F|_{C_{≤Y}}) induced by restriction along the inclusion C_{≤Y} ⊆ C_{≤Y′}. Moreover, we have a cohomology spectral sequence starting at the E_{2}-page

H^{p}(P; F^{q}) ⇒ H^{p+q}(C; F) .

Proof. The existence of a unique functor F^{q} as described in the statement is clear. Let I be an injective resolution of F in F(C; k). Then, for any object Y in P, the restriction I|_{C_{≤Y}} of I to the left ideal C_{≤Y} is still an injective resolution of the restriction F|_{C_{≤Y}}, by 3.6.(ii). By the previous Lemma we have Φ^{Y} ≅ C_{≤Y}, and thus, by the explicit formula for Kan extensions in 2.4, we get

R^{q}Φ_{!}(F)(Y) = H^{q}(lim_{⟵}_{Φ^{Y}}(I^{Y})) = H^{q}(lim_{⟵}_{C_{≤Y}}(I|_{C_{≤Y}})) = H^{q}(C_{≤Y}; F|_{C_{≤Y}}) = F^{q}(Y) .

The result follows from the base change spectral sequence 2.6. □

Theorem 6.5 applies to all canonical functors in the diagram 5.2 except the canonical functor S(C) → C. When applied to the case P = [C] and the canonical functor C → [C] this yields Theorem 6.1. Two other interesting special cases arise when Theorem 6.5 is applied to S(C) instead of C and to the canonical functors from S(C) to the posets [C] and [C]^{op} sending a chain X_{0} → ⋯ → X_{m} in S(C) to [X_{m}] and [X_{0}], respectively:

Corollary 6.6. Let C be an EI-category, let k be a commutative ring and let F be a functor in F(S(C); k). There are two cohomology spectral sequences starting at the E_{2}-page

H^{p}([C]; [X] ↦ H^{q}(S(C_{≤X}); F)) ⇒ H^{p+q}(S(C); F) ;
H^{p}([C]^{op}; [X] ↦ H^{q}(S(C_{≥X}); F)) ⇒ H^{p+q}(S(C); F) .

Proof. Let Φ : S(C) → [C] be the canonical functor sending a chain X_{0} → X_{1} → ⋯ → X_{m} in S(C) to [X_{m}]. Then, for any object X in C one verifies easily that S(C)_{≤[X]} = S(C_{≤X}). Hence the functor F_{q} as defined in 6.5 sends [X] to H^{q}(S(C_{≤X}); F). This yields the first of the two spectral sequences in 6.5. Similarly, if instead we take for Φ the canonical functor from S(C) to [C]^{op} sending a chain X_{0} → X_{1} → ⋯ → X_{m} in S(C) to [X_{0}] then, for any object X in C one verifies that this time S(C)_{≤[X]} = S(C_{≥X}), yielding the second spectral sequence. □

Another similar pair of spectral sequences arises with S(C) replaced by [S(C)] (the proof being strictly analogous to that of 6.6 is omitted):

Corollary 6.7. Let C be an EI-category, let k be a commutative ring and let F be a functor in F([S(C)]; k). There are two cohomology spectral sequences starting at the E_{2}-page

H^{p}([C]; [X] ↦ H^{q}([S(C_{≤X})]; F)) ⇒ H^{p+q}([S(C)]; F) ;
H^{p}([C]^{op}; [X] ↦ H^{q}([S(C_{≥X})]; F)) ⇒ H^{p+q}([S(C)]; F) .

§7 Comparing contravariant functors over C to covariant functors over [C]

Theorem 1.7 states the existence of a spectral sequence relating the cohomology of a contravariant functor F on C to that of covariant functors, abusively denoted by F^{q} again, on [C]:

Theorem 7.1. Let C be an EI-category and let k be a commutative ring. For any contravariant functor F : C → Mod(k) there is a cohomology spectral sequence

E_{2}^{p,q} = H^{p}([C]^{op}; F^{q}) ⇒ H^{p+q}(C; F) ,

where F^{q} : [C] → Mod(k) is a covariant functor satisfying F^{q}([X]) = H^{q}(C_{≥X}; F|_{C_{≥X}}) for any object X in C and any integer q ≥ 0.

The notation is as in the previous section; that is, C_{≥X} denotes the full subcategory of C of all objects Y such that Hom_{C}(X, Y) ≠ ∅ . If [X] ≤ [Y] then C_{≥Y} ⊆ C_{≥X}, so this time F^{q} maps the morphism [X] ≤ [Y] in [C] to the restriction map H^{q}(C_{≥X}; F|_{C_{≥X}}) → H^{q}(C_{≤Y}; F|_{C_{≤Y}}). This Theorem is proved in 7.6 in a slightly more general version. We start with some basic observations regarding the poset of right ideals in an EI-category C of the form C_{≥X}, with X running over the set of objects in C.

Lemma 7.2. Let C be an EI-category and let P be the poset of right ideals P = {C_{≥X} | X ∈ C}. The map sending C_{≥X} to [X] is an isomorphism of posets P ≅ [C]^{op}.

Proof. This is a trivial consequence of 6.3: if X → Y is a morphism in C then C_{≥Y} ⊆ C_{≥X}, hence if [X] ≤ [Y] then C_{≥Y} ⊆ C_{≥X}. The Lemma follows. □

Any set of right ideals in C is a poset, with respect to the inclusion. Given a poset P of right ideals in an EI-category C and an object X in C the subposet P_{X} = {D ∈ P | X ∈ D} consisting of all right ideals D in the poset P which contain X is itself a right ideal in P. Furthermore, the poset of right ideals P = {C_{≥X} | X ∈ C} has the following contractibility property: for any object X in C, the subposet P_{X} = {D ∈ P | X ∈ D} is non empty and has a unique minimal element, namely C_{≥X}. In particular, P_{X} is connected.

Given a right ideal D in an EI-category C and a functor F in F(C; A), where A is an abelian category, we denote as in 3.5 by F_{D} the unique functor in F(C; A) which coincides with F on D and vanishes outside D. The next result shows that under suitable hypotheses, the limit functor

lim_{⟵}_{P} : F(P; F(C; A)) ⟶ F(C; A)

splits.

Proposition 7.3. Let C be an EI-category and let P be a poset of right ideals in C such that for any object X in C the subposet P_{X} = {D ∈ P | X ∈ D} is non empty connected. Let A be an abelian category having small limits. There is a covariant functor

Π : \{ \begin{matrix}F(C; A) & ⟶ F(P; F(C; A)) \\ F & ↦ (D ↦ F_{D}) \end{matrix}

satisfying

lim_{⟵}_{P} ∘ Π = Id_{F(C;A)} .

Proof. If D, E are right ideals in C such that E ⊆ D then, by 3.5 (ii), there is a canonical natural transformation F_{D} → F_{E}, making the assignment D ↦ F_{D} contravariant functorial. Thus Π becomes a covariant functor in the obvious way. In order to verify the splitting property we compute, for any object X in C, the limit (lim_{⟵}_{P}(Π(F)))(X) = lim_{⟵}_{P}(D ↦ F_{D}(X)), where this equality comes from 4.4 applied to the

functor D ↦ F_{D} instead of G. Now the functor D ↦ F_{D}(X) on P is constant equal to F(X) when restricted to the subposet P_{X} and zero outside of P_{X}. Since P_{X} is non empty connected by the assumptions, we have lim_{⟵}_{P}(D ↦ F_{D}(X)) = F(X). The result

follows. □

Corollary 7.4. Let C be an EI-category and let P be a poset of right ideals in C such that for any object X in C the subposet P_{X} = {D ∈ P | X ∈ D} is non empty connected. Let A be an abelian category having small limits. We have a commutative diagram of covariant functors

\xymatrix{F(C; A) \ar[d]_{F(P;lim_{⟵}_{C})∘Π} \ar[dr]^{lim_{⟵}_{C}} \\ F(P; A) \ar[r]_{lim_{⟵}_{P}} & A}

Proof. Combine 7.3 and 4.1. □

Under suitable hypotheses the first of the two spectral sequences in 4.6 collapses:

Proposition 7.5. Let C be an EI-category, let P be a poset of right ideals in C such that for any object X in C the subposet P_{X} = {D ∈ P| X ∈ D} is non empty having a unique minimal element and let k be a commutative ring. Let Π be as in 7.3. For any functor F in F(C; k) the functor Π(F) in F(P; F(C; k)) is lim_{⟵}_{P}-acyclic.

Proof. Let i be a positive integer. By 4.5 (and with the notation of 4.5) we have (lim_{⟵}_{P}^{i}(Π(F))(X) ≅ H^{i}(P; Π(F)_{X}), where X is an object in C and where Π(F)_{X} is the

functor mapping D ∈ P to F(X) if X ∈ D and to zero, otherwise. In other words, Π(F)_{X} is the functor which takes the constant value F(X) on the right ideal P_{X} in P. By 3.7 we get that H^{i}(P; Π(F)_{X}) ≅ H^{i}(P_{X}; \underline{F(X)}), where \underline{F(X)} is the constant contravariant functor with value F(X). Now the right ideal P_{X} has a unique minimal element by assumption, and thus 2.8 applies, showing that this cohomology vanishes for positive i. □

Theorem 7.6. Let C be an EI-category, let P be a poset of right ideals in C such that for any object X in C the subposet P_{X} = {D ∈ P| X ∈ D} is non empty having a unique minimal element and let k be a commutative ring. Let F be a functor in F(C; k). For any integer q ≥ 0 denote by F^{q} the functor in F(P; k) sending D ∈ P to H^{q}(D; F|_{D}). There is a cohomology spectral sequence starting at the E_{2}-page

H^{p}(P; F^{q}) ⇒ H^{p+q}(C; F) .

Proof. We consider the two spectral sequences in 4.6 applied to the functor Π(F) in F(P; F(C; k)). By 7.5 the first of the two spectral sequences in 4.6 collapses to H^{p}(C; X ↦ H^{0}(P; Π(F)_{X})) ≅ (lim_{⟵}_{C} ∘ lim_{⟵}_{P})^{p}(Π(F)). The left side in this isomorphism is

isomorphic to H^{p}(C; F) because H^{0}(P; Π(F)_{X}) ≅ lim_{⟵}_{P}(Π(F))(X) ≅ F(X), where the

last isomorphism comes from 7.3. Moreover, by 3.7, we have H^{q}(D; F|_{D}) ≅ H^{q}(C; F_{D}), and thus the second spectral sequence in 4.6 takes the form as claimed. □

Proof of Theorem 7.1. Let P = {C_{≥X} | X ∈ C}. Let X be an object in C. Then P_{X} = {C_{≥Y} | X ∈ C_{≥Y} } has a unique minimal element, namely C_{≥X}. By 7.2 we have an isomorphism of posets P ≅ [C]^{op} sending P_{X} onto [C_{≤X}]^{op}. Thus 7.6 applies and yields the spectral sequence as stated in Theorem 7.1. □

Remark 7.7. The Mayer-Vietoris long exact sequence in 3.9 can be interpreted as the particular case of 7.6 applied to a poset of the form P = {D, E, D ∩ E}, where D, E are right ideals in C such that D ∪ E = C. The spectral sequence in 7.6 collapses then at the E_{2}-page, having at most two non zero columns, namely for p = 0 and p = 1.

§8 Constant coefficients over regular EI-categories

For a constant functor F on a regular EI-category it is possible to compute the functors F^{q} arising in the spectral sequence in 7.1 in terms of the cohomology of automorphism group.

Theorem 8.1. Let C be a regular EI-category, let X be an object in C, let k be a commutative ring and and let A be a k-module. The restriction from C_{≥X} to the full subcategory {X} induces an isomorphism

H^{∗}(C_{≥X}; A) ≅ H^{∗}(Aut_{C}(X); A) .

Here - as before - the notation H^{∗}(C_{≥X}; A) means the cohomology of the constant contravariant functor with value A on C_{≥X}. The proof of 8.1 is based on the following Lemma which is a sufficient criterion for the inclusion functor {X} → C_{≥X} to have a right adjoint.

Lemma 8.2. Let C be a small category, let X be an object in C and denote by D the full subcategory of C having X as unique object. Let Φ : D → C be the inclusion functor. Suppose that for any object Y in C there is a morphism ι_{Y} : X → Y in C with the property that the map given by composition with ι_{Y}

\{ \begin{matrix}Aut_{C}(X) & ⟶ Hom_{C}(X, Y) \\ α & ↦ ι_{Y} ∘ α \end{matrix}

is a bijection. Then the functor Φ has a right adjoint Ψ : C → D.

Proof. On objects, there is no choice but to have Ψ send any object Y in C to the unique object X in D. For a morphism φ : Y → Z we define the morphism Ψ(φ) : X → X in D as follows. The morphisms φ ∘ ι_{Y} and ι_{Z} are two morphisms in C from X to Z. Since composition with ι_{Z} is a bijection Aut_{C}(X) ≅ Hom_{C}(X, Z) there is a unique β ∈ Aut_{C}(X) such that

φ ∘ ι_{Y} = ι_{Z} ∘ β .

We define then Ψ(φ) = β. A trivial verification shows that this assignment is functorial. To see that the functor Ψ constructed in this way is right adjoint to Φ we observe that

Hom_{D}(X, Ψ(Y)) = Hom_{C}(X, X) ≅ Hom_{C}(X, Y) = Hom_{C}(Φ(X), Y)

for any object Y in C, where the bijection in the middle is given by composition with ι_{Y}. Another trivial verification shows that the bijections thus obtained are natural in X and Y, and hence Ψ is right adjoint to Φ. □

Proof of Theorem 8.1. By definition of C_{≥X}, for any object Y in C_{≥X} there is at least one morphism ι_{Y} : X → Y in C. Since C is regular, for any other morphism ψ : X → Y there is a unique automorphism α ∈ Aut_{C}(X) such that ψ = ι_{Y} ∘ α, or equivalently, composition with ι_{Y} is a bijection Aut_{C}(X) ≅ Hom_{C}(X, Y). Thus Lemma 8.2 applies to the category C_{≥X} and the object X, and hence the inclusion functor Φ : {X} → C_{≥X} has a right adjoint Ψ : C_{≥X} → {X}. Then Ψ^{∗} is right adjoint to Φ^{∗}, hence Φ_{!} = Ψ^{∗} is exact. It follows from 2.7 that H^{∗}(Aut_{C}(X); A) ≅ H^{∗}(C; Ψ^{∗}(A)) = H^{∗}(C; A). □

The spectral sequence 7.1 takes for constant functors on regular EI-categories the following form:

Theorem 8.3. Let C be a regular EI-category, let k be a commutative ring and let A be a k-module. We have a cohomology spectral sequence starting at the E_{2}-page

H^{p}([C]^{op}; [X] ↦ H^{q}(Aut_{C}(X); A)) ⇒ H^{p+q}(C; A) .

Proof. This is the spectral sequence 7.5 applied to the poset P = {C_{≥X} |X ∈ C} ≅ [C]^{op} (cf. 7.1), combined with 8.1. □

Theorem 8.4. Let C be a finite regular EI-category and let k be an algebraically closed field. There is a group homomorphism H^{2}([C]^{op}; k^{×}) → H^{2}(C; k^{×}) whose kernel and cokernel are both finite abelian groups; if in addition char(k) = ℓ > 0 the kernel and cokernel of this group homomorphism have orders prime to ℓ.

Proof. The spectral sequence 8.3 applied to k = ℤ and A = k^{×} yields a group homomorphism

E_{2}^{2,0} = H^{2}([C]^{op}; k^{×}) ⟶ H^{2}(C; k^{×}) .

The kernel of this group homomorphism is a homomorphic image of

E_{2}^{0,1} = H^{0}([C]^{op}; [X] ↦ H^{1}(Aut_{C}(X); k^{×})) .

The cokernel of this group homomorphism is filtered by subquotients of the two groups

E_{2}^{0,2} = H^{0}([C]^{op}; [X] ↦ H^{2}(Aut_{C}(X); k^{×})) ,
E_{2}^{1,1} = H^{1}([C]^{op}; [X] ↦ H^{1}(Aut_{C}(X); k^{×})) .

Thus both the kernel and cokernel of this group homomorphism are finite: since C is finite so are the groups H^{1}(Aut_{C}(X); k^{×}) and H^{2}(Aut_{C}(X); k^{×}) and taking H^{0} or H^{1} over the finite poset [C]^{op} of a functor with values in finite groups is obviously a finite group again. If char(k) = ℓ > 0 then all the occurring finite groups have order prime to ℓ. □

§9 Invariance under passage to subdivisions and proof of Theorem 1.3

For the computation of constant functors we may always replace an EI-category by its subdivision; this is a variation of the well-known principle that in order to compute cohomology of constant functors one may replace the category by its nerve. The following Theorem is equivalent to a Theorem of Słomińska [21, 1.5] (cf. Remark 9.3 below); for the convenience of the reader we include a detailed proof, as well as further invariance results for constant coefficients.

Theorem 9.1. Let C be an EI-category, let k be a commutative ring and let A be a k-module. The canonical functor S(C) → C induces an isomorphism H^{∗}(C; A) ≅ H^{∗}(S(C); A), with A viewed as constant contravariant functor on C and S(C), respectively.

The principal tool to prove 9.1 is the base change spectral sequence 2.6, together with 2.9 which describes the functors R^{q}Φ_{!} appearing in this spectral sequence in terms of the cohomology of functors on the categories Φ^{X}. The necessary routine calculations are contained in the following technical statement:

Proposition 9.2. Let C be an EI-category, let Φ : S(C) → C be the canonical covariant functor and let k be a commutative ring. Let X be an object in C. Denote by I^{X} the full subcategory of Φ^{X} consisting of all objects in Φ^{X} of the form (σ, φ), where σ = (X_{0} → X_{1} → ⋯ → X_{m}) is an object in S(C) and where φ : X_{m} → X is an isomorphism in C.

(i) The category I^{X} has (X, Id_{X}) as initial object. (ii) The inclusion functor I^{X} ↪ Φ^{X} has a left adjoint. (iii) For any contravariant functor F : S(C) → Mod(k) and any integer q ≥ 0 we have a natural isomorphism R^{q}Φ_{!}(F)(X) ≅ H^{q}(I^{X}; F^{X}|_{I^{X}}).

Proof. (i) Given an object (σ, φ) in I^{X}, the unique morphism from (X, Id_{X}) to (σ, φ) is given by φ^{−1} : X → X_{m}. (ii) We define a left adjoint T : Φ^{X} → I^{X} to the inclusion I^{X} ↪ Φ^{X} as follows. Given an object (σ, φ) in Φ^{X} we set

T (σ, φ) = (σ, φ)

if φ is an isomorphism, and

T (σ, φ) = ((X_{0} → ⋯ → X_{m} →^{φ} X), Id_{X})

if φ is not an isomorphism. Given a morphism α : (σ, φ) → (τ, ψ) we define T (α) as follows: if φ (and hence ψ) is an isomorphism we take T (α) = α, if none of φ, ψ is an isomorphism, we take for T (α) the obvious morphism T (σ, φ) → T (τ, ψ) obtained from extending α by Id_{X}, and if ψ is an isomorphism but φ is not, we take for T (α) the morphism obtainded from extending α by ψ^{−1}. It is easy to see that T is well-defined and left adjoint to the inclusion T^{X} ↪ Φ^{X}. (iii) By III.4.5 we have R^{q}Φ_{!}(F)(X) = H^{q}(Φ^{X}; F^{X}), and by III.4.3 together with (ii) this is isomorphic to H^{q}(I^{X}; F^{X}|_{I^{X}}) □

Proof of Theorem 9.1. By 9.2.(iii), for any positive integer q we have R^{q}Φ_{!}(A)(X) = H^{q}(I^{X}; A) = 0 because I^{X} has an initial object by 9.2.(i). Thus the base change spectral sequence 2.6 collapses in this case to the isomorphism stated in 9.1. □

Remark 9.3. One can show that 9.1 is equivalent to a result of Słomińska [21, 1.5] as follows. Since functor cohomology is invariant under equivalences of categories we may assume that each isomorphism class of C has a single object. It is not difficult to show that then S(C) is equivalent to the opposite category of the category s(C) as defined in [21, 1.1]. Since cohomology with constant coefficients is invariant under passage to opposite categories, Theorem 9.1 follows from [21, 1.5].

The following result is a generalisation of [2, 1.3] and will be useful for calculations involving groups of the form H^{2}(C; k^{×}) for right ideals C in fusion systems and an alegbraically closed field k. As before, whenever using the notation H^{∗}(C; A) for C a small category and A an abelian group, we consider implicitly A as constant contravariant functor on C.

Theorem 9.4. Let C, D be EI-categories, let Φ : C → D be a regular functor and suppose that every morphism in C is a monomorphism. Let A be an abelian group and for any object X in C set K(X) = ker(Aut_{C}(X) →^{Φ} Aut_{D}(Φ(X)). Suppose that H^{q}(K(X); A) = {0} for any positive integer q and any object X in C, with the trivial action of K(X) on A. Then Φ induces an isomorphism

H^{∗}(D; A) ≅ H^{∗}(C; A) .

Proof. We will show that under the given hypotheses, the base change spectral sequence H^{p}(D; R^{q}Φ_{!}(A)) ⇒ H^{p+q}(C; A) in 2.6 collapses. Let Y be an object in D. By 2.9 we have

R^{q}(Φ_{!}(A))(Y) ≅ H^{q}(Φ^{Y}; A)

for q ≥ 0. By 5.13, the category Φ^{Y} is a regular EI-category. By 5.14.(i) the associated poset [Φ^{Y}] has a unique maximal element. And by 8.3, applied to the regular category Φ^{Y}, there is a spectral sequence of the form

H^{p}([Φ^{Y}]^{op}; [(Z, ψ)] ↦ H^{q}(Aut_{Φ^{Y}} (Z, ψ); A)) ⇒ H^{p+q}(Φ^{Y}; A) .

Now Aut_{Φ^{Y}} (Z, ψ) = K(Z) by 5.15.(ii), and hence H^{q}(Aut_{Φ^{Y}} (Z, ψ); A) = {0} by the assumptions. Thus this spectral sequence collapses to

H^{p}([Φ^{Y}]^{op}; A) ≅ H^{p}(Φ^{Y}; A)

and since [Φ^{Y}] has a unique maximal element, [Φ^{Y}]^{op} has a unique minimal element and so we get H^{p}(Φ^{Y}; A) = {0} for p > 0 by 2.8. Thus R^{q}Φ_{!}(A) is the zero functor for q > 0, which implies that the base change spectral sequence at the beginning of this proof collapse to the required isomorphism

H^{p}(D; A) ≅ H^{p}(C; A)

for all p ≥ 0. □

Remark 9.5. Combining 9.1 with the second spectral sequence in 6.6 we can give an alternative proof of the spectral sequence for constant functors in 8.3 without using 7.1. If F is a constant functor on a regular EI-category C with value A then the second spectral sequence in 6.6 reads

H^{p}([C]^{op}; [X] ↦ H^{q}(S(C_{≥X}); A)) ⇒ H^{p+q}(S(C); A)

By 9.1 we have H^{q}(S(C_{≥X}); A) ≅ H^{q}(C_{≥X}; A) and H^{p+q}(S(C); A) ≅ H^{p+q}(C; A). By 8.1 we have H^{q}(C_{≥X}; A) ≅ H^{q}(Aut_{C}(X); A). This is the spectral sequence in 8.4.

Remark 9.6. A conjecture of Fei Xu [25, §6.2] asserts that for any finite EI-category C and a field k the quotient of the cohomology algebra H^{∗}(C; k) modulo nilpotent elements should finitely generated as k-algebra. It follows from 9.1 that in order to prove this conjecture it suffices to prove it for finite regular EI-categories, or for finite EI-categories with subobjects by 5.16.

Combining 8.4 and 9.1 yields a proof of 1.3:

Proof of Theorem 1.3. Since S(C) is regular, we have, by Theorem 8.4, a group homomorphism H^{2}([S(C)]^{op}; k^{×}) → H^{2}(S(C); k^{×}) with finite kernel and cokernel. By 9.1 we have H^{2}(S(C); k^{×}) ≅ H^{2}(C; k^{×}), whence the result. □

§10 Orbit spaces with vanishing low degree cohomlogy

Theorem 1.2 is an immediate consequence of Theorem 1.3. We will show that Theorem 1.2 is also a consequence of a more precise result which implies Theorem 1.4 and which requires the vanishing of the cohmology with coefficients in k^{×} in degree 2, 3 and 4.

Theorem 10.1. Let C be a finite EI-category and let k be an algebraically closed field. Suppose that H^{2}([S(C)]; k^{×}) = H^{3}([S([C)]; k^{×}) = H^{4}([S(C)]; k^{×}) = {0}. There is an exact sequence of abelian groups

0 → H^{1}([S(C)]; A^{1}) → H^{2}(C; k^{×}) → H^{0}([S(C)]; A^{2}) → H^{2}([S(C)]; A^{1}) → H^{3}(C; k^{×}) where A^{i} : [S(C)] → Mod(ℤ) is a covariant functor sending [σ] ∈ [S(C)] to H^{i}(Aut_{S(C)}(σ); k^{×}) for i = 1, 2. In particular, the abelian group H^{2}(C; k^{×}) is finite. Moreover, if in addition char(k) = ℓ > 0 for some prime number ℓ (not necessarily different from p) then the order of H^{2}(C; k^{×}) is prime to ℓ.

Proof. Denote by A : C → ℤ the constant contravariant functor sending every object in C to the abelian group k^{×}. Use the same letter A for its restriction to the constant functor on S(C) with value k^{×} through the canonical functor S(C) → C. By Theorem 9.1 we have

H^{∗}(C; A) ≅ H^{∗}(S(C); A)

Since S(C) is regular, the spectral sequence from 7.1 takes the form as described in 8.3; that is, we have a spectral sequence

E_{2}^{p,q} = H^{p}([S(C])^{op}; A^{q}) ⇒ H^{p+q}(S(C); A)

where A^{q} : [S(C)] → Mod(ℤ) is a covariant functor sending [σ] ∈ [S(C)] to H^{q}(Aut_{S(C)}(σ); k^{×}). Note that in particular A^{0} is the constant covariant functor on [S(C)] with value k^{×}. By the assumptions we have E_{2}^{0,0} = k^{×} and E_{2}^{2,0} = E_{2}^{3,0} = E_{2}^{4,0} = {0}. Therefore the E_{2}-page of this spectral sequence looks as follows:

⋯  ⋯  ⋯  ⋯
H^{0}([S(C)]; A^{2}) H^{1}([S(C)]; A^{2}) H^{2}([S(C)]; A^{2}) H^{3}([S(C)]; A^{2}) ⋯
H^{0}([S(C)]; A^{1}) H^{1}([S(C)]; A^{1}) H^{2}([S(C)]; A^{1}) H^{3}([S(C)]; A^{1}) ⋯
k^{×}  H^{1}([S(C)]; k^{×})  0  0  0 ⋯

This spectral sequence approximates H^{∗}(S(C); A) ≅ H^{∗}(C; A). There is no non zero differential ending or starting at E_{2}^{1,1} = H^{1}([S(C)]; A^{1}), and so we get an injective map

H^{1}([S(C)]; A^{1}) → H^{2}(C; A)

The cokernel of this map is the kernel of the differential

E_{2}^{0,2} = H^{0}([S(C)]; A^{2}) → E_{2}^{2,1} = H^{2}([S(C)]; A^{1})

because all differentials starting at (0, 2) from page 3 onwards are zero. Since from page 3 onwards there is no differential ending or starting at (2, 1) it follows that the cokernel of the last map is a subgroup of H^{3}(C; A). This proves the exactness of the sequence as stated. The groups A^{1}([σ]) = H^{1}(Aut_{S(C)}(σ); k^{×}) and A^{2}([σ]) = H^{2}(Aut_{S(C)}(σ); k^{×}) are finite groups (of order prime to ℓ if char(k) = ℓ > 0). Thus H^{1}([S(C]); A^{1}) is finite (of order prime to ℓ) because even the group Z^{1}([S(C)]; A^{1}) consisting of all maps from the finite morphism set of S(C) to finite groups (of order prime to ℓ) is a finite group (of order prime to ℓ). Moreover, the group H^{0}([S(C)]; A^{2}) is an inverse limit of finite abelian groups (of order prime to ℓ), hence a finite abelian group itself (of order prime to ℓ). This proves the finiteness of H^{2}(C; k^{×}) and that this group has order prime to ℓ if char(k) = ℓ > 0. □

We will observe that this Theorem applies to right ideals in fusion systems; we briefly recall the relevant definitions and some basic properties of fusion systems. Let p be a prime.

Definition 10.2. Given a finite p-group P, a category on P is a category F whose objects are the subgroups of P and whose morphism sets Hom_{F}(Q, R) are sets of injective group homomorphisms from Q to R, where Q, R are subgroups of P, with the following properties:

(i) the composition of morphisms in F is given by the usual composition of group homomorphisms; (ii) if Q ⊆ R then the inclusion map from Q to R is a morphism in F; (iii) if φ : Q → R is a morphism in F then so is the induced isomorphism Q ≅ φ(Q) and its inverse.

If Q, R are subgroups of P we denote by Hom_{P} (Q, R) the set of group homomorphisms from Q to R induced by conjugation with elements in P, and we write Aut_{P} (Q) = Hom_{P} (Q, Q).

Definition 10.3. Let F be a category on a finite p-group P and let Q be a subgroup of P. We say that Q is fully F-normalised if |N_{P} (Q)| ≥ |N_{P} (Q′)| for any subgroup Q′ of P such that there is an isomorphism Q′ ≅ Q in F. We say that Q is fully F-centralised if |C_{P} (Q)| ≥ |C_{P} (Q′)| for any subgroup Q′ of P such that there is an isomorphism Q′ ≅ Q in F. We say that Q is F-centric if C_{P} (Q′) = Z(Q′) for every subgroup Q′ of P such that there is an isomorphism Q′ ≅ Q in F.

Following the notation in [3], if φ : Q → P is a morphism in F we denote by N_{φ} the subgroup of all elements y in N_{Q}(P) for which there exists an element z ∈ N_{P} (φ(Q)) such that φ(yuy^{−1}) = zφ(u)z^{−1} for all u ∈ Q. Note that QC_{P} (Q) ⊆ N_{φ} ⊆ N_{P} (Q).

Definition 10.4. A category F on a finite p-group P is called a fusion system if Hom_{P} (Q, R) ⊆ Hom_{F}(Q, R) for all subgroups Q, R of P and if in addition the two following properties hold: (I-S) Aut_{P} (P) is a Sylow-p-subgroup of Aut_{F}(P); (II-S) every morphism φ : Q → P such that φ(Q) is fully F-normalised extends to a morphism ψ : N_{φ} → P in F (that is, ψ|_{Q} = φ).

The concept of a fusion system on a finite p-group is due to L. Puig; the above axioms (I-S) and (II-S) appear in Stancu [23]. As observed in [23] - see also [15, §1] for proofs - the axioms I-S and II-S are equivalent to the a priori stronger axioms used by Broto, Levi and Oliver in [3, 1.2] (I-BLO) if Q is a fully F-normalised subgroup of P then Q is fully F-centralised and Aut_{P} (Q) is a Sylow-p-subgroup of Aut_{F}(Q); (II-BLO) given any subgroup Q of P, every morphism φ : Q → P such that φ(Q) is fully F-centralised extends to a morphism ψ : N_{φ} → P in F (that is, ψ|_{Q} = φ).

Theorem 1.1 is an immediate consequence of 1.2 and the following result from [16, 5.11, 5.12], which shows that the hypotheses in 1.2 and 10.1 are fulfilled for right ideals in fusion systems:

Theorem 10.5. Let C be a right ideal in a fusion system F on a finite p-group P. Then [S(C)] is acyclic.

Remark 10.6. Let F be a fusion systems of a p-block b of a finite group G over an algebraically closed field k of characteristic p on a defect group P of b. Let C = F^{c} be the right ideal of F-centric subgroups of P. Since [S(C)] is contractible the hypotheses of 10.1 are satisfied. It follows from [12, 1.8] that b determines a “compatible” family of classes α_{Q} ∈ H^{2}(Aut_{F}(Q); k^{×}) for any F-centric subgroup Q of P. It is easy to see that “compatible” means that the family α_{Q} determines an element α^{0} in H^{0}([S(C)]; A^{2}). Let γ ∈ H^{2}([S(C)]; A^{1}) be the image of α^{0} under the map occuring in the exact sequence in Theorem 10.1. The gluing problem [13, 4.2] has a solution if and only if γ = 0 because that is equivalent to the existence of an α ∈ H^{2}(S(C); k^{×}) mapping onto α^{0}. The group H^{1}([S(C)]; A^{1}) acts then regularly on the set of such solutions via the injective map H^{1}([S(C)]; A^{1}) → H^{2}(C; k^{×}) from 10.1. In particular, such a solution, if it exists, is unique if and only if H^{1}([S(C)]; A^{1}) = {0}. Thus, in order to solve this gluing problem, one needs to calculate the cohomology of A^{1}. By results in [16, §4] one can play this problem back to normal chains; that is, by [16, 4.7, 4.10] there is an isomorphism

H^{∗}([S(C)]; A^{q}) ≅ H^{∗}([S_{◁}(C)]; A^{q})

for q ≥ 0, where the notation is as in [16, 4.1]. In particular, if

H^{1}([S_{◁}(C)]; A^{1}) = H^{2}([S_{◁}(C)]; A^{1}) = {0}

then

H^{2}(C; k^{×}) ≅ lim_{⟵}_{[S_{◁}(C)]} ([σ] ↦ H^{2}(Aut_{S(C)}(σ); k^{×}))

and so in that case any gluing problem of the form [13, 4.2] would have a unique solution. We do not know whether the functor A^{1} has vanishing first and second cohomology groups, in general. This is what motivates the spectral sequence 1.8: the vanishing of H^{2}([S(C)]; A^{1}) can be played back to the vanishing of the first cohomology groups of certain functors on fusion systems of finite groups by means of statement (iii) in Theorem 1.8.

§11 Further invariance properties of H^{∗}(C; k^{×}) for fusion systems

We will show in this section that applying 9.4 to right ideals of centric subgroups in a fusion system yields the invariance of H^{∗}(C; k^{×}) with respect to central extensions and passage to the orbit category and centric linking system of a fusion system. If F is a fusion system on a finite p-group P and Z is a subgroup of Z(P) then the equality F = C_{F}(Z) means that every morphism φ : Q → R in F can be extended to a morphism ψ : QZ → RZ in F such that ψ|_{Z} = Id_{Z}. If F = C_{F}(Z) then F induces a fusion system on P/Z, denoted by F/Z. Fusion systems of the form C_{F}(Z) and F/Z are particular cases of more general constructions of normalisers, centralisers and quotients in fusion systems due to Puig; see e.g. [3, Appendix] or [17] for details. Given a fusion system F on a finite p-group P, we denote by F^{c} the full subcategory in F of all F-centric subgroups of P; this is a right ideal in F (this is well-known - see e.g. [15, §1] or [17]). We denote by F̄ the quotient category of F having as objects the subgroups of P and as morphism sets the orbits

Aut_{R}(R)\Hom_{F}(Q, R)

of the morphism set Hom_{F}(Q, R) with respect to the action of the group of inner automorphisms Aut_{R}(R) of R via composition of group homomorphisms, called the orbit category of F. Denote by F̄^{c} the image of F^{c} in F̄. While all morphisms in F are monomorphisms, this is no longer true in F̄; however, all morphisms in F̄^{c} are epimorphisms (see e.g. [15, §1]). For the centric linking system L of F we refer to [3].

Theorem 11.1. Let F be a fusion system on a finite p-group P such that F = C_{F}(Z) for some subgroup Z of Z(P) and let C be a right ideal in F^{c} consisting of subgroups of P containing Z such that the image C/Z of C in F/Z is contained in (F/Z)^{c}. Let k be an algebraically closed field of characteristic k. The canonical functor C → C/Z is regular and induces an isomorphism H^{∗}(C; k^{×}) ≅ H^{∗}(C/Z; k^{×}).

Proof. Let Q, R be subgroups of P belonging to C. Then Z is contained in Q, R by the assumptions. The morphism set Hom_{F/Z}(Q/Z, R/Z) is the canonical image of the morphism set Hom_{F}(Q, R); in particular, the canonical functor C → C/Z is surjective on morphisms as required in part (b) of 5.10. This implies also that if Q/Z ≅ R/Z in C/Z then Q ≅ R in C, and hence we have the isomorphism of posets [C] ≅ [C/Z] as required in part (a) of 5.10. If φ, ψ : Q → R are two morphisms in C whose images in C/Z are equal, then in particular, φ(Q) = ψ(Q) ⊆ R, and with the obvious abuse of notation we get an automorphism κ = ψ^{−1} ∘ φ of Q whose image in C/Z is the identity on Q/Z. Thus κ is the unique element of the group

K(Q) = ker(Aut_{F}(Q) → Aut_{F/Z}(Q))

satisfying φ = ψ ∘ κ. This shows that the canonical functor C → C/Z is regular. Moreover, the group K(Q) is an abelian p-group by [15, §1], and since k is algebraically closed of characteristic p this implies that

H^{q}(K(Q); k^{×}) = {0}

for all q > 0. Thus 9.4 applies, proving the Theorem. □

Theorem 11.2. Let F be a fusion system on a finite group P, let C be a right ideal in F^{c} and let C̄ be the canonical image of C in the orbit category F̄. Let k be an algebraically closed fields of characteristic p. The canonical functor C → C̄ induces a regular functor S(C) → S(C̄) and an isomorphism H^{∗}(C; k^{×}) ≅ H^{∗}(C̄; k^{×}).

Proof. Two subgroups Q, R of P are isomorphic in F if and only if they are isomorphic in F̄ because morphisms in F̄ are induced by those in F. Thus in particular the canonical functor C → C̄ sends non-isomorphisms to non-isomorphisms, hence induces a functor S(C) → S(C̄). For φ : Q → R a morphism in C we denote by φ̄ : Q → R its image in C, and for σ an object in S(C) we denote by σ̄ its image in S(C̄). We show first that property (b) form definition 5.10 holds for the functor S(C) → S(C̄); that is, this functor is surjective on morphisms. To see this, let

σ = Q_{0} ⟶^{φ_{0}} Q_{1} ⟶^{φ_{1}} ⋯ ⟶^{φ_{m−1}} Q_{m}
τ = R_{0} ⟶^{ψ_{0}} R_{1} ⟶^{ψ_{1}} ⋯ ⟶^{ψ_{n−1}} R_{n}

be two objects in S(C), and let Let μ = (μ_{i})_{0≤i≤m} be a family of morphisms in C such that μ̄ = (μ̄_{i})_{0≤i≤m} is a morphism in S(C̄) from σ̄ to τ̄. That is, μ is a family μ = (μ_{i})_{0≤i≤m} where for each i there is α(i) ∈ {0, 1, .., n} such that μ_{i} : Q_{i} → R_{α(i)}) is an isomorphism in C which makes the obvious diagrams in C̄ commutative; more precisely,

μ̄_{i+1} ∘ φ̄_{i} = ψ̄_{α(i+1)−1} ∘ ⋯ ∘ ψ̄_{α(i)+1} ∘ ψ̄_{α(i)} ∘ μ̄_{i}

for any i ∈ {0, 1, .., m − 1}. Thus the lifts in C of these morphisms coicide up to conjugation by an inner automorphism of R_{α(i+1)}. In other words, there are elements v_{i} ∈ R_{α(i)} for 1 ≤ i ≤ m such that

c_{v_{i+1}} ∘ μ_{i+1} ∘ φ̄_{i} = ψ_{α(i+1)−1} ∘ ⋯ ∘ ψ_{α(i)+1} ∘ ψ_{α(i)} ∘ μ_{i}

for 0 ≤ i ≤ m − 1, where here c_{v_{i+1}} is the inner automorphism of R_{α(i+1)} given by conjugation with v_{i+1}. Thus we get a morphism ν = (ν_{i})_{0≤i≤m} from σ to τ in S(C) by setting

ν_{0} = μ_{0} ,
ν_{i} = c_{w_{i}} ∘ μ_{i}

for some suitable w_{i} ∈ R_{i}, where 1 ≤ i ≤ m. By construction, the image of ν in S(C̄) is equal to μ̄. This shows that 5.10.(b) holds for the functor S(C) → S(C̄). Since a morphism in C is an isomorphism in C if and only if its image in C̄ is an isomorphism, it follows that the functor S(C) → S(C̄) induces an isomophism of posets [S(C)] ≅ [S(C̄)], hence 5.10.(a) holds for this functor. Property 5.10.(c) follows from the fact that S(C) is regular: if μ, μ′ are two morphisms in S(C) from σ to τ then there is a unique automorphism τ of σ in S(C) such that μ′ = μ ∘ τ. Thus if in addition the images μ̄ and μ̄′ of μ, μ′ in S(C̄) are equal then the equality μ̄ = μ̄′ ∘ τ̄ = μ̄ ∘ τ̄ forces τ̄ = Idσ̄ because any morphism in S(C̄) is a monomorphism. Thus indeed

τ̄ ∈ K(σ) = ker(Aut_{S(C)}(σ) → Aut_{S(C̄)}(σ̄)) ,

which shows that property (c) of 5.10 holds as well. It follows that the functor S(C) → S(C̄) is regular. Also, the group K(σ) is a p-group because it consists of inner automorphisms of the subgroups of P occurring in σ. Thus H^{q}(K(σ); k^{×}) = {0} for q > 0 and hence Theorem 9.4 yields an isomorphism H^{∗}(S(C); k^{×}) ≅ H^{∗}(S(C̄); k^{×}). Applying 9.1 twice yields the isomorphism H^{∗}(C; k^{×}) ≅ H^{∗}(C̄; k^{×}) as stated. □

Theorem 11.3. Let F be a fusion system on a finite p-group P having a centric linking system L with structural functor π : L → F^{c}. Let C be a right ideal in L and let D be its image in F under π. Let k be an algebraically closed field of characteristic p. The functor π induces an isomorphism H^{∗}(C; k^{×}) ≅ H^{∗}(D; k^{×}) .

Proof. The centric linking system L is an extension of F^{c} by the centre functor Z : F^{c} → ℤ_{(p)} sending an F-centric subgroup Q of P to its centre Z(Q). Thus π induces a regular functor C → D and for any F-centric subgroup of P we have

K(Q) = ker(Aut_{L}(Q) → Aut_{F}(Q)) ≅ Z(Q) ,

hence H^{q}(K(Q); k^{×}) = {0} for q > 0. Thus Theorem 9.4 applies and yields the isomorphism as stated. □

The following result - another easy consequence of 8.2 - is well-known, at least for p-local coefficients, but we state this here for the sake of completeness and in view of our interest in coefficients with value k^{×} for k an algebraically closed field.

Theorem 11.4. Let F be a fusion system on a finite p-group P, let Q be a normal subgroup of P such that F = N_{F}(Q) and let C = F_{≥Q} be the right ideal consisting of all subgroups of P containing Q. Let k be a commutative ring and let A be a k-module. Restriction induces an isomorphism H^{∗}(C; A) ≅ H^{∗}(Aut_{F}(Q); A) .

Proof. Let D be the full subcategory of C having Q as unique object and let Φ : D → C be the inclusion functor. For any subgroup R belonging to C denote by ι_{R} : Q ⊆ R the inclusion homomorphism. For any morphism φ : Q → R we have φ(Q) = Q because F = N_{F}(Q). Thus composition with the inclusion morphism ι_{R} induces a bijection Aut_{F}(Q) ≅ Hom_{F}(Q, R). It follows from 8.2 that Φ has a right adjoint Ψ : C → D. Then Ψ^{∗} is right adjoint to Φ^{∗}; in other words, Φ_{!} = Ψ^{∗} is exact, and so by 2.7, we have H^{∗}(D; A) = H^{∗}(C; Ψ^{∗}(A)) = H^{∗}(C; A). □

§12 On H^{1}(C; k^{×})

Let k be an algebraically closed field. For G a finite group we have H^{1}(G; k^{×}) = Hom(G, k^{×}). More generally, for C a small category, we have H^{1}(C; k^{×}) = Z^{1}(C; k^{×})/B^{1}(C; k^{×}), where Z^{1}(C; k^{×}) is the group of all maps ζ : Mor(C) → k^{×} satisfying ζ(ψ ∘ φ) = ζ(ψ)ζ(φ) for any two composable morphisms φ, ψ in C, and where B^{1}(C; k^{×}) is the group of all maps τ : Mor(C) → k^{×} for which there exists a map β : Ob(C) → k^{×} such that τ(φ) = β(Y)β(X)^{−1} for any morphism φ : X → Y in C. We restate and prove Theorem 1.5.

Theorem 12.1. Let p be a prime, let P be a finite p-group, let F be a fusion system on P and let C be a right ideal in F. Let k be an algebraically closed field of characteristic p. The group homomorphism H^{1}(C; k^{×}) ⟶ H^{1}(Aut_{F}(P); k^{×}) induced by restriction is injective.

Proof. We proceed by induction over the number of isomorphism classes of subgroups in C. If P is the unique object of C then H^{∗}(C; k^{×}) = H^{∗}(Aut_{F}(P); k^{×}), so there is nothing to prove. Suppose that C has an object different from P. Let Q be a subgroup of minimal order of P belonging to C. Then [Q] is minimal in the poset [C]. Let D = C −[Q] be the right ideal in F obtained from removing the isomorphism class of Q from C. By induction it suffices to show that the map H^{1}(C; k^{×}) → H^{1}(D; k^{×}) induced by restriction along the inclusion D ⊆ C is injective. Let ζ ∈ Z^{1}(C; k^{×}) such that the restriction of ζ to D represents the zero class in H^{1}(D; k^{×}). We show first that we can actually assume that ζ takes the constant value 1 on all morphisms in D. Since the image of ζ in H^{1}(D; k^{×}) is trivial, there is a map β : Ob(D) → k^{×} such that ζ(φ) = β(S)β(R)^{−1} for any morphism φ : R → S in D. Extend β to a map, still called β, from Ob(C) to k^{×} by setting β(Q′) = 1 for any subgroup Q′ of P isomorphic to Q in C. Define ζ′ ∈ Z^{1}(C; k^{×}) by setting

ζ′(φ) = ζ(φ)β(S)^{−1}β(R)

for any morphism φ : R → S in C. Then ζ and ζ′ represent the same class in H^{1}(C; k^{×}) and by construction, ζ′(φ) = 1 for any morphism φ in D. Thus, after replacing ζ by ζ′ we may (and do) assume that ζ(φ) = 1 for any morphism φ in D.

Up to replacing Q by a suitable isomorphic subgroup we also may assume that Q is fully F-normalised; that is, Aut_{P} (Q) is a Sylow-p-subgroup of Aut_{F}(Q). Restriction of ζ to Aut_{F}(Q) yields a group homomorphism Aut_{F}(Q) → k^{×}, and since char(k) = p, the kernel of this group homomorphism contains the normal subgroup A generated by Aut_{P} (Q). Set N = N_{Aut_{F} (Q)}(Aut_{P} (Q)). By Frattini’s argument we have

Aut_{F}(Q) = A · N .

By the extension axiom, any morphism φ ∈ N extends to a morphism ψ ∈ Aut_{F}(N_{P} (Q)); that is, we have a commutative diagram of the form

N_{P} (Q) ⟶^{ψ} N_{P} (Q)
ι↑  ↑^{ι}
Q ⟶_{φ} Q

Thus ζ(φ)ζ(ι) = ζ(ι)ζ(ψ), and hence ζ(φ) = ζ(ψ) = 1 by our choice of ζ. This shows that ζ restricted to N is also 1, and hence ζ restricted to Aut_{F}(Q) is constant 1. Define now a map γ : Ob(C) → k^{×} by setting γ(R) = ζ(ι_{R}), where ι_{R} : R → P is the inclusion homomorphism. Thus if R ≅ Q̸ in C then γ(R) = 1. We now show that ζ represents the trivial class by observing the equality

ζ(φ) = γ(R)γ(S)^{−1}

for any morphism φ : R → S in C. This is clear if R is not isomorphic to Q, because then both sides are equal to 1. If this equality holds for two composable morphisms then also for their composition. Since every morphism in F is a composition of an isomorphism followed by an inclusion, it suffices to consider the cases where φ is either an inclusion or an isomorphism. If R ⊆ S and φ : R → S is the inclusion then ι_{s} ∘ φ = ι_{R}, hence ζ(φ) = ζ(ι_{R})ζ(ι_{S})^{−1} = γ(R)γ(S)^{−1} as required. Assume that φ : R → S is an isomorphism. Since Q is fully F-normalised there are isomorphisms ρ : R ≅ Q and σ : S ≅ Q which can be extended to N_{P} (R) and N_{P} (S), respectively. Note that σ∘φ∘ρ^{−1} is an automorphism of Q, hence ζ(σ ∘ φ ∘ ρ^{−1}) = 1, or equivalently, ζ(φ) = ζ(ρ)ζ(σ)^{−1}. Since ρ extends to N_{P} (R) there is a commutative diagram of the form

N_{P} (R) ⟶^{τ} P
κ↑  ↑^{ι_{Q}}
R ⟶_{ρ} Q

Thus ζ(ρ)ζ(ι_{Q}) = ζ(κ)ζ(τ), and since R is a proper subgroup of N_{P} (R) we get ζ(τ) = 1. Also, ζ(ι_{Q}) = γ(Q) and since κ is an inlusion, we have ζ(κ) = γ(R)γ(N_{P} (R)^{−1} = γ(R). Together we get ζ(ρ) = γ(R)γ(Q)^{−1}. The same argument yields ζ(σ) = γ(S)γ(Q)^{−1}, and now ζ(φ) = ζ(ρ)ζ(σ)^{−1} = γ(R)γ(S)^{−1} as required. □

Combining the spectral sequence 7.1 and Theorem 9.1 yields another possible way to compute H^{1}(C; k^{×}), where C is a finite EI-category with acyclic orbit space. The map sending an object X of C to H^{1}(Aut_{C}(X); k^{×}) = Hom(Aut_{C}(X), k^{×}) is not functorial, in general, because taking automorphism groups is not. It becomes functorial, though, upon taking subdivisions, and then the inverse limit of this functor is precisely H^{1}(C; k^{×}). This is the content of the next result, which holds more generally for arbitrary constant coefficients.

Theorem 12.2. Let C be a finite EI-category and let A be an abelian group such that H^{1}([S(C)]; A) = H^{2}([S(C)]; A) = {0}. There is a natural isomorphism

H^{1}(C; A) ≅ lim_{⟵}_{[S(C)]} ([σ] ↦ Hom(Aut_{S(C)}(σ), A)) .

Proof. The proof uses the same spectral sequence argument as that of 10.1. The category S(C) is regular. Thus, by 8.3, the spectral sequence 7.1 applied to S(C) and the constant contravariant functor with value A has an E_{2}-page of the form

\begin{matrix}⋯ & ⋯ & ⋯ & ⋯ \\ H^{0}([S(C)]; A^{1}) & H^{1}([S(C)]; A^{1}) & ⋯ & ⋯ \\ H^{0}([S(C)]; A^{0}) & H^{1}([S(C)]; A^{0}) & H^{2}([S(C)]; A^{0}) & ⋯ \end{matrix}

where A^{q} : [S(C)] → Ab is a covariant functor sending the isomorphism class [σ] of a chain σ ∈ S(C) to H^{q}(Aut_{S(C)}(σ); A), for q ≥ 0 (see [16, 4.10]). The functor A^{0} is the constant covariant functor on [S(C)] taking the value A. By the assumptions, the last row of this E_{2}-page is equal to

A 0 0 H^{3}([S(C)]; A) ⋯

Since this spectral sequence approximates H^{∗}(S(C); A) we get in particular an isomorphism

H^{1}(S(C); A) ≅ H^{0}([S(C)]; A^{1}) .

The left side is canonically isomorphic to H^{1}(C; A) by Theorem 9.1. The right side is equal to lim_{⟵}_{[S(C)]} (A^{1}), and the functor A^{1} sends the isomorphism class [σ] of a chain

σ ∈ S(C) to H^{1}(Aut_{S(C)}(σ), A) = Hom(Aut_{S(C)}(σ), A), which is what appears on the right side in the statement. □

Remark 12.3. A part of the Remark in 10.6 applies again: the functors A^{q} appearing in the proof of 12.2 depend only on the groups Aut_{S(C)}(σ) rather than the class [σ] itself. Thus, if C is a right ideal in a fusion system F on a finite p-group P then by [16, 4.7, 4.10, 4.11], denoting S_{◁}(C) is the full subcategory of S(C) of all normal chains Q_{0} < Q_{1} < ⋯ < Q_{m} (that is, Q_{i} is normal in Q_{m} for 0 ≤ i ≤ m), then restriction to this subcategory induces an isomorphism on cohomology H^{∗}([S(C)]; A^{q}) ≅ H^{∗}([S_{◁}(C)]; A^{q}).

§13 Proof of Theorem 1.8

Let F be a fusion system on a finite p-group P, where p is a prime, and let C be a right ideal in F. Following the notation in [16] we denote as above by S_{◁}(C) the full subcategory of S(C) consisting of all chains

σ = Q_{0} < Q_{1} < ⋯ < Q_{m}

of subgroups Q_{i} of P belonging to C such that Q_{i} is normal in Q_{m} for 0 ≤ i ≤ m. Chains of this type were introduced in [11]. Given such a chain σ = Q_{0} < Q_{1} < ⋯ < Q_{m} we denote, for any integer i such that 0 ≤ i ≤ m, by σ\i the chain

σ\i = Q_{0} < ⋯ Q_{i−1} < Q_{i+1} < ⋯ < Q_{m}

obtained from removing Q_{i}. Inclusion of chains yields a canonical morphism σ\i ⟶ σ in S_{◁}(C). For any fully F-normalised subgroup R of P in C we denote by S_{R◁}(C) the full subcategory of S_{◁}(C) consisting of all chains σ = Q_{0} < Q_{1} < ⋯ < Q_{m} with Q_{0} ≅ R in F and m ≥ 1. We denote by C_{R} the right ideal in N_{F}(R)/R consisting of all non-trivial subgroups of N_{P} (R)/R. We keep this notation throughout this Section.

Lemma 13.1. Let σ = (Q_{0} < Q_{1} < ⋯ < Q_{m}) be a chain in S_{◁}(F) and let R be a fully F-normalised subgroup of P such that R ≅ Q_{0} in F. Then there is a chain τ = (R_{0} < R_{1} < ⋯ < R_{m}) in S_{◁}(F) such that τ ≅ σ and R_{0} = R.

Proof. By standard properties of fusion systems there is a morphism φ : N_{P} (Q_{0}) → P such that φ(Q_{0}) = R_{0}, because R_{0} is fully F-normalised. Taking τ = φ(σ) = (φ(Q_{0}) < φ(Q_{1}) < ⋯ < φ(Q_{m})) proves the Lemma. □

Lemma 13.2. Let σ = (Q_{0} < Q_{1} < ⋯ < Q_{m}) and τ = (R_{0} < R_{1} < ⋯ < R_{m}) be chains in S_{◁}(F) such that Q = Q_{0} = R_{0} is fully F-normalised. Suppose that m > 0. Then σ ≅ τ in S_{◁}(F) if and only if σ\0 ≅ τ\0 in S_{◁}(N_{F}(Q)).

Proof. Any isomorphism σ ≅ τ induces an automorphism on Q = Q_{0} = R_{0}, and hence an isomorphism σ\0 ≅ τ\0 in S_{◁}(N_{F}(Q)). Conversely, any isomorphism σ\0 ≅ τ\0 in S_{◁}(N_{F}(Q)) induces an automorphism on Q, hence an isomorphism σ ≅ τ. □

Lemma 13.3. Suppose that F = N_{F}(Q) for some normal subgroup Q of P. Let σ = (Q_{0} < Q_{1} < ⋯ < Q_{m}) and τ = (R_{0} < R_{1} < ⋯ < R_{m}) be two chains in S_{◁}(F) such that Q ⊆ Q_{0} and Q ⊆ R_{0}. Then the chains σ̄ = (Q_{0}/Q < Q_{1}/Q < ⋯ < Q_{m}/Q) and τ̄ = (R_{0}/Q < R_{1}/Q < ⋯ < R_{m}/Q) belong to the category S_{◁}(F/Q), and we have σ ≅ τ in S_{◁}(F) if and only if σ̄ ≅ τ̄ in S_{◁}(F/Q).

Proof. Clearly the chains σ̄, τ̄ are in S_{◁}(F/Q), and if σ ≅ τ then σ̄ ≅ τ̄. Suppose conversely that we have an isomorphism μ̄ = (μ̄_{i})_{0≤i≤m} : σ̄ ≅ τ̄ given by a family of isomorphisms μ̄_{i} : Q_{i}/Q ≅ R_{i}/Q in F/Q. Then any representative μ_{m} of μ̄_{m} is an isormophism Q_{m} ≅ R_{m} which sends Q_{i} to R_{i}, for 0 ≤ i ≤ m, and hence setting μ_{i} = μ_{m}|_{Q_{i}} yields an isomorphism σ ≅ τ. □

Proposition 13.4. Let Q be a fully F-normalised subgroup of P. The map sending a chain σ = (Q_{0} < Q_{1} < ⋯ < Q_{m}) of positive length m in S_{R◁}(C) to the chain σ̄\0 = (Q_{1}/R < Q_{2}/R < ⋯ < Q_{m}/R) in S_{◁}(C_{R}) induces an isomorphism of posets

[S_{R◁}(C)] ≅ [S_{◁}(C_{R})] .

Proof. This follows from combining the three previous Lemmas. □

Proof of Theorem 1.8. We use the notation from the statement of Theorem 1.8; in particular, {R_{q} | 0 ≤ q ≤ r} is a system of representatives of the isomorphism classes in C with all R_{q} fully F-normalised and ordered in such a way that |R_{q}| ≥ |R_{q+1}| for 0 ≤ q < r. Note that since C is a right ideal in F this implies in particular that R_{0} = P.

By [16, 3.7] applied to the left ideal [S_{◁}(C)] in [S(C)], the cohomology of A is that of a cochain complex of k-modules C_{◁}(A) which is in degree n ≥ 0 equal to

C_{◁}(A)^{n} = ⊕_{[σ]∈[S_{◁}(C)]}_{|σ|=n} A([σ]) .

The differential of this complex is an alternating sum of maps A([σ] < [τ]) : A([σ]) → A([τ]), where [σ], [τ] are isomorphism classes of chains σ, τ in S_{◁}(C) such that [σ] < [τ] and such that |σ| + 1 = |τ|.

For any integer q such that 0 ≤ q ≤ r we define a full subcategory S_{◁}^{(q)}(C) of S_{◁}(C) as follows: a chain σ = (Q_{0} < Q_{1} < ⋯ < Q_{m}) in S_{◁}(C) belongs to S_{◁}^{(q)}(C) if and only if Q_{0} ≅ R_{j} for some j ≥ q. The subcategory S_{◁}^{(q)}(C) is in fact a right ideal in S_{◁}(C): if there is a morphism from a chain σ = (Q_{0} < Q_{1} < ⋯ < Q_{m}) in S_{◁}^{(q)}(C) to a chain σ′ = (Q′_{0} < Q′_{1} < ⋯ < Q′_{m′}) in S_{◁}(C) then in particular either Q_{0} ≅ Q′_{0} or |Q′_{0}| < |Q_{0}|. Since Q_{0} ≅ R_{j} for some j ≥ q this implies Q′_{0} ≅ R_{j′} for some j′ ≥ j. Thus σ′ belongs to the category S_{◁}^{(q)}(C) as well.

It follows that for 0 ≤ q ≤ r there is a subcomplex C_{◁}^{(q)}(A) of C_{◁}(A) defined by

C_{◁}^{(q)}(A)^{n} = ⊕_{σ∈[S_{◁}^{(q)}(C)]}_{|σ|=n} A([σ])

for n ≥ 0. Since the category S_{◁}(C) is filtered by the subcategories

S_{◁}^{(r)}(C) ⊆ S_{◁}^{(r−1)}(C) ⊆ ⋯ ⊆ S_{◁}^{(0)}(C) = S_{◁}(C)

it follows that the cochain complex C_{◁}(A) has a filtration of the form

{0} ⊆ C_{◁}^{(r)}(A) ⊆ C_{◁}^{(r−1)}(A) ⊆ ⋯ ⊆ C_{◁}^{(0)}(A) = C_{◁}(A) .

For notational convenience we set C_{◁}^{(q)}(A) = 0 for q > r. By [19, 2.6], the spectral sequence associated by this filtration takes the form

E_{1}^{p,q} = H^{p+q}(C_{◁}^{(q)}(A)/C_{◁}^{(q+1)}(A)) ⇒ H^{p+q}(C_{◁}(A)) .

The quotient complexes of this filtration are zero unless 0 ≤ q ≤ r, and they vanish in negative degrees; thus we get E_{1}^{p,q} = {0} for q < 0 or q > r or p + q < 0 as claimed in (ii). The right side in this spectral sequence is H^{p+q}(C_{◁}(A)) = H^{p+q}([S_{◁}(C)]; A) by [16, 3.7] applied to the left ideal [S_{◁}(C)] in [S(C)]. We need to identify E_{1}^{p,q} in the remaining cases. For 1 ≤ q ≤ r we set D_{q} = S_{R_{q}◁}(C), the full subcategory of S_{◁}(C) consisting of all chains σ = (Q_{0} < Q_{1} < ⋯ < Q_{m}) of positive length m such that Q_{0} ≅ R_{q}. By convention, D_{0} is the empty category. That is, we have a disjoint union of full subcategories

S_{◁}^{(q)}(C) = S_{◁}^{(q+1)}(C) ∪ D_{q} ∪ [R_{q}] .

Thus, for 0 ≤ q ≤ r, the quotient complex

C_{◁}^{(q)}(A)/C_{◁}^{(q+1)}(A)

is equal to

(C_{◁}^{(q)}(A)/C_{◁}^{(q−1)}(A))^{n} ≅ ⊕_{[σ]∈[D_{q}]}_{|σ|=n} A([σ])

for n > 0, and

(C_{◁}^{(q)}(A)/C_{◁}^{(q−1)}(A))^{0} ≅ A([R_{q}])

with [R_{q}] viewed as isomorphism class of chains of length zero. Note that by our conventions, for q = 0 this quotient complex is concentrated in degree zero, where it is equal to A([P]). Thus in particular E_{1}^{p,0} = {0} for p ≠ 0 and E_{1}^{0,0} = A([P]) which completes the proof of (ii).

Set C_{q} = C_{R_{q}} for 1 ≤ q ≤ r; that is, C_{q} is the full subcategory of N_{F}(R_{q})/R_{q} consisting of all non trivial subgroups of N_{P} (R_{q})/R_{q}; by convention, C_{0} is empty. By Proposition 13.4 we have an isomorphism of posets [D_{q}] ≅ [S_{◁}(C_{R_{q}})], where 1 ≤ q ≤ r. This isomorphism induces a functor

A^{q} : [S_{◁}(C_{q})] ⟶ Mod(k)

such that

A^{q}([Q_{1}/R_{q} < Q_{2}/R_{q} < ⋯ < Q_{m}/R_{q}]) = A([R_{q} < Q_{1} < Q_{2} < ⋯ < Q_{m}]) ,

where Q_{1} < Q_{2} < ⋯ < Q_{m} is a chain in S_{◁}(N_{F}(R_{q})) with R_{q} < Q_{1} and m ≥ 1. Note that the isomorphism of posets [D_{q}] ≅ [S_{◁}(C_{q})] sends a chain of positive length m to a chain of length m−1; this accounts for the degree shift in the statement of 1.8. Thus we can reformulate our description of the quotient complex C_{◁}^{(q)}(A)/C_{◁}^{(q−1)}(A) as follows: in degree n > 0 we have

(C_{◁}^{(q)}(A)/C_{◁}^{(q+1)}(A))^{n} ≅ ⊕_{[τ]∈[S_{◁}(C_{q})]}_{|τ|=n−1} A^{q}([τ]) ,

(which by our conventions is zero if q = 0 and n > 0) and degree zero term equal to A([R_{q}]), as pointed out before. It follows again from the description of the cohomology of posets associated with subdivisions in [16, §3] that for n > 0 we have

(C_{◁}^{(q)}(A)/C_{◁}^{(q+1)}(A))^{n} ≅ C_{◁}(A^{q})^{n−1} ,

where C_{◁}(A^{q}) is the complex which computes the cohomology of the functor A_{q} as in [16, 3.1], with C_{q} instead of C. In other words, (C_{◁}^{(q)}(A)/C_{◁}^{(q+1)}(A)) is the mapping cone of a chain map of the form A([R_{q}]) ⟶ C_{◁}(A^{q}), with A([R_{q}]) viewed as complex concentrated in degree zero. in particular, we have

H^{n}(C_{◁}^{(q)}(A)/C_{◁}^{(q+1)}(A)) ≅ H^{n−1}([S_{◁}(C_{q})]; A^{q})

for n ≥ 2 and 1 ≤ q ≤ r. This completes the proof of (i).

Furthermore, if all functors A^{q}, for 1 ≤ q ≤ r, have vanishing cohomology in a fixed positive degree m, then the cohomology of C_{◁}(A) vanishes in degree m + 1. Statement (iii) follows.

Finally, if A vanishes on chains of length zero, then

C_{◁}^{(q)}(A)/C_{◁}^{(q+1)}(A) = C_{◁}(A^{q})[−1]

where 0 ≤ q ≤ r, and so in this case we get that

H^{n}(C_{◁}^{(q)}(A)/C_{◁}^{(q+1)}(A)) ≅ H^{n−1}([S_{◁}(C_{q})]; A^{q})

for all integers n. This completes the proof of (iv). □

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Markus Linckelmann Department of Mathematical Sciences Meston Building Aberdeen, AB24 3UE

United Kingdom