Comparison of spectral sequences involving bifunctors

Matthias Künzer

June 8, 2009

Abstract

Suppose given functors A × A′ ^{\overline{}^{F}} B ^{\overline{}^{G}} C between abelian categories, an object X in A and an object X′ in A′ such that F (X, −), F (−, X′) and G are left exact, and such that further conditions hold. We show that, E_{1}-terms exempt, the Grothendieck spectral sequence of the composition of F (X, −) and G evaluated at X′ is isomorphic to the Grothendieck spectral sequence of the composition of F (−, X′) and G evaluated at X. The respective E_{2}-terms are a priori seen to be isomorphic. But instead of trying to compare the differentials and to proceed by induction on the pages, we rather compare the double complexes that give rise to these spectral sequences.

Contents

0 Introduction  2
0.1 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.2 First comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.3 Second comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.4 Results of Beyl and Barnes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1 Double and triple complexes  6
1.1 Double complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.2 Applying H in different directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Concentrated double complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.4 Row- and columnwise notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.5 Horizontally and vertically split acyclic double complexes . . . . . . . . . . . . . . . . . . 7
1.1.6 Total complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.7 The homotopy category of first quadrant double complexes as a quotient . . . . . . . . . . 9
1.2 Triple complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2 Planewise total complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Cartan-Eilenberg resolutions  10
2.1 A remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Exact categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 An exact category structure on C(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 An exact category structure on C^{[0}(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 The Cartan-Eilenberg resolution of a quasiisomorphism . . . . . . . . . . . . . . . . . . . . . . . 15

MSC 2000: 18G40.

3 Formalism of spectral sequences  17
3.1 Pointwise split and pointwise finitely filtered complexes . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Spectral objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Spectral sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 A short exact sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5 Classical indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.6 Comparing proper spectral sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.7 The first spectral sequence of a double complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Grothendieck spectral sequences  23
4.1 Certain quasiisomorphisms are preserved by a left exact functor . . . . . . . . . . . . . . . . . . . 23
4.2 Definition of the Grothendieck spectral sequence functor . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Haas transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3.1 Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3.2 Construction of the first Haas transformation . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3.3 Construction of the second Haas transformation . . . . . . . . . . . . . . . . . . . . . . . 29
5 The first comparison  30
5.1 The first comparison isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.2 Naturality of the first comparison isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6 The second comparison  32
6.1 The second comparison isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.2 Naturality of the second comparison isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
7 Acyclic CE-resolutions  36
7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7.2 A theorem of Beyl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
8 Applications  38
8.1 A Hopf algebra lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
8.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
8.1.2 Some basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
8.1.3 Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
8.1.4 Some remarks and a lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
8.2 Comparing Hochschild-Serre-Hopf with Grothendieck . . . . . . . . . . . . . . . . . . . . . . . . . 46
8.3 Comparing Lyndon-Hochschild-Serre with Grothendieck . . . . . . . . . . . . . . . . . . . . . . . 48
8.4 Comparing Hochschild-Serre with Grothendieck . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
8.5 Comparing two spectral sequences for a change of rings . . . . . . . . . . . . . . . . . . . . . . . 49
8.6 Comparing two spectral sequences for Ext and ⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8.7 Comparing two spectral sequences for Ext of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . 51

0 Introduction

To calculate Ext^{∗}(X, Y), one can either resolve X projectively or Y injectively; the result is, up to isomorphism, the same. To show this, one uses the double complex arising when one resolves both X and Y; cf. [5, Chap. V, Th. 8.1].

Two problems in this spirit occur in the context of Grothendieck spectral sequences; cf. §§ 0.2, 0.3.

0.1 Language

In §3, we give a brief introduction to the Deligne-Verdier spectral sequence language; cf. [17, II.§4], [6, App.]; or, on a more basic level, cf. [11, Kap. 4]. This language amounts to considering a diagram E(X) containing all the images between the homology groups of the subquotients of a given filtered complex X, instead of, as is classical, only selected ones. This helps to gain some elbow room in practice : to govern the objects of the diagram E(X) we can make use of a certain short exact sequence; cf. §3.4.

Dropping the E_{1}-terms and similar ones, we obtain the proper spectral sequence Ė(X) of our filtered complex X. Amongst others, it contains all E_{k}-terms for k ⩾ 2 in the classical language; cf. §§ 3.6, 3.5.

0.2 First comparison

Suppose given abelian categories A, A′ and B with enough injectives and an abelian category C. Suppose given objects X ∈ Ob A and X′ ∈ Ob A′. Let A × A′ ^{\overline{}^{F}} B be a biadditive functor such that F (X, −) and F (−, X′) are left exact. Let B ^{\overline{}^{G}} C be a left exact functor. Suppose further conditions to hold; see §5.1.

We have a Grothendieck spectral sequence for the composition G∘F (X, −) and a Grothendieck spectral sequence for the composition G∘F (−, X′). We evaluate the former at X′ and the latter at X.

In both cases, the E_{2}-terms are (R^{i}G)(R^{j}F)(X, X′). Moreover, they both converge to (R^{i+j}(G ∘ F))(X, X′). So the following assertion is well-motivated.

Theorem 31. The proper Grothendieck spectral sequences just described are isomorphic; i.e.

Ė_{F (X,−),G}^{Gr}(X′) ≃ Ė_{F (−,X′),G}^{Gr}(X) .

So instead of “resolving X′ twice”, we may just as well “resolve X twice”.

In fact, the underlying double complexes are connected by a chain of double homotopisms, i.e. isomorphisms in the homotopy category as defined in [5, IV.§4], and rowwise homotopisms (the proof uses a chain • ^{\overline{}^{double}} • ^{\overline{}^{roww.}} • ^{\overline{}^{roww.}} • ^{\overline{}^{double}} •). These morphisms then induce isomorphisms on the associated proper first spectral sequences.

0.3 Second comparison

Suppose given abelian categories A and B′ with enough injectives and abelian categories B and C. Suppose given objects X ∈ Ob A and Y ∈ Ob B. Let A ^{\overline{}^{F}} B′ be a left exact functor. Let B × B′ ^{\overline{}^{G}} C be a biadditive functor such that G(Y, −) is left exact.

Let B ∈ Ob C^{[0}(B) be a resolution of Y, i.e. a complex B admitting a quasiisomorphism Conc Y \overline{}B. Suppose that G(B^{k}, −) is exact for all k ⩾ 0. Let A ∈ Ob C^{[0}(A) be, say, an injective resolution of X. Suppose further conditions to hold; see §6.1.

We have a Grothendieck spectral sequence for the composition G(Y, −) ∘ F, which we evaluate at X. On the other hand, we can consider the double complex G(B, F A), where the indices of B count rows and the indices of A count columns. To the first filtration of its total complex, we can associate the proper spectral sequence Ė_{I}(G(B, F A)).

If B has enough injectives and B is an injective resolution of Y, then in both cases the E_{2}-terms are a priori seen to be (R^{i}G)(Y, (R^{j}F)(X)). So also the following assertion is well-motivated.

Theorem 34. We have Ė_{F,G(Y,−)}^{Gr}(X) ≃ Ė_{I}(G(B, F A)) .

So instead of “resolving X twice”, we may just as well “resolve X once and Y once”.

The left hand side spectral sequence converges to (R^{i+j}(G(Y, −) ∘ F))(X). By this theorem, so does the right hand side one.

The underlying double complexes are connected by two morphisms of double complexes (in the directions • \overline{}• \overline{}•) that induce isomorphisms on the associated proper spectral sequences.

Of course, Theorems 31 and 34 have dual counterparts.

0.4 Results of Beyl and Barnes

Let R be a commutative ring. Let G be a group. Let N ⊴ G be a normal subgroup. Let M be an RG-module.

Beyl generalises Grothendieck’s setup, allowing for a variant of a Cartan-Eilenberg resolution that consists of acyclic, but no longer necessarily injective objects [4, Th. 3.4]. We have documented Beyl’s Theorem as Theorem 40 in our framework, without claiming originality.

Beyl uses his Theorem to prove that, from the E_{2}-term on, the Grothendieck spectral

sequence for RG -Mod ^{\overline{}^{(−)^{N}}} RN -Mod^{\overline{}^{(−)^{G/N}}} R -Mod at M is isomorphic to the Lyndon-Hochschild-Serre spectral sequence, i.e. the spectral sequence associated to the double complex _{RG}(Bar_{G/N;R} ⊗_{R} Bar_{G;R} , M); cf. [4, Th. 3.5], [3, §3.5]. This is now also a consequence of Theorems 31 and 34, as explained in §§ 8.2, 8.3.

Barnes works in a slightly different setup. He supposes given a commutative ring R, abelian categories A, B and C of R-modules, and left exact functors F : A \overline{}B and G : B \overline{}C, where F is supposed to have an exact left adjoint J : B \overline{}A that satisfies F ∘ J = 1_{B}. Moreover, he assumes A to have ample injectives and C to have enough injectives. In this setup, he obtains a general comparison theorem. See [2, Sec. X.5, Def. X.2.5, Th. X.5.4].

Beyl [4] and Barnes [2] also consider cup products; in this article, we do not.

0.5 Acknowledgements

Results of Beyl and Haas are included for sake of documentation that they work within our framework; cf. Theorem 40 and §4. No originality from my part is claimed.

I thank B. Keller for directing me to [12, XII.§11]. I thank the referee for helping to considerably improve the presentation, and for suggesting Lemma 47 and §8.2. I thank G. Carnovale and G. Hiß for help with Hopf algebras.

Conventions

Throughout these conventions, let C and D be categories, let A be an additive category, let B and B′ be abelian categories, and let E be an exact category in which all idempotents split.

• For a, b ∈ Z, we write [a, b] := {c ∈ Z : a ⩽ c ⩽ b}, [a, b[ := {c ∈ Z : a ⩽ c < b}, etc.

• Given I ⊆ Z and i ∈ Z, we write I_{⩾i} := {j ∈ I : j ⩾ i} and I_{<i} := {j ∈ I : j < i}.

• The disjoint union of sets A and B is denoted by A ⊔ B.

• Composition of morphisms is written on the right, i.e. ^{\overline{}^{a}} ^{\overline{}^{b}} = ^{\overline{}^{ab}} .

• Functors act on the left. Composition of functors is written on the left, i.e. ^{\overline{}^{F}} ^{\overline{}^{G}} = ^{\overline{}^{G∘F}} • Given objects X, Y in C, we denote the set of morphisms from X to Y by _{C}(X, Y).

• The category of functors from C to D and transformations between them is denoted by C, D .

• Denote by C(A) the category of complexes

X = (⋯ ^{\overline{}^{d}} X^{i−1} ^{\overline{}^{d}} X^{i} ^{\overline{}^{d}} X^{i+1} ^{\overline{}^{d}} ⋯)

with values in A. Denote by C^{[0}(A) the full subcategory of C(A) consisting of complexes X with X^{i} = 0

for i < 0. We have a full embedding A ^{\overline{}^{Conc}} C^{[0}(A), where, given X ∈ Ob A, the complex Conc X has entry X at position 0 and zero elsewhere.

• Given a complex X ∈ Ob C(A) and k ∈ Z, we denote by X^{•+k} the complex that has differential

X^{i+k} ^{\overline{}^{(−1)^{k}d}} X^{i+1+k} between positions i and i + 1. We also write X^{•−1} := X^{•+(−1)} etc.

• Suppose given a full additive subcategory M ⊆ A. Then A/M denotes the quotient of A by M, which has the same objects as A, and which has as morphisms residue classes of morphisms of A, where two morphisms are in the same residue class if their difference factors over an object of M.

• A morphism in A is split if it isomorphic, as a diagram on • \overline{}•, to a morphism of the form

X ⊕ Y ^{\overline{}^{( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} )}} X ⊕ Z. A complex X ∈ Ob C(A) is split if all of its differentials are split.

• An elementary split acyclic complex in C(A) is a complex of the form

⋯ \overline{}0 \overline{}T ^{\overline{}^{1}} T \overline{}0 \overline{}⋯,

where the entry T is at positions k and k + 1 for some k ∈ Z. A split acyclic complex is a complex isomorphic to a direct sum of elementary split acyclic complexes, i.e. a complex isomorphic to a complex of the form

⋯ ^{\overline{}^{( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )}} T^{i} ⊕ T^{i+1} ^{\overline{}^{( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )}} T^{i+1} ⊕ T^{i+2} ^{\overline{}^{( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )}} T^{i+2} ⊕ T^{i+3} ^{\overline{}^{( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )}} ⋯

Let C_{sp ac}(A) ⊆ C(A) denote the full additive subcategory of split acyclic complexes. Let K(A) := C(A)/C_{sp ac}(A) denote the homotopy category of complexes with values in A. Let K^{[0}(A) denote the image of C^{[0}(A) in K(A). A morphism in C(A) is a homotopism if its image in K(A) is an isomorphism.

• We denote by Inj B ⊆ B the full subcategory of injective objects.

• Concerning exact categories, introduced by Quillen [14, p. 15], we use the conventions of [10, Sec. A.2]. In particular, a commutative quadrangle in E being a pullback is indicated by

[figure: A; B; C; D ,]

a commutative quadrangle being a pushout by

[figure: A; B; C; D .]

• Given X ∈ Ob C(E) with pure differentials, and given k ∈ Z, we denote by Z^{k}X the kernel of the differential X^{k} \overline{}X^{k+1}, by Z^{′k}X the cokernel of the differential X^{k−1} \overline{}X^{k}, and by B^{k}X the image of the differential X^{k−1} \overline{}X^{k}. Furthermore, we have pure short exact sequences B^{k}X [figure] Z^{k}X \overline{}H^{k}X and H^{k}X [figure] Z^{′k}X \overline{}B^{k+1}X.

• A morphism X \overline{}Y in C(E) between complexes X and Y with pure differentials is a quasiisomorphism if H^{k} applied to it yields an isomorphism for all k ∈ Z. A complex X with pure differentials is acyclic if H^{k}X ≃ 0 for all k ⩾ 0. Such a complex is also called a purely acyclic complex.

• Suppose that B has enough injectives. Given a left exact functor B ^{\overline{}^{F}} B′, an object X ∈ Ob B is F -acyclic if R^{i}F X ≃ 0 for all i ⩾ 1. In other words, X is F -acyclic if for an injective resolution I ∈ C^{[0}(Inj B) of X (and then for all such injective resolutions), we have H^{i}F I ≃ 0 for all i ⩾ 1.

• By a module, we understand a left module, unless stated otherwise. If A is a ring, we abbreviate A(−, =) := _{A -Mod}(−, =) = Hom_{A}(−, =).

1 Double and triple complexes

We fix some notations and sign conventions.

Let A and B be additive categories. Let C(A) ^{\overline{}^{H}} B be an additive functor.

1.1 Double complexes

1.1.1 Definition

A double complex with entries in A is a diagram

\xymatrix{ &  & ⋮ & ⋮ & ⋮ \\  & ⋯ \ar[r]^{d} & X^{i+2,j} \ar[r]^{d} \ar[u]^{∂} & X^{i+2,j+1} \ar[r]^{d} \ar[u]^{∂} & X^{i+2,j+2} \ar[r]^{d} \ar[u]^{∂} & ⋯ \\ X = & ⋯ \ar[r]^{d} & X^{i+1,j} \ar[r]^{d} \ar[u]^{∂} & X^{i+1,j+1} \ar[r]^{d} \ar[u]^{∂} & X^{i+1,j+2} \ar[r]^{d} \ar[u]^{∂} & ⋯ \\  & ⋯ \ar[r]^{d} & X^{i,j} \ar[r]^{d} \ar[u]^{∂} & X^{i,j+1} \ar[r]^{d} \ar[u]^{∂} & X^{i,j+2} \ar[r]^{d} \ar[u]^{∂} & ⋯ \\  &  & ⋮ \ar[u]^{∂} & ⋮ \ar[u]^{∂} & ⋮ \ar[u]^{∂}}

in A such that dd = 0, ∂∂ = 0 and d∂ = ∂d everywhere. As morphisms between double complexes, we take all diagram morphisms. Let CC(A) denote the category of double complexes. We may identify CC(A) = C(C(A)).

The double complexes considered in this §1.1 are stipulated to have entries in A.

Let CC^{⌞}(A) := C^{[0}(C^{[0}(A)) be the category of first quadrant double complexes, consisting of double complexes X such that X^{i,j} = 0 whenever i < 0 or j < 0.

Given a double complex X and i ∈ Z, we let X^{i,∗} ∈ Ob C(A) denote the complex that has entry X^{i,j} at position j ∈ Z, the differentials taken accordingly; X^{i,∗} is called the ith row of X.

Similarly, given j ∈ Z, X^{∗,j} ∈ Ob C(A) denotes the jth column of X.

1.1.2 Applying H in different directions

Given X ∈ Ob CC(A), we let H(X^{∗,−}) ∈ Ob C(A) denote the complex that has H(X^{∗,j}) at position j ∈ Z, and as differential H(X^{∗,j}) \overline{}H(X^{∗,j+1}) the image of the morphism X^{∗,j} \overline{}X^{∗,j+1} of complexes under H. Similarly, H(X^{−,∗}) ∈ Ob C(A) has H(X^{j,∗}) at position j ∈ Z.

In other words, a “∗” denotes the index direction to which H is applied, a “−” denotes the surviving index direction. For short, “∗” before “−”.

1.1.3 Concentrated double complexes

Given a complex U ∈ Ob C^{[0}(A), we denote by Conc_{2} U ∈ Ob CC^{⌞}(A) the double complex whose 0th row is given by U, and whose other rows are zero; i.e. given j ∈ Z, then (Conc_{2} U)^{i,j} equals U^{j} if i = 0, and 0 otherwise, the differentials taken accordingly. Similarly, Conc_{1} U ∈ Ob CC^{⌞}(B) denotes the double complex whose 0th column is given by U, and whose other columns are zero.

1.1.4 Row- and columnwise notions

A morphism X ^{\overline{}^{f}} Y of double complexes is called a rowwise homotopism if X^{i,∗} ^{\overline{}^{f^{i,∗}}} Y^{i,∗} is a homotopism for all i ∈ Z. Provided A is abelian, it is called a rowwise quasiisomorphism if X^{i,∗} ^{\overline{}^{f^{i,∗}}} Y^{i,∗} is a quasiisomorphism for all i ∈ Z.

A morphism X ^{\overline{}^{f}} Y of double complexes is called a columnwise homotopism if X^{∗,j} ^{\overline{}^{f^{∗,j}}} Y^{∗,j} is a homotopism for all j ∈ Z. Provided A is abelian, it is called a columnwise quasiisomorphism if X^{∗,j} ^{\overline{}^{f^{∗,j}}} Y^{∗,j} is a quasiisomorphism for all j ∈ Z.

Provided A is abelian, a double complex X is called rowwise split if X^{i,∗} is split for all i ∈ Z; a short exact sequence X′ \overline{}X \overline{}X′′ of double complexes is called rowwise split short exact if X^{′i,∗} \overline{}X^{i,∗} \overline{}X^{′′i,∗} is split short exact for all i ∈ Z.

A double complex X is called rowwise split acyclic if X^{i,∗} is a split acyclic complex for all i ∈ Z. It is called columnwise split acyclic if X^{∗,j} is a split acyclic complex for all j ∈ Z.

1.1.5 Horizontally and vertically split acyclic double complexes

An elementary horizontally split acyclic double complex is a double complex of the form

[figure: ⋮; ⋮; ⋮; ⋮; ⋯; 0; T^{i+1}; T^{i+1}; 0; ⋯; ⋯; 0; T^{i}; ∂; T^{i}; ∂; 0; ⋯; ⋮; ⋮; ⋮; ⋮; .]

A horizontally split acyclic double complex is a double complex isomorphic to a direct sum of elementary horizontally split acyclic double complexes, i.e. to one of the form

\xymatrix{ & ⋮ & ⋮ \\ ⋯ \ar[r] & T^{i+1,j} ⊕T^{i+1,j+1} \ar[r]^{_{(\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix})}} \ar[u] & T^{i+1,j+1}⊕T^{i+1,j+2} \ar[r] \ar[u] & ⋯ \\ ⋯ \ar[r] & T^{i,j} ⊕T^{i,j+1} \ar[r]^{_{(\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix})}} \ar[u]^{_{(\begin{matrix}∂ & 0 \\ 0 & ∂ \end{matrix} )}} & T^{i,j+1}⊕T^{i,j+2} \ar[r] \ar[u]^{_{(\begin{matrix}∂ & 0 \\ 0 & ∂ \end{matrix} )}} & ⋯ \\  & ⋮ \ar[u] & ⋮ \ar[u] & .}

An elementary vertically split acyclic double complex is a double complex of the form

[figure: ⋮; ⋮; ⋯; 0; 0; ⋯; ⋯; T^{i}; d; T^{i+1}; ⋯; ⋯; T^{i}; d; T^{i+1}; ⋯; ⋯; 0; 0; ⋯; ⋮; ⋮; .]

A vertically split acyclic double complex is a double complex isomorphic to a direct sum of elementary vertically split acyclic double complexes, i.e. to one of the form

\xymatrix{ & ⋮ & ⋮ \\ ⋯ \ar[r] & T^{i+1,j} ⊕T^{i+2,j} \ar[r]^{_{(\begin{matrix}d & 0 \\ 0 & d \end{matrix})}} \ar[u] & T^{i+1,j+1}⊕T^{i+2,j+1} \ar[r] \ar[u] & ⋯ \\ ⋯ \ar[r] & T^{i,j} ⊕T^{i+1,j} \ar[r]^{_{(\begin{matrix}d & 0 \\ 0 & d \end{matrix})}} \ar[u]^{_{(\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix})}} & T^{i,j+1}⊕T^{i+1,j+1} \ar[r] \ar[u]^{_{(\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix})}} & ⋯ \\  & ⋮ \ar[u] & ⋮ \ar[u] & .}

A horizontally split acyclic double complex is in particular rowwise split acyclic. A vertically split acyclic double complex is in particular columnwise split acyclic.

A double complex is called split acyclic if it is isomorphic to the direct sum of a horizontally and a vertically split acyclic double complex. Let CC_{sp ac}(A) denote the full additive subcategory of split acyclic double complexes. Let

KK(A) := CC(A)/CC_{sp ac}(A) ;

cf. [5, IV.§4]. A morphism in CC(A) that is mapped to an isomorphism in KK(A) is called a double homotopism.

A speculative aside. The category K(A) is Heller triangulated; cf. [10, Def. 1.5.(i), Th. 4.6]. Such a Heller triangulation hinges on two induced shift functors, one of them induced by the shift functor on K(A). Now KK(A) carries two shift functors, and so there might be more isomorphisms between induced shift functors one can fix. How can the formal structure of KK(A) be described?

1.1.6 Total complex

Let KK^{⌞}(A) be the full image of CC^{⌞}(A) in KK(A).

The total complex tX of a double complex X ∈ Ob CC^{⌞}(A) is given by the complex

tX = (X^{0,0} ^{\overline{}^{(^{d ∂})}} X^{0,1}⊕X^{1,0} ^{\overline{}^{(\begin{matrix}d & ∂ & 0 \\ 0 & −d & −∂ \end{matrix})}} X^{0,2}⊕X^{1,1}⊕X^{2,0} ^{\overline{}^{(\begin{matrix}d & ∂ & 0 & 0 \\ 0 & −d & −∂ & 0 \\ 0 & 0 & d & ∂ \end{matrix})}} X^{0,3}⊕X^{1,2}⊕X^{2,1}⊕X^{3,0} \overline{}⋯)

in Ob C^{[0}(A). Using the induced morphisms, we obtain a total complex functor CC^{⌞}(A) ^{\overline{}^{t}} C^{[0}(A). Since t maps elementary horizontally or vertically split acyclic double complexes to split acyclic complexes, it induces a functor KK^{⌞}(A) ^{\overline{}^{t}} K^{[0}(A). If, in addition, A is abelian, the total complex functor maps rowwise quasiisomorphisms and columnwise quasiisomorphisms to quasiisomorphisms, as one sees using the long exact homology sequence and induction on a suitable filtration.

Note that we have an isomorphism U ^{\overline{}^{∼}} t Conc_{1} U, natural in U ∈ Ob C^{[0}(A), having entries 1_{U_{0}}, 1_{U_{1}}, −1_{U_{2}}, −1_{U_{3}}, 1_{U_{4}}, etc. Moreover, U = t Conc_{2} U, natural in U ∈ Ob C^{[0}(A).

1.1.7 The homotopy category of first quadrant double complexes as a quotient

Lemma 1 The residue class functor CC(A) \overline{}KK(A), restricted to CC^{⌞}(A) \overline{}KK^{⌞}(A), induces an equivalence

CC^{⌞}(A)/(CC_{sp ac}(A) ∩ CC^{⌞}(A)) ^{\overline{}^{∼}} KK^{⌞}(A) .

Proof. We have to show faithfulness; i.e. that if a morphism X \overline{}Y in CC^{⌞}(A) factors over a split acyclic double complex, then it factors over a split acyclic double complex that lies in Ob CC^{⌞}(A). By symmetry and additivity, it suffices to show that if a morphism X \overline{}Y in CC^{⌞}(A) factors over a horizontally split acyclic double complex, then it factors over a horizontally split acyclic double complex that lies in Ob CC^{⌞}(A). Furthermore, we may assume X \overline{}Y to factor over an elementary horizontally split acyclic double complex S concentrated in the columns k and k + 1 for some k ∈ Z. We may assume that S^{i,j} = 0 for i < 0 and j ∈ Z. If k < 0, and in particular, if k = −1, then X \overline{}Y is zero because S \overline{}Y is zero, so that in this case we may assume S = 0. On the other hand, if k ⩾ 0, then S ∈ Ob CC^{⌞}(A).

Cf. also the similar Remark 2.

1.2 Triple complexes

1.2.1 Definition

Let CCC(A) := C(C(C(A))) be the category of triple complexes. A triple complex Y has entries Y^{k,ℓ,m} for k, ℓ, m ∈ Z.

We denote the differentials in the three directions by Y^{k,ℓ,m} ^{\overline{}^{d_{1}}} Y^{k+1,ℓ,m}, Y^{k,ℓ,m} ^{\overline{}^{d_{2}}} Y^{k,ℓ+1,m} and Y^{k,ℓ,m} ^{\overline{}^{d_{3}}} Y^{k,ℓ,m+1}, respectively.

Let k, ℓ, m ∈ Z. We shall use the notation Y^{−,ℓ,=} for the double complex having at position (k, m) the entry Y^{k,ℓ,m}, differentials taken accordingly. Similarly the complex Y^{k,ℓ,∗} etc.

Given a triple complex Y ∈ Ob CCC(A), we write HY^{−,=,∗} ∈ Ob CC(A) for the double complex having at position (k, ℓ) the entry H(Y^{k,ℓ,∗}), differentials taken accordingly.

Denote by CCC (A) ⊆ CCC(A) the full subcategory of first octant triple complexes; i.e. triple complexes Y having Y^{k,ℓ,m} = 0 whenever k < 0 or ℓ < 0 or m < 0.

1.2.2 Planewise total complex

For Y ∈ Ob CCC (A) we denote by t_{1,2}Y ∈ Ob CC^{⌞}(A) the planewise total complex of Y, defined for m ∈ Z as

(t_{1,2}Y)^{∗,m} := t(Y^{−,=,m}) ,

with the differentials of t_{1,2}Y in the horizontal direction being induced by the differentials in the third index direction of Y, and with the differentials of t_{1,2}Y in the vertical direction being given by the total complex differentials. Explicitly, given k, ℓ ⩾ 0, we have

(t_{1,2}Y)^{k,ℓ} = ⨁_{i, j ⩾ 0, i+j = k} Y^{i,j,ℓ} .

By means of induced morphisms, this furnishes a functor

[figure: CCC (A); t_{1,2}; CC^{⌞}(A); Y; t_{1,2}Y .]

2 Cartan-Eilenberg resolutions

We shall use Quillen’s language of exact categories [14, p. 15] to deal with Cartan-Eilenberg resolutions [5, XVII.§1], as it has been done by Mac Lane already before this language was available; cf. [12, XII.§11]. The assertions in this section are for the most part wellknown.

2.1 A remark

Remark 2 Let A be an additive category. Then C^{[0}(A)/(C^{[0}(A) ∩ C_{sp ac}(A)) \overline{}K^{[0}(A) is an equivalence.

Proof. Faithfulness is to be shown. A morphism X \overline{}Y in C^{[0}(A) that factors over an elementary split acyclic complex of the form (⋯ \overline{}0 \overline{}T T \overline{}0 \overline{}⋯) with T in positions k and k + 1 is zero, provided k < 0.

2.2 Exact categories

Concerning the terminology of exact categories, introduced by Quillen [14, p. 15], we refer to [10, Sec. A.2].

Let E be an exact category in which all idempotents split. An object I ∈ Ob E is called relatively injective, or a relative injective (relative to the set of pure short exact sequences, that is), if _{E}(−, I) maps pure short exact sequences of E to short exact sequences. We say that E has enough relative injectives, if for all X ∈ Ob E, there exists a relative injective I and a pure monomorphism X [figure] I.

In case E is an abelian category, with all short exact sequences stipulated to be pure, then we omit “relative” and speak of “injectives” etc.

Definition 3 Suppose given a complex X ∈ Ob C^{[0}(E) with pure differentials. A relatively injective complex resolution of X is a complex I ∈ Ob C^{[0}(E), together with a quasiisomorphism X \overline{}I, such that the following properties are satisfied.

(1) The object entries of I are relatively injective.

(2) The differentials of I are pure.

(3) The quasiisomorphism X \overline{}I consists of pure monomorphisms.

We often refer to such a relatively injective complex resolution just by I.

A relatively injective object resolution, or just a relatively injective resolution, of an object Y ∈ Ob E is a relatively injective complex resolution of Conc Y.

A relatively injective resolution is the complex of a relatively injective object resolution of some object in E.

Remark 4 Suppose that E has enough relative injectives. Every complex X ∈ Ob C^{[0}(E) with pure differentials has a relatively injective complex resolution I ∈ Ob C^{[0}(E).

In particular, every object Y ∈ Ob E has a relatively injective resolution J ∈ Ob C^{[0}(E).

Proof. Let X^{0} [figure] I^{0} be a pure monomorphism into a relatively injective object I^{0}. Forming a pushout along X^{0} [figure] I^{0}, we obtain a pointwise purely monomorphic morphism of complexes X \overline{}X′ with X^{′0} = I^{0} and X^{′k} = X^{k} for k ⩾ 2. By considering its cokernel, we see that it is a quasiisomorphism. So we may assume X^{0} to be relatively injective.

Let X^{1} [figure] I^{1} be a pure monomorphism into a relatively injective object I^{1}. Form a pushout along X^{1} [figure] I^{1} etc.

Remark 5 Suppose given X ∈ Ob C^{[0}(E) with pure differentials such that H^{k}X ≃ 0 for k ⩾ 1. Suppose given I ∈ Ob C^{[0}(E) such that I^{k} is purely injective for k ⩾ 0, and such that the differential I^{0} ^{\overline{}^{d}} I^{1} has a kernel in E. Then the map

K^{[0}(E)(X, I) \overline{}_{E}( Kern(X^{0} ^{\overline{}^{d}} X^{1}), Kern(I^{0} ^{\overline{}^{d}} I^{1}))

that sends a representing morphism of complexes to the morphism induced on the mentioned kernels, is bijective.

Suppose E to have enough relative injectives. Let I ⊆ E denote the full subcategory of relative injectives. Let C^{[0, res}(I) denote the full subcategory of C^{[0}(I) consisting of complexes X with pure differentials such that H^{k}X ≃ 0 for k ⩾ 1. Let K^{[0, res}(I) denote the image of C^{[0, res}(I) in K(E).

Remark 6 The functor C^{[0, res}(I) \overline{}E, X \overline{}H^{0}(X), induces an equivalence

K^{[0, res}(I) ^{\overline{}^{∼}} E .

Proof. This functor is dense by Remark 4, and full and faithful by Remark 5.

Remark 7 (exact Horseshoe Lemma) Given a pure short exact sequence X′ \overline{}X \overline{}X′′ and relatively injective resolutions I′ of X′ and I′′ of X′′, there exists a relatively injective resolution I of X and a pointwise split short exact sequence I′ \overline{}I \overline{}I′′ that maps under H^{0} to X′ \overline{}X \overline{}X′′.

Proof. Choose pure monomorphisms X′ [figure] I^{′0} and X′′ [figure] I^{′′0} into relative injectives I^{′0} and I^{′′0}. Embed them into a morphism from the pure short exact sequence X′ [figure] X \overline{}X′′ to

the split short exact sequence I′ ^{\overline{}^{(^{1 0})}} I′ ⊕ I′′ ^{\overline{}^{(\begin{matrix}0 \\ 1 \end{matrix})}} I′′. Insert the pushout T of X′ [figure] X along X′ [figure] I^{′0} and the pullback of I^{′0} ⊕ I^{′′0} \overline{}I^{′′0} along X′′ [figure] I^{′′0} to see that X \overline{}I^{′0} ⊕ I^{′′0} is purely monomorphic. So we can take the cokernel B^{1}I′ \overline{}B^{1}I \overline{}B^{1}I′′ of this morphism of pure short exact sequences. Considering the cokernels on the commutative triangle (X, T, I^{′0} ⊕ I^{′′0}) of pure monomorphisms, we obtain a bicartesian square (T, I^{′0}⊕I^{′′0}, B^{1}I′, B^{1}I) and conclude that the sequence of cokernels is itself purely short exact. So we can iterate.

2.3 An exact category structure on C(A)

Let A be an abelian category with enough injectives.

Remark 8 The following conditions on a short exact sequence X′ \overline{}X \overline{}X′′ in C(A) are equivalent.

(1) All connectors in its long exact homology sequence are equal to zero.

(2) The sequence B^{k}X′ \overline{}B^{k}X \overline{}B^{k}X′′ is short exact for all k ∈ Z.

(3) The morphism Z^{k}X \overline{}Z^{k}X′′ is epimorphic for all k ∈ Z.

(3′) The morphism Z^{′k}X′ \overline{}Z^{′k}X is monomorphic for all k ∈ Z.

(4) The diagram

\xymatrix{B^{k}X′ \ar[r] \ar[d] & Z^{k}X′ \ar[r] \ar[d] & H^{k}X′ \ar[d] \\ B^{k}X \ar[r] \ar[d] & Z^{k}X \ar[r] \ar[d] & H^{k}X \ar[d] \\ B^{k}X′′ \ar[r] & Z^{k}X′′ \ar[r] & H^{k}X′′}

has short exact rows and short exact columns for all k ∈ Z.

Proof. We consider the diagram in (4) as a (horizontal) short exact sequence of (vertical) complexes and regard its long exact homology sequence. Taking into account that all assertions are supposed to hold for all k ∈ Z, we can employ the long exact homology sequence on X′ \overline{}X \overline{}X′′ to prove the equivalence of (1), (2), (3) and (4).

Now the assertion (1) ⟺ (3) is dual to the assertion (1) ⟺ (3′).

Remark 9 The category C(A), equipped with the set of short exact sequences that have zero connectors on homology as pure short exact sequences, is an exact category with enough relatively injective objects in which all idempotents split. With respect to this exact category structure on C(A), a complex is relatively injective if and only if it is split and has injective object entries.

Cf. [12, XII.§11], where pure short exact sequences are called proper. A relatively injective object in C(A) is also referred to as an injectively split complex. To a relatively injective resolution of a complex X ∈ Ob C(A), we also refer as a Cartan-Eilenberg-resolution, or, for short, as a CE-resolution of X; cf. [5, XVII.§1]. A CE-resolution is a CE-resolution of some complex. Considered as a double complex, it is in particular rowwise split and has injective object entries.

Given a morphism X ^{\overline{}^{f}} X′ in C(A), CE-resolutions J of X and J′ of X′, a morphism J ^{\overline{}^{f̂}} J′ in CC(A) such that (J^{i,j} ^{\overline{}^{f̂^{i,j}}} J^{′i,j}) = (0 \overline{}0) for i < 0 and such that

H^{0}(J^{∗,−} ^{\overline{}^{f̂^{∗,−}}} J^{′∗,−}) = (X ^{\overline{}^{f}} X′)

is called a CE-resolution of X ^{\overline{}^{f}} X′. By Remarks 9 and 6, each morphism in C(A) has a CE-resolution.

Proof of Remark 9. We claim that C(A), equipped with the said set of short exact sequences, is an exact category. We verify the conditions (Ex 1, 2, 3) listed in [10, Sec. A.2]. The conditions (Ex 1^{∘}, 2^{∘}, 3^{∘}) then follow by duality.

Note that by Remark 8.(3′), a monomorphism X \overline{}Y in C(A) is pure if and only if Z^{′k}(X \overline{}Y) is monomorphic in A for all k ∈ Z.

Ad (Ex 1). To see that a split monomorphism is pure, we may use additivity of the functor Z^{′k} for k ∈ Z.

Ad (Ex 2). To see that the composition of two pure monomorphisms is pure, we may use Z^{′k} being a functor for k ∈ Z.

Ad (Ex 3). Suppose given a commutative triangle

[figure: Y; X; •; Z ,]

in C(A). Applying the functor Z^{′k} to it, for k ∈ Z, we conclude that Z^{′k}(X \overline{}Y) is monomorphic, whence X \overline{}Y is purely monomorphic. So we may complete to in C(A) with (X, Y, B) and (A, Y, Z) pure short exact sequences. Applying Z^{′k} to this diagram, we conclude that Z^{′k}(A \overline{}B) is a monomorphism for k ∈ Z, whence A \overline{}B is a pure monomorphism.

[figure: A; •; B; Y; X; •; •; Z]

This proves the claim.

Note that idempotents in C(A) are split since C(A) is also an abelian category.

We claim relative injectivity of complexes with split differentials and injective object entries. By a direct sum decomposition, and using the fact that any monomorphism from an elementary split acyclic complex with injective entries to an arbitrary complex is split, we are reduced to showing that a pure monomorphism from a complex with a single nonzero injective entry, at position 0, say, to an arbitrary complex is split. So suppose given I ∈ Ob Inj A, X ∈ Ob C(A) and a pure monomorphism Conc I [figure] X. Using Remark 8.(3′), we may choose a retraction to the composite (I \overline{}X^{0} \overline{}Z^{′0}X). This yields a retraction to I \overline{}X^{0} that composes to 0 with X^{−1} \overline{}X^{0}, which can be employed for the sought retraction X \overline{}Conc I. This proves the claim.

Let X ∈ Ob C(A). We claim that there exists a pure monomorphism from X to a relatively injective complex. Since A has enough injectives, by a direct sum decomposition we are reduced to finding a pure monomorphism from X to a split complex. Consider the following morphism φ_{k} of complexes for k ∈ Z,

\xymatrix{⋯ \ar[r] & 0 \ar[r] & X^{k} \ar[r]^{^{(^{1 0})}} & X^{k} ⊕ Z^{′k}X \ar[r] & 0 \ar[r] & ⋯ \\ ⋯ \ar[r] & X^{k−2} \ar[r]^{d} \ar[u] & X^{k−1} \ar[r]^{d} \ar[u]^{d} & X^{k} \ar[r]^{d} \ar[u]_{_{(_{1 p})}} & X^{k+1} \ar[r] \ar[u] & ⋯,}

where X^{k} ^{\overline{}^{p}} Z^{′k}X is taken from X. The functor Z^{′k} maps it to the identity. We take the direct sum of the upper complexes over k ∈ Z and let the morphisms φ_{k} be the components of a morphism φ from X to this direct sum. At position k, this morphism φ is monomorphic because φ_{k} is. Moreover, Z^{′k}(φ) is a monomorphism because Z^{′k}(φ_{k}) is. Hence φ is purely monomorphic by condition (3′) of Remark 8. This proves the claim.

Remark 10 Write E := C(A). Given ℓ ⩾ 0, we have a homology functor E ^{\overline{}^{H^{ℓ}}} A, which induces a functor C(E)^{\overline{}^{C(H^{ℓ})}} C(A). Suppose given a purely acyclic complex X ∈ Ob C(E). Then C(H^{ℓ})X ∈ Ob C(A) is acyclic.

Proof. This follows using the definition of pure short exact sequences, i.e. Remark 8.(1).

2.4 An exact category structure on C^{[0}(A)

Write CC^{⌞, CE}(Inj A) for the full subcategory of CC^{⌞}(A) whose objects are CE-resolutions. Write KK^{⌞, CE}(Inj A) for the full subcategory of KK^{⌞}(A) whose objects are CE-resolutions.

Remark 11 The category C^{[0}(A), equipped with the short exact sequences that lie in C^{[0}(A) and that are pure in C(A) in the sense of Remark 9 as pure short exact sequences, is an exact category wherein idempotents are split. It has enough relative injectives, viz. injectively split complexes that lie in C^{[0}(A).

Proof. To show that it has enough relative injectives, we replace φ_{0} in the proof of Remark 9 by X ^{\overline{}^{φ′_{0}}} Conc X^{0}, defined by X_{0} ^{\overline{}^{1_{X_{0}}}} X_{0} at position 0.

2.5 The Cartan-Eilenberg resolution of a quasiisomorphism

Abbreviate E := C(A), which is an exact category as in Remark 9. Consider CC^{⌞}(A) ⊆ C^{[0}(E), where the second index of X ∈ Ob CC^{⌞}(A) counts the positions in E = C(A); i.e. when X is viewed as a complex with values in E, its entry at position k is given by X^{k,∗} ∈ E = C(A).

Remark 12 Suppose given a split acyclic complex X ∈ Ob C^{[0}(A). There exists a horizontally split acyclic CE-resolution J ∈ Ob CC^{⌞, CE}(Inj A) of X.

Proof. This holds for an elementary split acyclic complex, and thus also in the general case by taking a direct sum.

Lemma 13 Suppose given X ∈ Ob CC^{⌞}(A) with pure differentials when considered as an object of C^{[0}(E), and with H^{k}(X^{∗,−}) ≃ 0 in C^{[0}(A) for k ⩾ 1.

Suppose given J ∈ Ob CC^{⌞}(Inj A) with split rows J^{k,∗} for k ⩾ 1. In other words, J is supposed to consist of relative injective object entries when considered as an object of C^{[0}(E).

Then the map

(∗)  _{KK^{⌞}(A)}(X, J) ^{\overline{}^{H^{0}((−)^{∗,−})}} _{K^{[0}(A)}(H^{0}(X^{∗,−}), H^{0}(J^{∗,−}))

is bijective.

Proof. First, we observe that by Remark 5, we have

(∗∗)  _{K^{[0}(E)}(X, J)  \overline{}^{H^{0}((−)^{∗,−})}_{∼}  _{E}(H^{0}(X^{∗,−}), H^{0}(J^{∗,−})) .

So it remains to show that (∗) is injective. Let X ^{\overline{}^{f}} J be a morphism that vanishes under (∗). Then H^{0}(X^{∗,−}) \overline{}H^{0}(J^{∗,−}) factors over a split acyclic complex S ∈ Ob C^{[0}(A); cf. Remark 2. Let K be a horizontally split acyclic CE-resolution of S; cf. Remark 12. By Remark 5, we obtain a morphism X \overline{}K that lifts H^{0}(X^{∗,−}) \overline{}S and a morphism K \overline{}J that lifts S \overline{}H^{0}(J^{∗,−}). The composite X \overline{}K \overline{}J vanishes in KK^{⌞}(A). The difference

(X ^{\overline{}^{f}} J) − (X \overline{}K \overline{}J)

lifts H^{0}(X^{∗,−}) ^{\overline{}^{0}} H^{0}(J^{∗,−}). Hence by (∗∗), it vanishes in K^{[0}(E) and so a fortiori in KK^{⌞}(A). Altogether, X ^{\overline{}^{f}} J vanishes in KK^{⌞}(A).

Proposition 14 The functor CC^{⌞, CE}(Inj A) ^{\overline{}^{H^{0}((−)^{∗,−})}} C^{[0}(A) induces an equivalence

KK^{⌞, CE}(Inj A)  \overline{}^{H^{0}((−)^{∗,−})}_{∼}  K^{[0}(A) .

Proof. By Lemma 13, this functor is full and faithful. By Remark 4, it is dense.

Corollary 15 Suppose given X, X′ ∈ Ob C^{[0}(A). Let J be a CE-resolution of X. Let J′ be a CE-resolution of X′. If X and X′ are isomorphic in K^{[0}(A), then J and J′ are isomorphic in KK^{⌞}(A).

The following lemma is to be compared to Remark 12.

Lemma 16 Suppose given an acyclic complex X ∈ Ob C^{[0}(A). There exists a rowwise split acyclic CE-resolution J of X. Each CE-resolution of X is isomorphic to J in KK^{⌞}(A).

Proof. By Corollary 15, it suffices to show that there exists a rowwise split acyclic CE-resolution of X. Recall that a CE-resolution of an arbitrary complex Y ∈ Ob C^{[0}(A) can be constructed by a choice of injective resolutions of H^{k}Y and B^{k}Y for k ∈ Z, followed by an application of the abelian Horseshoe Lemma to the short exact sequences B^{k}Y \overline{}Z^{k}Y \overline{}H^{k}Y for k ∈ Z and then to Z^{k}Y \overline{}Y^{k} \overline{}B^{k+1}Y for k ∈ Z; cf. [5, Chap. XVII, Prop. 1.2]. Since H^{k}X = 0 for k ∈ Z, we may choose the zero resolution for it. Applying this construction, we obtain a rowwise split acyclic CE-resolution.

Given X ^{\overline{}^{f}} X′ in C^{[0}(A), a morphism J ^{\overline{}^{f̂}} J′ in CC^{⌞}(A) is called a CE-resolution of X ^{\overline{}^{f}} X′ if H^{0}(f̂^{∗,−}) ≃ f, as diagrams of the form • \overline{}•. By Remark 5, given CE-resolutions J of X and J′ of X′, there exists a CE-resolution J ^{\overline{}^{f̂}} J′ of X ^{\overline{}^{f}} X′.

Proposition 17 Let X ^{\overline{}^{f}} X′ be a quasiisomorphism in C^{[0}(A). Let J ^{\overline{}^{f̂}} J′ be a CE-resolution of X ^{\overline{}^{f}} X′. Then f̂ can be written as a composite in CC^{⌞, CE}(Inj A) of a rowwise homotopism, followed by a double homotopism.

Proof. Choose a pointwise split monomorphism X ^{\overline{}^{a}} A into a split acyclic complex X. We can factor

(X ^{\overline{}^{f}} X′) = (X ^{\overline{}^{(^{f a})}} X′ ⊕ A ^{\overline{}^{(\begin{matrix}1 \\ 0 \end{matrix})}} X′) ,

so that (^{f a}) is a pointwise split monomorphism. Let B be a CE-resolution of A. Choosing a CE-resolution b of a, we obtain the factorisation

(J ^{\overline{}^{f̂}} J′) = (J \overline{}^{(^{f̂b})} J′ ⊕ B ^{\overline{}^{(\begin{matrix}1 \\ 0 \end{matrix})}} J′) .

Since X′ ⊕A ^{\overline{}^{(\begin{matrix}1 \\ 0 \end{matrix})}} X′ is a homotopism, J′ ⊕B ^{\overline{}^{(\begin{matrix}1 \\ 0 \end{matrix})}} J is a double homotopism; cf. Corollary 15. Hence f̂ is a composite of a rowwise homotopism and a double homotopism if and only if this holds for (f̂^{b}). So we may assume that f is pointwise split monomorphic, so in particular, monomorphic.

By Proposition 14, we may replace the given CE-resolution f̂ by an arbitrary CE-resolution of f between J and an arbitrarily chosen CE-resolution of X′ without changing the property of being a composite of a rowwise homotopism and a double homotopism for this newly chosen CE-resolution of f.

Let X ^{\overline{}^{f}} X′ \overline{}X̄ be a short exact sequence in C^{[0}(A). Since f is a quasiisomorphism, X̄ ∈ Ob C^{[0}(A) is acyclic. Let J̄ be a rowwise split acyclic CE-resolution of X̄; cf. Lemma 16. The short exact sequence X ^{\overline{}^{f}} X′ \overline{}X̄ is pure by acyclicity of X̄; cf. Remark 8.(1). Hence by the exact Horseshoe Lemma, there exists a rowwise split short exact sequence J \overline{}J̃′ \overline{}J̄ of CE-resolutions that maps to X ^{\overline{}^{f}} X′ \overline{}X̄ under H^{0}((−)^{∗,−}); cf. Remark 7. Since J̄ is rowwise split acyclic and since the sequence J \overline{}J̃′ \overline{}J̄ is rowwise split short exact, J \overline{}J̃′ is a rowwise homotopism. Since J \overline{}J̃′ is a CE-resolution of X ^{\overline{}^{f}} X′, this proves the proposition.

3 Formalism of spectral sequences

We follow essentially Verdier [17, II.4]; cf. [6, App.]; on a more basic level, cf. [11, Kap. 4].

Let A be an abelian category.

3.1 Pointwise split and pointwise finitely filtered complexes

Let Z_{∞} := {−∞}⊔Z⊔{∞}, considered as a linearly ordered set, and thus as a category. Write ]α, β] := {σ ∈ Z_{∞} : α < σ ⩽ β} for α, β ∈ Z_{∞} such that α ⩽ β; etc.

Given X ∈ Ob Z_{∞}, C(A) , the morphism of X on α ⩽ β in Z_{∞} shall be denoted by X(α) ^{\overline{}^{x}} X(β).

An object X ∈ Ob Z_{∞}, C(A) is called a pointwise split and pointwise finitely filtered complex (with values in A), provided (SFF 1, 2, 3) hold.

(SFF 1) We have X(−∞) = 0.

(SFF 2) The morphism X(α)^{i} ^{\overline{}^{x^{i}}} X(β)^{i} is split monomorphic for all i ∈ Z and all α ⩽ β in Z_{∞}.

(SFF 3) For all i ∈ Z, there exist β_{0}, α_{0} ∈ Z such that X(α)^{i} ^{\overline{}^{x^{i}}} X(β)^{i} is an identity whenever α ⩽ β ⩽ β_{0} or α_{0} ⩽ α ⩽ β in Z_{∞}.

The pointwise split and pointwise finitely filtered complexes with values in A form a full subcategory SFFC(A) ⊆ Z_{∞}, C(A) .

Suppose given a pointwise split and pointwise finitely filtered complex X with values in A for the rest of the present §3.

Let α ∈ Z_{∞}. Write X̄(α) := Cokern (X(α − 1) \overline{}X(α)) for α ∈ Z. Given i ∈ Z, we obtain X(α)^{i} ≃ ⨁_{σ∈]−∞,α]} X̄(σ)^{i}, which is a finite direct sum. We identify along this isomorphism. In particular, we get as a matrix representation for the differential

(X(α)^{i} ^{\overline{}^{d}} X(α)^{i+1}) = ( ⨁_{σ∈]−∞,α]} X̄(σ)^{i}  ^{\overline{}^{(d_{σ,τ}^{i})_{σ,τ}}}  ⨁_{τ∈]−∞,α]} X̄(τ)^{i+1}) ,

where d_{σ,τ}^{i} = 0 whenever σ < τ; a kind of lower triangular matrix.

3.2 Spectral objects

Let Z̄_{∞} := Z_{∞} × Z. Write α^{+k} := (α, k), where α ∈ Z_{∞} and k ∈ Z. Let α^{+k} ⩽ β^{+ℓ} in Z̄_{∞} if k < ℓ or (k = ℓ and α ⩽ β), i.e. let Z̄_{∞} be linearly ordered via a lexicographical ordering. We have an automorphism α^{+k} \overline{}α^{+k+1} of the poset Z̄_{∞}, to which we refer as shift. Note that −∞^{+k} = (−∞)^{+k}.

We have an order preserving injection Z_{∞} \overline{}Z̄_{∞}, α \overline{}α^{+0}. We use this injection as an identification of Z_{∞} with its image in Z̄_{∞}, i.e. we sometimes write α := α^{+0} by abuse of notation.

Let Z̄_{∞}^{#} := {(α, β) ∈ Z̄_{∞} × Z̄_{∞} : β^{−1} ⩽ α ⩽ β ⩽ α^{+1}}. We usually write β/α := (α, β) ∈ Z̄_{∞}^{#}; reminiscent of a quotient. The set Z̄_{∞}^{#} is partially ordered by β/α ⩽ β′/α′ :⟺ (β ⩽ β′ and α ⩽ α′). We have an automorphism β/α \overline{}(β/α)^{+1} := α^{+1}/β of the poset Z̄_{∞}^{#}, to which, again, we refer as shift.

We write Z_{∞}^{#} := {β/α ∈ Z̄_{∞}^{#} : −∞ ⩽ α ⩽ β ⩽ ∞}. Note that any element of Z̄_{∞}^{#} can uniquely be written as (β/α)^{+k} for some β/α ∈ Z_{∞}^{#} and some k ∈ Z.

We shall construct the spectral object Sp(X) ∈ Ob Z̄_{∞}^{#}, K(A) . The morphism of Sp(X) on β/α ⩽ β′/α′ in Z̄_{∞}^{#} shall be denoted by X(β/α) ^{\overline{}^{x}} X(β′/α′).

We require that

(X((β/α)^{+k}) ^{\overline{}^{x}} X((β′/α′)^{+k})) = (X(β/α) ^{\overline{}^{x}} X(β′/α′))^{•+k}

for β/α ⩽ β′/α′ in Z̄_{∞}^{#}; i.e., roughly put, that Sp(X) be periodic up to shift of complexes.

Define

X(β/α) := Cokern (X(α) ^{\overline{}^{x}} X(β))

for β/α ∈ Z_{∞}^{#}. By periodicity, we conclude that X(α/α) = 0 and X(α^{+1}/α) = 0 for all α ∈ Z̄_{∞}.

Write

D_{β/α, β′/α′}^{i} := (d_{σ,τ}^{i})_{σ∈]α,β], τ∈]α′,β′]} : X(β/α)^{i} \overline{}X(β′/α′)^{i+1}

for i ∈ Z and β/α, β′/α′ ∈ Z_{∞}^{#}.

Given −∞ ⩽ α ⩽ β ⩽ γ ⩽ ∞ and i ∈ Z, we let

\begin{aligned}(X(β/α)^{i} ^{\overline{}^{x^{i}}} X(γ/α)^{i}) & := & (X(β/α)^{i} ^{\overline{}^{(^{1 0})}} X(β/α)^{i} ⊕ X(γ/β)^{i}) \\ (X(γ/α)^{i} ^{\overline{}^{x^{i}}} X(γ/β)^{i}) & := & (X(β/α)^{i} ⊕ X(γ/β)^{i} ^{\overline{}^{(\begin{matrix}0 \\ 1 \end{matrix})}} X(γ/β)^{i}) \\ (X(γ/β)^{i} ^{\overline{}^{x^{i}}} X(α^{+1}/β)^{i}) & := & (X(γ/β)^{i} ^{\overline{}^{D_{γ/β, β/α}^{i}}} X(β/α)^{i+1}) . \end{aligned}

By periodicity up to shift of complexes, this defines Sp(X). The construction is functorial in X ∈ Ob SFFC(A).

3.3 Spectral sequences

Let Z̄_{∞}^{##} := {(γ/α, δ/β) ∈ Z̄_{∞}^{#}×Z̄_{∞}^{#} : δ^{−1} ⩽ α ⩽ β ⩽ γ ⩽ δ ⩽ α^{+1}}. Given (γ/α, δ/β) ∈ Z̄_{∞}^{##}, we usually write δ/β//γ/α := (γ/α, δ/β). The set Z̄_{∞}^{##} is partially ordered by

δ/β//γ/α ⩽ δ′/β′//γ′/α′ :⟺ (γ/α ⩽ γ′/α′ and δ/β ⩽ δ′/β′) .

Define the spectral sequence E(X) ∈ Ob Z̄_{∞}^{##}, A of X by letting its value on

δ/β//γ/α ⩽ δ′/β′//γ′/α′

in Z̄_{∞}^{##} be the morphism that appears in the middle column of the diagram

[figure: H^{0}(X(γ/α)); H^{0}(x); E(δ/β//γ/α)(X); •; e; H^{0}(X(δ/β)); H^{0}(x); H^{0}(X(γ′/α′)); E(δ′/β′//γ′/α′)(X); •; H^{0}(X(δ′/β′)) .]

Given δ/β//γ/α ∈ Z̄_{∞}^{##} and k ∈ Z, we also write

E(δ/β//γ/α)^{+k}(X) := E((δ/β)^{+k}//(γ/α)^{+k})(X) .

Altogether,

[figure: Z_{∞}, C(A); ⊇; SFFC(A); Z̄_{∞}^{#}, K(A); Z̄_{∞}^{##}, A]
[figure: X; Sp(X); E(X) .]

3.4 A short exact sequence

Lemma 18 Given ε^{−1} ⩽ α ⩽ β ⩽ γ ⩽ δ ⩽ ε ⩽ α^{+1} in Z̄_{∞}, we have a short exact sequence

E(ε/β//γ/α)(X) [figure: e] E(ε/β//δ/α)(X) ^{\overline{}^{e}} E(ε/γ//δ/α)(X) .

Proof. See [10, Lem. 3.9].

Lemma 19 Given ε^{−1} ⩽ α ⩽ β ⩽ γ ⩽ δ ⩽ ε ⩽ α^{+1} in Z̄_{∞}, we have a short exact sequence

E(ε/γ//δ/α)(X) [figure: e] E(ε/γ//δ/β)(X) ^{\overline{}^{e}} E(α^{+1}/γ//δ/β)(X) .

Proof. Apply the functor induced by β/α \overline{}α^{+1}/β to Sp(X). Then apply [10, Lem. 3.9].

The short exact sequence in Lemma 18 is called a fundamental short exact sequence (in first notation), the short exact sequence in Lemma 19 is called a fundamental short exact sequence (in second notation). They will be used without further comment.

3.5 Classical indexing

Let 1 ⩽ r ⩽ ∞ and let p, q ∈ Z. Denote

E_{r}^{p,q} = E_{r}^{p,q}(X) := E(−p − 1 + r/−p − 1//−p/−p − r)^{+p+q}(X) ,

where i + ∞ := ∞ and i − ∞ := −∞ for all i ∈ Z.

Example 20 The short exact sequences in Lemmata 18, 19 allow to derive the exact couples of Massey. Write D_{r}^{i,j} = D_{r}^{i,j}(X) := E(−i/−∞//−i−r+1/−∞)^{+i+j}(X) for i, j ∈ Z and r ⩾ 1. We obtain an exact sequence

D_{r}^{i, j} ^{\overline{}^{e}} D_{r}^{i−1, j+1} ^{\overline{}^{e}} E_{r}^{i+r−2, j−r+2} ^{\overline{}^{e}} D_{r}^{i+r−1, j−r+2} ^{\overline{}^{e}} D_{r}^{i+r−2, j−r+3}

by Lemmata 18, 19.

3.6 Comparing proper spectral sequences

Let X ^{\overline{}^{f}} Y be a morphism in SFFC(A), i.e. a morphism of pointwise split and pointwise finitely filtered complexes with values in A. Write E(X) ^{\overline{}^{E(f)}} E(Y) for the induced morphism on the spectral sequences.

For α, β ∈ Z̄_{∞}, we write α ˙< β if

(α < β) or (α = β and α ∈ {∞^{+k} : k ∈ Z} ∪ {−∞^{+k} : k ∈ Z}) .

We write

˙Z̄_{∞}^{##} := {δ/β//γ/α ∈ Z̄_{∞}^{##} : δ^{−1} ⩽ α ˙< β ⩽ γ ˙< δ ⩽ α^{+1}} .

We write

Ė = Ė(X) := E(X)|_{˙Z̄_{∞}^{##}} ∈ Ob ˙ Z̄_{∞}^{##}, A

for the proper spectral sequence of X; analogously for the morphisms.

Lemma 21 If E(α + 1/α − 1//α/α − 2)^{+k}(f) is an isomorphism for all α ∈ Z and all k ∈ Z, then Ė(f) is an isomorphism.

Proof. Claim 1. We have an isomorphism E(γ/β − 1//β/β − 2)^{+k}(f) for all k ∈ Z, all β ∈ Z and all γ ∈ Z such that γ > β. We have an isomorphism E(β + 1/β − 1//β/α − 1)^{+k}(f) for all k ∈ Z, all β ∈ Z and all α ∈ Z such that α < β.

The assertions follow by induction using the exact sequences

E(γ + 2/γ//γ + 1/β)^{+k−1} ^{\overline{}^{e}} E(γ/β − 1//β/β − 2)^{+k} ^{\overline{}^{e}} E(γ + 1/β − 1//β/β − 2)^{+k} \overline{}0

and

0 \overline{}E(β+1/β−1//β/α−2)^{+k} ^{\overline{}^{e}} E(β+1/β−1//β/α−1)^{+k} ^{\overline{}^{e}} E(β−1/α−2//α−1/α−3)^{+k+1} .

Claim 2. We have an isomorphism E(γ/β − 1//β/α − 1)^{+k}(f) for all k ∈ Z and all α, β, γ ∈ Z such that α < β < γ.

We proceed by induction on γ − α. By Claim 1, we may assume that α < β − 1 < β + 1 < γ. Consider the image diagram

E(γ − 1/β − 1//β/α − 1)^{+k} ^{\overline{}^{e}} E(γ/β − 1//β/α − 1)^{+k} [figure: e] E(γ/β − 1//β/α)^{+k} .

Claim 3. We have an isomorphism E(δ/β//γ/α)^{+k}(f) for all k ∈ Z and all α, β, γ, δ ∈ Z such that α < β ⩽ γ < δ.

We may assume that γ − β ⩾ 1, for E(δ/β//β/α)^{+k} = 0. We proceed by induction on γ − β. By Claim 2, we may assume that γ − β ⩾ 2. Consider the short exact sequence

E(δ/β//γ − 1/α)^{+k} [figure: e] E(δ/β//γ/α)^{+k} ^{\overline{}^{e}} E(δ/γ − 1//γ/α)^{+k} .

Claim 4. We have an isomorphism E(δ/β//γ/α)^{+k}(f) for all k ∈ Z and all α, β, γ, δ ∈ Z_{∞} such that α < β ⩽ γ < δ.

In view of Claim 3, it suffices to choose α̃ ∈ Z small enough such that E(δ/β//γ/α̃)^{+k}(f) = E(δ/β//γ/−∞)^{+k}(f); etc.

Claim 5. We have an isomorphism E(δ/β//γ/α)^{+k}(f) for all k ∈ Z and all α, β, γ, δ ∈ Z_{∞} such that α ˙< β ⩽ γ ˙< δ.

In view of Claim 4, it suffices to choose β̃ ∈ Z small enough such that E(δ/β̃//γ/−∞)^{+k}(f) = E(δ/−∞//γ/−∞)^{+k}(f); etc.

Claim 6. We have an isomorphism E(δ/β//γ/α)^{+k}(f) for all k ∈ Z and all α, β, γ, δ ∈ Z̄_{∞} such that −∞ ⩽ δ^{−1} ⩽ α ˙< β ⩽ γ ⩽ ∞ < −∞^{+1} ⩽ δ ⩽ α^{+1}.

In view of Claim 5, it suffices to consider the short exact sequence

E(∞/β//γ/δ^{−1})^{+k} [figure: e] E(∞/β//γ/α)^{+k} ^{\overline{}^{e}} E(δ/β//γ/α)^{+k} .

Claim 7. The morphism Ė(f) is an isomorphism.

Suppose given α, β, γ, δ ∈ Z̄_{∞} such that δ^{−1} ⩽ α ˙< β ⩽ γ ˙< δ ⩽ α^{+1}. Via a shift, we may assume that we are in the situation of Claim 5 or of Claim 6.

3.7 The first spectral sequence of a double complex

Let A be an abelian category. Let X ∈ Ob CC^{⌞}(A). Given n ∈ Z_{∞}, we write X^{[n,∗} for the double complex arising from X by replacing X^{i,j} by 0 for all i ∈ [0, n[. We define a pointwise split and pointwise finitely filtered complex t_{I}X, called the first filtration of tX, by letting t_{I}X(α) := tX^{[−α,∗} for α ∈ Z_{∞}; and by letting t_{I}X(α) \overline{}t_{I}X(β) be the pointwise split inclusion tX^{[−α,∗} \overline{}tX^{[−β,∗} for α, β ∈ Z_{∞} such that α ⩽ β. Let E_{I} = E_{I}(X) := E(t_{I}X). This construction is functorial in X ∈ Ob CC^{⌞}(A). Note that \overline{t_{I}X}(α) = X^{−α,k+α}.

We record the following wellknown lemma in the language we use here.

Lemma 22 Let α ∈]−∞, 0]. Let k ∈ Z such that k ⩾ −α. We have

\begin{aligned}E_{I}(α/α − 1//α/α − 1)^{+k}(X) & = & H^{k+α}(X^{−α,∗}) \\ E_{I}(α + 1/α − 1//α/α − 2)^{+k}(X) & = & H^{−α}(H^{k+α}(X^{−,∗})) , \end{aligned}

naturally in X ∈ Ob CC^{⌞}(A).

Proof. The first equality follows by E_{I}(α/α − 1//α/α − 1)^{+k} = H^{k}t_{I}X(α/α − 1) = H^{k+α}(X^{−α,∗}).

The morphism t_{I}X(α/α − 1) \overline{}t_{I}X((α − 2)^{+1}/α − 1) = t_{I}X(α − 1/α − 2)^{•+1} from Sp(t_{I}X) is at position k ⩾ 0 given by

\overline{t_{I}X}(α)^{k} = X^{−α,k+α} ^{^{(−1)^{α} ∂}\overline{}}X^{−α+1,k+α} = \overline{t_{I}X}(α − 1)^{k+1} ;

cf. §1.1.6. In particular, the morphisms

E_{I}(α + 1/α//α + 1/α)^{+k−1} ^{\overline{}^{e}} E_{I}(α/α − 1//α/α − 1)^{+k} ^{\overline{}^{e}} E_{I}(α − 1/α − 2//α − 1/α − 2)^{+k+1}

are given by

H^{k+α}(X^{−α−1,∗}) ^{\overline{}^{(−1)^{α+1}H^{k+α}(∂)}} H^{k+α}(X^{−α,∗}) ^{\overline{}^{(−1)^{α}H^{k+α}(∂)}} H^{k+α}(X^{−α+1,∗}) .

Now the second equality follows by the diagram

[figure: E_{I}(α + 1/α − 1//α/α − 2)^{+k}; •; e; E_{I}(α/α − 1//α/α − 2)^{+k}; •; e; e; E_{I}(α + 1/α − 1//α/α − 1)^{+k}; E_{I}(α + 1/α//α + 1/α)^{+k−1}; e; E_{I}(α/α − 1//α/α − 1)^{+k}; e; e; E_{I}(α − 1/α − 2//α − 1/α − 2)^{+k+1} .]

Remark 23  Let X ^{\overline{}^{f}} Y be a rowwise quasiisomorphism in CC^{⌞}(A).  Then

E_{I}(δ/β//γ/α)^{+k}(f) is an isomorphism for δ^{−1} ⩽ α ⩽ β ⩽ γ ⩽ δ ⩽ α^{+1} in Z̄_{∞} and k ∈ Z.

Proof. It suffices to show that the morphism Sp(t_{I}f) in Z̄_{∞}^{#}, K(A) is pointwise a quasiisomorphism. To have this, it suffices to show that tf^{[k,∗} is a quasiisomorphism for k ⩾ 0. But f^{[k,∗} is a rowwise quasiisomorphism for k ⩾ 0; cf. §1.1.6.

Lemma 24 The functor CC^{⌞}(A) ^{\overline{}^{Ė_{I}}} ˙ Z̄_{∞}^{##}, A factors over

KK^{⌞}(A) ^{\overline{}^{Ė_{I}}} ˙ Z̄_{∞}^{##}, A .

Proof. By Lemma 1, we have to show that Ė_{I} annihilates all elementary horizontally split acyclic double complexes in Ob CC^{⌞}(A) and all elementary vertically split acyclic double complexes in Ob CC^{⌞}(A).

Let U ∈ Ob CC^{⌞}(A) be an elementary vertically split acyclic double complex concentrated in rows i and i + 1, where i ⩾ 0. Let V ∈ Ob CC^{⌞}(A) be an elementary horizontally split acyclic double complex concentrated in columns j and j + 1, where j ⩾ 0.

Since V is rowwise acyclic, E_{I} annihilates V by Remark 23, whence so does Ė_{I}.

Suppose given

(∗)  −∞ ⩽ α ˙< β ⩽ γ ˙< δ ⩽ ∞

in Z̄_{∞} and k ∈ Z. We claim that the functor E_{I}(δ/β//γ/α)^{+k} annihilates U. We may assume that β < γ. Note that E_{I}(δ/β//γ/α)^{+k}(U) is the image of

H^{k}(t_{I}U(γ/α)) \overline{}H^{k}(t_{I}U(δ/β)) .

The double complex U^{[−δ,∗}/U^{[−β,∗} is columnwise acyclic except possibly if −β = i + 1 or if −δ = i + 1. The double complex U^{[−γ,∗}/U^{[−α,∗} is columnwise acyclic except possibly if −α = i + 1 or if −γ = i + 1. All three remaining combinations of these exceptional cases are excluded by (∗), however. Hence E_{I}(δ/β//γ/α)^{+k}(U) = 0. This proves the claim.

Suppose given

(∗∗)  δ^{−1} ⩽ α ˙< β ⩽ γ ⩽ ∞ ⩽ −∞^{+1} ⩽ δ ⩽ α^{+1} .

in Z̄_{∞} and k ∈ Z. We claim that the functor E_{I}(δ/β//γ/α)^{+k} annihilates U. We may assume that β < γ and that δ^{−1} < α. Note that E_{I}(δ/β//γ/α)^{+k}(U) is the image of

H^{k}(t_{I}U(γ/α)) \overline{}H^{k+1}(t_{I}U(β/δ^{−1})) .

The double complex U^{[−β,∗}/U^{[−(δ^{−1}),∗} is columnwise acyclic except possibly if −(δ^{−1}) = i + 1 or if −β = i + 1. The double complex U^{[−γ,∗}/U^{[−α,∗} is columnwise acyclic except possibly if −γ = i + 1 or if −α = i + 1. Both remaining combinations of these exceptional cases are excluded by (∗∗), however. Hence E_{I}(δ/β//γ/α)^{+k}(U) = 0. This proves the claim.

Both claims taken together show that Ė_{I} annihilates U.

4 Grothendieck spectral sequences

4.1 Certain quasiisomorphisms are preserved by a left exact functor

Suppose given abelian categories A, B, and suppose that A has enough injectives. Let A ^{\overline{}^{F}} B be a left exact functor.

Remark 25 Suppose given an F -acyclic object X ∈ Ob A and an injective resolution I ∈ Ob C^{[0}(Inj A) of X. Let Conc X ^{\overline{}^{f}} I be its quasiisomorphism. Then Conc F X ^{\overline{}^{F f}} F I is a quasiisomorphism.

Proof. This follows since F is left exact and since H^{i}(F I) ≃ (R^{i}F)X ≃ 0 for i ⩾ 1.

Remark 26 Suppose given a complex U ∈ Ob C^{[0}(A) consisting of F -acyclic objects. There exists an injective complex resolution I ∈ Ob C^{[0}(Inj A) of U such that its quasiisomorphism U ^{\overline{}^{f}} I maps to a quasiisomorphism F U ^{\overline{}^{F f}} F I.

Proof. Let J ∈ Ob CC^{⌞, CE}(Inj A) be a CE-resolution of U; cf. Remark 9. Since the morphism of double complexes Conc_{2} U \overline{}J is a columnwise quasiisomorphism consisting of monomorphisms, taking the total complex, we obtain a quasiisomorphism U \overline{}tJ consisting of monomorphisms. By F -acyclicity of the entries of U, the image Conc_{2} F U \overline{}F J under F is a columnwise quasiisomorphism, too; cf. Remark 25. Hence F maps the quasiisomorphism U \overline{}tJ to the quasiisomorphism F U \overline{}F tJ. So we may take I := tJ.

Lemma 27 Suppose given a complex U ∈ Ob C^{[0}(A) consisting of F -acyclic objects and an injective complex resolution I ∈ Ob C^{[0}(Inj A) of U. Let U ^{\overline{}^{f}} I be its quasiisomorphism. Then F U ^{\overline{}^{F f}} F I is a quasiisomorphism.

Proof. Let U \overline{}I′ be a quasiisomorphism to an injective complex resolution I′ that is mapped to a quasiisomorphism by F; cf. Remark 26. Since U \overline{}I′ is a quasiisomorphism, the induced map _{K(A)}(U, I) \overline{}_{K(A)}(I′, I) is surjective, so that there exists a morphism I′ \overline{}I such that (U \overline{}I′ \overline{}I) = (U ^{\overline{}^{f}} I) in K(A). Since, moreover, U ^{\overline{}^{f}} I is a quasiisomorphism, I′ \overline{}I is a homotopism. Since F U \overline{}F I′ is a quasiisomorphism and F I′ \overline{}F I is a homotopism, we conclude that F U \overline{}F I is a quasiisomorphism.

4.2 Definition of the Grothendieck spectral sequence functor

Suppose given abelian categories A, B and C, and suppose that A and B have enough injectives. Let A ^{\overline{}^{F}} B and B ^{\overline{}^{G}} C be left exact functors.

A (F, G)-acyclic resolution of X ∈ Ob A is a complex A ∈ Ob C^{[0}(A), together with a quasiisomorphism Conc X \overline{}A, such that the following hold.

(A 1) The object A^{i} is F -acyclic for i ⩾ 0.

(A 2) The object A^{i} is (G ∘ F)-acyclic for i ⩾ 0.

(A 3) The object F A^{i} is G-acyclic for i ⩾ 0.

An object X ∈ Ob A that possesses an (F, G)-acyclic resolution is called (F, G)-acyclicly resolvable. The full subcategory of (F, G)-acyclicly resolvable objects in A is denoted by A_{(F,G)}.

A complex A ∈ Ob C^{[0}(A), together with a quasiisomorphism Conc X \overline{}A, is called an F-acyclic resolution of X ∈ Ob A if (A 2) holds.

Remark 28 If F carries injective objects to G-acyclic objects, then (A 1) and (A 3) imply (A 2).

Proof. Given i ⩾ 0, we let I be an injective resolution of A^{i}, and Ĩ the acyclic complex obtained by appending A^{i} to I in position −1. Since A^{i} is F -acyclic, the complex F Ĩ is acyclic; cf. Remark 25. Note that F B^{0}Ĩ ≃ F A^{i} is G-acyclic by assumption. Since

(R^{k}G)F Ĩ^{j} \overline{}(R^{k}G)F B^{j+1}Ĩ \overline{}(R^{k+1}G)F B^{j}Ĩ

is exact in the middle for j ⩾ 0 and k ⩾ 1, we may conclude by induction on j and by G-acyclicity assumption on F Ĩ^{j} that F B^{j}Ĩ is G-acyclic for j ⩾ 0. In particular, we have (R^{1}G)(F B^{j}Ĩ) ≃ 0 for j ⩾ 0, whence

GF B^{j}Ĩ \overline{}GF Ĩ^{j} \overline{}GF B^{j+1}Ĩ

is short exact for j ⩾ 0. We conclude that (G ∘ F)Ĩ is acyclic. Hence A^{i} is (G ∘ F)-acyclic.

To see Remark 28, one could also use a Grothendieck spectral sequence, once established.

Remark 29 Suppose given X ∈ Ob A, an injective resolution I of X and an F -acyclic resolution A of X. Then there exists a quasiisomorphism A \overline{}I that is mapped to 1_{X} by H^{0}. Moreover, any morphism A ^{\overline{}^{u}} I that is mapped to 1_{X} by H^{0} is a quasiisomorphism and is mapped to a quasiisomorphism F A ^{\overline{}^{F u}} F I by F.

Proof. Let I′ be an injective complex resolution of A such that its quasiisomorphism A \overline{}I′ is mapped to a quasiisomorphism by F; cf. Remark 26. We use the composite quasiisomorphism Conc X \overline{}A \overline{}I′ to resolve X by I′.

To prove the first assertion, note that there is a homotopism I′ \overline{}I resolving 1_{X}; whence the composite (A \overline{}I′ \overline{}I) is a quasiisomorphism resolving 1_{X}.

To prove the second assertion, note that the induced map _{K(A)}(A, I) \overline{}_{K(A)}(I′, I) is surjective, whence there is a factorisation (A \overline{}I′ \overline{}I) = (A ^{\overline{}^{u}} I) in K(A) for some morphism I′ \overline{}I, which, since resolving 1_{X} as well, is a homotopism. In particular, A ^{\overline{}^{u}} I is a quasiisomorphism. Finally, since F I′ \overline{}F I is a homotopism, also F A ^{\overline{}^{F u}} F I is a quasiisomorphism.

Alternatively, in the last step of the preceding proof we could have invoked Lemma 27.

The following construction originates in [5, XVII.§7] and [7, Th. 2.4.1]. In its present form, it has been carried out by Haas in the classical framework [8]. We do not claim any originality. I do not know whether the use of injectives in A in the following construction can be avoided; in any case, it would be desirable to do so.

We set out to define the proper Grothendieck spectral sequence functor

A_{(F,G)} ^{\overline{}^{Ė_{F,G}^{Gr}}} ˙ Z̄_{∞}^{##}, C .

We define Ė_{F,G}^{Gr} on objects. Suppose given X ∈ Ob A_{(F,G)}. Choose an (F, G)-acyclic resolution A_{X} ∈ Ob C^{[0}(A) of X. Choose a CE-resolution J_{X} ∈ Ob CC^{⌞}(Inj B) of F A_{X}. Let E_{F,G}^{Gr}(X) := E_{I}(GJ_{X}) = E(t_{I}GJ_{X}) ∈ Ob Z̄_{∞}^{##}, C be the Grothendieck spectral sequence of X with respect to F and G. Accordingly, let

Ė_{F,G}^{Gr}(X) := Ė_{I}(GJ_{X}) = Ė(t_{I}GJ_{X}) ∈ Ob ˙ Z̄_{∞}^{##}, C

be the proper Grothendieck spectral sequence of X with respect to F and G.

We define Ė_{F,G}^{Gr} on morphisms. Suppose given X ∈ Ob A_{(F,G)}, and let A_{X} and J_{X} be as above. Choose an injective resolution I_{X} ∈ Ob C^{[0}(Inj A) of X. Choose a quasiisomorphism A_{X} ^{\overline{}^{p_{X}}} I_{X} that is mapped to 1_{X} by H^{0} and to a quasiisomorphism by F; cf. Remark 29. Choose a CE-resolution K_{X} ∈ Ob CC^{⌞}(Inj B) of F I_{X}. Choose a morphism J_{X} ^{\overline{}^{q_{X}}} K_{X} in CC^{⌞}(Inj B) that is mapped to F p_{X} by H^{0}((−)^{∗,−}); cf. Remark 6.

Note that J_{X} ^{\overline{}^{q_{X}}} K_{X} can be written as a composite in CC^{⌞, CE}(Inj B) of a rowwise homotopism, followed by a double homotopism; cf. Proposition 17. Hence, so can GJ_{X} ^{\overline{}^{Gq_{X}}} GK_{X}. Thus

Ė_{I}(GJ_{X}) ^{\overline{}^{Ė_{I}(Gq_{X})}} Ė_{I}(GK_{X}) is an isomorphism; cf. Remark 23, Lemma 24.

Suppose given X ^{\overline{}^{f}} Y in A_{(F,G)}. Choose a morphism I_{X} ^{\overline{}^{f′}} I_{Y} in C^{[0}(A) that is mapped to f by H^{0}. Choose a morphism K_{X} ^{\overline{}^{f′′}} K_{Y} in CC^{⌞}(Inj B) that is mapped to F f′ by H^{0}((−)^{∗,−}); cf. Remark 6. Let

Ė_{F,G}^{Gr}(X ^{\overline{}^{f}} Y) := (Ė_{I}(GJ_{X})  \overline{}^{Ė_{I}(Gq_{X})}_{∼}  Ė_{I}(GK_{X}) ^{\overline{}^{Ė_{I}(Gf′′)}} Ė_{I}(GK_{Y}) \overline{}^{Ė_{I}(Gq_{Y} )}_{∼}  Ė_{I}(GJ_{Y})) .

The procedure can be adumbrated as follows.

[figure: X; f; Y; A_{X}; p_{X}; I_{X}; f′; A_{Y}; p_{Y}; I_{Y}; J_{X}; q_{X}; K_{X}; f′′; J_{Y}; q_{Y}; K_{Y}]

We show that this defines a functor Ė_{F,G}^{Gr} : A_{(F,G)} \overline{}˙Z̄_{∞}^{##}, C . We need to show independence of the construction from the choices of f′ and f′′, for then functoriality follows by appropriate choices.

Let I_{X} ^{\overline{}^{f̃′}} I_{Y} and K_{X} ^{\overline{}^{f̃′′}} K_{Y} be alternative choices. The residue classes of f′ and f̃′ in K^{[0}(A) coincide, whence so do the residue classes of F f′ and F f̃′ in K^{[0}(B). Therefore, the residue classes of f′′ and f̃′′ in KK^{⌞}(B) coincide; cf. Proposition 14. Hence, so do the residue classes of Gf′′ and Gf̃′′ in KK^{⌞}(C). Thus Ė_{I}(Gf′′) = Ė_{I}(Gf̃′′); cf. Lemma 24.

We show that alternative choices of A_{X}, I_{X} and p_{X}, and of J_{X}, K_{X} and q_{X}, yield isomorphic proper Grothendieck spectral sequence functors.

Let Ã_{X} ^{\overline{}^{p̃_{X}}} Ĩ_{X} and J̃_{X} ^{\overline{}^{q̃_{X}}} K̃_{X} be alternative choices, where X runs through Ob A_{(F,G)}.

Suppose given X ^{\overline{}^{f}} Y in A_{(F,G)}. We resolve the commutative quadrangle

[figure: X; f; Y; X; f; Y]

in A to a commutative quadrangle

\xymatrix{I_{X} \ar[r]^{f′} \ar[d]_{u_{X}} & I_{Y} \ar[d]^{u_{Y}} \\ Ĩ_{X} \ar[r]^{f̃′} & Ĩ_{Y}}

in K^{[0}(A), in which u_{X} and u_{Y} are homotopisms; cf. Remark 6. Then we resolve the commutative quadrangle

\xymatrix{F I_{X} \ar[r]^{F f′} \ar[d]_{F u_{X}} & F I_{Y} \ar[d]^{F u_{Y}} \\ F Ĩ_{X} \ar[r]^{F f̃′} & F Ĩ_{Y}}

in K^{[0}(B) to a commutative quadrangle

\xymatrix{K_{X} \ar[r]^{f′′} \ar[d]_{v_{X}} & K_{Y} \ar[d]^{v_{Y}} \\ K̃_{X} \ar[r]^{f̃′′} & K̃_{Y}}

in KK^{⌞}(B); cf. Proposition 14. Therein, v_{X} and v_{Y} are each composed of a rowwise homotopism, followed by a double homotopism; cf. Proposition 17. So are Gv_{X} and Gv_{Y}. An application of Ė_{I}(G(−)) yields the sought isotransformation, viz.

(Ė_{I}(GJ_{X})  \overline{}^{Ė_{I}(Gq_{X})}_{∼}  Ė_{I}(GK_{X})  \overline{}^{Ė_{I}(Gv_{X})}_{∼}  Ė_{I}(GK̃_{X}) \overline{}^{Ė_{I}(Gq̃_{X})}_{∼}  Ė_{I}(GJ̃_{X}))

at X ∈ Ob A_{(F,G)}; cf. Remark 23, Lemma 24.

Finally, we recall the starting point of the whole enterprise.

Remark 30 ([5, XVII.§7], [7, Th. 2.4.1]) Suppose given X ∈ Ob A_{(F,G)} and k, ℓ ∈ Z_{⩾0}. We have

Ė_{F,G}^{Gr}(−k + 1/−k − 1//−k/−k − 2)^{+k+ℓ}(X) ≃ (R^{k}G)(R^{ℓ}F)(X)
Ė_{F,G}^{Gr}(∞/−∞//∞/−∞)^{+k+ℓ}(X)  ≃ (R^{k+ℓ}(G ∘ F))(X) ,

naturally in X.

Proof. Keep the notation of the definition of Ė_{F,G}^{Gr} . We shall prove the first isomorphism. By Lemma 22, we have

Ė_{F,G}^{Gr}(−k + 1/−k − 1//−k/−k − 2)^{+k+ℓ}(X) ≃ H^{k}(H^{ℓ}(GJ_{X}^{−,∗})) .

Since J_{X} is rowwise split, we have H^{ℓ}(GJ_{X}^{−,∗}) ≃ G(H^{ℓ}J_{X}^{−,∗}). Note that H^{ℓ}J_{X}^{−,∗} is an injective resolution of H^{ℓ}F A_{X}; cf. Remark 8.(1). By Remark 29, H^{ℓ}F A_{X} \overline{}^{H^{ℓ}F p_{X}}_{∼} H^{ℓ}F I_{X} ≃ (R^{ℓ}F)(X). So

H^{k}(H^{ℓ}(GJ_{X}^{−,∗})) ≃ H^{k}(G(H^{ℓ}J_{X}^{−,∗})) ≃ (R^{k}G)(H^{ℓ}F A_{X}) ≃ (R^{k}G)(R^{ℓ}F)(X) . We shall prove naturality of the first isomorphism. Suppose given X ^{\overline{}^{f}} Y in A_{(F,G)}. Consider the following commutative diagram. Abbreviate E := Ė(−k + 1/−k − 1//−k/−k − 2)^{+k+ℓ}.

\xymatrix{E(t_{I}GJ_{X}) \ar[r]^{E(t_{I}Gq_{X})}_{∼} \ar[d]^{≀} & E(t_{I}GK_{X}) \ar[r]^{E(t_{I}Gf′′)} \ar[d]^{≀} & E(t_{I}GK_{Y}) \ar[d]^{≀} & E(t_{I}GJ_{Y}) \ar[l]_{E(t_{I}Gq_{Y} )}^{∼} \ar[d]^{≀} \\ H^{k}H^{ℓ}GJ_{X}^{−,∗} \ar[r]^{H^{k}H^{ℓ}Gq_{X}^{−,∗}}_{∼} \ar[d]^{≀} & H^{k}H^{ℓ}GK_{X}^{−,∗} \ar[r]^{H^{k}H^{ℓ}Gf^{′′−,∗}} \ar[d]^{≀} & H^{k}H^{ℓ}GK_{Y}^{−,∗} \ar[d]^{≀} & H^{k}H^{ℓ}GJ_{Y}^{−,∗} \ar[l]_{H^{k}H^{ℓ}Gq_{Y}^{−,∗}}^{∼} \ar[d]^{≀} \\ H^{k}GH^{ℓ}J_{X}^{−,∗} \ar[r]^{H^{k}GH^{ℓ}q_{X}^{−,∗}}_{∼} \ar[d]^{≀} & H^{k}GH^{ℓ}K_{X}^{−,∗} \ar[r]^{H^{k}GH^{ℓ}f^{′′−,∗}} \ar[d]^{≀} & H^{k}GH^{ℓ}K_{Y}^{−,∗} \ar[d]^{≀} & H^{k}GH^{ℓ}J_{Y}^{−,∗} \ar[l]_{H^{k}GH^{ℓ}q_{Y}^{−,∗}}^{∼} \ar[d]^{≀} \\ (R^{k}G)H^{ℓ}F A_{X} \ar[r]^{(R^{k}G)H^{ℓ}F p_{X}}_{∼} & (R^{k}G)H^{ℓ}F I_{X} \ar[r]^{(R^{k}G)H^{ℓ}F f′} \ar[d]^{≀} & (R^{k}G)H^{ℓ}F I_{Y} \ar[d]^{≀} & (R^{k}G)H^{ℓ}F A_{Y} \ar[l]_{(R^{k}G)H^{ℓ}F p_{Y}}^{∼} \\  & (R^{k}G)(R^{ℓ}F)(X) \ar[r]^{(R^{k}G)(R^{ℓ}F)(f)} & (R^{k}G)(R^{ℓ}F)(Y)}

We shall prove the second isomorphism. By Lemma 27, the quasiisomorphism F A_{X} \overline{}tJ_{X} maps to a quasiisomorphism GF A_{X} \overline{}tGJ_{X} ≃ GtJ_{X}. By Lemma 27, the quasiisomorphism A_{X} ^{\overline{}^{p_{X}}} I_{X} maps to a quasiisomorphism GF A_{X} ^{\overline{}^{GF p_{X}}} GF I_{X}. So

Ė_{F,G}^{Gr}(∞/−∞//∞/−∞)^{+k+ℓ}(X) ≃ H^{k+ℓ}(tGJ_{X}) ≃ H^{k+ℓ}(GtJ_{X}) ≃ H^{k+ℓ}(GF A_{X})
≃ H^{k+ℓ}(GF I_{X}) ≃ (R^{k+ℓ}(G ∘ F))(X) .

We shall prove naturality of the second isomorphism. Consider the following diagram. Abbreviate Ẽ := Ė_{F,G}^{Gr}(∞/−∞//∞/−∞)^{+k+ℓ}.

\xymatrix{Ẽ(t_{I}GJ_{X}) \ar[r]^{Ẽ(t_{I}Gq_{X})}_{∼} \ar[d]^{≀} & Ẽ(t_{I}GK_{X}) \ar[r]^{Ẽ(t_{I}Gf′′)} \ar[d]^{≀} & Ẽ(t_{I}GK_{Y}) \ar[d]^{≀} & Ẽ(t_{I}GJ_{Y}) \ar[l]_{Ẽ(t_{I}Gq_{Y} )}^{∼} \ar[d]^{≀} \\ H^{k+ℓ}tGJ_{X} \ar[r]^{H^{k+ℓ}tGq_{X}}_{∼} \ar[d]^{≀} & H^{k+ℓ}tGK_{X} \ar[r]^{H^{k+ℓ}tGf′′} \ar[d]^{≀} & H^{k+ℓ}tGK_{Y} \ar[d]^{≀} & H^{k+ℓ}tGJ_{Y} \ar[l]_{H^{k+ℓ}tGq_{Y}}^{∼} \ar[d]^{≀} \\ H^{k+ℓ}GtJ_{X} \ar[r]^{H^{k+ℓ}Gtq_{X}}_{∼} & H^{k+ℓ}GtK_{X} \ar[r]^{H^{k+ℓ}Gtf′′} & H^{k+ℓ}GtK_{Y} & H^{k+ℓ}GtJ_{Y} \ar[l]_{H^{k+ℓ}Gtq_{Y}}^{∼} \\ H^{k+ℓ}GF A_{X} \ar[r]^{H^{k+ℓ}GF p_{X}}_{∼} \ar[u]_{≀} & H^{k+ℓ}GF I_{X} \ar[r]^{H^{k+ℓ}GF f′} \ar[d]^{≀} \ar[u] & H^{k+ℓ}GF I_{Y} \ar[d]^{≀} \ar[u] & H^{k+ℓ}GF A_{Y} \ar[l]_{H^{k+ℓ}GF p_{Y}}^{∼} \ar[u]_{≀} \\  & (R^{k+ℓ}(G ∘ F))(X) \ar[r]^{(R^{k+ℓ}(G∘F))(f)} & (R^{k+ℓ}(G ∘ F))(Y)}

4.3 Haas transformations

The following transformations have been constructed in the classical framework by Haas [8]. We do not claim any originality.

4.3.1 Situation

Consider the following diagram of abelian categories, left exact functors and transformations,

[figure: A; F; U; B; G; V; C; W; A′; F′; B′; G′; C′ ,; μ; ν]

i.e. F′ ∘ U ^{\overline{}^{μ}} V ∘ F and G′ ∘ V ^{\overline{}^{ν}} W ∘ G. Suppose that the conditions (1, 2, 3) hold.

(1) The categories A, B, A′ and B′ have enough injectives.

(2) The functors U and V carry injectives to injectives.

(3) The functor F carries injective to G-acyclic objects. The functor F′ carries injective to G′-acyclic objects.

We have A_{(F,G)} = A since an injective resolution is an (F, G)-acyclic resolution. Likewise, we have A′_{(F′,G′)} = A′.

Note in particular the case U = 1_{A}, V = 1_{B} and W = 1_{C}.

We set out to define the Haas transformations

Ė_{F′,G′}^{Gr}(U(−)) ^{\overline{}^{h^{I}^{μ}}}  Ė_{F,G′∘V}^{Gr} ( − ) ^{\overline{}^{h{II \atop ν}}}  Ė_{F,W ∘G}^{Gr}( − ) ,

where h_{μ}^{I} depends on F, F′, G′, U, V and μ, and where h_{ν}^{II} depends on F, G, G′, V, W and ν.

4.3.2 Construction of the first Haas transformation

Given T ∈ Ob A, we let Ė_{F,G}^{Gr}(T) be defined via an injective resolution I_{T} of T and via a CE-resolution J_{T} of F I_{T}; cf. §4.2.

Given T′ ∈ Ob A′, we let Ė_{F′,G′}^{Gr}(T′) be defined via an injective resolution I′_{T′} of T′ and via a CE-resolution J′_{T′} of F′I′_{T′}; cf. §4.2.

We define h_{μ}^{I}. Let X ∈ Ob A. By Remark 5, there is a unique morphism I′_{UX} ^{\overline{}^{h′X}} UI_{X} in K^{[0}(A′) that maps to 1_{UX} under H^{0}. Let J′_{UX} ^{\overline{}^{h′′X}} V J_{X} be the unique morphism in KK^{⌞}(B′) that maps

to the composite morphism (F′I′_{UX} ^{\overline{}^{F′h′X}} F′UI_{X} ^{\overline{}^{μ}} V F I_{X}) in K^{[0}(B′) under H^{0}((−)^{∗,−}); cf. Lemma 13. Let the first Haas transformation be defined by

(Ė_{F′,G′}^{Gr}(UX) ^{\overline{}^{h_{μ}^{I}X}} Ė_{F,G′∘V}^{Gr} (X)) := (E_{I}(G′J′_{UX}) ^{\overline{}^{E_{I}(G′h′′X)}} E_{I}(G′V J_{X})) .

We show that h_{μ}^{I} is a transformation. Let X ^{\overline{}^{f}} Y be a morphism in A. Let I_{X} ^{\overline{}^{f′}} I_{Y} resolve X ^{\overline{}^{f}} Y. Let J_{X} ^{\overline{}^{f′′}} J_{Y} resolve F I_{X} ^{\overline{}^{f′}} F I_{Y}. Let I′_{UX} ^{\overline{}^{f̃′}} I′_{UY} resolve UX ^{\overline{}^{Uf}} UY. Let J′_{UX} ^{\overline{}^{f̃′′}} J_{UY} resolve F′I_{UX} ^{\overline{}^{F′f̃′}} F′I_{UY}. The quadrangle

[figure: UX; Uf; UX; Uf; UY; UY]

commutes in A′. Hence, by Remark 5, applied to I′_{UX} and UI_{Y}, the resolved quadrangle

\xymatrix{I′_{UX} \ar[r]^{h′X} \ar[d]_{f̃′} & UI_{X} \ar[d]^{Uf′} \\ I′_{UY} \ar[r]_{h′Y} & UI_{Y}}

commutes in K^{[0}(A′). Hence both quadrangles in

\xymatrix{F′I′_{UX} \ar[r]^{F′h′X} \ar[d]_{F′f̃′} & F′UI_{X} \ar[d]^{F′Uf′} \ar[r]^{μ} & V F I_{X} \ar[d]^{V F f′} \\ F′I′_{UY} \ar[r]_{F′h′Y} & F′UI_{Y} \ar[r]_{μ} & V F I_{Y}}

commute in K^{[0}(B′). By Lemma 13, applied to J′_{UX} and V J_{Y}, the outer quadrangle in the latter diagram can be resolved to the commutative quadrangle

\xymatrix{J′_{UX} \ar[r]^{h′′X} \ar[d]_{f̃′′} & V J_{X} \ar[d]^{V f′′} \\ J′_{UY} \ar[r]_{h′′Y} & V J_{Y}}

in KK^{⌞}(B′). Applying E_{I}(G′(−)) and employing the definitions of Ė_{F′,G′}^{Gr} , Ė_{F,G′∘V}^{Gr} and h_{μ}^{I} , we obtain the sought commutative diagram

\xymatrix{Ė_{F′,G′}^{Gr}(UX) \ar[r]^{h_{μ}^{I}X} \ar[d]_{Ė_{F′,G′}^{Gr}(Uf)} & Ė_{F,G′∘V}^{Gr} (X) \ar[d]^{Ė_{F,G′∘V}^{Gr} (f)} \\ Ė_{F′,G′}^{Gr}(UY) \ar[r]_{h_{μ}^{I}Y} & Ė_{F,G′∘V}^{Gr} (Y)}

in ˙ Z̄_{∞}^{##}, C′ .

4.3.3 Construction of the second Haas transformation

We maintain the notation of §4.3.2. Given X ∈ Ob A, we let the second Haas transformation be defined by

(Ė_{F,G∘V}^{Gr} (X) ^{\overline{}^{h_{ν}^{II}X}} Ė_{F,W ∘G}^{Gr}(X)) := (Ė_{I}(G′V J_{X}) ^{\overline{}^{Ė_{I}(ν)}} Ė_{I}(W GJ_{X})) .

It is a transformation since ν is.

5 The first comparison

5.1 The first comparison isomorphism

Suppose given abelian categories A, A′ and B with enough injectives and an abelian category C.

Let A × A′ ^{\overline{}^{F}} B be a biadditive functor. Let B ^{\overline{}^{G}} C be an additive functor.

Suppose given objects X ∈ Ob A and X′ ∈ Ob A′. Suppose the following properties to hold.

(a) The functor F (−, X′) : A \overline{}B is left exact.

(a′) The functor F (X, −) : A′ \overline{}B is left exact.

(b) The functor G is left exact.

(c) The object X possesses a (F (−, X′), G)-acyclic resolution A ∈ Ob C^{[0}(A).

(c′) The object X′ possesses a (F (X, −), G)-acyclic resolution A′ ∈ Ob C^{[0}(A′).

Moreover, the resolutions appearing in (c) and (c′) are stipulated to have the following properties.

(d) For all k ⩾ 0, the quasiisomorphism Conc X \overline{}A is mapped to a quasiisomorphism Conc F (X, A^{′k}) \overline{}F (A, A^{′k}) under F (−, A^{′k}).

(d′) For all k ⩾ 0, the quasiisomorphism Conc X′ \overline{}A′ is mapped to a quasiisomorphism Conc F (A^{k}, X′) \overline{}F (A^{k}, A′) under F (A^{k}, −).

The conditions (d, d′) are e.g. satisfied if F (−, A^{′k}) and F (A^{k}, −) are exact for all k ⩾ 0.

Theorem 31 (first comparison) The proper Grothendieck spectral sequence for the functors F (X, −) and G, evaluated at X′, is isomorphic to the proper Grothendieck spectral sequence for the functors F (−, X′) and G, evaluated at X; i.e.

Ė_{F (X,−),G}^{Gr}(X′) ≃ Ė_{F (−,X′),G}^{Gr}(X)

in ˙ Z̄_{∞}^{##}, C .

Proof. Let J_{A} , J_{A′} , J_{A,A′} ∈ Ob CC^{⌞}(Inj B) be CE-resolutions of the complexes F (A, X′), F (X, A′), tF (A, A′) ∈ Ob C^{[0}(B), respectively.

The quasiisomorphism Conc X \overline{}A induces a morphism F (Conc X, A′) \overline{}F (A, A′), yielding F (X, A′) \overline{}tF (A, A′), which is a quasiisomorphism since Conc F (X, A^{′k}) \overline{}F (A, A^{′k}) is a quasiisomorphism for all k ⩾ 0 by (d).

Choose a CE-resolution J_{A′} \overline{}J_{A,A′} of F (X, A′) \overline{}tF (A, A′); cf. Remark 6. Since the morphism F (X, A′) \overline{}tF (A, A′) is a quasiisomorphism, J_{A′} \overline{}J_{A,A′} is a composite in CC^{⌞, CE}(Inj B) of a rowwise homotopism and a double homotopism; cf. Proposition 17. So is GJ_{A′} \overline{}GJ_{A,A′}. Hence, by Remark 23 and by Lemma 24, we obtain an isomorphism of the proper spectral sequences of the first filtrations of the total complexes,

Ė_{F (X,−),G}^{Gr}(X′) = Ė_{I}(GJ_{A′}) ^{\overline{}^{∼}} Ė_{I}(GJ_{A,A′}) .

Likewise, we have an isomorphism

Ė_{F (−,X′),G}^{Gr}(X) = Ė_{I}(GJ_{A}) ^{\overline{}^{∼}} Ė_{I}(GJ_{A,A′}) .

We compose to an isomorphism Ė_{F (X,−),G}^{Gr}(X′) ^{\overline{}^{∼}} Ė_{F (−,X′),G}^{Gr}(X) as sought.

5.2 Naturality of the first comparison isomorphism

We narrow down the assumptions just as we have done for the introduction of the Haas transformations in §4.3.1 in order to be able to express, in this narrower case, a naturality of the first comparison isomorphism from Theorem 31.

Suppose given abelian categories A, A′ and B with enough injectives and an abelian category C. Let A × A′ ^{\overline{}^{F}} B be a biadditive functor. Let B ^{\overline{}^{G}} C be an additive functor. Suppose that the following properties hold.

(a) The functor F (−, X′) : A \overline{}B is left exact for all X′ ∈ Ob A′. (a′) The functor F (X, −) : A′ \overline{}B is left exact for all X ∈ Ob A.

(b) The functor G is left exact.

(c) For all X′ ∈ Ob A′, the functor F (−, X′) carries injective objects to G-acyclic objects. (c′) For all X ∈ Ob A, the functor F (X, −) carries injective objects to G-acyclic objects.

(d) The functor F (I, −) is exact for all I ∈ Ob Inj A. (d′) The functor F (−, I′) is exact for all I′ ∈ Ob Inj A′.

Proposition 32 Suppose given X ^{\overline{}^{x}} X̃ in A and X′ ∈ Ob A′. Note that we have a transformation F (x, −) : F (X, −) \overline{}F (X̃, −). The following quadrangle, whose vertical isomorphisms are given by the construction in the proof of Theorem 31, commutes.

\xymatrix{Ė_{F (X,−),G}^{Gr}(X′) \ar[r]^{h_{F (x,−)}^{I}X′} \ar[d]_{≀} & Ė_{F (X̃,−),G}^{Gr}(X′) \ar[d]^{≀} \\ Ė_{F (−,X′),G}^{Gr}(X) \ar[r]^{Ė_{F (−,X′),G}^{Gr}(x)} & Ė_{F (−,X′),G}^{Gr}(X̃)}

For the definition of the first Haas transformation h_{F (x,−)}^{I}, see §4.3.2. An analogous assertion holds with interchanged roles of A and A′. Proof of Proposition 32. Let I resp. Ĩ be an injective resolution of X resp. X̃ in A. Let I ^{\overline{}^{x̂}} Ĩ be a resolution of X ^{\overline{}^{x}} X̃. Let I′ be an injective resolution of X′ in A′. Let J_{I′}^{(X)} resp. J_{I′}^{(X̃)} be a CE-resolution of F (X, I′) resp. F (X̃, I′). Let J_{I,I′} resp. J_{Ĩ,I′} be a CE-resolution of tF (I, I′) resp. tF (Ĩ, I′).

Let J_{I} resp. JĨ be a CE-resolution of F (I, X′) resp. F (Ĩ, X′). We have a commutative diagram

\xymatrix{F (X, I′) \ar[r]^{F (x,I′)} \ar[d] & F (X̃, I′) \ar[d] \\ tF (I, I′) \ar[r]^{tF (x̂,I′)} & tF (Ĩ, I′) \\ F (I, X′) \ar[r]^{F (x̂,X′)} \ar[u] & F (Ĩ, X′) \ar[u]}

in C^{[0}(B), hence in K^{[0}(B). By Proposition 14, it can be resolved to a commutative diagram

\xymatrix{J_{I′}^{(X)} \ar[r] \ar[d] & J_{I′}^{(X̃)} \ar[d] \\ J_{I,I′} \ar[r] & J_{Ĩ,I′} \\ J_{I} \ar[r] \ar[u] & JĨ \ar[u]}

in KK^{⌞}(B). Application of Ė_{I}(G(−)) yields the result; cf. Lemma 24.

We refrain from investigating naturality of the first comparison isomorphism in G.

6 The second comparison

6.1 The second comparison isomorphism

Suppose given abelian categories A and B′ with enough injectives, and abelian categories B and C. Let A ^{\overline{}^{F}} B′ be an additive functor. Let B × B′ ^{\overline{}^{G}} C be a biadditive functor. Suppose given objects X ∈ Ob A and Y ∈ Ob B. Let B ∈ Ob C^{[0}(B) be a resolution of Y, i.e. suppose a quasiisomorphism Conc Y \overline{}B to exist. Suppose the following properties to hold.

(a) The functor F is left exact.

(b) The functor G(Y, −) is left exact.

(c) The object X possesses an (F, G(Y, −))-acyclic resolution A ∈ Ob C^{[0}(A).

(d) The functor G(B^{k}, −) is exact for all k ⩾ 0.

(e) The functor G(−, I′) is exact for all I′ ∈ Ob Inj B′.

Remark 33 Suppose given a morphism D ^{\overline{}^{f}} D′ in CC^{⌞}(C). If H^{ℓ}(f^{−,∗}) is a quasiisomorphism for all ℓ ⩾ 0, then f induces an isomorphism

Ė_{I}(D) ^{\overline{}^{Ė_{I}(f)}} Ė_{I}(D′)

of proper spectral sequences.

Proof. By Lemma 21, it suffices to show that E_{I}(α + 1/α − 1//α/α − 2)^{+k}(f) is an isomorphism for all α ∈ Z and all k ∈ Z. By Lemma 22, this amounts to isomorphisms H^{k}H^{ℓ}(f^{−,∗}) for all k, ℓ ⩾ 0, i.e. to quasiisomorphisms H^{ℓ}(f^{−,∗}) for all ℓ ⩾ 0.

Consider the double complex G(B, FA) ∈ Ob CC^{⌞}(C), where the indices of B count rows and the indices of A count columns. To the first filtration of its total complex, we can associate the proper spectral sequence Ė_{I}(G(B, FA)) ∈ Ob ˙ Z̄_{∞}^{##}, C .

Theorem 34 (second comparison) The proper Grothendieck spectral sequence for the functors F and G(Y, −), evaluated at X, is isomorphic to Ė_{I}(G(B, FA)); i.e.

Ė_{F,G(Y,−)}^{Gr}(X) ≃ Ė_{I}(G(B, FA))

in ˙ Z̄_{∞}^{##}, C .

Proof. Let J′ ∈ Ob CC^{⌞}(Inj B′) be a CE-resolution of FA. By definition, Ė_{F,G(Y,−)}^{Gr}(X) = Ė_{I}(G(Y, J′)). By Remark 33, it suffices to find D ∈ Ob CC^{⌞}(C) and two morphisms of double complexes

G(B, FA) ^{\overline{}^{u}} D ^{\overline{}^{v}} G(Y, J′)

such that H^{ℓ}(u^{−,∗}) and H^{ℓ}(v^{−,∗}) are quasiisomorphisms for all ℓ ⩾ 0.

Given a complex U ∈ Ob C^{[0}(B), recall that we denote by Conc_{2}U ∈ Ob CC^{⌞}(B) the double complex whose row number 0 is given by U, and whose other rows are zero.

We have a diagram

G(B, Conc_{2} FA) \overline{}G(B, J′) \overline{}G(Conc Y, J′)

in CCC (C). Let ℓ ⩾ 0. Application of H^{ℓ}((−)^{−,=,∗}) yields a diagram

(∗) H^{ℓ}(G(B, Conc_{2} FA)^{−,=,∗}) \overline{}H^{ℓ}(G(B, J′)^{−,=,∗}) \overline{}H^{ℓ}(G(Conc Y, J′)^{−,=,∗})

in CC^{⌞}(C). We have

H^{ℓ}(G(B, Conc_{2} FA)^{−,=,∗}) ≃ G(B , H^{ℓ}((Conc_{2} F A)^{−,∗})) = G(B, Conc H^{ℓ}(FA))

and

H^{ℓ}(G(B, J′)^{−,=,∗}) ≃ G(B, H^{ℓ}(J^{′−,∗})) ,

since the functor G(B^{k}, −) is exact for all k ⩾ 0 by (d), or, since the CE-resolution J is rowwise split. Since the CE-resolution J′ is rowwise split, we moreover have

H^{ℓ}(G(Conc Y, J′)^{−,=,∗}) ≃ G( Conc Y, H^{ℓ}(J^{′−,∗})) .

So the diagram (∗) is isomorphic to the diagram

(∗∗)  G(B, Conc H^{ℓ}(FA)) \overline{}G(B, H^{ℓ}(J^{′−,∗})) \overline{}G( Conc Y, H^{ℓ}(J^{′−,∗})) ,

whose left hand side morphism is induced by the quasiisomorphism Conc H^{ℓ}(FA) \overline{}H^{ℓ}(J^{′−,∗}), and whose right hand side morphism is induced by the quasiisomorphism Conc Y \overline{}B.

By exactness of G(B^{k}, −) for k ⩾ 0, the left hand side morphism of (∗∗) is a rowwise quasiisomorphism. Since H^{ℓ}(J^{′k,∗}) is injective, the functor G(−, H^{ℓ}(J^{′k,∗})) is exact by (e), and therefore the right hand side morphism of (∗∗) is a columnwise quasiisomorphism. Thus an application of t to (∗∗) yields two quasiisomorphisms; cf. §1.1.6. Hence, also an application of t to (∗) yields two quasiisomorphisms in the diagram

tH^{ℓ}(G(B, Conc_{2} FA)^{−,=,∗}) \overline{}tH^{ℓ}(G(B, J′)^{−,=,∗}) \overline{}tH^{ℓ}(G(Conc Y, J′)^{−,=,∗}) .

Note that t ∘ H^{ℓ}((−)^{−,=,∗}) = H^{ℓ}((−)^{−,∗}) ∘ t_{1,2}, where t_{1,2} denotes taking the total complex in the first and the second index of a triple complex; cf. §1.2.2. Hence we have a diagram

H^{ℓ}((t_{1,2}G(B, Conc_{2} FA))^{−,∗}) \overline{}H^{ℓ}((t_{1,2}G(B, J′))^{−,∗}) \overline{}H^{ℓ}((t_{1,2}G(Conc Y, J′))^{−,∗})

consisting of two quasiisomorphisms. This diagram in turn, is isomorphic to

H^{ℓ}(G(B, FA)^{−,∗}) \overline{}H^{ℓ}((t_{1,2}G(B, J′))^{−,∗}) \overline{}H^{ℓ}((G(Y, J′))^{−,∗}) ,

where the left hand side morphism is obtained by precomposition with the isomorphism G(B, F A^{k}) ^{\overline{}^{∼}} t Conc_{1} G(B, F A^{k}) = (t_{1,2}G(B, Conc_{2} FA))^{−,k}, where k ⩾ 0; cf. §1.1.6.

Hence we may take

(G(B, FA) ^{\overline{}^{u}} D ^{\overline{}^{v}} G(B, J′)) := (G(B, FA) \overline{}t_{1,2}G(B, J′) \overline{}G(Y, J′)) .

6.2 Naturality of the second comparison isomorphism

Again, we narrow down the assumptions just as we have done for the introduction of the Haas transformations in §4.3.1 to express a naturality of the second comparison isomorphism from Theorem 34.

Suppose given abelian categories A and B′ with enough injectives, and abelian categories B and C. Suppose given additive functors A [figure: F; F̃] B′ and a transformation F ^{\overline{}^{φ}} F̃. Let B × B′ ^{\overline{}^{G}} C be a biadditive functor.

Suppose given a morphism X ^{\overline{}^{x}} X̃ in A and an object Y ∈ Ob B. Let B ∈ Ob C^{[0}(B) be a resolution of Y, i.e. suppose a quasiisomorphism Conc Y \overline{}B to exist. Suppose the following properties to hold.

(a) The functors F and F̃ are left exact and carry injective to G(Y, −)-acyclic objects.

(b) The functor G(Y, −) is left exact.

(c) The functor G(B^{k}, −) is exact for all k ⩾ 0.

(d) The functor G(−, I′) is exact for all I′ ∈ Ob Inj B′.

Let A ^{\overline{}^{a}} Ã in C^{[0}(Inj A) be an injective resolution of X ^{\overline{}^{x}} X̃ in A. Note that we have a commutative quadrangle

\xymatrix{G(B, F A) \ar[r]^{G(B,φA)} \ar[d]_{G(B,F a)} & G(B, F̃A) \ar[d]^{G(B,F̃a)} \\ G(B, F Ã) \ar[r]^{G(B,φÃ)} & G(B, F̃Ã)}

in CC^{⌞}(C).

Note that once chosen injective resolutions A of X and Ã of X̃, the image of G(B, F a) in KK^{⌞}(C) does not depend on the choice of the resolution A ^{\overline{}^{a}} Ã of X ^{\overline{}^{x}} X̃, for C^{[0}(A) ^{\overline{}^{G(B,F (−))}} CC^{⌞}(C) maps an elementary split acyclic complex to an elementary horizontally split acyclic complex.

Lemma 35 The quadrangle

\xymatrix{Ė_{F,G(Y,−)}^{Gr}(X) \ar[r]^{Ė_{F,G(Y,−)}^{Gr}(x)} \ar[d]_{≀} & Ė_{F,G(Y,−)}^{Gr}(X̃) \ar[d]^{≀} \\ Ė_{I}(G(B, F A)) \ar[r]^{Ė_{I}(G(B,F a))} & Ė_{I}(G(B, F Ã))}

commutes, where the vertical isomorphisms are those constructed in the proof of Theorem 34.

Proof. Let J′ ^{\overline{}^{â}} J̃′ be a CE-resolution of F A ^{\overline{}^{F a}} F Ã. Consider the following commutative diagram in CC^{⌞}(C).

\xymatrix{G(Y, J′) \ar[r]^{G(Y,â)} \ar[d] & G(Y, J̃′) \ar[d] \\ t_{1,2}G(B, J′) \ar[r]^{t_{1,2}G(B,â)} & t_{1,2}G(B, J̃′) \\ G(B, F A) \ar[r]^{G(B,F a)} \ar[u] & G(B, F Ã) \ar[u]}

An application of Ė_{I} yields the result.

Lemma 36 The quadrangle

\xymatrix{Ė_{F,G(Y,−)}^{Gr}(X) \ar[r]^{h_{φ}^{I}X} \ar[d]_{≀} & Ė_{F̃,G(Y,−)}^{Gr}(X) \ar[d]^{≀} \\ Ė_{I}(G(B, F A)) \ar[r]^{Ė_{I}(G(B,φA))} & Ė_{I}(G(B, F̃A))}

commutes, where the vertical morphisms are those constructed in the proof of Theorem 34.

For the definition of the first Haas transformation h_{F (x,−)}^{I}, see §4.3.2.

Proof. Let J′ ^{\overline{}^{φ̂}} J̆′ be a CE-resolution of F A ^{\overline{}^{F φ}} F̃A. Consider the following commutative diagram in CC^{⌞}(C).

\xymatrix{G(Y, J′) \ar[r]^{G(Y,φ̂)} \ar[d] & G(Y, J̆′) \ar[d] \\ t_{1,2}G(B, J′) \ar[r]^{t_{1,2}G(B,φ̂)} & t_{1,2}G(BJ̆′) \\ G(B, F A) \ar[r]^{G(B,φA)} \ar[u] & G(B, F̃A) \ar[u]}

An application of Ė_{I} yields the result.

We refrain from investigating naturality of the second comparison isomorphism in Y.

7 Acyclic CE-resolutions

We record Beyl’s Theorem [4, Th. 3.4] (here Theorem 40) in order to document that it fits in our context. The argumentation is entirely due to Beyl [4, Sec. 3], so we do not claim any originality.

Let A, B and C be abelian categories. Suppose A and B to have enough injectives. Let A ^{\overline{}^{F}} B ^{\overline{}^{G}} C be left exact functors.

7.1 Definition

Let T ∈ Ob C^{[0}(B). In this §7, a CE-resolution of T will synonymously (and not quite correctly) be called an injective CE-resolution, to emphasise the fact that its object entries are injective.

We regard C^{[0}(B) as an exact category as in Remarks 9 and 11.

Definition 37 A double complex B ∈ CC^{⌞}(B) is called a G-acyclic CE-resolution of T if the following conditions are satisfied.

(1) We have H^{0}(B^{∗,−}) ≃ T and H^{k}(B^{∗,−}) ≃ 0 for all k ⩾ 1.

(2) The morphism of complexes B^{k,∗} \overline{}B^{k+1,∗}, consisting of vertical differentials of B, is a pure morphism for all k ⩾ 0.

(3) The object B^{ℓ}(B^{k,∗}) is G-acyclic for all k, ℓ ⩾ 0 .

(4) The object Z^{ℓ}(B^{k,∗}) is G-acyclic for all k, ℓ ⩾ 0 .

A G-acyclic CE-resolution is a G-acyclic CE-resolution of some T ∈ Ob C^{[0}(B).

From (3, 4) and the short exact sequence Z^{ℓ}(B^{k,∗}) \overline{}B^{k,ℓ} \overline{}B^{ℓ+1}(B^{k,∗}), we conclude that B^{k,ℓ} is G-acyclic for all k, ℓ ⩾ 0 .

From (3, 4) and the short exact sequence B^{ℓ}(B^{k,∗}) \overline{}Z^{ℓ}(B^{k,∗}) \overline{}H^{ℓ}(B^{k,∗}), we conclude that H^{ℓ}(B^{k,∗}) is G-acyclic for all k, ℓ ⩾ 0 .

Example 38 An injective CE-resolution of T is in particular a G-acyclic CE-resolution of T.

Note that given Y ∈ Ob C(B) and ℓ ∈ Z, we have Z^{ℓ}GY ≃ GZ^{ℓ}Y, whence the universal property of the cokernel H^{ℓ}GY of GY^{ℓ−1} \overline{}Z^{ℓ}GY induces a morphism H^{ℓ}GY \overline{}GH^{ℓ}Y. This furnishes a transformation H^{ℓ}(GX^{k,∗}) ^{\overline{}^{θX}} GH^{ℓ}(X^{k,∗}), natural in X ∈ Ob CC^{⌞}(B).

Remark 39 If B is a G-acyclic CE-resolution, then H^{ℓ}(GB^{−,∗}) ^{\overline{}^{θB}} GH^{ℓ}(B^{−,∗}) is an isomorphism for all ℓ ⩾ 0.

Proof. The sequences

[figure: GB^{ℓ}(B^{k,∗}); GZ^{ℓ}(B^{k,∗}); GH^{ℓ}(B^{k,∗}); GZ^{ℓ−1}(B^{k,∗}); GB^{k,ℓ−1}; GB^{ℓ}(B^{k,∗})]

are short exact for k, ℓ ⩾ 0 by G-acyclicity of B^{ℓ}(B^{k,∗}) resp. of Z^{ℓ−1}(B^{k,∗}). In particular, the cokernel of GB^{k,ℓ−1} \overline{}GZ^{ℓ}(B^{k,∗}) is given by GH^{ℓ}(B^{k,∗}).

7.2 A theorem of Beyl

Let X ∈ Ob A_{(F,G)}. Let A ∈ Ob C^{[0}(A) be a (F, G)-acyclic resolution of X. Let B ∈ CC^{⌞}(B) be a G-acyclic CE-resolution of FA.

Theorem 40 (Beyl, [4, Th. 3.4]) We have an isomorphism of proper spectral sequences

Ė_{F,G}^{Gr}(X) ≃ Ė_{I}(GB)

in ˙ Z̄_{∞}^{##}, C .

Proof. Since the proper Grothendieck spectral sequence is, up to isomorphism, independent of the choice of an injective CE-resolution, as pointed out in §4.2, our assertion is equivalent to the existence of an injective CE-resolution J of FA such that Ė_{I}(GJ) ≃ Ė_{I}(GB). So by Remark 33, it suffices to show that there exists an injective CE-resolution J of FA and a morphism B \overline{}J that induces a quasiisomorphism H^{ℓ}(GB^{−,∗}) \overline{}H^{ℓ}(GJ^{−,∗}) for all ℓ ⩾ 0. By Remark 39 and Example 38, it suffices to show that GH^{ℓ}(B^{−,∗}) \overline{}GH^{ℓ}(J^{−,∗}) is a quasiisomorphism for all ℓ ⩾ 0.

By the conditions (1, 2) on B and by G-acyclicity of H^{ℓ}(B^{k,∗}) for k, ℓ ⩾ 0, the complex H^{ℓ}(B^{−,∗}) is a G-acyclic resolution of H^{ℓ}(FA); cf. Remark 10.

By Remark 4, there exists J ∈ Ob CC^{⌞}(Inj B) with vertical pure morphisms and split rows, and a morphism B \overline{}J consisting rowwise of pure monomorphisms such that H^{k}(B^{∗,−}) \overline{}H^{k}(J^{∗,−}) is an isomorphism of complexes for all k ⩾ 0. In particular, the composite (Conc_{2} FA \overline{}B \overline{}J) turns J into an injective CE-resolution of FA.

Let ℓ ⩾ 0. Since B is a G-acyclic and J an injective CE-resolution of FA, both Conc H^{ℓ}(FA) \overline{}H^{ℓ}(B^{−,∗}) and Conc H^{ℓ}(FA) \overline{}H^{ℓ}(J^{−,∗}) are quasiisomorphisms. Hence H^{ℓ}(B^{−,∗}) \overline{}H^{ℓ}(J^{−,∗}) is a quasiisomorphism, too. Now Lemma 27 shows that GH^{ℓ}(B^{−,∗}) \overline{}GH^{ℓ}(J^{−,∗}) is a quasiisomorphism as well.

8 Applications

We will apply Theorems 31 and 34 in various algebraic situations. In particular, we will re-prove a theorem of Beyl; viz. Theorem 53 in §8.3.

In several instances below, we will make tacit use of the fact that a left exact functor between abelian categories respects injectivity of objects provided it has an exact left adjoint.

8.1 A Hopf algebra lemma

We will establish Lemma 47 in §8.1.4, needed to prove an acyclicity that enters the proof of the comparison result Theorem 52 in §8.2 for Hopf algebra cohomology, which in turn allows to derive comparison results for group cohomology and Lie algebra cohomology; cf. §§ 8.3, 8.4.

8.1.1 Definition

Let R be a commutative ring. Write ⊗ := ⊗_{R}. A Hopf algebra over R is an R-algebra H together with R-algebra morphisms H ^{\overline{}^{ε}} R (counit) and H ^{\overline{}^{Δ}} H ⊗H (comultiplication), and an R-linear map H ^{\overline{}^{S}} H (antipode) such that the following conditions (i–iv) hold.

Write xΔ = ∑_{i} xu_{i} ⊗ xv_{i} for x ∈ H, where u_{i} and v_{i} are chosen maps from H to H, and where i runs over a suitable indexing set. Note that ∑_{i}(r · x + s · y)u_{i} ⊗ (r · x + s · y)v_{i} = r · (∑_{i} xu_{i} ⊗ xv_{i}) + s · (∑_{i} yu_{i} ⊗ yv_{i}) for x, y ∈ H and r, s ∈ R, whereas u_{i} and v_{i} are not necessarily R-linear maps.

The elegant Sweedler notation [15, §1.2] for the images under Δ(Δ ⊗ 1) etc. led the author, being new to Hopf algebras, to confusion in a certain case. So we will express them in these more naive terms.

Write H ⊗ H ^{\overline{}^{∇}} H, x ⊗ y \overline{}x · y and R ^{\overline{}^{η}} H, r \overline{}r · 1_{H}. Write H ⊗ H ^{\overline{}^{τ}} H ⊗ H, x ⊗ y \overline{}y ⊗ x.

(i) We have Δ(ε ⊗ id_{H}) = (x \overline{}1_{R} ⊗ x), i.e. ∑_{i} xu_{i}ε · xv_{i} = x for x ∈ H.

(i′) We have Δ(id_{H} ⊗ε) = (x \overline{}x ⊗ 1_{R}), i.e. ∑_{i} xu_{i} · xv_{i}ε = x for x ∈ H.

(ii) We have Δ(id_{H} ⊗Δ) = Δ(Δ⊗id_{H}), i.e. ∑_{i,j} xu_{i} ⊗xv_{i}u_{j} ⊗xv_{i}v_{j} = ∑_{i,j} xu_{i}u_{j} ⊗xu_{i}v_{j} ⊗xv_{i} for x ∈ H.

(iii) We have Δ(S ⊗ id_{H})∇ = εη, i.e. ∑_{i} xu_{i}S · xv_{i} = xε · 1_{H} for x ∈ H.

(iii′) We have Δ(id_{H} ⊗S)∇ = εη, i.e. ∑_{i} xu_{i} · xv_{i}S = xε · 1_{H} for x ∈ H.

(iv) We have S^{2} = id_{H}.

In particular, imposing (iv), we stipulate a Hopf algebra to have an involutive antipode.

8.1.2 Some basic properties

In an attempt to be reasonably self-contained, we recall some basic facts on Hopf algebras needed for Lemma 47 below; cf. [15, Ch. IV], [1, §2], [13, §§1-3]. In doing so, we shall use direct arguments.

Suppose given a Hopf algebra H over R.

Remark 41 ([15, Prop. 4.0.1], [1, Th. 2.1.4], [13, 3.4.2]) The following hold.

(1) We have ∑_{i}(x · y)u_{i} ⊗ (x · y)v_{i} = ∑_{i,j}(xu_{i} · yu_{j}) ⊗ (xv_{i} · yv_{j}) for x, y ∈ H.

(2) We have 1_{H}S = 1_{H}.

(3) We have (x · y)S = yS · xS for x, y ∈ H.

(4) We have Sε = ε.

(5) We have Δ(S ⊗ S)τ = SΔ, i.e. ∑_{i} xu_{i}S ⊗ xv_{i}S = ∑_{i} xSv_{i} ⊗ xSu_{i} for x ∈ H.

(6) We have x · y = ∑_{i} (∑_{j}(xu_{i})u_{j} · y · (xu_{i})v_{j}S) · xv_{i} for x, y ∈ H. (6′) We have y · x = ∑_{i} xu_{i} · (∑_{j}(xv_{i})u_{j}S · y · (xv_{i})v_{j}) for x, y ∈ H.

(7) We have ∑_{i} xv_{i} · xu_{i}S = xε · 1_{H} for x ∈ H. (7′) We have ∑_{i} xv_{i}S · xu_{i} = xε · 1_{H} for x ∈ H.

Proof. Ad (1). Given x, y ∈ H, we obtain ∑i (xy)u_{i} ⊗ (xy)v_{i} = (xy)Δ = xΔ · yΔ = _{∑i,j} (xu_{i} · yu_{j}) ⊗ (xv_{i} · yv_{j}) .

Ad (2). Remarking that 1_{H}Δ = 1_{H} ⊗ 1_{H}, we obtain

1_{H}S = 1_{H}Δ(S ⊗ id_{H})∇ =^{(iii)} 1_{H}ε · 1_{H} = 1_{H} .

Ad (3). Given x, y ∈ H, we obtain

(x · y)S =^{2 × (i′)} ∑_{i,k}(xu_{i} · xv_{i}ε · yu_{k} · yv_{k}ε)S
=^{(iii′)} ∑_{i,j,k}(xu_{i} · yu_{k} · yv_{k}ε)S · xv_{i}u_{j} · xv_{i}v_{j}S
=^{(iii′)} ∑_{i,j,k,ℓ}(xu_{i} · yu_{k})S · xv_{i}u_{j} · yv_{k}u_{ℓ} · yv_{k}v_{ℓ}S · xv_{i}v_{j}S
=^{2 × (ii)} ∑_{i,j,k,ℓ}(xu_{i}u_{j} · yu_{k}u_{ℓ})S · xu_{i}v_{j} · yu_{k}v_{ℓ} · yv_{k}S · xv_{i}S
=^{(1)} ∑_{i,j,k}(xu_{i} · yu_{k})u_{j}S · (xu_{i} · yu_{k})v_{j} · yv_{k}S · xv_{i}S
=^{(iii)} ∑_{i,k}(xu_{i} · yu_{k})ε · yv_{k}S · xv_{i}S
= ∑_{i,k}(yu_{k}ε · yv_{k})S · (xu_{i}ε · xv_{i})S
=^{2 × (i)} yS · xS .

Ad (4). Note that (yε · z)ε = yε · zε = (y · z)ε for y, z ∈ H. Given x ∈ H, we obtain

xSε =^{(i)} (_{∑i} xu_{i}ε · xv_{i})Sε = (_{∑i} xu_{i}ε · xv_{i}S)ε = (_{∑i} xu_{i} · xv_{i}S)ε =^{(iii′)} (xε · 1_{H})ε = xε .

Ad (5). Given x ∈ H, we obtain

xΔ(S ⊗ S)τ =^{(i)} ∑_{i}(xu_{i}ε · xv_{i})Δ(S ⊗ S)τ
= ∑_{i}(xu_{i}ε · 1_{H})Δ · xv_{i}Δ(S ⊗ S)τ
=^{(iii)} ∑_{i,j}(xu_{i}u_{j}S · xu_{i}v_{j})Δ · xv_{i}Δ(S ⊗ S)τ
= ∑_{i,j} xu_{i}u_{j}SΔ · xu_{i}v_{j}Δ · xv_{i}Δ(S ⊗ S)τ
=^{(ii)} ∑_{i,j} xu_{i}SΔ · xv_{i}u_{j}Δ · xv_{i}v_{j}Δ(S ⊗ S)τ
= ∑_{i,j,k,ℓ} xu_{i}SΔ · (xv_{i}u_{j}u_{k} ⊗ xv_{i}u_{j}v_{k}) · (xv_{i}v_{j}v_{ℓ}S ⊗ xv_{i}v_{j}u_{ℓ}S)
= ∑_{i,j,k,ℓ} xu_{i}SΔ · (xv_{i}u_{j}u_{k} · xv_{i}v_{j}v_{ℓ}S ⊗ xv_{i}u_{j}v_{k} · xv_{i}v_{j}u_{ℓ}S)
=^{(ii)} ∑_{i,j,k,ℓ} xu_{i}SΔ · (xv_{i}u_{j} · xv_{i}v_{j}v_{k}v_{ℓ}S ⊗ xv_{i}v_{j}u_{k} · xv_{i}v_{j}v_{k}u_{ℓ}S)
=^{(ii)} ∑_{i,j,k,ℓ} xu_{i}SΔ · (xv_{i}u_{j} · xv_{i}v_{j}v_{k}S ⊗ xv_{i}v_{j}u_{k}u_{ℓ} · xv_{i}v_{j}u_{k}v_{ℓ}S)
=^{(iii′)} ∑_{i,j,k} xu_{i}SΔ · (xv_{i}u_{j} · xv_{i}v_{j}v_{k}S ⊗ xv_{i}v_{j}u_{k}ε · 1_{H})
= ∑_{i,j,k} xu_{i}SΔ · (xv_{i}u_{j} · (xv_{i}v_{j}v_{k} · xv_{i}v_{j}u_{k}ε)S ⊗ 1_{H})
=^{(i)} ∑_{i,j} xu_{i}SΔ · (xv_{i}u_{j} · xv_{i}v_{j}S ⊗ 1_{H})
=^{(iii′)} ∑_{i} xu_{i}SΔ · (xv_{i}ε · 1_{H} ⊗ 1_{H})
= ∑_{i}(xu_{i} · xv_{i}ε)SΔ
=^{(i′)} xSΔ .

Ad (6). Given x, y ∈ H, we obtain

x · y =^{(i′)} _{∑i} xu_{i} · y · xv_{i}ε =^{(iii)} _{∑i,j} xu_{i} · y · xv_{i}u_{j}S · xv_{i}v_{j} =^{(ii)} _{∑i,j} xu_{i}u_{j} · y · xu_{i}v_{j}S · xv_{i} .

Ad (6′). Given x ∈ H, we obtain

y · x =^{(i)} _{∑i} xu_{i}ε · y · xv_{i} =^{(iii′)} _{∑i,j} xu_{i}u_{j} · xu_{i}v_{j}S · y · xv_{i} =^{(ii)} _{∑i,j} xu_{i} · xv_{i}u_{j}S · y · xv_{i}v_{j} .

Ad (7). Given x ∈ H, we have

∑i xv_{i}·xu_{i}S =^{(iv)} _{∑i} xS^{2}v_{i}·xS^{2}u_{i}S =^{(5)} _{∑i} xSu_{i}S·xSv_{i}S^{2} =^{(iv)} _{∑i} xSu_{i}S·xSv_{i} =^{(iii)} xSε·1_{H} =^{(4)} xε·1_{H} .

Ad (7′). Given x ∈ H, we have

∑i xv_{i}S·xu_{i} =^{(iv)} _{∑i} xS^{2}v_{i}S·xS^{2}u_{i} =^{(5)} _{∑i} xSu_{i}S^{2}·xSv_{i}S =^{(iv)} _{∑i} xSu_{i}·xSv_{i}S =^{(iii′)} xSε·1_{H} =^{(4)} xε·1_{H} .

In the present §8.1, we shall refer to the assertions Remark 41.(1–7′) just by (1–7′).

8.1.3 Normality

Suppose given a Hopf algebra H over R, and an R-subalgebra K ⊆ H. Suppose H and K to be flat as modules over R.

Note that K ⊗ K \overline{}H ⊗ H is injective. We will identify K ⊗ K with its image.

The R-subalgebra K ⊆ H is called a Hopf-subalgebra if KΔ ⊆ K ⊗ K and KS ⊆ K. In this case, we may and will suppose the maps u_{i} and v_{i} to restrict to maps from K to K.

Suppose K ⊆ H to be a Hopf-subalgebra. It is called normal, if for all a ∈ K and all x ∈ H, we have

∑i xu_{i} · a · xv_{i}S ∈ K  and  _{∑i} xu_{i}S · a · xv_{i} ∈ K .

An ideal I ⊆ H is called a Hopf ideal if IΔ ⊆ I ⊗ H + H ⊗ I (where we have identified I ⊗ H and H ⊗ I with their images in H ⊗ H), Iε = 0 and IS ⊆ I. In this case, the quotient H/I carries a Hopf algebra structure via

[figure: H/I; ε; R ,; x + I; xε; H/I; Δ; H/I ⊗ H/I ,; x + I; ∑_{i}(xu_{i} + I) ⊗ (xv_{i} + I); H/I; S; H/I ,; x + I; xS + I .]

Suppose K ⊆ H to be a normal Hopf subalgebra. Write K^{+} := Kern(K ^{\overline{}^{ε}} R). By (6, 6′, 3, 4) and by writing

kΔ = (_{∑i}(ku_{i} − ku_{i}ε) ⊗ kv_{i}) + 1 ⊗ k

for k ∈ K^{+}, the ideal HK^{+} = K^{+}H is a Hopf ideal in H.

8.1.4 Some remarks and a lemma

Suppose given a Hopf algebra H over R and a normal Hopf-subalgebra K ⊆ H. Suppose H and K to be flat as modules over R.

Write H̄ := H/HK^{+}. Given x ∈ H, write x̄ := x + HK^{+} ∈ H̄ for its residue class.

Let N′, N, M, M′ and Q be H-modules. Let P be an H̄-module, which we also consider as an H-module via H \overline{}H̄, x \overline{}x̄.

We write _{K}(N, M) = _{K}(N|_{K}, M|_{K}) for the R-module of K-linear maps from N to M.

Remark 42 Given f ∈ _{R}(N, M) and x ∈ H, we define x · f ∈ _{R}(N, M) by

[n](x · f) := _{∑i} xu_{i} · [xv_{i}S · n]f

for n ∈ N. This defines a left H-module structure on _{R}(N, M).

Formally, squared brackets mean the same as parentheses. Informally, squared brackets are to accentuate the arguments of certain maps.

Proof. We claim that x′ · (x · f) = (x′ · x) · f for x, x′ ∈ H. Suppose given n ∈ N. We obtain

[n](x′ · (x · f)) = ∑_{i} x′u_{i} · [x′v_{i}S · n](x · f)
= ∑_{i,j} x′u_{i} · xu_{j} · [xv_{j}S · x′v_{i}S · n]f
=^{(3)} ∑_{i,j}(x′u_{i} · xu_{j}) · [(x′v_{i} · xv_{j})S · n]f
=^{(1)} ∑_{i}(x′ · x)u_{i} · [(x′ · x)v_{i}S · n]f
= [n]((x′ · x) · f) .

We claim that 1_{H} · f = f. Suppose given n ∈ N. We obtain

[n](1_{H} · f) = _{∑i} 1_{H}u_{i} · [1_{H}v_{i}S · n]f = 1_{H} · [1_{H}S · n]f =^{(2)} [n]f ,

remarking that 1_{H}Δ = 1_{H} ⊗ 1_{H} .

I owe to G. Hiß the hint to improve a previous weaker version of Corollary 45 below by means of the following Remark 43.

Denote by

M^{K} := {m ∈ M : a · m = aε · m for all a ∈ K}

the fixed point module of M under K. Remark 43 Letting x̄ · m := x · m for x ∈ H and m ∈ M^{K}, we define an H̄-module structure on M^{K}. Proof. The value of the product x̄ · m does not depend on the chosen representative x of x̄ since, given y ∈ H, a ∈ K^{+} and m ∈ M^{K}, we have

y · a · m = y · aε · m = 0 .

It remains to be shown that given x ∈ H and m ∈ M^{K}, the element x · m lies in M^{K}. In fact, given a ∈ K, we obtain

a · x · m =^{(6′)} ∑_{i} xu_{i} · (∑_{j}(xv_{i})u_{j}S · a · (xv_{i})v_{j}) · m
= ∑_{i} xu_{i} · (∑_{j}(xv_{i})u_{j}S · a · (xv_{i})v_{j})ε · m
= ∑_{i,j} xu_{i} · xv_{i}u_{j}Sε · aε · xv_{i}v_{j}ε · m
=^{(4)} ∑_{i,j} xu_{i} · xv_{i}u_{j}ε · aε · xv_{i}v_{j}ε · m
=^{(ii)} ∑_{i,j} xu_{i}u_{j} · xu_{i}v_{j}ε · aε · xv_{i}ε · m
=^{(i′)} ∑_{i} xu_{i} · aε · xv_{i}ε · m
=^{(i′)} aε · x · m .

Remark 44 We have (_{R}(N, M))^{K} = _{K}(N, M), as subsets of _{R}(N, M).

Proof. The module (_{R}(N, M))^{K} consists of the R-linear maps N ^{\overline{}^{f}} M that satisfy

∑i xu_{i} · [xv_{i}S · n]f = xε · [n]f .

for x ∈ H and n ∈ N. The module _{K}(N, M) consists of the R-linear maps N ^{\overline{}^{f}} M that satisfy

[x · n]f = x · [n]f

for x ∈ H and n ∈ N. By (iii′), we have (_{R}(N, M))^{K} ⊇ _{K}(N, M). It remains to show that (_{R}(N, M))^{K} ⊆ _{K}(N, M). Given f ∈ (_{R}(N, M))^{K}, x ∈ H and n ∈ N, we obtain

x · [n]f =^{(i′)} ∑_{i} xu_{i} · xv_{i}ε · [n]f
= ∑_{i} xu_{i} · [xv_{i}ε · n]f
=^{(iii)} ∑_{i,j} xu_{i} · [xv_{i}u_{j}S · xv_{i}v_{j} · n]f
=^{(ii)} ∑_{i,j} xu_{i}u_{j} · [xu_{i}v_{j}S · xv_{i} · n]f
= ∑_{i} xu_{i}ε · [xv_{i} · n]f
=^{(i)} [x · n]f .

Corollary 45 Given f ∈ _{K}(N, M) and x ∈ H, we define x̄ · f ∈ _{K}(N, M) by

[n](x̄ · f) := _{∑i} xu_{i} · [xv_{i}S · n]f

for n ∈ N. This defines a left H̄-module structure on _{K}(N, M). Proof. By Remark 42, we may apply Remark 43 to _{R}(N, M). By Remark 44, the assertion follows. Remark 46 Given f ∈ _{K}(N, M), x ∈ H, and H-linear maps N′ ^{\overline{}^{ν}} N, M ^{\overline{}^{μ}} M′, we obtain

ν(x̄ · f)μ = x̄ · (νfμ) .

Proof. Given n′ ∈ N′, we obtain [n′](ν(x̄ · f)μ) = (_{∑i} xu_{i} · [xv_{i}S · n′ν]f)μ = _{∑i} xu_{i} · [xv_{i}S · n′](νfμ) = [n′](x̄ · (νfμ)) .

The following Lemma 47 has been suggested by the referee, and has been achieved with the help of G. Carnovale. It is reminiscent of [16, Cor. 4.3], but easier. It resembles a bit a Fourier inversion.

Note that the right H̄-module structure on H̄ induces a left H̄-module structure on _{R}(H̄, M). Lemma 47 We have the following mutually inverse isomorphisms of H̄-modules.

\begin{matrix}_{K}(H, M) & \overline{}^{Φ}_{∼} & _{R}(H̄, M) \\ f & \overline{} & (x̄ \overline{}∑_{i} xu_{i} · [xv_{i}S]f) \\ _{K}(H, M) & \overline{}^{Ψ}_{∼} & _{R}(H̄, M) \\ (x \overline{}∑_{j} xv_{j} · [ \overline{xu_{j}S}]g) & \overline{} & g \end{matrix}

Proof. We claim that Φ is a welldefined map. We have to show that fΦ is welldefined, i.e. that its value at x̄ does not depend on the representing element x. Suppose given y ∈ H and a ∈ K^{+}. We obtain

∑_{i}(ya)u_{i} · [(ya)v_{i}S]f =^{(1)} ∑_{i,j} yu_{i} · au_{j} · [(yv_{i} · av_{j})S]f
=^{(3)} ∑_{i,j} yu_{i} · au_{j} · [av_{j}S · yv_{i}S]f
= ∑_{i,j} yu_{i} · au_{j} · av_{j}S · [yv_{i}S]f
=^{(iii′)} ∑_{i} yu_{i} · aε · [yv_{i}S]f
= 0 .

We claim that Φ is H̄-linear. Suppose given y ∈ H and x ∈ H. We obtain

[x̄]((ȳf)Φ) = ∑_{i} xu_{i} · [xv_{i}S](ȳf)
= ∑_{i,j} xu_{i} · yu_{j} · [yv_{j}S · xv_{i}S]f
=^{(3)} ∑_{i,j} xu_{i} · yu_{j} · [(xv_{i} · yv_{j})S]f
=^{(1)} ∑_{i}(x · y)u_{i} · [(x · y)v_{i}S]f
= [x̄](ȳ(fΦ)) .

We claim that Ψ is a welldefined map. We have to show that gΨ is K-linear. Suppose given a ∈ K and x ∈ H. Note that au_{i} ∈ K for all i, whence also au_{i}S ∈ K, and therefore au_{i}S ≡_{HK^{+}} au_{i}Sε · 1_{H}. We obtain

[a · x](gΨ) = ∑_{j}(a · x)v_{j} · [ \overline{(a · x)u_{j}S}]g
=^{(1)} ∑_{i,j} av_{i} · xv_{j} · [ \overline{(au_{i} · xu_{j})S}]g
=^{(3)} ∑_{i,j} av_{i} · xv_{j} · [ \overline{xu_{j}S} · \overline{au_{i}S}]g
= ∑_{i,j} av_{i} · xv_{j} · [ \overline{xu_{j}S} · \overline{au_{i}Sε}]g
=^{(4)} ∑_{i,j} au_{i}ε · av_{i} · xv_{j} · [ \overline{xu_{j}S}]g
=^{(i)} ∑_{j} a · xv_{j} · [ \overline{xu_{j}S}]g
= a · [x](gΨ) .

We claim that ΦΨ = id_{_{K}(H,M)}. Suppose given x ∈ H. We obtain

[x](fΦΨ) = ∑_{j} xv_{j} · [ \overline{xu_{j}S}](fΦ)
= ∑_{i,j} xv_{j} · xu_{j}Su_{i} · [xu_{j}Sv_{i}S]f
=^{(5)} ∑_{i,j} xv_{j} · xu_{j}v_{i}S · [xu_{j}u_{i}S^{2}]f
=^{(iv)} ∑_{i,j} xv_{j} · xu_{j}v_{i}S · [xu_{j}u_{i}]f
=^{(ii)} ∑_{i,j} xv_{j}v_{i} · xv_{j}u_{i}S · [xu_{j}]f
=^{(7)} ∑_{j} xv_{j}ε · [xu_{j}]f
=^{(i)} [x]f .

We claim that ΨΦ = id_{R}_{(H̄,M)}. Suppose given x ∈ H. We obtain

[x̄](gΨΦ) = ∑_{i} xu_{i} · [xv_{i}S](gΨ)
= ∑_{i,j} xu_{i} · xv_{i}Sv_{j} · [ \overline{xv_{i}Su_{j}S}]g
=^{(5)} ∑_{i,j} xu_{i} · xv_{i}u_{j}S · [ \overline{xv_{i}v_{j}S^{2}} ]g
=^{(iv)} ∑_{i,j} xu_{i} · xv_{i}u_{j}S · [ \overline{xv_{i}v_{j}}]g
=^{(ii)} ∑_{i,j} xu_{i}u_{j} · xu_{i}v_{j}S · [ \overline{xv_{i}}]g
=^{(iii′)} ∑_{i} xu_{i}ε · [ \overline{xv_{i}}]g
=^{(i)} [x̄]g .

Finally, it follows by H̄-linearity of Φ and by Ψ = Φ^{−1} that Ψ is H̄-linear.

The tensor product N ⊗ M is an H-module via Δ. Note that R is an H-module via ε. Note that R ⊗ M ≃ M ≃ M ⊗ R as H-modules by (i, i′).

Remark 48 (cf. [3, Lemma 3.5.1]) We have mutually inverse isomorphisms of R-modules

\begin{matrix}H̄(P, _{K}(Q, M)) & \overline{}^{α}_{∼} & _{H}(P ⊗ Q, M) \\ f & \overline{} & (p ⊗ q \overline{}[q](pf)) \\ H̄(P, _{K}(Q, M)) & \overline{}^{β}_{∼} & _{H}(P ⊗ Q, M) \\ (p \overline{}(q \overline{}[p ⊗ q]g)) & \overline{} & g , \end{matrix}

natural in P ∈ Ob H̄-Mod, Q ∈ Ob H-Mod and M ∈ Ob H-Mod.

Proof. We claim that α is welldefined. We have to show that fα is H-linear. Suppose given x ∈ H. We obtain

x · (p ⊗ q) = ∑_{i} \overline{xu_{i}} · p ⊗ xv_{i} · q
\overline{}^{fα} ∑_{i}[xv_{i} · q]((\overline{xu_{i}} · p)f)
= ∑_{i}[xv_{i} · q](\overline{xu_{i}} · (pf))
= ∑_{i,j} xu_{i}u_{j} · [xu_{i}v_{j}S · xv_{i} · q](pf)
=^{(ii)} ∑_{i,j} xu_{i} · [xv_{i}u_{j}S · xv_{i}v_{j} · q](pf)
=^{(iii)} ∑_{i} xu_{i} · [xv_{i}ε · q](pf)
=^{(i′)} x · [q](pf)
= x · [p ⊗ q](fα) .

We claim that β is welldefined. First, we have to show that [p](gβ) is K-linear. Suppose given a ∈ K. We obtain

a · q ^{\overline{}^{[p](gβ)}} [p ⊗ a · q]g =^{(i)} _{∑i} [\overline{au_{i}ε} · p ⊗ av_{i} · q]g = _{∑i} [\overline{au_{i}} · p ⊗ av_{i} · q]g = a · [p ⊗ q]g .

Second, we have to show that gβ is H̄-linear. Suppose given x ∈ H. We obtain

x̄ · p ^{\overline{}^{gβ}} (q \overline{}[x̄ · p ⊗ q]g)
=^{(i)} (q \overline{}∑_{i}[\overline{xu_{i} · xv_{i}ε} · p ⊗ q)]g)
=^{(iii′)} (q \overline{}∑_{i,j}[\overline{xu_{i}} · p ⊗ xv_{i}u_{j} · xv_{i}v_{j}S · q)]g)
=^{(ii)} (q \overline{}∑_{i,j}[\overline{xu_{i}u_{j}} · p ⊗ xu_{i}v_{j} · xv_{i}S · q)]g)
= (q \overline{}∑_{i} xu_{i} · [p ⊗ xv_{i}S · q]g)
= x̄ · (q \overline{}[p ⊗ q]g) .

Finally, α and β are mutually inverse.

Corollary 49 We have H̄(P, M^{K}) ≃ H̄(P, _{K}(R, M)) ≃ _{H}(P, M) as R-modules, natural in P and M.

Proof. Note that M ≃ _{R}(R, M) as H-modules, whence M^{K} ≃ _{K}(R, M) as H̄-modules by Remarks 43, 44. Now the assertion follows from Remark 48, letting Q = R.

8.2 Comparing Hochschild-Serre-Hopf with Grothendieck

Let R be a commutative ring. Suppose given a Hopf algebra H over R (with involutive antipode) and a normal Hopf-subalgebra K ⊆ H; cf. §8.1.3. Write H̄ := H/HK^{+}. Suppose H, K and H̄ to be projective as modules over R. Suppose H to be projective as a module over K.

Let B ∈ Ob C(H-Mod) be a projective resolution of R over H. Let B̄ ∈ Ob C(H̄-Mod) be a projective resolution of R over H̄. Note that since H̄ is projective over R, B̄|_{R} ∈ Ob C(R-Mod) is a projective resolution of R over R. Let M be an H-module.

By Corollary 45 and by Remark 46, we have a biadditive functor

(H-Mod)^{∘} × H-Mod ^{\overline{}^{U}}  H̄-Mod
(X , X′)  \overline{}U(X, X′) := _{K}(X, X′) .

Write

(H̄-Mod)^{∘} × H̄-Mod ^{\overline{}^{V}} R-Mod
(Y , Y′)  \overline{}V (Y, Y′) := H̄(Y, Y′)

for the usual Hom-functor.

In particular, we shall consider the functors

[figure: H-Mod; U(R,−); H̄-Mod; V (R,−); R-Mod]
X  \overline{}U(R, X) ≃ X^{K}
Y  \overline{}V (R, Y) ≃ Y H̄ .

On the other hand, we shall consider the double complex

D(M) = D^{−,=}(M) := V (B̄_{−} , U(B_{=} , M)) = H̄(B̄_{−} , _{K}(B_{=} , M)) .

Note that D(M) is isomorphic in CC^{⌞}(R-Mod) to _{H}(B̄_{−} ⊗_{R} B_{=} , M), naturally in M; cf. Remark 48.

Lemma 50 The H̄-module U(H, M) is V (R, −)-acyclic.

Proof. By Lemma 47, this amounts to showing that _{R}(H̄, M) is V (R, −)-acyclic, which in turn amounts to showing that V (B̄ , _{R}(H̄, M)) = H̄(B̄ , _{R}(H̄, M)) has vanishing cohomology in degrees ⩾ 1. Now,

H̄(B̄ , _{R}(H̄, M)) ≃ _{R}(H̄ ⊗H̄ B̄ , M) ≃ _{R}(B̄ , M) ,

whose cohomology in degree i ⩾ 1 is Ext_{R}^{i}(R, M) ≃ 0.

Lemma 51 Given a projective H-module P, the H̄-module U(P, M) is V (R, −)-acyclic.

Proof. It suffices to show that U(∐_{Γ} H, M) ≃ ∏_{Γ} U(H, M) is V (R, −)-acyclic for any indexing set Γ. By Lemma 50, it remains to be shown that R^{i}V (R, ∏_{Γ} Y) is isomorphic to ∏_{Γ} R^{i}V (R, Y) for a given H̄-module Y and for i ⩾ 1. Having chosen an injective resolution J of Y, we may choose the injective resolution ∏_{Γ} J of ∏_{Γ} Y. Then

R^{i}V (R, ∏_{Γ} Y) ≃ H^{i}V (R, ∏_{Γ} J) ≃ H^{i} ∏_{Γ} V (R, J) ≃ ∏_{Γ} H^{i}V (R, J) ≃ ∏_{Γ} R^{i}V (R, Y) .

Theorem 52 The proper spectral sequences

Ė_{I}(D(M))  and  Ė_{U(R,−), V (R,−)}^{Gr}(M)

are isomorphic (in ˙ Z̄_{∞}^{##}, R-Mod ), naturally in M ∈ Ob H-Mod.

Proof. To apply Theorem 31 with, in the notation of §5.1,

(A × A′ ^{^{F}\overline{}}B ^{\overline{}^{G}} C) = ((H-Mod)^{∘} × H-Mod ^{\overline{}^{U}} H̄-Mod ^{\overline{}^{V (R,−)}} R-Mod ) ,

and with X = R and X′ = M, we verify the conditions (a–d′) of loc. cit. in this case.

Ad (c). We claim that B is a (U(−, M), V (R, −))-acyclic resolution of R. We have to show that U(B_{i}, M) is V (R, −)-acyclic for i ⩾ 0; cf. §4.2. Since B_{i} is projective over H, this follows by Lemma 51. This proves the claim.

Ad (c′). Let I be an injective resolution of M over H. We claim that I is a (U(R, −), V (R, −))- acyclic resolution of M. We have to show that U(R, I^{i}) is V (R, −)-acyclic for i ⩾ 0. In fact, by Corollary 49, U(R, I^{i}) is an injective H̄-module. This proves the claim.

Ad (d, d′). We claim that U(B_{i}, −) and U(−, I^{i}) are exact for i ⩾ 0; cf. §5.1. The former follows from H being projective over K. The latter is a consequence of I^{i}|_{K} being injective in K-Mod by exactness of K-Mod ^{\overline{}^{H⊗_{K}−}} H-Mod. This proves the claim.

So an application of Theorem 31 yields

Ė_{U(R,−),V (R,−)}^{Gr}(M) ≃ Ė_{U(−,M),V (R,−)}^{Gr}(R) .

To apply Theorem 34 with, in the notation of §6.1,

(A ^{^{F}\overline{}}B′ , B ×B′ ^{\overline{}^{G}} C) = ((H-Mod)^{∘} ^{\overline{}^{U(−,M)}} H̄-Mod , (H̄-Mod)^{∘} ×H̄-Mod ^{^{V}\overline{}}C) , and with X = R and Y = R, we verify the conditions (a–e) of loc. cit. in this case.

Ad (c). We have already remarked that B is a (U(−, M), V (R, −))-acyclic resolution of R.

Ad (d). As a resolution of R over H̄, we choose B̄.

So an application of Theorem 34 yields

Ė_{U(−,M),V (R,−)}^{Gr}(R) ≃ Ė_{I}(V(B̄_{−} , U(B_{=} , M))) .

Naturality in M ∈ Ob H-Mod remains to be shown. Suppose given M ^{\overline{}^{m}} M̃ in H-Mod. Note that the requirements of §5.2 are met. By Proposition 32, with roles of A and A′ interchanged, we have the following commutative quadrangle.

\xymatrix{Ė_{U(R,−),V (R,−)}^{Gr}(M) \ar[r]^{Ė_{U(R,−),V (R,−)}^{Gr}(m)} & Ė_{U(R,−),V (R,−)}^{Gr}(M̃) \\ Ė_{U(−,M),V (R,−)}^{Gr}(R) \ar[r]^{h_{U(−,m)}^{I}R} \ar[u]^{≀} & Ė_{U(−,M̃),V (R,−)}^{Gr}(R) \ar[u]_{≀}}

Note that the requirements of §6.2 are met. By Lemma 36, we have the following commutative quadrangle.

\xymatrix{Ė_{U(−,M),V (R,−)}^{Gr}(R) \ar[r]^{h_{U(−,m)}^{I}R} \ar[d]_{≀} & Ė_{U(−,M̃),V (R,−)}^{Gr}(R) \ar[d]^{≀} \\ Ė_{I}(V(B̄_{−} , U(B_{=} , M))) \ar[r]^{Ė_{I}(V (B̄_{−}, U(B_{=},m)))} & Ė_{I}(V(B̄_{−} , U(B_{=} , M̃)))}

8.3 Comparing Lyndon-Hochschild-Serre with Grothendieck

Let R be a commutative ring. Let G be a group and let N ⊴ G be a normal subgroup. Write Ḡ := G/N. Let M be an RG-module. Write Bar_{G;R} ∈ Ob C(RG -Mod) for the bar resolution of R over RG, having (Bar_{G;R})_{i} = RG^{⊗(i+1)} for i ⩾ 0, the tensor product being taken over R.

Note that RG is a Hopf algebra over R via

[figure: RG; Δ; RG ⊗ RG ,; g; g ⊗ g; RG; S; RG ,; g; g^{−1}; RG; ε; R ,; g; 1 ,]

where g ∈ G; cf. §8.1.1. Moreover, RN is a normal Hopf subalgebra of RG such that RG/(RG)(RN)^{+} ≃ RḠ; cf. §8.1.3.

Note that RG, RN and RḠ are projective over R, and that RG is projective over RN.

We have functors RG -Mod ^{\overline{}^{(−)^{N}}} RḠ -Mod ^{\overline{}^{(−)Ḡ}} R-Mod, taking respective fixed points.

Theorem 53 (Beyl, [4, Th. 3.5]) The proper spectral sequences

Ė_{(−)^{N} , (−)Ḡ}^{Gr}(M)  and  Ė_{I}(_{RG}((Bar_{Ḡ;R})_{−} ⊗_{R} (Bar_{G;R})_{=} , M))

are isomorphic (in ˙ Z̄_{∞}^{##}, R-Mod ), naturally in M ∈ Ob RG -Mod.

Beyl uses his Theorem 40 to prove Theorem 53. We shall re-derive it from Theorem 52, which in turn relies on the Theorems 31 and 34.

Proof. This follows by Theorem 52.

8.4 Comparing Hochschild-Serre with Grothendieck

Let R be a commutative ring. Let 𝔤 be a Lie algebra over R that is free as an R-module. Let 𝔫 ⊴ 𝔤 be an ideal such that 𝔫 and 𝔤̄ := 𝔤/𝔫 are free as R-modules. Let M be a 𝔤-module, i.e. a U(𝔤)-module. Write Bar_{𝔤;R} ∈ Ob C(U(𝔤) -Mod) for the Chevalley-Eilenberg resolution of R over U(𝔤), having (Bar_{𝔤;R})_{i} = U(𝔤) ⊗_{R} ∧^{i}𝔤 for i ⩾ 0; cf. [5, XIII.§7] or [18, Th. 7.7.2].

Note that U(𝔤) is a Hopf algebra over R via

[figure: U(𝔤); Δ; U(𝔤) ⊗ U(𝔤) ,; g; g ⊗ 1 + 1 ⊗ g; U(𝔤); S; U(𝔤) ,; g; −g; U(𝔤); ε; R ,; g; 0 ,]

where g ∈ 𝔤; cf. §8.1.1.

Note that U(𝔤), U(𝔫) and U(𝔤̄) are projective over R, and that U(𝔤) is projective over U(𝔫); cf. [18, Cor. 7.3.9].

We have functors U(𝔤) -Mod ^{\overline{}^{(−)^{𝔫}}} U(𝔤̄) -Mod ^{\overline{}^{(−)𝔤̄}} R-Mod, taking respective annihilated submodules; cf. [18, p. 221].

Theorem 54 The proper spectral sequences

Ė_{(−)^{𝔫}, (−)𝔤̄}^{Gr}(M)  and  Ė_{I}(_{U(𝔤)}((Bar_{𝔤̄;R})_{−} ⊗_{R} (Bar_{𝔤;R})_{=} , M))

are isomorphic (in ˙ Z̄_{∞}^{##}, R-Mod ), naturally in M ∈ Ob U(𝔤) -Mod.

Cf. Barnes, [2, Sec. IV.4, Ch. VII].

Proof. This follows by Theorem 52.

8.5 Comparing two spectral sequences for a change of rings

The following application is taken from [5, XVI.§6].

Let R be a commutative ring. Let A ^{\overline{}^{φ}} B be a morphism of R-algebras. Consider the functors A-Mod ^{\overline{}^{^{A}(B,−)}} B-Mod and (B-Mod)^{∘} × B-Mod ^{\overline{}^{^{B}(−,=)}} R-Mod.

Let X be an A-module, let Y be a B-module.

We shall compare two spectral sequences with E_{2}-terms Ext_{B}^{i}(Y, Ext_{A}^{j}(B, X)), converging to Ext_{A}^{i+j}(Y, X). If one views X⇑_{A}^{B} := _{A}(B, X) as a way to induce from A-Mod to B-Mod, this measures the failure of the Eckmann-Shapiro-type formula Ext_{B}^{i}(Y, X⇑_{A}^{B}) ≃^{?} Ext_{A}^{i}(Y, X), which holds if B is projective over A.

Let I ∈ Ob C^{[0}(A-Mod) be an injective resolution of X. Let P ∈ Ob C^{[0}(B-Mod) be a projective resolution of Y.

Proposition 55 The proper spectral sequences

Ė_{_{A}(B,−), _{B}(Y,−)}^{Gr}(X)  and  Ė_{I}(_{B}(P_{−}, _{A}(B, I^{=})))

are isomorphic (in ˙ Z̄_{∞}^{##}, R-Mod ).

Proof. To apply Theorem 34, if suffices to remark that for each injective A-module I′, the B-module _{A}(B, I′) is injective, and thus _{B}(Y, −)-acyclic.

Remark 56 The functor _{A}(B, −) can be replaced by _{A}(M, −), where M is an A-B-bimodule that is flat over B.

8.6 Comparing two spectral sequences for Ext and ⊗

Let R be a commutative ring. Let S be a ring. Let A be an R-algebra. Let M be an R-S-bimodule. Let X and X′ be A-modules. Assume that X is flat over R. Assume that Ext_{R}^{i}(M, X′) ≃ 0 for i ⩾ 1.

Example 57 Let T be a discrete valuation ring, with maximal ideal generated by t. Let R = T/t^{ℓ} for some ℓ ⩾ 1. Let S = T/t^{k}, where 1 ⩽ k ⩽ ℓ. Let G be a finite group, and let A = RG. Let M = S. Let X and X′ be RG-modules that are both finitely generated and free over R.

Consider the functors

(A-Mod)^{∘} × A-Mod ^{\overline{}^{^{A}(−,=)}} R-Mod ^{\overline{}^{^{R}(M,−)}} S -Mod

Proposition 58 The proper Grothendieck spectral sequences

Ė_{_{A}(X,−), _{R}(M,−)}^{Gr}(X′)  and  Ė_{_{A}(−,X′), _{R}(M,−)}^{Gr}(X)

are isomorphic (in ˙ Z̄_{∞}^{##}, S -Mod ).

Both have E_{2}-terms Ext_{R}^{i}(M, Ext_{A}^{j}(X, X′)) and converge to Ext_{A}^{i+j}(X ⊗_{R}M, X′). In particular, in the situation of Example 57, both have E_{2}-terms Ext_{R}^{i}(S, Ext_{RG}^{j}(X, X′)) and converge to Ext_{RG}^{i+j}(X/t^{k}, X′).

Proof of Proposition 58. To apply Theorem 31, we comment on the conditions in §5.1.

(c) Given a projective A-module P, we want to show that the R-module _{A}(P, X′) is R(M, −)-acyclic. We may assume that P = A, which is to be viewed as an A-R-bimodule. Now, we have Ext_{R}^{i}(M, _{A}(A, X′)) ≃ Ext_{R}^{i}(M, X′) ≃ 0 for i ⩾ 1 by assumption.

(c′) Given an injective A-module I′, the R-module _{A}(X, I′) is injective since X is flat over R by assumption.

8.7 Comparing two spectral sequences for Ext of sheaves

Let T ^{\overline{}^{f}} S be a flat morphism of ringed spaces, i.e. suppose that

O_{T} ⊗_{f^{−1}O_{S}} − : f^{−1}O_{S} -Mod \overline{}O_{T} -Mod

is exact. Consequently, f^{∗} : O_{S} -Mod \overline{}O_{T} -Mod is exact.

Given O_{S}-modules F and F′, we abbreviate _{O_{S}}(F, F′) := Hom_{O_{S}}(F, F′) ∈ Ob R-Mod and O_{S}((F, F′)) := Hom_{O_{S}}(F, F′) ∈ Ob O_{S} -Mod.

Let F be an O_{S}-module that has a locally free resolution B ∈ Ob C(O_{S} -Mod); cf. [9, Prop. III.6.5]. Let G ∈ Ob O_{T} -Mod. Let A ∈ Ob C^{[0}(O_{T} -Mod) be an injective resolution of G.

Consider the functors O_{T} -Mod ^{\overline{}^{f_{∗}}} O_{S} -Mod and (O_{S} -Mod)^{∘} × O_{S} -Mod ^{\overline{}^{^{O_{S}}((−,=))}} O_{S} -Mod.

Proposition 59 The proper spectral sequences

Ė_{f_{∗}, _{O_{S}}((F,−))}^{Gr}(G)  and  Ė_{I}(_{O_{S}}((B_{−}, f_{∗}A^{=})))

are isomorphic (in ˙ Z̄_{∞}^{##}, O_{S} -Mod ).

In particular, both spectral sequences have E_{2}-terms Ext_{O_{S}}^{i}(F, (R^{j}f_{∗})(G)) and converge to (R^{i+j}IΓ_{F})(G), where IΓ_{F}(−) := _{O_{S}}((F, f_{∗}(−))) ≃ f_{∗} _{O_{T}}((f^{∗}F, −)). For example, if S = {∗} is a one-point-space and if we write R := O_{S}(S), then we can identify O_{S} -Mod = R-Mod. If, in this case, F = R/rR for some r ∈ R, then IΓ_{R/rR}(G) ≃ Γ(T, G)[r] := {g ∈ G(T) : rg = 0}.

Proof of Proposition 59. To apply Theorem 34, we comment on the conditions in §6.1.

(c) Since f_{∗} maps injective O_{T} -modules to injective O_{S}-modules by flatness of T ^{\overline{}^{f}} S, the complex A is an (f_{∗}, _{O_{S}}((F, −)))-acyclic resolution of G.

(e) If I is an injective O_{S}-module and U ⊆ S is an open subset, then I|_{U} is an injective O_{U}-module; cf. [9, Lem. III.6.1]. Hence _{O_{S}}((−, I)) turns a short exact sequence of O_{S}- modules into a sequence that is short exact as a sequence of abelian presheaves, and hence a fortiori short exact as a sequence of O_{S}-modules. In other words, the functor _{O_{S}}((−, I)) is exact.

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Matthias Künzer

Lehrstuhl D für Mathematik

RWTH Aachen Templergraben 64 D-52062 Aachen

kuenzer@math.rwth-aachen.de www.math.rwth-aachen.de/∼kuenzer