Chapter 0_III

§11. Complements on homological algebra

Translator's note. In the OCR'd French source, the content of §11 is housed inside the file named for §10 (03-c0-s10-complements-modules-plats.md, lines 266–1485). It is given its own translated file here.

11.1. Recall on spectral sequences

11.1.1.

In what follows we shall use a notion of spectral sequence more general than the one defined in (T, 2.4); keeping the notation of (T, 2.4), we shall call a spectral sequence in an abelian category $C$ a system $E$ consisting of the following data:

a) A family $(E_{r}^{p, q})$ of objects of $C$ defined for $p \in Z$ , $q \in Z$ , and $r \geq 2$ .

b) A family of morphisms $d_{r}^{p, q} : E_{r}^{p, q} \to E_{r}^{p + r, q - r + 1}$ such that $d_{r}^{p + r, q - r + 1} \circ d_{r}^{p, q} = 0$ .

We set $Z_{r + 1} (E_{2}^{p, q}) = Ker (d_{r}^{p, q})$ , $B_{r + 1} (E_{2}^{p, q}) = I m (d_{r}^{p - r, q + r - 1})$ , so that

$B_{r + 1} (E_{2}^{p, q}) \subset Z_{r + 1} (E_{2}^{p, q}) \subset E_{2}^{p, q} .$

c) A family of isomorphisms $α_{r}^{p, q} : Z_{r} (E_{2}^{p, q}) / B_{r} (E_{2}^{p, q}) \sim E_{r}^{p, q}$ .

One then defines, for $k \geq r + 1$ , by induction on $k$ , the subobjects $B_{k} (E_{2}^{p, q})$ and $Z_{k} (E_{2}^{p, q})$ as the inverse images, under the canonical morphism $E_{2}^{p, q} \to E_{2}^{p, q} / B_{r} (E_{2}^{p, q})$ , of the subobjects of this quotient identified by $α_{r}^{p, q}$ with the subobjects $B_{k} (E_{r}^{p, q})$ and $Z_{k} (E_{r}^{p, q})$ respectively. It is clear that one then has, up to isomorphism,

$Z_{k} (E_{2}^{p, q}) / B_{k} (E_{2}^{p, q}) = E_{k}^{p, q} (11.1.1.1)$

for $k \geq r + 1$ , and, if we further set $B_{1} (E_{2}^{p, q}) = 0$ and $Z_{1} (E_{2}^{p, q}) = E_{2}^{p, q}$ , one has the inclusion relations

  0 = B_1(E_2^{p,q}) ⊂ B_2(E_2^{p,q}) ⊂ B_3(E_2^{p,q}) ⊂ …
       … ⊂ Z_3(E_2^{p,q}) ⊂ Z_2(E_2^{p,q}) ⊂ Z_1(E_2^{p,q}) = E_2^{p,q}.       (11.1.1.2)

The remaining data of $E$ are then:

d) Two subobjects $B_{\infty} (E_{2}^{p, q})$ and $Z_{\infty} (E_{2}^{p, q})$ of $E_{2}^{p, q}$ such that one has $B_{\infty} (E_{2}^{p, q}) \subset Z_{\infty} (E_{2}^{p, q})$ and, for every $k \geq 2$ ,

  B_k(E_2^{p,q}) ⊂ B_∞(E_2^{p,q})    and    Z_∞(E_2^{p,q}) ⊂ Z_k(E_2^{p,q}).

One sets

$E_{\infty}^{p, q} = Z_{\infty} (E_{2}^{p, q}) / B_{\infty} (E_{2}^{p, q}) . (11.1.1.3)$

e) A family $(E^{n})$ of objects of $C$ , each equipped with a decreasing filtration $(F^{p} (E^{n}))_{p \in Z}$ . As usual we denote by $g r (E^{n})$ the graded object associated to the filtered object $E^{n}$ , the direct sum of the $g r_{p} (E^{n}) = F^{p} (E^{n}) / F^{p + 1} (E^{n})$ .

f) For every pair $(p, q) \in Z \times Z$ , an isomorphism $β^{p, q} : E_{\infty}^{p, q} \sim g r_{p} (E^{p + q})$ .

The family $(E^{n})$ , without the filtrations, is called the abutment of the spectral sequence $E$ .

Suppose either that the category $C$ admits infinite direct sums, or that for every $r \geq 2$ and every $n \in Z$ , the pairs $(p, q)$ such that $p + q = n$ and $E_{r}^{p, q} \neq = 0$ are finite in number (it suffices that this hold for $r = 2$ ). Then the $E_{r}^{(n)} = \oplus_{p + q = n} E_{r}^{p, q}$ are defined, and denoting by $d_{r}^{(n)}$ the morphism $E_{r}^{(n)} \to E_{r}^{(n + 1)}$ whose restriction to $E_{r}^{p, q}$ is $d_{r}^{p, q}$ for each pair $(p, q)$ with $p + q = n$ , one has $d_{r}^{(n + 1)} \circ d_{r}^{(n)} = 0$ ; in other words, $(E_{r}^{(n)})_{n \in Z}$ is a complex $E_{r}^{(∙)}$ in $C$ , with derivation operator of degree +1, and it follows from c) that $H^{∙} (E_{r}^{(∙)})$ is isomorphic to $E_{r + 1}^{(∙)}$ for every $r \geq 2$ .

11.1.2.

A morphism $u : E \to E^{'}$ from a spectral sequence $E$ to a spectral sequence $E^{'} = (E_{r}^{' p, q}, E^{' n})$ consists of systems of morphisms $u_{r}^{p, q} : E_{r}^{p, q} \to E_{r}^{' p, q}$ , $u^{n} : E^{n} \to E^{' n}$ , the $u^{n}$ being compatible with the filtrations of $E^{n}$ and $E^{' n}$ , the diagrams

$d_{r}^{p, q} E_{r}^{p, q} \to E_{r}^{p + r, q - r + 1} ↓↓ u_{r}^{p, q} u_{r}^{p + r, q - r + 1} ↓↓ E_{r}^{' p, q} \to E_{r}^{' p + r, q - r + 1} d_{r}^{' p, q}$

being commutative; moreover, on passing to quotients, $u_{r}^{p, q}$ induces a morphism $\overset{u}{ˉ}_{r}^{p, q} : Z_{r + 1} (E_{2}^{p, q}) / B_{r + 1} (E_{2}^{p, q}) \to Z_{r + 1} (E_{2}^{' p, q}) / B_{r + 1} (E_{2}^{' p, q})$ , and one must have $α_{r + 1}^{' p, q} \circ \overset{u}{ˉ}_{r}^{p, q} = u_{r + 1}^{p, q} \circ α_{r + 1}^{p, q}$ ; finally, one must have $u_{2}^{p, q} (B_{\infty} (E_{2}^{p, q})) \subset B_{\infty} (E_{2}^{' p, q})$ and $u_{2}^{p, q} (Z_{\infty} (E_{2}^{p, q})) \subset Z_{\infty} (E_{2}^{' p, q})$ ; on passing to quotients, $u_{2}^{p, q}$ then gives a morphism $u_{\infty}^{p, q} : E_{\infty}^{p, q} \to E_{\infty}^{' p, q}$ , and the diagram

$u_{\infty}^{p, q} E_{\infty}^{p, q} \to E_{\infty}^{' p, q} ↓↓ β^{p, q} β^{' p, q} ↓↓ g r_{p} (E^{p + q}) \to g r_{p} (E^{' p + q}) g r_{p} (u^{p + q})$

must be commutative.

The preceding definitions show, by induction on $r$ , that if the $u_{2}^{p, q}$ are isomorphisms, then so are the $u_{r}^{p, q}$ for $r \geq 2$ ; if one further knows that $u_{2}^{p, q} (B_{\infty} (E_{2}^{p, q})) = B_{\infty} (E_{2}^{' p, q})$ and $u_{2}^{p, q} (Z_{\infty} (E_{2}^{p, q})) = Z_{\infty} (E_{2}^{' p, q})$ and that the $u^{n}$ are isomorphisms, then one can conclude that $u$ is an isomorphism.

11.1.3.

Recall that if $(F^{p} (X))_{p \in Z}$ is a (decreasing) filtration of an object $X \in C$ , the filtration is said to be separated if $in f_{p} (F^{p} (X)) = 0$ , discrete if there exists $p$ such that $F^{p} (X) = 0$ , exhaustive (or co-separated) if $sup_{p} (F^{p} (X)) = X$ , co-discrete if there exists $p$ such that $F^{p} (X) = X$ .

We shall say that a spectral sequence $E = (E_{r}^{p, q}, E^{n})$ is weakly convergent if one has $B_{\infty} (E_{2}^{p, q}) = sup_{k} (B_{k} (E_{2}^{p, q}))$ , $Z_{\infty} (E_{2}^{p, q}) = in f_{k} (Z_{k} (E_{2}^{p, q}))$ (in other words, the objects $B_{\infty} (E_{2}^{p, q})$ and $Z_{\infty} (E_{2}^{p, q})$ are determined by data a) through c) of the spectral sequence $E$ ). We shall say that the spectral sequence $E$ is regular if it is weakly convergent and moreover:

1° For every pair $(p, q)$ , the decreasing sequence $(Z_{k} (E_{2}^{p, q}))_{k \geq 2}$ is stationary; the hypothesis that $E$ is weakly convergent then implies $Z_{\infty} (E_{2}^{p, q}) = Z_{k} (E_{2}^{p, q})$ for $k$ sufficiently large (depending on $p$ and $q$ ).

2° For every $n$ , the filtration $(F^{p} (E^{n}))_{p \in Z}$ of $E^{n}$ is discrete and exhaustive.

One says that the spectral sequence $E$ is co-regular if it is weakly convergent and moreover:

3° For every pair $(p, q)$ , the increasing sequence $(B_{k} (E_{2}^{p, q}))_{k \geq 2}$ is stationary, which entails $B_{\infty} (E_{2}^{p, q}) = B_{k} (E_{2}^{p, q})$ , and consequently $E_{\infty}^{p, q} = in f_{k} E_{k}^{p, q}$ .

4° For every $n$ , the filtration of $E^{n}$ is co-discrete.

Finally, one says that $E$ is biregular if it is both regular and co-regular; in other words, if the following conditions hold:

a) For every pair $(p, q)$ , the sequences $(B_{k} (E_{2}^{p, q}))_{k \geq 2}$ and $(Z_{k} (E_{2}^{p, q}))_{k \geq 2}$ are stationary, and one has $B_{\infty} (E_{2}^{p, q}) = B_{k} (E_{2}^{p, q})$ and $Z_{\infty} (E_{2}^{p, q}) = Z_{k} (E_{2}^{p, q})$ for $k$ sufficiently large (which entails $E_{\infty}^{p, q} = E_{k}^{p, q}$ ).

b) For every $n$ , the filtration $(F^{p} (E^{n}))_{p \in Z}$ is discrete and co-discrete (which one also expresses by saying that it is finite).

The spectral sequences defined in (T, 2.4) are thus the biregular spectral sequences.

11.1.4.

Suppose that in the category $C$ filtered inductive limits exist and that the functor $lim$ is exact (which is equivalent to saying that axiom AB 5) of (T, 1.5) is verified (cf. T, 1.8)). The condition that the filtration $(F^{p} (X))_{p \in Z}$ of an object $X \in C$ be exhaustive then also reads $lim_{p \to - \infty} F^{p} (X) = X$ . If a spectral sequence $E$ is weakly convergent, one has $B_{\infty} (E_{2}^{p, q}) = lim_{k \to \infty} B_{k} (E_{2}^{p, q})$ ; if moreover $u : E \to E^{'}$ is a morphism of $E$ into a weakly convergent spectral sequence $E^{'}$ of $C$ , one has $u_{2}^{p, q} (B_{\infty} (E_{2}^{p, q})) = B_{\infty} (E_{2}^{' p, q})$ by exactness of $lim$ . Moreover:

Proposition (11.1.5).

Let $C$ be an abelian category in which filtered inductive limits are exact, let $E$ , $E^{'}$ be two regular spectral sequences in $C$ , and let $u : E \to E^{'}$ be a morphism of spectral sequences. If the $u_{2}^{p, q}$ are isomorphisms, then so is $u$ .

Proof. We already know (11.1.2) that the $u_{r}^{p, q}$ are isomorphisms and that

$u_{2}^{p, q} (B_{\infty} (E_{2}^{p, q})) = B_{\infty} (E_{2}^{' p, q});$

the hypothesis that $E$ and $E^{'}$ are regular also implies $u_{2}^{p, q} (Z_{\infty} (E_{2}^{p, q})) = Z_{\infty} (E_{2}^{' p, q})$ , and since $u_{2}^{p, q}$ is an isomorphism, so is $u_{\infty}^{p, q}$ ; one therefore concludes that $g r_{p} (u^{p + q})$ is also an isomorphism. But since the filtrations of the $E^{n}$ and the $E^{' n}$ are discrete and exhaustive, this entails that the $u^{n}$ are also isomorphisms (Bourbaki, Alg. comm., chap. III, § 2, n° 8, th. 1).

11.1.6.

It follows from (11.1.1.2) and from definition (11.1.1.3) that if, in a spectral sequence $E$ , one has $E_{r}^{p, q} = 0$ , then one has $E_{k}^{p, q} = 0$ for $k \geq r$ and $E_{\infty}^{p, q} = 0$ . One says that a spectral sequence is degenerate if there exists an integer $r \geq 2$ and, for every integer $n \in Z$ , an integer $q (n)$ such that $E_{r}^{n - q, q} = 0$ for every $q \neq = q (n)$ . From the preceding remark one first deduces that one also has $E_{k}^{n - q, q} = 0$ for $k \geq r$ (including $k = \infty$ ) and $q \neq = q (n)$ . Moreover, the definition of $E_{r + 1}^{p, q}$ shows that one has $E_{r + 1}^{n - q (n), q (n)} = E_{r}^{n - q (n), q (n)}$ ; if $E$ is weakly convergent, one therefore also has $E_{\infty}^{n - q (n), q (n)} = E_{r}^{n - q (n), q (n)}$ ; in other words, for every $n \in Z$ , $g r_{p} (E^{n}) = 0$ for $p \neq = q (n)$ and $g r_{q (n)} (E^{n}) = E_{r}^{n - q (n), q (n)}$ . If moreover the filtration of $E^{n}$ is discrete and exhaustive, the sequence $E$ is regular and one has $E^{n} = E_{r}^{n - q (n), q (n)}$ up to isomorphism.

11.1.7.

Suppose that in the category $C$ filtered inductive limits exist and are exact, and let $(E_{λ}, u_{λ μ})$ be an inductive system (over a filtered index set) of spectral sequences in $C$ . Then the inductive limit of this inductive system exists in the additive category of spectral sequences of objects of $C$ : to see this it suffices to define $E_{r}^{p, q}$ , $d_{r}^{p, q}$ , $α_{r}^{p, q}$ , $B_{k} (E_{2}^{p, q})$ , $Z_{k} (E_{2}^{p, q})$ , $E^{n}$ , $F^{p} (E^{n})$ and $β^{p, q}$ as the respective inductive limits of $E_{r, λ}^{p, q}$ , $d_{r, λ}^{p, q}$ , $α_{r, λ}^{p, q}$ , $B_{k} (E_{2, λ}^{p, q})$ , $Z_{k} (E_{2, λ}^{p, q})$ , $E_{λ}^{n}$ , $F^{p} (E_{λ}^{n})$ and $β_{λ}^{p, q}$ ; the verification of the conditions of (11.1.1) follows from the exactness of the functor $lim$ in $C$ .

Remark (11.1.8).

Suppose that the category $C$ is the category of $A$ -modules over a Noetherian ring $A$ (resp. over a ring $A$ ). Then the definitions of (11.1.1) show that if, for a given $r$ , the $E_{r}^{p, q}$ are $A$ -modules of finite type (resp. of finite length), then so are all the modules $E_{s}^{p, q}$ for $s \geq r$ , as well as the $E_{\infty}^{p, q}$ . If moreover the filtration of the abutment $(E^{n})$ is discrete and co-discrete for every $n$ , one concludes that each of the $E^{n}$ is also an $A$ -module of finite type (resp. of finite length).

11.1.9.

We shall have to consider conditions ensuring that a spectral sequence $E$ is biregular "uniformly" in $p + q = n$ . We shall then use the following lemma:

Lemma (11.1.10).

Let $(E_{r}^{p, q})$ be a family of objects of $C$ linked by data a), b), c) of (11.1.1). For a fixed integer $n$ , the following properties are equivalent:

a) There exists an integer $r (n)$ such that for $r \geq r (n)$ , $p + q = n$ or $p + q = n - 1$ , the morphisms $d_{r}^{p, q}$ are all zero.

b) There exists an integer $r (n)$ such that for $p + q = n$ or $p + q = n + 1$ , one has $B_{s} (E_{2}^{p, q}) = B_{r} (E_{2}^{p, q})$ for $s \geq r \geq r (n)$ .

c) There exists an integer $r (n)$ such that for $p + q = n$ or $p + q = n - 1$ , one has $Z_{s} (E_{2}^{p, q}) = Z_{r} (E_{2}^{p, q})$ for $s \geq r \geq r (n)$ .

d) There exists an integer $r (n)$ such that for $p + q = n$ , one has $B_{s} (E_{2}^{p, q}) = B_{r} (E_{2}^{p, q})$ and $Z_{s} (E_{2}^{p, q}) = Z_{r} (E_{2}^{p, q})$ for $s \geq r \geq r (n)$ .

Proof. Indeed, by conditions a), b), c) of (11.1.1), to say that $Z_{r + 1} (E_{2}^{p, q}) = Z_{r} (E_{2}^{p, q})$ is equivalent to saying that $d_{r}^{p, q} = 0$ , and to say that $B_{r + 1} (E_{2}^{p + r, q - r + 1}) = B_{r} (E_{2}^{p + r, q - r + 1})$ is also equivalent to saying that $d_{r}^{p, q} = 0$ ; the lemma follows at once from this remark.

11.2. The spectral sequence of a filtered complex

11.2.1.

Given an abelian category $C$ , we shall use notation such as $K^{∙}$ for complexes $(K^{i})_{i \in Z}$ of objects of $C$ in which the derivation operator is of degree +1, and notation such as $K_{∙}$ for complexes $(K_{i})_{i \in Z}$ of objects of $C$ in which the derivation operator is of degree $- 1$ . To any complex $K^{∙} = (K^{i})$ whose derivation operator $d$ is of degree +1, one can associate a complex $K_{∙}^{'} = (K_{i}^{'})$ by setting $K_{i}^{'} = K^{- i}$ , the derivation operator $K_{i}^{'} \to K_{i - 1}^{'}$ being the operator $d : K^{- i} \to K^{- i + 1}$ ; and conversely, which will allow us, according to circumstances, to consider one type of complex or the other and to translate every result for one type into a result for the other. We shall denote similarly by notation such as $K^{∙, ∙} = (K^{i, j})$ (resp. $K_{∙, ∙} = (K_{i, j})$ ) bicomplexes of objects of $C$ in which the two derivation operators are of degree +1 (resp. $- 1$ ); one again passes from one type to the other by changing the signs of the indices, and one has analogous notation and remarks for arbitrary multicomplexes. The notation $K^{∙}$ or $K_{∙}$ will also be used for graded objects of $C$ , of type $Z$ , which are not necessarily complexes (or which one may consider as such with zero derivation operators); for instance, we shall write $H^{∙} (K^{∙}) = (H^{i} (K^{∙}))_{i \in Z}$ for the cohomology of a complex $K^{∙}$ whose derivation operator is of degree +1, and $H_{∙} (K_{∙}) = (H_{i} (K_{∙}))_{i \in Z}$ for the homology of a complex $K_{∙}$ whose derivation operator is of degree $- 1$ ; when one passes from $K^{∙}$ to $K_{∙}^{'}$ by the operation described above, one has $H_{i} (K_{∙}^{'}) = H^{- i} (K^{∙})$ .

Recall in this connection that for a complex $K^{∙}$ (resp. $K_{∙}$ ), we shall generally write $Z^{i} (K^{∙}) = Ker (K^{i} \to K^{i + 1})$ ("object of cocycles") and $B^{i} (K^{∙}) = I m (K^{i - 1} \to K^{i})$ ("object of coboundaries") (resp. $Z_{i} (K_{∙}) = Ker (K_{i} \to K_{i - 1})$ ("object of cycles") and $B_{i} (K_{∙}) = I m (K_{i + 1} \to K_{i})$ ("object of boundaries")), so that $H^{i} (K^{∙}) = Z^{i} (K^{∙}) / B^{i} (K^{∙})$ (resp. $H_{i} (K_{∙}) = Z_{i} (K_{∙}) / B_{i} (K_{∙})$ ).

If $K^{∙} = (K^{i})$ (resp. $K_{∙} = (K_{i})$ ) is a complex in $C$ , and $T : C \to C^{'}$ a functor from $C$ to an abelian category $C^{'}$ , we shall denote by $T (K^{∙})$ (resp. $T (K_{∙})$ ) the complex $(T (K^{i}))$ (resp. $(T (K_{i}))$ ) in $C^{'}$ .

We do not return to the definition of $\partial$ -functors (T, 2.1), except to point out that we shall also say $\partial$ -functor in place of $\partial^{*}$ -functor when the morphism $\partial$ lowers the degree by one unit; the context should make this clear when there could be doubt.

Finally, we shall say that a graded object $(A^{i})_{i \in Z}$ of $C$ is bounded below (resp. bounded above) if there exists $i_{0}$ such that $A^{i} = 0$ for $i < i_{0}$ (resp. $i > i_{0}$ ).

11.2.2.

Let $K^{∙}$ be a complex in $C$ whose derivation operator $d$ is of degree +1, and suppose it equipped with a filtration $F (K^{∙}) = (F^{p} (K^{∙}))_{p \in Z}$ formed of graded subobjects of $K^{∙}$ ,

in other words $F^{p} (K^{∙}) = (K^{i} \cap F^{p} (K^{∙}))_{i \in Z}$ ; moreover, we assume that $d (F^{p} (K^{∙})) \subset F^{p} (K^{∙})$ for every $p \in Z$ . We now briefly recall how one functorially defines a spectral sequence $E (K^{∙})$ from $K^{∙}$ (M, XV, 4 and G, I, 4.3). For $r \geq 2$ , the canonical morphism $F^{p} (K^{∙}) / F^{p + r} (K^{∙}) \to F^{p} (K^{∙}) / F^{p + 1} (K^{∙})$ defines a morphism in cohomology

$H^{p + q} (F^{p} (K^{∙}) / F^{p + r} (K^{∙})) \to H^{p + q} (F^{p} (K^{∙}) / F^{p + 1} (K^{∙})) .$

One denotes by $Z_{r}^{p, q} (K^{∙})$ the image of this morphism. Similarly, from the exact sequence

$0 \to F^{p} (K^{∙}) / F^{p + 1} (K^{∙}) \to F^{p - r + 1} (K^{∙}) / F^{p + 1} (K^{∙}) \to F^{p - r + 1} (K^{∙}) / F^{p} (K^{∙}) \to 0$

one deduces, by the long exact sequence of cohomology, a morphism

$H^{p + q - 1} (F^{p - r + 1} (K^{∙}) / F^{p} (K^{∙})) \to H^{p + q} (F^{p} (K^{∙}) / F^{p + 1} (K^{∙}))$

and one denotes by $B_{r}^{p, q} (K^{∙})$ the image of this morphism; one shows that $B_{r}^{p, q} (K^{∙}) \subset Z_{r}^{p, q} (K^{∙})$ and one sets $E_{r}^{p, q} (K^{∙}) = Z_{r}^{p, q} (K^{∙}) / B_{r}^{p, q} (K^{∙})$ ; we shall not specify the definitions of the $d_{r}^{p, q}$ or the $α_{r}^{p, q}$ .

We note here that all the $Z_{r}^{p, q} (K^{∙})$ and $B_{r}^{p, q} (K^{∙})$ , for $p$ , $q$ fixed, are subobjects of the same object $H^{p + q} (F^{p} (K^{∙}) / F^{p + 1} (K^{∙}))$ , which one denotes $Z_{1}^{p, q} (K^{∙})$ ; one sets $B_{1}^{p, q} (K^{∙}) = 0$ , so that the preceding definitions for $Z_{r}^{p, q} (K^{∙})$ and $B_{r}^{p, q} (K^{∙})$ also apply for $r = 1$ ; one further sets $E_{1}^{p, q} (K^{∙}) = Z_{1}^{p, q} (K^{∙})$ . One also defines the $d_{1}^{p, q}$ and $α_{1}^{p, q}$ so that the conditions of (11.1.1) are satisfied for $r = 1$ . One defines on the other hand the subobjects $Z_{\infty}^{p, q} (K^{∙})$ , image of the morphism

$H^{p + q} (F^{p} (K^{∙})) \to H^{p + q} (F^{p} (K^{∙}) / F^{p + 1} (K^{∙})) = E_{1}^{p, q} (K^{∙})$

and $B_{\infty}^{p, q} (K^{∙})$ , image of the morphism

$H^{p + q - 1} (K^{∙} / F^{p} (K^{∙})) \to H^{p + q} (F^{p} (K^{∙}) / F^{p + 1} (K^{∙})) = E_{1}^{p, q} (K^{∙})$

obtained as above from a long exact sequence of cohomology. One takes for $Z_{\infty} (E_{2}^{p, q} (K^{∙}))$ and $B_{\infty} (E_{2}^{p, q} (K^{∙}))$ the canonical images in $E_{2}^{p, q} (K^{∙})$ of $Z_{\infty}^{p, q} (K^{∙})$ and $B_{\infty}^{p, q} (K^{∙})$ .

Finally, one denotes by $F^{p} (H^{n} (K^{∙}))$ the image in $H^{n} (K^{∙})$ of the morphism $H^{n} (F^{p} (K^{∙})) \to H^{n} (K^{∙})$ coming from the canonical injection $F^{p} (K^{∙}) \to K^{∙}$ ; by the long exact sequence of cohomology, it is also the kernel of the morphism $H^{n} (K^{∙}) \to H^{n} (K^{∙} / F^{p} (K^{∙}))$ . One thus defines a filtration on $E^{n} (K^{∙}) = H^{n} (K^{∙})$ ; we shall not give the definition of the isomorphisms $β^{p, q}$ here either.

11.2.3.

The functorial character of $E (K^{∙})$ is to be understood as follows: given two filtered complexes $K^{∙}$ , $K^{' ∙}$ of $C$ and a morphism of complexes $u : K^{∙} \to K^{' ∙}$ compatible with the filtrations, one deduces from it in an obvious way the morphisms $u_{r}^{p, q}$ (for $r \geq 1$ ) and $u^{n}$ , and one shows that these morphisms are compatible with the $d_{r}^{p, q}$ , $α_{r}^{p, q}$ and $β^{p, q}$ in the sense of (11.1.2), so that they indeed define a morphism $E (u) : E (K^{∙}) \to E (K^{' ∙})$ of spectral sequences. Moreover, one shows that if $u$ and $v$ are morphisms $K^{∙} \to K^{' ∙}$ of the preceding type, homotopic of order $\leq k$ , then $u_{r}^{p, q} = v_{r}^{p, q}$ for $r > k$ and $u^{n} = v^{n}$ for every $n$ (M, XV, 3.1).

11.2.4.

Suppose that in $C$ filtered inductive limits are exact. Then, if the filtration $(F^{p} (K^{∙}))$ of $K^{∙}$ is exhaustive, so is the filtration $(F^{p} (H^{n} (K^{∙})))$ for every $n$ , since by hypothesis $K^{∙} = lim_{p \to - \infty} F^{p} (K^{∙})$ and the hypothesis on $C$ implies that cohomology commutes with inductive limits. Moreover, for the same reason, one has $B_{\infty} (E_{2}^{p, q} (K^{∙})) = sup_{k} B_{k} (E_{2}^{p, q} (K^{∙}))$ . One says that the filtration $(F^{p} (K^{∙}))$ of $K^{∙}$ is regular if for every $n$ there exists an integer $u (n)$ such that $H^{n} (F^{p} (K^{∙})) = 0$ for $p > u (n)$ . This holds in particular when the filtration of $K^{∙}$ is discrete. When the filtration of $K^{∙}$ is regular and exhaustive, and filtered inductive limits in $C$ are exact, one shows (M, XV, 4) that the spectral sequence $E (K^{∙})$ is regular.

11.3. The spectral sequences of a bicomplex

11.3.1.

As regards the conventions on bicomplexes, we follow those of (T, 2.4) rather than those of (M), the two derivations $d^{'}$ , $d^{''}$ (of degree +1) of such a bicomplex $K^{∙, ∙} = (K^{i, j})$ being therefore assumed to commute. Suppose that one of the two following conditions is verified: 1° infinite direct sums exist in $C$ ; 2° for every $n \in Z$ , there are only finitely many pairs $(p, q)$ such that $p + q = n$ and $K^{p, q} \neq = 0$ . Then the bicomplex $K^{∙, ∙}$ defines a (simple) complex $(K^{(n)})_{n \in Z}$ with $K^{(n)} = \oplus_{i + j = n} K^{i, j}$ , the derivation operator $d$ (of degree +1) of this complex being given by $d x = d^{'} x + (- 1)^{i} d^{''} x$ for $x \in K^{i, j}$ . Whenever in what follows we speak of the (simple) complex defined by a bicomplex $K^{∙, ∙}$ , it will always be understood that one of the preceding conditions is satisfied. One adopts analogous conventions for multicomplexes.

We denote by $K^{i, ∙}$ (resp. $K^{∙, j}$ ) the simple complex $(K^{i, j})_{j \in Z}$ (resp. $(K^{i, j})_{i \in Z}$ ), and by $Z_{II}^{q} (K^{i, ∙})$ , $B_{II}^{q} (K^{i, ∙})$ , $H_{II}^{q} (K^{i, ∙})$ (resp. $Z_{I}^{p} (K^{∙, j})$ , $B_{I}^{p} (K^{∙, j})$ , $H_{I}^{p} (K^{∙, j})$ ) its $q$ th (resp. $p$ th) objects of cocycles, coboundaries, and cohomology respectively; the derivation $d^{'} : K^{i, ∙} \to K^{i + 1, ∙}$ is a morphism of complexes, which therefore gives an operator on the cocycles, coboundaries, and cohomology,

$d^{'} : Z_{II}^{q} (K^{i, ∙}) \to Z_{II}^{q} (K^{i + 1, ∙}) d^{'} : B_{II}^{q} (K^{i, ∙}) \to B_{II}^{q} (K^{i + 1, ∙}) d^{'} : H_{II}^{q} (K^{i, ∙}) \to H_{II}^{q} (K^{i + 1, ∙})$

and it is clear that for these operators, $(Z_{II}^{q} (K^{i, ∙}))_{i \in Z}$ , $(B_{II}^{q} (K^{i, ∙}))_{i \in Z}$ , and $(H_{II}^{q} (K^{i, ∙}))_{i \in Z}$ are complexes; we shall denote the complex $(H_{II}^{q} (K^{i, ∙}))_{i \in Z}$ by $H_{II}^{q} (K^{∙, ∙})$ , and its $p$ th objects of cocycles, coboundaries, and cohomology by $Z_{I}^{p} (H_{II}^{q} (K^{∙, ∙}))$ , $B_{I}^{p} (H_{II}^{q} (K^{∙, ∙}))$ , and $H_{I}^{p} (H_{II}^{q} (K^{∙, ∙}))$ . One defines similarly the complexes $H_{I}^{p} (K^{∙, ∙})$ and their objects of cohomology $H_{II}^{q} (H_{I}^{p} (K^{∙, ∙}))$ . Recall on the other hand that $H^{n} (K^{∙, ∙})$ denotes the $n$ th object of cohomology of the (simple) complex defined by $K^{∙, ∙}$ .

11.3.2.

On the complex defined by a bicomplex $K^{∙, ∙}$ , one may consider two canonical filtrations $(F_{I}^{p} (K^{∙, ∙}))$ and $(F_{II}^{p} (K^{∙, ∙}))$ given by

  F_I^p(K^{•,•}) = (⊕_{i+j=n, i ≥ p} K^{i,j})_{n ∈ ℤ},   F_{II}^p(K^{•,•}) = (⊕_{i+j=n, j ≥ p} K^{i,j})_{n ∈ ℤ}.   (11.3.2.1)

which are by definition graded subobjects of the (simple) complex defined by $K^{∙, ∙}$ , and thus make this complex into a filtered complex; moreover, it is clear that these filtrations are exhaustive and separated.

To each of these filtrations there corresponds a spectral sequence (11.2.2); we shall denote by $^{'} E (K^{∙, ∙})$ and $^{''} E (K^{∙, ∙})$ the spectral sequences corresponding to $(F_{I}^{p} (K^{∙, ∙}))$ and $(F_{II}^{p} (K^{∙, ∙}))$ respectively, called the spectral sequences of the bicomplex $K^{∙, ∙}$ , both having as abutment the cohomology $(H^{n} (K^{∙, ∙}))$ . One shows moreover (M, XV, 6) that

  'E_2^{p,q}(K^{•,•}) = H_I^p(H_{II}^q(K^{•,•})),    ″E_2^{p,q}(K^{•,•}) = H_{II}^p(H_I^q(K^{•,•})).   (11.3.2.2)

Any morphism $u : K^{∙, ∙} \to K^{' ∙, ∙}$ of bicomplexes is ipso facto compatible with the filtrations of the same type on $K^{∙, ∙}$ and $K^{' ∙, ∙}$ , and thus defines a morphism for each of the two spectral sequences; moreover, two homotopic morphisms define a homotopy of order $\leq 1$ of the corresponding (simple) filtered complexes, hence the same morphism for each of the two spectral sequences (M, XV, 6.1).

Proposition (11.3.3).

Let $K^{∙, ∙} = (K^{i, j})$ be a bicomplex in an abelian category $C$ .

(i) If there exist $i_{0}$ and $j_{0}$ such that $K^{i, j} = 0$ for $i < i_{0}$ or $j < j_{0}$ (resp. $i > i_{0}$ or $j > j_{0}$ ), the two spectral sequences $^{'} E (K^{∙, ∙})$ and $^{''} E (K^{∙, ∙})$ are biregular.

(ii) If there exist $i_{0}$ and $i_{1}$ such that $K^{i, j} = 0$ for $i < i_{0}$ or $i > i_{1}$ (resp. if there exist $j_{0}$ and $j_{1}$ such that $K^{i, j} = 0$ for $j < j_{0}$ or $j > j_{1}$ ), the two spectral sequences $^{'} E (K^{∙, ∙})$ and $^{''} E (K^{∙, ∙})$ are biregular.

Suppose moreover that in $C$ filtered inductive limits exist and are exact. Then:

(iii) If there exists $i_{0}$ such that $K^{i, j} = 0$ for $i > i_{0}$ (resp. if there exists $j_{0}$ such that $K^{i, j} = 0$ for $j < j_{0}$ ), the sequence $^{'} E (K^{∙, ∙})$ is regular.

(iv) If there exists $i_{0}$ such that $K^{i, j} = 0$ for $i < i_{0}$ (resp. if there exists $j_{0}$ such that $K^{i, j} = 0$ for $j > j_{0}$ ), the sequence $^{''} E (K^{∙, ∙})$ is regular.

Proof. The proposition follows at once from the definitions (11.1.3) and from (11.2.4), together with the following observations concerning the filtration F_I (and the analogous observations one deduces for $F_{II}$ by exchanging the roles of the two indices in $K^{∙, ∙}$ ):

1° If there exists $i_{0}$ such that $K^{i, j} = 0$ for $i > i_{0}$ , the filtration $F_{I} (K^{∙, ∙})$ is discrete.

2° If there exists $i_{0}$ such that $K^{i, j} = 0$ for $i < i_{0}$ , the filtration $F_{I} (K^{∙, ∙})$ is co-discrete. One deduces at once that the same holds for the corresponding filtration $F_{I} (H^{n} (K^{∙, ∙}))$ for every $n$ ; moreover, the definition of $B_{r}^{p, q}$ corresponding to the filtration $F_{I} (K^{∙, ∙})$ (11.2.2) shows that for every pair $(p, q)$ , the sequence $(B_{r}^{p, q})_{r \geq 2}$ is stationary.

3° If there exists $j_{0}$ such that $K^{i, j} = 0$ for $j < j_{0}$ , one has

$F_{I}^{p + r} (K^{∙, ∙}) \cap (\oplus_{i + j = n} K^{i, j}) = 0$

whenever $p + r + j_{0} > n$ , hence $Z_{r}^{p, q} = Z_{\infty} (E_{2}^{p, q})$ for $r > q - j_{0} + 1$ ; on the other hand, $H^{n} (F_{I}^{p} (K^{∙, ∙})) = 0$ for $p > n - j_{0} + 1$ .

4° If there exists $j_{0}$ such that $K^{i, j} = 0$ for $j > j_{0}$ , one has

  F_I^{p-r+1}(K^{•,•}) ∩ (⊕_{i+j=n} K^{i,j}) = ⊕_{i+j=n} K^{i,j}

whenever $p - r + 1 + j_{0} < n$ , hence $B_{r}^{p, q} = B_{\infty} (E_{2}^{p, q})$ for $r < j_{0} - q + 1$ ; on the other hand, $H^{n} (F_{I}^{p} (K^{∙, ∙})) = H^{n} (K^{∙, ∙})$ for $p + j_{0} < n - 1$ .

11.3.4.

Suppose that the bicomplex $K^{∙, ∙} = (K^{i, j})$ is such that $K^{i, j} = 0$ for $i < 0$ or $j < 0$ . It is known that one can then define for every $p \in Z$ a canonical "edge homomorphism"

$^{'} E_{2}^{p, 0} (K^{∙, ∙}) \to H^{p} (K^{∙, ∙}) (11.3.4.1)$

(M, XV, 6). We briefly recall that this is due, on the one hand, to the fact that one has $Z_{r}^{p, 0} = Z_{\infty} (E_{2}^{p, 0})$ in the spectral sequence $^{'} E (K^{∙, ∙})$ for $2 \leq r \leq + \infty$ , and, on the other hand, to the fact that $H^{p} (F_{I}^{p + 1} (K^{∙, ∙})) = 0$ , so that the isomorphism $β^{p, 0} :^{'} E_{\infty}^{p, 0} \sim H^{p} (F_{I}^{p}) / H^{p} (F_{I}^{p + 1})$ gives a homomorphism $^{'} E_{\infty}^{p, 0} \to H^{p} (F_{I}^{p} (K^{∙, ∙})) \to H^{p} (K^{∙, ∙})$ ; the equality of all the $Z_{r}^{p, 0}$ then makes it possible to define canonical homomorphisms $^{'} E_{r}^{p, 0} \to^{'} E_{s}^{p, 0}$ for $r \leq s$ , in particular a homomorphism $^{'} E_{2}^{p, 0} \to^{'} E_{\infty}^{p, 0}$ , whence by composition the edge homomorphism $^{'} E_{2}^{p, 0} \to H^{p} (K^{∙, ∙})$ ; moreover, one verifies at once that, to the class mod. $B_{\infty}^{p, 0}$ of an element $z \in Z_{II}^{0} (K^{∙, ∙}) \subset K^{p, 0}$ such that $d^{'} z = 0$ , the edge homomorphism so defined associates in $^{'} E_{\infty}^{p, 0}$ the class of $z$ mod. $B_{\infty}^{p, 0}$ , and then, to this last, the cohomology class of $z$ in $H^{p} (K^{∙, ∙})$ . One thus sees finally that the edge homomorphism (11.3.4.1) comes, by passage to cohomology, from the canonical injection $Z_{II}^{0} (K^{∙, ∙}) \to K^{∙, ∙}$ (where $K^{∙, ∙}$ is considered as a simple complex). One similarly interprets the edge homomorphism

$^{''} E_{2}^{p, 0} (K^{∙, ∙}) \to H^{p} (K^{∙, ∙}) (11.3.4.2)$

as coming from the canonical injection $Z_{I}^{0} (K^{∙, ∙}) \to K^{∙, ∙}$ .

11.3.5.

Let now $K_{∙, ∙} = (K_{i, j})$ be a bicomplex in $C$ whose two derivation operators are of degree $- 1$ . We shall then write $K_{i, ∙}$ (resp. $K_{∙, j}$ ) for the simple complex $(K_{i, j})_{j \in Z}$ (resp. $(K_{i, j})_{i \in Z}$ ), $H_{p}^{II} (K_{i, ∙})$ (resp. $H_{p}^{I} (K_{∙, j})$ ) for its $p$ th object of homology, $H_{p}^{II} (K_{∙, ∙})$ (resp. $H_{p}^{I} (K_{∙, ∙})$ ) for the complex $(H_{p}^{II} (K_{i, ∙}))_{i \in Z}$ (resp. $(H_{p}^{I} (K_{∙, j}))_{j \in Z}$ ), $H_{q}^{I} (H_{p}^{II} (K_{∙, ∙}))$ (resp. $H_{q}^{II} (H_{p}^{I} (K_{∙, ∙}))$ ) for its $q$ th object of homology; analogous notation for the objects of cycles and boundaries; finally, $H_{n} (K_{∙, ∙})$ will denote (when it exists) the $n$ th object of homology of the simple complex (with derivation operator of degree $- 1$ ) defined by $K_{∙, ∙}$ .

Let $K^{∙, ∙} = (K^{i, j})$ with $K^{i, j} = K_{- i, - j}$ be the bicomplex with derivation operators of degree +1 associated to $K_{∙, ∙}$ ; by definition, the spectral sequences of $K_{∙, ∙}$ are those of $K^{∙, ∙}$ , which one writes $^{'} E (K_{∙, ∙})$ and $^{''} E (K_{∙, ∙})$ , where one changes the notation, however, by setting

  'E_{p,q}^r(K_{•,•}) = 'E_r^{-p,-q}(K^{•,•}),    ″E_{p,q}^r(K_{•,•}) = ″E_r^{-p,-q}(K^{•,•})

for $2 \leq r \leq \infty$ . With this notation, one has

  'E_{p,q}^2(K_{•,•}) = H_p^I(H_q^{II}(K_{•,•})),    ″E_{p,q}^2(K_{•,•}) = H_p^{II}(H_q^I(K_{•,•})).

To avoid sign errors, it will generally be preferable, for the relations between these spectral sequences and their abutment, to return to the complex $K^{∙, ∙}$ . Let us note nonetheless the criteria corresponding to (11.3.3):

11.3.6.

The spectral sequences $^{'} E (K_{∙, ∙})$ and $^{''} E (K_{∙, ∙})$ are biregular in the following cases: a) There exist $i_{0}$ and $j_{0}$ such that $K_{i, j} = 0$ for $i > i_{0}$ or for $j > j_{0}$ (resp. for $i < i_{0}$

or for $j < j_{0}$ ); b) There exist $i_{0}$ and $i_{1}$ such that $K_{i, j} = 0$ for $i < i_{0}$ and $i > i_{1}$ ; c) There exist $j_{0}$ and $j_{1}$ such that $K_{i, j} = 0$ for $j < j_{0}$ and $j > j_{1}$ .

The sequence $^{'} E (K_{∙, ∙})$ is regular if there exists $i_{0}$ such that $K_{i, j} = 0$ for $i < i_{0}$ , or if there exists $j_{0}$ such that $K_{i, j} = 0$ for $j > j_{0}$ .

The sequence $^{''} E (K_{∙, ∙})$ is regular if there exists $i_{0}$ such that $K_{i, j} = 0$ for $i > i_{0}$ , or if there exists $j_{0}$ such that $K_{i, j} = 0$ for $j < j_{0}$ .

11.4. Hypercohomology of a functor with respect to a complex $K^{∙}$

11.4.1.

Let $C$ be an abelian category; recall that one calls a right resolution (or cohomological resolution) of an object $A$ of $C$ a complex of objects of $C$ , whose derivation operator is of degree +1,

$0 \to L^{0} \to L^{1} \to L^{2} \to \dots$

equipped with a morphism $ϵ : A \to L^{0}$ called the augmentation of the resolution (and which one can consider as a morphism of complexes

  0 → A → 0 → 0 → …
        ↓   ↓   ↓
  0 → L^0 → L^1 → L^2 → … )

such that the sequence

  0 → A →^ε L^0 → L^1 → …

is exact; similarly, a left resolution (or homological resolution) of $A$ is a complex $0 \leftarrow L_{0} \leftarrow L_{1} \leftarrow \dots$ of objects of $C$ whose derivation operator is of degree $- 1$ , equipped with an augmentation $ϵ : L_{0} \to A$ , such that the sequence

  0 ← A ←^ε L_0 ← L_1 ← …

is exact.

When a right resolution $(L^{i})_{i \geq 0}$ of an object $A$ is such that $L^{i} = 0$ for $i \geq n + 1$ , one says that this resolution is of length $\leq n$ . One defines similarly a left resolution of length $\leq n$ . A resolution that is of length $\leq n$ for some integer $n$ is said to be finite.

A resolution of $A$ is called projective (resp. injective) if the objects of $C$ other than $A$ that compose it are projective (resp. injective). When $C$ is the category of modules (left modules, say) over a ring, one will say similarly that a resolution of $A$ is flat (resp. free) when the modules other than $A$ that compose it are flat (resp. free).

11.4.2.

Let $K^{∙} = (K^{i})_{i \in Z}$ be a complex of objects of $C$ whose derivation operator is of degree +1.

We call a right Cartan–Eilenberg resolution of $K^{∙}$ the pair consisting of a bicomplex $L^{∙, ∙} = (L^{i, j})$ with derivation operators of degree +1, with $L^{i, j} = 0$ for $j < 0$ , and a morphism of simple complexes $ϵ : K^{∙} \to L^{∙, 0}$ , such that the following conditions are satisfied:

(i) For each index $i$ , the sequences

  0 → K^i →^ε L^{i,0} → L^{i,1} → …
  0 → B^i(K^•) →^ε B_I^i(L^{•,0}) → B_I^i(L^{•,1}) → …
  0 → Z^i(K^•) →^ε Z_I^i(L^{•,0}) → Z_I^i(L^{•,1}) → …
  0 → H^i(K^•) →^ε H_I^i(L^{•,0}) → H_I^i(L^{•,1}) → …

are exact; in other words, $(L^{i, ∙})$ , $(B_{I}^{i} (L^{∙, ∙}))$ , $(Z_{I}^{i} (L^{∙, ∙}))$ and $(H_{I}^{i} (L^{∙, ∙}))$ are respectively resolutions of $K^{i}$ , $B^{i} (K^{∙})$ , $Z^{i} (K^{∙})$ and $H^{i} (K^{∙})$ .

(ii) For each $j$ , the simple complex $L^{∙, j}$ is split; in other words, the exact sequences

$0 \to B_{I}^{i} (L^{∙, j}) \to Z_{I}^{i} (L^{∙, j}) \to H_{I}^{i} (L^{∙, j}) \to 0 (11.4.2.1) 0 \to Z_{I}^{i} (L^{∙, j}) \to L^{i, j} \to B_{I}^{i + 1} (L^{∙, j}) \to 0 (11.4.2.2)$

are split.

One proves (M, XVII, 1.2) that if every object of $C$ is a subobject of an injective object, every complex $K^{∙}$ of $C$ admits an injective Cartan–Eilenberg resolution, that is, one formed of injective objects $L^{i, j}$ (condition (ii) above then entails that the $B_{I}^{i} (L^{∙, j})$ , $Z_{I}^{i} (L^{∙, j})$ and $H_{I}^{i} (L^{∙, j})$ are also injective objects). Moreover, for every morphism $f : K^{∙} \to K^{' ∙}$ of complexes of $C$ , every Cartan–Eilenberg resolution $L^{∙, ∙}$ of $K^{∙}$ and every injective Cartan–Eilenberg resolution $L^{' ∙, ∙}$ of $K^{' ∙}$ , there exists a morphism of bicomplexes $F : L^{∙, ∙} \to L^{' ∙, ∙}$ compatible with $f$ and the augmentations; and if $f$ and $g$ are two homotopic morphisms from $K^{∙}$ to $K^{' ∙}$ , the corresponding morphisms from $L^{∙, ∙}$ to $L^{' ∙, ∙}$ are homotopic (loc. cit.).

When $K^{∙}$ is bounded below (resp. bounded above), one may take $L^{∙, ∙}$ such that $L^{i, ∙} = 0$ for $i < i_{0}$ (resp. $i > i_{0}$ ) if $K^{i} = 0$ for $i < i_{0}$ (resp. $i > i_{0}$ ) (M, XVII, 1.3).

Suppose on the other hand that there exists an integer $n$ such that every object of $C$ admits an injective resolution of length $\leq n$ ; then one may suppose that $L^{i, j} = 0$ for $j > n$ (M, XVII, 1.4).

11.4.3.

Let now $T$ be a covariant additive functor from $C$ to an abelian category $C^{'}$ . Given a complex $K^{∙}$ of $C$ and an injective Cartan–Eilenberg resolution $L^{∙, ∙}$ of $K^{∙}$ , suppose that the (simple) complex defined by the bicomplex $T (L^{∙, ∙})$ exists (cf. 11.3.1); then the two spectral sequences $^{'} E (T (L^{∙, ∙}))$ and $^{''} E (T (L^{∙, ∙}))$ of this bicomplex are called the spectral sequences of hypercohomology of $T$ with respect to the complex $K^{∙}$ ; by virtue of (11.4.2) and (11.3.2), they in fact depend only on $K^{∙}$ and not on the chosen injective Cartan–Eilenberg resolution $L^{∙, ∙}$ ; moreover, they depend functorially on $K^{∙}$ . They have a common abutment $H^{∙} (T (L^{∙, ∙}))$ , called the hypercohomology of $T$ with respect to $K^{∙}$ , and denoted $R^{∙} T (K^{∙})$ . One shows that the terms E_2 of the two preceding spectral sequences are given by

$^{'} E_{2}^{p, q} = H^{p} (R^{qT} (K^{∙})) (11.4.3.1)^{''} E_{2}^{p, q} = R^{pT} (H^{q} (K^{∙})) (11.4.3.2)$

$R^{pT}$ denoting as usual the $p$ th derived functor of $T$ for $p \in Z$ ; $R^{∙} T (K^{∙})$ denotes the complex $(R^{pT} (K^{i}))_{p \in Z}$ . Unless expressly stated otherwise, we shall henceforth assume that every object of $C$ is a subobject of an injective object of $C$ , so that injective Cartan–Eilenberg resolutions exist for every complex of $C$ . Since $L^{i, j} = 0$ for $j < 0$ , the criteria of (11.3.3) show that the two hypercohomology spectral sequences of $T$ with respect to $K^{∙}$ exist and are biregular in each of the two following cases: 1° $K^{∙}$ is bounded below; 2° every object of $C$ admits an injective resolution of length at most equal to an integer $n$ (independent of the object considered). Indeed, in the first case, one may suppose (11.4.2) that there exists $i_{0}$ such that $L^{i, j} = 0$ for $i < i_{0}$ , and in the second that there exists $j_{1}$ such that $L^{i, j} = 0$ for $j > j_{1}$ ; in each of the two cases, it is moreover clear that for given $n$ , there are only finitely many pairs $(i, j)$ such that $L^{i, j} \neq = 0$ and $i + j = n$ , which establishes our assertions.

When one supposes that in $C^{'}$ filtered inductive limits exist and are exact (which implies in particular the existence in $C^{'}$ of infinite direct sums), then the complex defined by the bicomplex $T (L^{∙, ∙})$ exists, and criterion (11.3.3) shows that the spectral sequence $^{'} E (T (L^{∙, ∙}))$ is always regular.

11.4.4.

When $K^{∙}$ is a complex all of whose terms $K^{i}$ are zero except for a single $K^{i_{0}}$ , $R^{n T} (K^{∙})$ is isomorphic to $R^{n - i_{0}} T (K^{i_{0}})$ , as follows at once from the definitions on taking a Cartan–Eilenberg resolution $L^{∙, ∙}$ such that $L^{i, j} = 0$ for $i \neq = i_{0}$ .

If $K^{∙}$ and $K^{' ∙}$ are two complexes of $C$ , $f$ , $g$ two homotopic morphisms from $K^{∙}$ to $K^{' ∙}$ , then the morphisms $R^{∙} T (K^{∙}) \to R^{∙} T (K^{' ∙})$ deduced from $f$ and $g$ are identical, and likewise for the morphisms of the cohomology spectral sequences.

Proposition (11.4.5).

Suppose that in $C^{'}$ filtered inductive limits exist and are exact. If $R^{n T} (K^{i}) = 0$ for every $n > 0$ and every $i \in Z$ , one has functorial isomorphisms

$R^{i T} (K^{∙}) \sim H^{i} (T (K^{∙})) (11.4.5.1)$

for $i \in Z$ .

Proof. The only nonzero terms E_2 of the first spectral sequence (11.4.3.1) are then $^{'} E_{2}^{p, 0} = H^{p} (T (K^{∙}))$ ; in other words, this sequence is degenerate; since it is regular (11.4.4), the conclusion follows from (11.1.6).

11.4.6.

Consider now, for example, a covariant bifunctor $(M, N) \mapsto T (M, N)$ from $C \times C^{'}$ to $C^{''}$ , where $C$ , $C^{'}$ , $C^{''}$ are three abelian categories; one assumes, for simplicity, that $T$ is additive in each of its arguments, and moreover that every object of $C$ and every object of $C^{'}$ is a subobject of an injective object, and that filtered inductive limits exist in $C^{''}$ and are exact. One then defines the hypercohomology of $T$ with respect to two complexes $K^{∙}$ , $K^{' ∙}$ of $C$ and $C^{'}$ respectively, with derivation operators of degree +1, by taking for $K^{∙}$ (resp. $K^{' ∙}$ ) an injective Cartan–Eilenberg resolution $L^{∙, ∙}$ (resp. $L^{' ∙, ∙}$ ); then $T (L^{∙, ∙}, L^{' ∙, ∙})$ is a quadricomplex of $C^{''}$ , which one regards as a bicomplex of $C^{''}$ by taking as degrees of $T (L^{i, j}, L^{' h, k})$ the integers $i + h$ and $j + k$ . The hypercohomology of $T$ with respect to $K^{∙}$ and $K^{' ∙}$ is by definition the cohomology $H^{∙} (T (L^{∙, ∙}, L^{' ∙, ∙}))$ of this bicomplex (in other words, that of the associated simple complex),

denoted $R^{∙} T (K^{∙}, K^{' ∙})$ ; it is the abutment of two spectral sequences whose terms E_2 are given by

  'E_2^{p,q} = H^p(R^qT(K^•, K'^•))                                            (11.4.6.1)
  ″E_2^{p,q} = ⊕_{q'+q″=q} R^pT(H^{q'}(K^•), H^{q″}(K'^•))   (cf. M, XVII, 2). (11.4.6.2)

Here $R^{∙} T (K^{∙}, K^{' ∙})$ is the bicomplex $(R^{∙} T (K^{i}, K^{' j}))_{(i, j) \in Z \times Z}$ and the second member of (11.4.6.1) is its cohomology when one regards it as a simple complex.

Moreover, the first spectral sequence is always regular, and the two spectral sequences are biregular when there exists $n$ such that every object of $C$ and every object of $C^{'}$ admits an injective resolution of length $\leq n$ , or when $K^{∙}$ and $K^{' ∙}$ are bounded below; in the latter case, one may moreover omit the hypothesis that inductive limits exist in $C$ and $C^{'}$ .

If $K_{1}^{∙}$ , $K_{1}^{' ∙}$ are two other complexes of $C$ and $C^{'}$ respectively, $f$ , $g$ two homotopic morphisms from $K^{∙}$ to $K_{1}^{∙}$ , $f^{'}$ , $g^{'}$ two homotopic morphisms from $K^{' ∙}$ to $K_{1}^{' ∙}$ , then the morphisms $R^{∙} T (K^{∙}, K^{' ∙}) \to R^{∙} T (K_{1}^{∙}, K_{1}^{' ∙})$ deduced from $f$ and $f^{'}$ on the one hand, from $g$ and $g^{'}$ on the other, are identical, and likewise for the morphisms of the hypercohomology spectral sequences.

One generalizes easily to any covariant additive multifunctor.

Proposition (11.4.7).

Suppose that for every injective object $I$ of $C$ (resp. $I^{'}$ of $C^{'}$ ), $A^{'} \mapsto T (I, A^{'})$ (resp. $A \mapsto T (A, I^{'})$ ) is an exact functor. Then, with the notation of (11.4.6), one has canonical isomorphisms

  R^•T(K^•, K'^•) ⥲ H^•(T(L^{•,•}, K'^•)) ⥲ H^•(T(K^•, L'^{•,•}))               (11.4.7.1)

where the last two terms are the cohomology of the simple complexes defined by the tricomplexes $T (L^{∙, ∙}, K^{' ∙})$ and $T (K^{∙}, L^{' ∙, ∙})$ respectively.

Proof. Let us define, for example, the first of these isomorphisms. The quadricomplex $T (L^{∙, ∙}, L^{' ∙, ∙})$ can be considered as a bicomplex, by taking as degrees of $T (L^{i, j}, L^{' h, k})$ the numbers $i + j + h$ and $k$ . As for each $h$ , $L^{' h, ∙}$ is a resolution of $K^{' h}$ , one has, for this bicomplex, by virtue of the hypothesis on $T$ , $H_{II}^{q} (T (L^{∙, ∙}, L^{' ∙, ∙})) = 0$ for $q \neq = 0$ and $H_{II}^{0} (T (L^{∙, ∙}, L^{' ∙, ∙})) = T (L^{∙, ∙}, K^{' ∙})$ ; the first spectral sequence of this bicomplex is therefore degenerate; since $L^{' h, k} = 0$ for $k < 0$ , this sequence is moreover regular (11.3.3), and the conclusion follows from (11.1.6).

One has analogous results for a covariant multifunctor in any number $n$ of arguments: in computing the hypercohomology, it is not necessary to replace all the complexes by a Cartan–Eilenberg resolution, but only $n - 1$ of them, provided that when one fixes any $n - 1$ arguments giving them as values injective objects, the covariant functor in the remaining argument is exact.

11.5. Passage to the inductive limit in hypercohomology

Lemma (11.5.1).

Let $K^{∙} = (K^{i})_{i \in Z}$ be a complex of $C$ , and for every integer $r \in Z$ , let $K_{(r)}^{∙}$ be the complex such that $K_{(r)}^{i} = 0$ for $i < r$ , $K_{(r)}^{i} = K^{i}$ for $i \geq r$ . Let $T$ be a covariant additive functor

from $C$ to $C^{'}$ , commuting with inductive limits (one assumes that filtered inductive limits exist and are exact in $C$ and $C^{'}$ ). Then $R^{∙} T (K^{∙})$ is isomorphic to the inductive limit $lim R^{∙} T (K_{(r)}^{∙})$ as $r$ tends to $- \infty$ .

Proof. The construction of an injective Cartan–Eilenberg resolution of $K^{∙}$ is performed by choosing arbitrarily, for each $i$ , an injective resolution $(X_{B}^{i, j})_{j \geq 0}$ of $B^{i} (K^{∙})$ and an injective resolution $(X_{H}^{i, j})_{j \geq 0}$ of $H^{i} (K^{∙})$ ; that done, the method of construction shows that the injective resolution $(L^{i, j})_{j \geq 0}$ of $K^{i}$ and the derivation operators $L^{i, j} \to L^{i + 1, j}$ depend only on the resolutions $(X_{B}^{i, ∙})$ , $(X_{H}^{i, ∙})$ and $(X_{B}^{i + 1, ∙})$ (M, XVII, 1.2). Now, it is clear that one has $B^{i} (K_{(r)}^{∙}) = B^{i} (K^{∙})$ and $H^{i} (K_{(r)}^{∙}) = H^{i} (K^{∙})$ for $i \geq r + 1$ . One has, on the other hand, for each $r$ , a canonical injection $ϕ_{r - 1, r} : K_{(r)}^{∙} \to K_{(r - 1)}^{∙}$ , $ϕ_{r - 1, r}$ being the identity for $i \neq = r - 1$ . The preceding remark shows that if $L^{∙, ∙} = (L^{i, j})$ is an injective Cartan–Eilenberg resolution of $K^{∙}$ , one can for each $r$ define an injective Cartan–Eilenberg resolution $L_{(r)}^{∙, ∙} = (L_{(r)}^{i, j})$ of $K_{(r)}^{∙}$ such that $L_{(r)}^{i, j} = 0$ for $i < r$ and $L_{(r)}^{i, j} = L^{i, j}$ for $i \geq r + 1$ . One can, on the other hand, define a morphism of bicomplexes $Φ_{r - 1, r}^{∙, ∙} : L_{(r)}^{∙, ∙} \to L_{(r - 1)}^{∙, ∙}$ corresponding to $ϕ_{r - 1, r}$ , and the method of definition of this morphism (loc. cit.) shows once again that one may construct it so that $Φ_{r - 1, r}^{i, j}$ is the identity for $i \neq = r$ and $i \neq = r - 1$ . One has thus defined an inductive system $(L_{(r)}^{∙, ∙})$ of bicomplexes of $C$ , of which $L^{∙, ∙}$ is evidently the inductive limit as $r$ tends to $- \infty$ ; by reason of the commutativity of direct sums and inductive limits, the simple complex associated to $L^{∙, ∙}$ is also the inductive limit of the simple complex associated to $L_{(r)}^{∙, ∙}$ . Since $T$ commutes by hypothesis with inductive limits, and the same holds for cohomology (by exactness of the functor $lim$ ), one indeed has $H^{∙} (T (L^{∙, ∙})) = lim H^{∙} (T (L_{(r)}^{∙, ∙}))$ up to isomorphism.

Lemma (11.5.1) allows one to extend to arbitrary complexes $K^{∙}$ , by passage to the inductive limit, properties valid for complexes bounded below. As a first example, we shall prove:

Proposition (11.5.2).

Under the hypotheses of (11.5.1) concerning $C$ , $C^{'}$ and $T$ , $R^{∙} T (K^{∙})$ is a cohomological functor in the abelian category of complexes of $C$ .

Proof. We show that we can reduce to the case of complexes bounded below: if one has an exact sequence $0 \to K^{' ∙} \to K^{∙} \to K^{'' ∙} \to 0$ of complexes, one evidently deduces from it for each $r$ an exact sequence $0 \to K_{(r)}^{' ∙} \to K_{(r)}^{∙} \to K_{(r)}^{'' ∙} \to 0$ , whence by hypothesis an exact sequence

  … → R^nT(K_{(r)}'^•) → R^nT(K_{(r)}^•) → R^nT(K_{(r)}″^•) →^∂ R^{n+1}T(K_{(r)}'^•) → …

these exact sequences forming an inductive system; lemma (11.5.1) and the exactness of the functor $lim$ show that one has an exact sequence

  … → R^nT(K'^•) → R^nT(K^•) → R^nT(K″^•) →^∂ R^{n+1}T(K'^•) → …

To deal with the case of complexes bounded below, we may confine ourselves to those for which $K^{i} = 0$ for $i < 0$ ; these evidently form an abelian category $K$ .

Lemma (11.5.2.1).

In $K$ , let $J$ be the set of complexes $Q^{∙} = (Q^{i})_{i \geq 0}$ having the

following properties: 1° Every $Q^{i}$ is an injective object of $C$ ; 2° For every $i \geq 0$ , one has $Z^{i} (Q^{∙}) = B^{i} (Q^{∙})$ , and $Z^{i} (Q^{∙})$ is a direct factor of $Q^{i}$ . Then:

(i) Every $Q^{∙} \in J$ is an injective object of $K$ .

(ii) Every object of $K$ is isomorphic to a subcomplex of a complex belonging to $J$ .

Proof. (i) Let $A^{∙} = (A^{i})$ be an object of $K$ , $A^{' ∙} = (A^{' i})$ a subobject of $A^{∙}$ , $Q^{∙} = (Q^{i})$ an object of $J$ , and suppose given a morphism $f = (f^{i}) : A^{' ∙} \to Q^{∙}$ , which we wish to extend to a morphism $g = (g^{i}) : A^{∙} \to Q^{∙}$ . We shall use the language of the category of modules for simplicity (cf. [27]).

We identify $Q^{i}$ with $B^{i} (Q^{∙}) \oplus B^{i + 1} (Q^{∙})$ ; we proceed by induction on $i$ , supposing therefore the $g^{j}$ defined for $j < i$ , compatible with the derivation operators $d^{j} : A^{j} \to A^{j + 1}$ and $d^{j} : Q^{j} \to Q^{j + 1}$ for $j < i - 1$ and such moreover that: 1° $g^{i - 1} (Z^{i - 1} (A^{∙})) \subset Z^{i - 1} (Q^{∙})$ ; 2° If one sets $C^{j} = (d^{j})^{- 1} (A^{' j + 1})$ for every $j$ , then $d^{i - 1} \circ g^{i - 1}$ coincides with $f^{i} \circ d^{i - 1}$ on $C^{i - 1}$ . The morphism $f^{i} : A^{' i} \to Q^{i}$ gives, by composition with the projections, two morphisms $f^{i^{'}} : A^{' i} \to B^{i} (Q^{∙})$ and $f^{ii} : A^{' i} \to B^{i + 1} (Q^{∙})$ . Since $d^{i - 1} \circ g^{i - 1}$ carries $A^{i - 1}$ into $B^{i} (Q^{∙})$ and vanishes on $Z^{i - 1} (A^{∙})$ , it defines a morphism $h^{i} : B^{i} (A^{∙}) \to B^{i} (Q^{∙})$ , and since $d^{i - 1} \circ g^{i - 1}$ coincides with $f^{i} \circ d^{i - 1}$ on $C^{i - 1}$ , $h^{i}$ coincides with $f_{1}^{i}$ on $B^{i} (A^{∙}) \cap A^{' i}$ . Since $B^{i} (Q^{∙})$ , a direct factor of $Q^{i}$ , is injective, there is a morphism $g^{i^{'}} : A^{i} \to B^{i} (Q^{∙})$ which coincides with $h^{i}$ on $B^{i} (A^{∙})$ and with $f^{i^{'}}$ on $A^{' i}$ . Consider on the other hand the morphism $f^{ii + 1} \circ d^{i} : C^{i} \to B^{i + 1} (Q^{∙})$ , which vanishes on $Z^{i} (A^{∙})$ ; since $B^{i + 1} (Q^{∙})$ is injective, there is a morphism $g^{ii} : A^{i} \to B^{i + 1} (Q^{∙})$ , which coincides with $f^{ii + 1} \circ d^{i}$ on $C^{i}$ and with 0 on $Z^{i} (A^{∙})$ . It suffices then to take $g^{i} = g^{i^{'}} + g^{ii}$ to be able to continue the induction.

(ii) To embed $A^{∙} = (A^{i})$ in a complex belonging to $J$ , one takes for each $i \geq 1$ an injective object $Q^{' i}$ of $C$ such that there exists an injection $f^{' i} : A^{i} \to Q^{' i}$ . Then set $Q^{i} = 0$ for $i < 0$ , $Q^{0} = Q^{' 1}$ and $Q^{i} = Q^{' i} \oplus Q^{' i + 1}$ for $i \geq 1$ , with the obvious derivation operator. Set $f^{i} = f^{' i + 1} + (f^{' i + 1} \circ d^{i})$ for every $i$ (with $f^{' i} = 0$ for $i \leq 0$ ); it is immediate that $f^{i}$ is injective for every $i$ , and that the $f^{i}$ are compatible with the derivation operators.

Corollary (11.5.2.2).

Every object $K^{∙}$ of $K$ admits a right resolution formed of objects of $J$ . If $L^{∙, ∙}$ is such a resolution, then for any resolution $L^{' ∙, ∙}$ of $K^{∙}$ formed of objects of $K$ , there is a morphism of bicomplexes $F : L^{' ∙, ∙} \to L^{∙, ∙}$ compatible with the augmentations, and any two such morphisms $F$ , $F^{'}$ are homotopic.

Proof. This is none other than (M, V, 1.1 a)) applied to the abelian category $K$ .

11.5.2.3.

These preliminaries laid, consider an injective Cartan–Eilenberg resolution $L^{' ∙, ∙}$ of $K^{∙}$ and a resolution $L^{∙, ∙}$ of $K^{∙}$ formed of objects of $J$ , and let us show that one has an isomorphism $H^{∙} (T (L^{' ∙, ∙})) \sim H^{∙} (T (L^{∙, ∙}))$ . Indeed, one deduces from (11.5.2.2) a morphism of bicomplexes $T (L^{' ∙, ∙}) \to T (L^{∙, ∙})$ , and consequently a morphism $^{'} E (T (L^{' ∙, ∙})) \to^{'} E (T (L^{∙, ∙}))$ of the first spectral sequences of these bicomplexes. Since by virtue of (11.3.3) these spectral sequences are regular, it suffices (11.1.5) to see that the preceding morphism is an isomorphism for the terms E_2, or, equivalently, that $H_{II}^{q} (T (L^{i, ∙}))$ is equal to $R^{qT} (K^{i})$ ; since $L^{i, ∙}$ is a right resolution of $K^{i}$ , one is reduced to proving the

Lemma (11.5.2.4).

If $L^{∙} = (L^{i})_{i \in Z}$ is a right resolution of an object $A$ of $C$ such that $R^{n T} (L^{i}) = 0$ for every $i \in Z$ and every $n > 0$ , then one has $H^{∙} T (L^{∙}) = R^{∙} T (A)$ .

Proof. This is a particular case of (T, 2.5.2).

11.5.2.5.

The proof of (11.5.2) now concludes at once, for $L^{' ∙, ∙}$ is an injective resolution of $K^{∙}$ in the abelian category $K$ ; in other words, $K^{∙} \mapsto H^{∙} (T (L^{' ∙, ∙}))$ is none other than the right-derived cohomological functor of $T$ in the category $K$ (T, 2.3).

Proposition (11.5.3).

Under the hypotheses of (11.5.1) concerning $C$ , $C^{'}$ and $T$ , let $L^{∙, ∙} = (L^{i, j})$ be a bicomplex such that $L^{i, j} = 0$ for $j < 0$ and such that, for every $i$ , $L^{i, ∙}$ is a resolution of $K^{i}$ ; suppose finally that $R^{n T} (L^{i, j}) = 0$ for every pair $(i, j)$ and every $n > 0$ . Then there exists a functorial isomorphism

$R^{∙} T (K^{∙}) \sim H^{∙} (T (L^{∙, ∙})) . (11.5.3.1)$

Proof. Let $L_{(r)}^{∙, ∙} = (L_{(r)}^{i, j})$ be the bicomplex such that $L_{(r)}^{i, j} = 0$ for $i < r$ , $L_{(r)}^{i, j} = L^{i, j}$ for $i \geq r$ ; it is immediate that $L^{∙, ∙}$ is the inductive limit of $L_{(r)}^{∙, ∙}$ as $r$ tends to $- \infty$ ; by virtue of the hypothesis on $T$ and of (11.5.1), it suffices therefore to prove the proposition when $K^{∙}$ is bounded below, for example $K^{i} = 0$ for $i < 0$ , and $L^{i, j} = 0$ for $i < 0$ . Let then $L^{' ∙, ∙} = (L^{' i, j})$ be a right resolution of $K^{∙}$ formed of objects of $J$ (11.5.2.2); there is a morphism of bicomplexes $L^{∙, ∙} \to L^{' ∙, ∙}$ compatible with the augmentations, whence a morphism $^{'} E (T (L^{∙, ∙})) \to^{'} E (T (L^{' ∙, ∙}))$ for the first spectral sequences; lemma (11.5.2.4) shows, as in (11.5.2.3), that this morphism is an isomorphism, whence the conclusion.

Remark (11.5.4).

The preceding arguments prove that the conclusions of (11.5.2) and (11.5.3) are valid in the category of complexes $K^{∙}$ bounded below, without supposing that $T$ commutes with filtered inductive limits. Moreover, when one considers only the category $K$ of complexes $K^{∙}$ such that $K^{i} = 0$ for $i < 0$ , the characterization of $R^{∙} T (K^{∙})$ as the system of right-derived functors of $T$ in $K$ shows that this cohomological functor is universal (T, 2.3).

Another case in which (11.5.2) is valid without making any additional hypothesis on $T$ is the case where there exists an integer $m > 0$ such that every object of $C$ admits an injective resolution of length $\leq m$ . Indeed, in the proof of (11.5.1), all the injective resolutions of objects of $C$ that intervene may be taken of length $\leq m$ , whence it follows at once that the terms of total degree $n$ of the bicomplex $T (L_{(r)}^{∙, ∙})$ are equal to those of $T (L^{∙, ∙})$ and finite in number, as soon as $r$ is sufficiently large; this entails that for every $n$ , $H^{n} (T (L^{∙, ∙})) = H^{n} (T (L_{(r)}^{∙, ∙}))$ as soon as $r$ is sufficiently large. With the notation of (11.5.2), one therefore also has $R^{n T} (K_{(r)}^{∙}) = R^{n T} (K^{∙})$ for $r$ sufficiently large (depending on $n$ ), and likewise for $K^{' ∙}$ and $K^{'' ∙}$ , whence the conclusion. In the same way, (11.5.3) is valid without additional condition on $T$ when $C$ satisfies the preceding hypothesis and one supposes that the resolutions $L^{i, ∙}$ are of length $\leq m$ .

11.5.5.

The result of (11.5.2) generalizes to covariant multifunctors. Consider for example the situation of (11.4.6), where one supposes that in $C$ , $C^{'}$ and $C^{''}$

filtered inductive limits exist and are exact, and that $T$ commutes with inductive limits. Then $R^{∙} T (K^{∙}, K^{' ∙})$ is a bifunctor cohomological in each of the complexes $K^{∙}$ , $K^{' ∙}$ ; to see this, one reduces as in (11.5.2) to the case where $K^{∙}$ and $K^{' ∙}$ are bounded below; taking then injective resolutions of $K^{∙}$ and $K^{' ∙}$ of the type described in (11.5.2.2), one is reduced to the general property proved in (M, V, 4.1).

11.5.6.

Similarly, the results of (11.4.7) and (11.5.3) generalize as follows. Suppose (under the hypotheses of (11.5.5)) that one has two bicomplexes $L^{∙, ∙} = (L^{i, j})$ , $L^{' ∙, ∙} = (L^{' i, j})$ such that $L^{i, j} = 0$ and $L^{' i, j} = 0$ for $j < 0$ , that for every $i$ , $L^{i, ∙}$ is a resolution of $K^{i}$ and $L^{' i, ∙}$ is a resolution of $K^{' i}$ , and finally that $R^{n T} (L^{i, j}, L^{' h, k}) = 0$ for $n > 0$ and for every system of indices $(i, j, h, k)$ . Then one has a functorial isomorphism in $K^{∙}$ and $K^{' ∙}$

  R^•T(K^•, K'^•) ⥲ H^•(T(L^{•,•}, L'^{•,•})).                                  (11.5.6.1)

This is established as in (11.5.3) by reducing to the case where $K^{∙}$ and $K^{' ∙}$ are bounded below.

Suppose moreover that for every pair $(i, j)$ and for every pair $(h, k)$ , the functors $A \mapsto T (A, L^{' h, k})$ and $A^{'} \mapsto T (L^{i, j}, A^{'})$ are exact in $C$ and $C^{'}$ respectively. Then one also has functorial isomorphisms

  R^•T(K^•, K'^•) ⥲ H^•(T(L^{•,•}, K'^•)) ⥲ H^•(T(K^•, L'^{•,•})).              (11.5.6.2)

The proof is similar to that of (11.4.7).

11.5.7.

One again notes that the results of (11.5.5) and (11.5.6) are valid without supposing that $T$ commutes with inductive limits, provided one restricts oneself to complexes $K^{∙}$ , $K^{' ∙}$ bounded below, or one supposes that every object of $C$ (resp. $C^{'}$ ) admits an injective resolution of bounded length, and that in (11.5.6) the bicomplexes $L^{∙, ∙}$ and $L^{' ∙, ∙}$ have their second degree bounded above.

11.6. Hyperhomology of a functor with respect to a complex $K_{∙}$

11.6.1.

Let $C$ be an abelian category, $K_{∙} = (K_{i})_{i \in Z}$ a complex of objects of $C$ whose derivation operator is of degree $- 1$ . A left Cartan–Eilenberg resolution of $K_{∙}$ consists of a bicomplex $L_{∙, ∙} = (L_{i, j})$ with derivation operators of degree $- 1$ with $L_{i, j} = 0$ for $j < 0$ , and a morphism of simple complexes $ϵ : L_{∙, 0} \to K_{∙}$ , such that the conditions obtained from those of (11.4.2) by "reversal of arrows" are satisfied. If every object of $C$ is a quotient of a projective object, every complex $K_{∙}$ of $C$ admits a projective Cartan–Eilenberg resolution, that is, one formed of projective objects $L_{i, j}$ , with functorial properties similar to those of (11.4.2). Moreover, if $K_{∙}$ is bounded below (resp. bounded above), one may suppose that $L_{i, j} = 0$ for $i < i_{0}$ (resp. $i > i_{0}$ ) if $K_{i} = 0$ for $i < i_{0}$ (resp. $i > i_{0}$ ). If every object of $C$ admits a projective resolution of length $\leq n$ , one may suppose that $L_{i, j} = 0$ for $j > n$ .

11.6.2.

Suppose that $T$ is a covariant additive functor from $C$ to an abelian category $C^{'}$ . The definition of the hyperhomology $L_{∙} T (K_{∙})$ and of the spectral sequences of hyperhomology of $T$ with respect to a complex $K_{∙}$ of $C$ (when they exist) is performed again from (11.4.3) by "reversal of arrows", the terms $E^{2}$ of the two spectral sequences thus obtained being

$^{'} E_{p, q}^{2} = H_{p} (L_{qT} (K_{∙})) (11.6.2.1)^{''} E_{p, q}^{2} = L_{pT} (H_{q} (K_{∙})) (11.6.2.2)$

where $L_{pT}$ denotes the $p$ th derived functor of $T$ for $p \geq 0$ , and 0 for $p < 0$ ; $L_{∙} T (K_{∙})$ denotes the complex $(L_{qT} (K_{i}))_{i \in Z}$ .

The properties of hyperhomology are not all deducible by simple "reversal of arrows" from those of hypercohomology (unless one makes additional hypotheses of type AB 5*) of T, 1.5 on the category $C^{'}$ ) because of the regularity conditions on the two preceding spectral sequences, to which one must this time apply the criteria of (11.3.4). These last show that when one supposes that in $C^{'}$ filtered inductive limits exist and are exact, then the complex defined by the bicomplex $T (L_{∙, ∙})$ exists, and the second spectral sequence $^{''} E (T (L_{∙, ∙}))$ is this time regular. If one supposes either that $K_{∙}$ is bounded below, or that there exists an integer $n$ such that every object of $C$ admits a projective resolution of length $\leq n$ , then the two hyperhomology spectral sequences exist (without hypothesis on $C^{'}$ ) and are biregular.

Proposition (11.6.3).

Let $C$ , $C^{'}$ be two abelian categories, $T$ a covariant additive functor from $C$ to $C^{'}$ . Then:

(i) Hyperhomology $L_{∙} T (K_{∙})$ is a homological functor in the abelian category of complexes of $C$ bounded below.

(ii) Let $K_{∙}$ be a complex of $C$ bounded below. If $L_{n T} (K_{i}) = 0$ for every $n > 0$ and every $i \in Z$ , one has functorial isomorphisms

$L_{i T} (K_{∙}) \sim H_{i} (T (K_{∙})) (11.6.3.1)$

for $i \in Z$ .

(iii) Let $K_{∙}$ be a complex of $C$ bounded below. Let $L_{∙, ∙} = (L_{i, j})$ be a bicomplex such that $L_{i, j} = 0$ for $j < 0$ and such that, for every $i$ , $L_{i, ∙}$ is a resolution of $K_{i}$ ; suppose finally that $L_{n T} (L_{i, j}) = 0$ for every pair $(i, j)$ and every $n > 0$ . Then there exists a functorial isomorphism

$L_{∙} T (K_{∙}) \sim H_{∙} (T (L_{∙, ∙})) . (11.6.3.2)$

The proofs proceed as those of (11.5.2), (11.4.5) and (11.5.3) in the case of complexes bounded below. We leave the details of these arguments to the reader.

11.6.4.

One has entirely analogous results for covariant multifunctors additive in each of the arguments. For example, for a bifunctor $T$ , one has the two hyperhomology spectral sequences with terms $E^{2}$ given by

  'E_{p,q}^2 = H_p(L_qT(K_•, K_•'))                                            (11.6.4.1)
  ″E_{p,q}^2 = ⊕_{q'+q″=q} L_pT(H_{q'}(K_•), H_{q″}(K_•')).                    (11.6.4.2)

Here too, it is the second spectral sequence which is regular, the two sequences being biregular when one deals with complexes $K_{∙}$ , $K_{∙}^{'}$ bounded below, or when the objects of the abelian categories considered have projective resolutions of fixed length.

Moreover:

Proposition (11.6.5).

Let $C$ , $C^{'}$ , $C^{''}$ be three abelian categories, $T$ a covariant bifunctor biadditive from $C \times C^{'}$ to $C^{''}$ .

(i) $L_{∙} T (K_{∙}, K_{∙}^{'})$ is a homological bifunctor in each of the complexes bounded below $K_{∙}$ , $K_{∙}^{'}$ (formed respectively of objects of $C$ and of objects of $C^{'}$ ).

(ii) Suppose $K_{∙}$ and $K_{∙}^{'}$ bounded below. Let $L_{∙, ∙} = (L_{i, j})$ , $L_{∙, ∙}^{'} = (L_{i, j}^{'})$ be two bicomplexes such that $L_{i, j} = 0$ and $L_{i, j}^{'} = 0$ for $j < 0$ , such that for every $i$ , $L_{i, ∙}$ is a resolution of $K_{i}$ and $L_{i, ∙}^{'}$ is a resolution of $K_{i}^{'}$ , and finally that $L_{n T} (L_{i, j}, L_{h, k}^{'}) = 0$ for $n > 0$ and every system $(i, j, h, k)$ . Then one has a functorial isomorphism

  L_•T(K_•, K_•') ⥲ H_•(T(L_{•,•}, L_{•,•}')).                                 (11.6.5.1)

(iii) Suppose moreover that for every pair $(i, j)$ and every pair $(h, k)$ , the functors $A \mapsto T (A, L_{h, k}^{'})$ and $A^{'} \mapsto T (L_{i, j}, A^{'})$ are exact in $C$ and $C^{'}$ respectively. Then one has functorial isomorphisms

  L_•T(K_•, K_•') ⥲ H_•(T(L_{•,•}, K_•')) ⥲ H_•(T(K_•, L_{•,•}')).             (11.6.5.2)

The proofs are analogous to those of (11.5.5) and (11.5.6).

11.7. Hyperhomology of a functor with respect to a bicomplex $K_{∙, ∙}$

11.7.1.

Let $C$ be an abelian category in which every object is a quotient of a projective object. Consider a bicomplex $K_{∙, ∙} = (K_{i, j})$ formed of objects of $C$ , and whose two degrees are bounded below; one may always restrict to the case where $K_{i, j} = 0$ for $i < 0$ or $j < 0$ , and this is what we shall do henceforth. One may consider $K_{∙, ∙}$ as a (simple) complex formed of objects $K_{i, ∙} = (K_{i, j})_{j \geq 0}$ of the abelian category $K$ of complexes of positive degrees of objects of $C$ . It follows from lemma (11.5.2.1) (or rather from the "dual" lemma obtained by "reversal of arrows") and from (M, V, 2.2) that $K_{∙, ∙}$ admits a projective Cartan–Eilenberg resolution in the category $K$ ; such a resolution is a tricomplex $M_{∙, ∙, ∙} = (M_{i, j, k})$ of $C$ , with all degrees $\geq 0$ , formed of projective objects, such that for every $i$ , $M_{i, ∙, ∙}$ , $B_{j}^{I} (M_{∙, ∙, ∙})$ , $Z_{j}^{I} (M_{∙, ∙, ∙})$ , $H_{j}^{I} (M_{∙, ∙, ∙})$ constitute projective resolutions of $K_{i, ∙}$ , $B_{j}^{I} (K_{∙, ∙})$ , $Z_{j}^{I} (K_{∙, ∙})$ , $H_{j}^{I} (K_{∙, ∙})$ respectively in the category $K$ ; in particular, for every pair $(i, j)$ , $M_{i, j, ∙}$ is a projective resolution of $K_{i, j}$ in $C$ .

Proposition (11.7.2).

Let $T$ be a covariant additive functor from $C$ to an abelian category $C^{'}$ . With the notation of (11.7.1), the homology $H_{∙} (T (M_{∙, ∙, ∙}))$ of the simple complex associated to the tricomplex $T (M_{∙, ∙, ∙})$ is canonically isomorphic to the hyperhomology $L_{∙} T (K_{∙})$ of the

simple complex associated to $K_{∙, ∙}$ (11.6.2), and is the common abutment of six biregular spectral sequences denoted $^{(t)} E$ (with $t = a$ , $b$ , $a^{'}$ , $b^{'}$ , $c$ , or $d$ ), whose terms $E^{2}$ are given by the formulas

$^{(a)} E_{p, q}^{2} = L_{pT} (H_{q}^{I} (K_{∙, ∙}))^{(b)} E_{p, q}^{2} = H_{p} (L_{q}^{II} T (K_{∙, ∙}))^{(a^{'})} E_{p, q}^{2} = L_{pT} (H_{q}^{II} (K_{∙, ∙}))^{(b^{'})} E_{p, q}^{2} = H_{p} (L_{q}^{I} T (K_{∙, ∙}))^{(c)} E_{p, q}^{2} = L_{pT} (H_{q} (K_{∙, ∙}))^{(d)} E_{p, q}^{2} = H_{p} (L_{q}^{I} T (K_{∙, ∙}))$

(Recall that we use the notation $F (A_{∙})$ to denote the complex of objects $F (A_{i})$ for every complex $A_{∙} = (A_{i})$ ; for example $L_{q}^{II} T (K_{∙, ∙})$ denotes the complex $(L_{q}^{II} T (K_{i, ∙}))_{i \geq 0}$ , where $L_{q}^{II} T (K_{i, ∙})$ is the hyperhomology of index $q$ of the functor $T$ with respect to the simple complex $K_{i, ∙}$ .)

Proof. Denote by $L_{∙}$ the simple complex associated to $K_{∙, ∙}$ , so that $L_{i} = \oplus_{r + s = i} K_{r, s}$ , and set $N_{i, ∙} = \oplus_{r + s = i} M_{r, s, ∙}$ ; it is clear that for each $i$ , $N_{i, ∙}$ is a projective resolution of $L_{i}$ in $C$ ; therefore by (11.6.3) and (11.6.4), one has a functorial isomorphism $L_{∙} T (L_{∙}) \sim H_{∙} (T (N_{∙, ∙}))$ ; as the simple complex associated to the bicomplex $T (N_{∙, ∙})$ is also associated to the tricomplex $T (M_{∙, ∙, ∙})$ , this proves the first assertion of the statement.

Moreover, $L_{∙} T (L_{∙})$ is the abutment of the two hyperhomology spectral sequences (11.6.2) of $T$ relative to the simple complex $L_{∙}$ , which are nothing other than the sequences $^{(b^{'})} E$ and $^{(b)} E$ respectively.

Consider now $M_{∙, ∙, ∙}$ as a bicomplex $U_{∙, ∙}$ with $U_{i, j} = \oplus_{r + s = j} M_{i, r, s}$ ; $H_{∙} (T (M_{∙, ∙, ∙}))$ is again the abutment of the two spectral sequences of the bicomplex $T (U_{∙, ∙})$ . Now, for every $i$ , $M_{i, ∙, ∙}$ is a bicomplex satisfying the conditions of (11.6.3) relative to the simple complex $K_{i, ∙}$ ; hence $H_{q}^{II} (T (U_{i, ∙})) = L_{q}^{II} T (K_{i, ∙})$ , and the first spectral sequence of $T (U_{∙, ∙})$ is nothing other than $^{(b)} E$ . On the other hand, for every $r$ , $M_{∙, r, ∙}$ is a Cartan–Eilenberg resolution of the simple complex $K_{∙, r}$ ; the calculation done in (M, XV, 2) shows that $H_{q}^{I} (T (M_{∙, r_{∙}, ∙})) = T (H_{q}^{I} (M_{∙, r_{∙}, ∙}))$ , hence $H_{q}^{I} (T (U_{∙, ∙})) = \oplus_{r + s = j} T (H_{q}^{I} (M_{∙, r, s}))$ ; in other words, the simple complex $H_{q}^{I} (T (U_{∙, ∙}))$ is nothing other than the simple complex associated to the bicomplex $T (H_{q}^{I} (M_{∙, ∙, ∙}))$ . Now, for every $q$ , $H_{q}^{I} (M_{∙, ∙, ∙})$ is a projective resolution of the simple complex $H_{q}^{I} (K_{∙, ∙})$ in the category $K$ ; applying (11.6.3), one sees that one has

$H_{p}^{II} (T (H_{q}^{I} (M_{∙, ∙, ∙}))) = L_{pT} (H_{q}^{I} (K_{∙, ∙})),$

and one thus obtains the sequence $^{(a)} E$ . Finally, the sequences $^{(a^{'})} E$ and $^{(c)} E$ are obtained by interchanging the roles of the indices $i$ and $j$ in the definition of the tricomplex $M_{∙, ∙, ∙}$ and applying to this new tricomplex the preceding arguments.

One says that $L_{∙} T (K_{∙, ∙})$ is the hyperhomology of $T$ relative to the bicomplex $K_{∙, ∙}$ .

Remarks (11.7.3).

(i) It follows from (11.6.3) that $L_{∙} T (K_{∙, ∙})$ is a homological functor in the category of bicomplexes $K_{∙, ∙}$ of $C$ bounded below in each of their degrees.

(ii) Let $M_{∙, ∙, ∙}$ be a tricomplex of $C$ such that for each pair $(r, s)$ , $M_{r, s, ∙}$ is a resolution of $K_{r, s}$ and that $L_{n T} (M_{ijk}) = 0$ for all triples $(i, j, k)$ and every $n > 0$ . Then one has an isomorphism $L_{∙} T (K_{∙, ∙}) \sim H_{∙} (T (M_{∙, ∙, ∙}))$ ; indeed, with the notation of the proof of (11.7.2), $N_{i, ∙}$ is a resolution of $L_{i}$ such that $L_{n T} (N_{i, j}) = 0$ for $n > 0$ and every pair $(i, j)$ , and it suffices to apply (11.6.3, (iii)).

(iii) One generalizes at once the results of (11.7.2) to covariant multifunctors; for example, let $C^{'}$ be a second abelian category in which every object is a quotient of a projective object, $K_{∙, ∙}^{'}$ a bicomplex of $C^{'}$ whose two degrees are bounded below, and $T$ a covariant additive bifunctor from $C \times C^{'}$ to an abelian category $C^{''}$ . If $L_{∙}$ and $L_{∙}^{'}$ are the simple complexes associated to $K_{∙, ∙}$ and $K_{∙, ∙}^{'}$ respectively, one defines the hyperhomology of $T$ with respect to the two bicomplexes $K_{∙, ∙}$ , $K_{∙, ∙}^{'}$ as the hyperhomology $L_{∙} T (L_{∙}, L_{∙}^{'})$ ; applying (11.6.4) and (11.6.5), one has, as in (11.7.2), six spectral sequences abutting to this hyperhomology, which we leave the reader to write out.

11.8. Complements on the cohomology of simplicial complexes

11.8.1.

Let $A$ be a finite set, $Σ (A)$ the set of finite sequences $σ = (α_{0}, \dots, α_{h})$ of elements of $A$ ("simplices" of $A$ ); one sets $∣ σ ∣ = α_{0}, \dots, α_{h}$ ; recall that the chain complex $C_{∙} (A)$ is the free graded abelian group generated by the elements of $Σ (A)$ , $(α_{0}, \dots, α_{h})$ being of degree $h$ , with a differential defined by

  d(α_0, …, α_h) = ∑_{i=0}^h (−1)^i (α_0, …, α̂_i, …, α_h).

The subgroup $D_{∙} (A)$ of $C_{∙} (A)$ generated by the chains $σ = (α_{0}, \dots, α_{h})$ for which two of the $α_{i}$ are equal, and by the chains $π (σ) - ϵ_{π} \cdot σ$ , where for every permutation $π$ , $π (σ) = (α_{π (0)}, \dots, α_{π (h)})$ and $ϵ_{π}$ is the signature of $π$ , is a subcomplex of $C_{∙} (A)$ whose elements are called degenerate chains; one sets $L_{∙} (A) = C_{∙} (A) / D_{∙} (A)$ , and one has a natural homomorphism of complexes $p : C_{∙} (A) \to L_{∙} (A)$ . One defines on the other hand a homomorphism of complexes $j : L_{∙} (A) \to C_{∙} (A)$ as follows: one totally orders $A$ ; to the class mod. $D_{∙} (A)$ of a simplex $σ = (α_{0}, \dots, α_{h})$ , one associates 0 if two of the $α_{i}$ are equal, and the sequence $(β_{0}, \dots, β_{h})$ of the $α_{i}$ arranged in increasing order otherwise. It is clear that $p \circ j$ is the identity of $L_{∙} (A)$ .

11.8.2.

Let $B$ be a second finite set; if $d^{'}$ , $d^{''}$ are the differentials of $C_{∙} (A)$ and $C_{∙} (B)$ , the tensor product complex $C_{∙} (A) \otimes C_{∙} (B)$ can be considered as the free abelian group generated by the elements of $Σ (A) \times Σ (B)$ , with the differential $d (σ, τ) = (d^{'} σ, τ) + (- 1)^{h + 1} (σ, d^{''} τ)$ if $C a r d ∣ σ ∣ = h + 1$ .

The natural homomorphisms $C_{∙} (A) \to L_{∙} (A)$ , $C_{∙} (B) \to L_{∙} (B)$ define a homomorphism $p : C_{∙} (A) \otimes C_{∙} (B) \to L_{∙} (A) \otimes L_{∙} (B)$ , this last tensor product being isomorphic to $(C_{∙} (A) \otimes C_{∙} (B)) / (C_{∙} (A) \otimes D_{∙} (B) + D_{∙} (A) \otimes C_{∙} (B))$ . Likewise, by means of the homomorphisms $L_{∙} (A) \to C_{∙} (A)$ and $L_{∙} (B) \to C_{∙} (B)$ defined in (11.8.1) (by means of total orders on $A$ and $B$ ), one defines a homomorphism $j : L_{∙} (A) \otimes L_{∙} (B) \to C_{∙} (A) \otimes C_{∙} (B)$ such that $p \circ j$ is the identity.

With this notation:

Proposition (11.8.3).

There exists a homotopy $h : C_{∙} (A) \otimes C_{∙} (B) \to C_{∙} (A) \otimes C_{∙} (B)$ such that $h (σ, τ)$ is a linear combination of pairs of simplices $(σ_{i}, τ_{i})$ with $∣ σ_{i} ∣ \subset ∣ σ ∣$ , $∣ τ_{i} ∣ \subset ∣ τ ∣$ , and such that for $f = j \circ p$ one has

  f − 1 = h ∘ d + d ∘ h.                                                       (11.8.3.1)

Proof. It suffices to define $h$ on each pair $(σ, τ)$ of simplices, reasoning by induction on the sum of the degrees of $σ$ and $τ$ , since one can take $h = 0$ when this sum is 0. Let $ω = f (σ, τ) - (σ, τ) - h (d (σ, τ))$ ; by the induction hypothesis and the definition of $d$ , one has $ω \in C_{∙} (∣ σ ∣) \otimes C_{∙} (∣ τ ∣)$ . One has

  dω = f(d(σ, τ)) − d(σ, τ) − d(h(d(σ, τ))) = h(d(d(σ, τ))) = 0

by virtue of (11.8.3.1) and the induction hypothesis. Now, one has $H_{q} (C_{∙} (A)) = 0$ for $q > 0$ (G, I, 3.7.4), hence also $H_{q} (C_{∙} (A) \otimes C_{∙} (B)) = 0$ for $q > 0$ , by virtue of Künneth's formula (G, I, 5.5.2). Applying this remark on replacing $A$ by $∣ σ ∣$ and $B$ by $∣ τ ∣$ , one sees that there exists an element $ω^{'}$ of $C_{∙} (∣ σ ∣) \otimes C_{∙} (∣ τ ∣)$ such that $ω = d ω^{'}$ ; taking $h (σ, τ) = ω^{'}$ , one verifies (11.8.3.1) for the pair $(σ, τ)$ and the induction can continue.

11.8.4.

The notation being that of (11.8.2), we shall set $(σ, τ) \leq (σ^{'}, τ^{'})$ if $∣ σ ∣ \subset ∣ σ^{'} ∣$ and $∣ τ ∣ \subset ∣ τ^{'} ∣$ . A system of coefficients $S$ on $Σ (A) \times Σ (B)$ consists of a family $(Γ_{σ, τ})$ of abelian groups, where $Γ_{σ, τ}$ depends only on the sets $∣ σ ∣$ and $∣ τ ∣$ , and a family of homomorphisms $Γ_{σ, τ} \to Γ_{σ^{'}, τ^{'}}$ for $(σ, τ) \leq (σ^{'}, τ^{'})$ , forming an inductive system for this preorder relation. One then defines a cochain complex $C^{∙} (A, B; S)$ as the set of families $λ = (λ (σ, τ))$ where $(σ, τ)$ runs over $Σ (A) \times Σ (B)$ , with $λ (σ, τ) \in Γ_{σ, τ}$ for every pair $(σ, τ)$ . The differential is given as follows: if $d (σ, τ) = \sum_{i} \pm (σ_{i}, τ_{i})$ , one has $∣ σ_{i} ∣ \subset ∣ σ ∣$ , $∣ τ_{i} ∣ \subset ∣ τ ∣$ for every $i$ , and one takes

  dλ(σ, τ) = ∑_i ± λ_i(σ_i, τ_i),

where $λ_{i} (σ_{i}, τ_{i})$ denotes the canonical image of $λ (σ_{i}, τ_{i})$ in $Γ_{σ, τ}$ .

We shall say that a cochain $λ \in C^{∙} (A, B; S)$ is bi-alternating if $λ (σ, τ) = 0$ whenever one of the two simplices $σ$ , $τ$ has two equal terms, and if $λ (π (σ), τ) = ϵ_{π} λ (σ, τ)$ and $λ (σ, π^{'} (τ)) = ϵ_{π^{'}} λ (σ, τ)$ for arbitrary permutations $π$ , $π^{'}$ of the indices. It is clear that these cochains generate a subcomplex $L^{∙} (A, B; S)$ of $C^{∙} (A, B; S)$ .

Proposition (11.8.5).

The canonical injection $L^{∙} (A, B; S) \to C^{∙} (A, B; S)$ defines an isomorphism for the cohomology of these two complexes.

Proof. Note that if $p$ and $j$ have the meaning defined in (11.8.2), the maps $^{t} p : λ \mapsto λ \circ p$ and $^{t} j : λ \mapsto λ \circ j$ are defined in $L^{∙} (A, B; S)$ and $C^{∙} (A, B; S)$ respectively, the first being none other than the canonical injection. Since $^{t} j \circ^{t} p$ is the identity, it suffices to show that $^{t} p \circ^{t} j$ is homotopic to the identity; now, by (11.8.3), $^{t} h : λ \mapsto λ \circ h$ is defined in $C^{∙} (A, B; S)$ and one can therefore transpose the identity (11.8.3.1), which yields the desired result.

11.8.6.

Proposition (11.8.5) reduces the computation of the cohomology of $L^{∙} (A, B; S)$ to that of the cohomology of $C^{∙} (A, B; S)$ . Recall on the other hand that this last is, by the Eilenberg–Zilber theorem (G, I, 3.10.2), canonically isomorphic to the cohomology of the cochain complex defined as follows: one forms the chain complex $P_{∙} (A, B)$ , consisting of the linear combinations of the $(σ, τ) \in Σ (A) \times Σ (B)$ such that $σ$ and $τ$ have the same degree; the differential of this complex is given by $d : (σ, τ) \mapsto \sum_{j, k} (- 1)^{j + k} (σ_{j}, τ_{k})$ if $d σ = \sum_{j} (- 1)^{j} σ_{j}$ and $d τ = \sum_{k} (- 1)^{k} τ_{k}$ ; one then has two canonical homomorphisms of complexes

  f : P_•(A, B) → C_•(A) ⊗ C_•(B),    g : C_•(A) ⊗ C_•(B) → P_•(A, B),

and one shows (loc. cit.) that there are homotopies $h$ , $h^{'}$ such that

  f ∘ g − 1 = d ∘ h + h ∘ d    and    g ∘ f − 1 = d ∘ h' + h' ∘ d.

Moreover, one has $f (σ, τ) \in C_{∙} (∣ σ ∣) \otimes C_{∙} (∣ τ ∣)$ and $g (σ, τ) \in P_{∙} (∣ σ ∣, ∣ τ ∣)$ and the homotopies $h$ , $h^{'}$ may be taken such that $h (σ, τ) \in C_{∙} (∣ σ ∣) \otimes C_{∙} (∣ τ ∣)$ and $h^{'} (σ, τ) \in P_{∙} (∣ σ ∣, ∣ τ ∣)$ . This point arises from the fact that the definition of $h (σ, τ)$ and $h^{'} (σ, τ)$ can be made by induction on the sum of the degrees of $σ$ and $τ$ , and from the fact that the $H^{q} (C_{∙} (∣ σ ∣) \otimes C_{∙} (∣ τ ∣))$ and $H^{q} (P_{∙} (∣ σ ∣, ∣ τ ∣))$ are zero for $q > 0$ (loc. cit.); one reasons then as in (11.8.3) and the conclusion follows.

One then defines $P^{∙} (A, B; S)$ as the set of families $λ = (λ (σ, τ))$ where $(σ, τ)$ runs over the pairs whose terms have the same degree, with $λ (σ, τ) \in Γ_{σ, τ}$ , and since one has $d σ = \sum_{j} (- 1)^{j} σ_{j} \in C_{∙} (∣ σ ∣)$ and $d τ = \sum_{k} (- 1)^{k} τ_{k} \in C_{∙} (∣ τ ∣)$ ,

  dλ(σ, τ) = ∑_{j, k} (−1)^{j+k} λ_{j,k}(σ_j, τ_k)

is defined and gives the differential of the complex $P^{∙} (A, B; S)$ . With this, the maps $^{t} f : λ \mapsto λ \circ f$ , $^{t} g : λ \mapsto λ \circ g$ , $^{t} h : λ \mapsto λ \circ h$ and $^{t} h^{'} : λ \mapsto λ \circ h^{'}$ are all defined by virtue of the preceding remarks; $^{t} f \circ^{t} g$ and $^{t} g \circ^{t} f$ are therefore homotopic to the identity, whence the desired isomorphism between the cohomology of $C^{∙} (A, B; S)$ and that of $P^{∙} (A, B; S)$ .

Remark (11.8.7).

The same reasoning as in (11.8.3), but applied to $C_{∙} (A)$ and $L_{∙} (A)$ , shows that if $j$ and $p$ are defined as in (11.8.1), $f = j \circ p$ verifies once more a relation (11.8.3.1), with $∣ h (σ) ∣ \subset ∣ σ ∣$ , whence one deduces as in (11.8.5) an isomorphism of the cohomology of $L^{∙} (A; S)$ onto that of $C^{∙} (A; S)$ , these two complexes being defined obviously. This is the result whose proof is sketched in (G, I, 3.8.1).

11.8.8.

Take up now the notation and hypotheses of (11.8.2), and consider a complex $S^{∙} = (S^{k})$ of systems of coefficients on $Σ (A) \times Σ (B)$ : for each $(σ, τ)$ , the $Γ_{σ, τ}^{k}$ therefore form a complex of abelian groups $(k \in Z)$ , and one has the commutative diagrams

$Γ_{σ, τ}^{k} \to Γ_{σ, τ}^{k + 1} ↓↓ Γ_{σ^{'}, τ^{'}}^{k} \to Γ_{σ^{'}, τ^{'}}^{k + 1}$

for $(σ, τ) \leq (σ^{'}, τ^{'})$ . Then one verifies at once that $C^{∙} (A, B; S^{∙}) = (C^{h} (A, B; S^{k}))_{(h, k) \in Z \times Z}$ is a bicomplex of abelian groups, and $L^{∙} (A, B; S^{∙}) = (L^{h} (A, B; S^{k}))$ is a sub-bicomplex of it.

Proposition (11.8.9).

The canonical injection $L^{∙} (A, B; S^{∙}) \to C^{∙} (A, B; S^{∙})$ defines an isomorphism for the cohomology of these two bicomplexes.

Proof. Set $C^{∙, ∙} = C^{∙} (A, B; S^{∙})$ and $L^{∙, ∙} = L^{∙} (A, B; S^{∙})$ for simplicity, and note that since $C^{hk} = L^{hk} = 0$ for $h < 0$ , the second spectral sequences of these bicomplexes are regular (11.3.3); the homomorphism $L^{∙, ∙} \to C^{∙, ∙}$ therefore gives a morphism of spectral sequences $^{''} E (L^{∙, ∙}) \to^{''} E (C^{∙, ∙})$ which, for the terms E_2, reduces to

$H_{II}^{p} (H_{I}^{q} (L^{∙, ∙})) \to H_{II}^{p} (H_{I}^{q} (C^{∙, ∙})) . (11.8.9.1)$

But for every $k \in Z$ , it follows from (11.8.3) that the homomorphism $H_{I}^{q} (L^{∙, k}) \to H_{I}^{q} (C^{∙, k})$ is bijective; the conclusion therefore follows from (11.1.5).

11.8.10.

Likewise, with the notation of (11.8.6), one has canonical homomorphisms of bicomplexes $C^{∙} (A, B; S^{∙}) \to P^{∙} (A, B; S^{∙})$ (with obvious notation), and the same reasoning as in (11.8.9), based this time on (11.8.6), shows that this homomorphism again gives an isomorphism in cohomology.

11.9. A lemma on complexes of finite type

Proposition (11.9.1).

Let $C$ be an abelian category, $K^{'}$ and $K^{''}$ parts of the set of objects of $C$ , such that $K^{''} \subset K^{'}$ , and verifying the following conditions:

(i) For every object $A^{'} \in K^{'}$ and every epimorphism $u : A \to A^{'}$ in $C$ , there exists an object $B \in K^{''}$ and a morphism $v : B \to A$ such that uv is an epimorphism.

(ii) For every pair of objects $A \in K^{''}$ , $B \in K^{'}$ and every epimorphism $u : A \to B$ , $Ker (u)$ belongs to $K^{'}$ .

(iii) The product of two objects of $K^{''}$ belongs to $K^{''}$ .

Let $P_{∙} = (P_{i})_{i \in Z}$ be a complex in $C$ , such that $H_{i} (P_{∙}) \in K^{'}$ for every $i$ , and such that there exists $d$ with $H_{i} (P_{∙}) = 0$ for $i < d$ . Then there exists a complex $Q_{∙} = (Q_{i})_{i \in Z}$ in $C$ such that $Q_{i} \in K^{''}$ for every $i$ and $Q_{i} = 0$ for $i < d$ , and a morphism $u : Q_{∙} \to P_{∙}$ of complexes such that the corresponding morphism $H_{∙} (Q_{∙}) \to H_{∙} (P_{∙})$ is an isomorphism.

Proof. Let us first prove the following consequence of property (i):

(i bis) Let $u : C \to B$ be an epimorphism in $C$ , $A$ an object of $K^{'}$ , $v : A \to B$ a morphism in $C$ ; then there exists an object $D \in K^{''}$ , an epimorphism $u^{'} : D \to A$ and a morphism $v^{'} : D \to C$ such that the diagram

                u'
            D    →    A
            ↓         ↓                                                        (11.9.1.1)
           v'         v
            ↓         ↓
            C    →    B
                u

is commutative.

Indeed, consider the fibre product $C \times_{B} A$ in $C$ and the canonical projections $p : C \times_{B} A \to C$ , $q : C \times_{B} A \to A$ , making the diagram

                          q
            C ×_B A    →    A
                ↓             ↓                                                (11.9.1.2)
                p             v
                ↓             ↓
                C    →     B
                       u

commutative.

It is known ([27], p. I-12) that the cokernel of $q$ is the quotient of $A$ by $v^{- 1} (u (C))$ ; since $u$ is an epimorphism, $u (C) = B$ and $v^{- 1} (u (C)) = A$ , hence $q$ is an epimorphism; it then suffices to apply (i) to the epimorphism $q : C \times_{B} A \to A$ : there is an object $D \in K^{''}$ and a morphism $w : D \to C \times_{B} A$ such that qw is an epimorphism; one takes $u^{'} = qw$ , $v^{'} = pw$ .

That being said, to prove the proposition, we proceed by induction. Suppose, for some $i \geq d - 1$ , we have constructed, for $j \leq i$ , the objects $Q_{j}$ , the morphisms $d_{j} : Q_{j} \to Q_{j - 1}$ and the morphisms $u_{j} : Q_{j} \to P_{j}$ so that $Q_{j} = 0$ for $j < d$ , that $d_{j - 1} \circ d_{j} = 0$ and $d_{j} \circ u_{j} = u_{j - 1} \circ d_{j}$ for $j \leq i$ ; in addition, we suppose verified the following conditions:

(I_i) One has $Q_{j} \in K^{''}$ for $j \leq i$ and $B_{j} (Q_{∙}) \in K^{'}$ for $j < i$ .

(II_i) For $j < i$ , the homomorphism $H_{j} (Q_{∙}) \to H_{j} (P_{∙})$ deduced from the family $(u_{k})_{k \leq i}$ is an isomorphism.

(III_i) The composite morphism $v_{i} : Z_{i} (Q_{∙}) \to Z_{i} (P_{∙}) \to H_{i} (P_{∙})$ (where the left arrow is the restriction of $u_{i}$ and the right arrow is the canonical morphism) is an epimorphism.

Note that, by (ii), $Z_{i} (Q_{∙})$ , the kernel of the epimorphism $Q_{i} \to B_{i - 1} (Q_{∙})$ , belongs to $K^{'}$ by virtue of hypothesis (I_i). One again deduces from (ii) that $N_{i} = Ker (v_{i})$ also belongs to $K^{'}$ , taking into account hypothesis (III_i). By virtue of (i bis), there exists a $Q_{i + 1}^{'} \in K^{''}$ , an epimorphism $d_{i + 1}^{'} : Q_{i + 1}^{'} \to N_{i}$ and a morphism $u_{i + 1}^{'} : Q_{i + 1}^{'} \to P_{i + 1}$ , such that the diagram

$d_{i + 1}^{'} Q_{i + 1}^{'} \to N_{i} ↓↓ (11.9.1.3) u_{i + 1}^{'} v_{i} ↓↓ P_{i + 1} \to B_{i} (P_{∙}) d_{i + 1}$

is commutative.

Since the canonical morphism $Z_{i + 1} (P_{∙}) \to H_{i + 1} (P_{∙})$ is an epimorphism and $H_{i + 1} (P_{∙}) \in K^{'}$ by hypothesis, it follows from (i) that there exists an object $Q_{i + 1}^{''} \in K^{''}$ and a morphism $u_{i + 1}^{''} : Q_{i + 1}^{''} \to Z_{i + 1} (P_{∙})$ such that the composite $Q_{i + 1}^{''} \to Z_{i + 1} (P_{∙}) \to H_{i + 1} (P_{∙})$ is an epimorphism. If one takes $d_{i + 1}^{''} : Q_{i + 1}^{''} \to N_{i}$ equal to 0, the diagram

$d_{i + 1}^{''} Q_{i + 1}^{''} \to N_{i} ↓↓ (11.9.1.4) u_{i + 1}^{''} v_{i} ↓↓ Z_{i + 1} (P_{∙}) \to P_{i} d_{i + 1}$

is commutative, the horizontal arrow at the bottom being 0. Then take $Q_{i + 1} = Q_{i + 1}^{'} \times Q_{i + 1}^{''}$ , which belongs to $K^{''}$ by virtue of (iii), and $d_{i + 1} = d_{i + 1}^{'} + d_{i + 1}^{''}$ , $u_{i + 1} = u_{i + 1}^{'} + u_{i + 1}^{''}$ . Since $d_{i + 1} (Q_{i + 1}) = d_{i + 1}^{'} (Q_{i + 1}^{'}) = N_{i} \subset Z_{i} (Q_{∙})$ , one has $d_{i} \circ d_{i + 1} = 0$ and, with the usual notation, $B_{i} (Q_{∙}) = N_{i}$ , which verifies (I_{i+1}). The commutativity of the diagrams (11.9.1.3) and (11.9.1.4) shows that $d_{i + 1} \circ u_{i + 1} = u_{i} \circ d_{i + 1}$ . By definition of $N_{i}$ , the morphism $H_{i} (Q_{∙}) = Z_{i} (Q_{∙}) / N_{i} \to H_{i} (P_{∙})$ deduced from the system of the $u_{k}$ ( $k \leq i + 1$ ) is the morphism deduced from $v_{i}$ by passage to quotients, hence it is an isomorphism since $v_{i}$ is an epimorphism, whence (II_{i+1}). Finally, one has $Q_{i + 1}^{''} \subset Z_{i + 1} (Q_{∙})$ by definition; the choice of $u_{i + 1}^{''}$ shows that the morphism $v_{i + 1} : Z_{i + 1} (Q_{∙}) \to Z_{i + 1} (P_{∙}) \to H_{i + 1} (P_{∙})$ is an epimorphism, its restriction to $Q_{i + 1}^{''}$ already being so, whence (III_{i+1}). The induction can therefore continue, and the proposition is proved.

Corollary (11.9.2).

Let $A$ be a Noetherian ring (not necessarily commutative), $P_{∙} = (P_{i})_{i \in Z}$ a complex of right $A$ -modules. Suppose that the $H_{i} (P_{∙})$ are $A$ -modules of finite type and that there exists $d$ such that $H_{i} (P_{∙}) = 0$ for $i < d$ . Then there exists a complex $Q_{∙} = (Q_{i})_{i \in Z}$ formed of right $A$ -modules free of finite rank, such that $Q_{i} = 0$ for $i < d$ , and a homomorphism $u : Q_{∙} \to P_{∙}$ of complexes, such that the homomorphism $H_{∙} (Q_{∙}) \to H_{∙} (P_{∙})$ corresponding to $u$ is bijective.

Proof. One applies (11.9.1) taking for $C$ the category of right $A$ -modules, for $K^{'}$ (resp. $K^{''}$ ) the set of $A$ -modules of finite type (resp. the set of $A$ -modules free of finite rank); verification of conditions (i), (ii) and (iii) of (11.9.1) is immediate, taking into account the hypothesis that $A$ is Noetherian.

Remarks (11.9.3).

(i) Under the conditions of (11.9.2), suppose moreover that the $P_{i}$ are flat right $A$ -modules. Then, for every left $A$ -module $M$ , the homomorphism of complexes $u \otimes 1 : Q_{∙} \otimes_{A} M \to P_{∙} \otimes_{A} M$ still defines an isomorphism $H_{∙} (Q_{∙} \otimes_{A} M) \sim H_{∙} (P_{∙} \otimes_{A} M)$ of the homology, as we shall see in chap. III.

(ii) The conclusion of (11.9.2) is no longer necessarily exact when one does not suppose $A$ Noetherian; indeed, applying it to a complex reduced to 0 except for a single term, one would conclude that every left $A$ -module of finite type admits a resolution by free modules of finite type, which is not true in general (cf. Bourbaki, Alg. comm., chap. I, § 2, exerc. 6).

However, instead of supposing $A$ Noetherian, one may suppose only that the $H_{i} (P_{∙})$ have an $\infty$ -presentation finite (cf. chap. IV).

11.10. Euler–Poincaré characteristic of a complex of modules of finite length

11.10.1.

Let $A$ be a ring (not necessarily commutative),

  M^• : 0 → M^0 → M^1 → … → M^n → 0                                            (11.10.1.1)

a complex of left $A$ -modules of finite length. One calls Euler–Poincaré characteristic of this complex the number

  χ(M^•) = ∑_{i=0}^n (−1)^i long(M^i).                                          (11.10.1.2)

Proposition (11.10.2).

For every finite complex $M^{∙}$ of left $A$ -modules of finite length, one has $χ (M^{∙}) = χ (H^{∙} (M^{∙}))$ ( $H^{∙} (M^{∙})$ being considered as a complex for the trivial derivation). In particular, if the sequence (11.10.1.1) is exact, one has $χ (M^{∙}) = 0$ .

Proof. Set, to abbreviate, $B^{i} = B^{i} (M^{∙})$ , $Z^{i} = Z^{i} (M^{∙})$ , $H^{i} = H^{i} (M^{∙}) = Z^{i} / B^{i}$ ; the $B^{i}$ , $Z^{i}$ , $H^{i}$ are of finite length. From the exact sequences

$0 \to B^{i} \to Z^{i} \to H^{i} \to 00 \to Z^{i} \to M^{i} \to B^{i + 1} \to 0$

one derives the relations

  long(Z^i) = long(H^i) + long(B^i)
  long(M^i) = long(Z^i) + long(B^{i+1})

whence

  long(M^i) − long(H^i) = long(B^{i+1}) + long(B^i)

Multiply this relation by $(- 1)^{i}$ and sum the relations obtained for $0 \leq i \leq n$ , noting that $B^{0} = B^{n + 1} = 0$ ; the desired equality follows.

Corollary (11.10.3).

Let $E = (E_{r}^{p, q})$ be a spectral sequence in the category of modules over a ring $A$ . Suppose that the $E_{2}^{p, q}$ are $A$ -modules of finite length and that there are only finitely many pairs $(p, q)$ such that $E_{2}^{p, q} \neq = 0$ . Then the Euler–Poincaré characteristics $χ (E_{r}^{(n)})$ of all the complexes $E_{r}^{(n)} = (E_{r}^{p, q})_{p + q = n}$ (11.1.1) are all equal. If, in addition, the sequence $E$ is weakly convergent and one sets $E_{\infty}^{(n)} = \oplus_{p + q = n} E_{\infty}^{p, q}$ for every $n \in Z$ , one also has $χ (E_{\infty}^{(∙)}) = χ (E_{2}^{(∙)})$ , $E_{\infty}^{(∙)} = (E_{\infty}^{(n)})_{n \in Z}$ being considered as a complex with trivial derivation.

Proof. Note first that if $E_{2}^{p, q} = 0$ , one has $E_{r}^{p, q} = 0$ for $2 \leq r \leq + \infty$ , so all the complexes $E_{r}^{(∙)}$ are finite and formed of $A$ -modules of finite length; the relation $χ (E_{r}^{(∙)}) = χ (E_{r + 1}^{(∙)})$ for every finite $r$ therefore follows from (11.10.2) and the isomorphism between $H^{∙} (E_{r}^{(∙)})$ and $E_{r + 1}^{(∙)}$ (as complexes with trivial derivation). The hypothesis that the $E_{2}^{p, q}$ are of finite length entails that for every pair $(p, q)$ , the sequences $(B_{k} (E_{2}^{p, q}))_{k \geq 2}$ and $(Z_{k} (E_{2}^{p, q}))_{k \geq 2}$ are stationary; the hypothesis that $E$ is weakly convergent and that $E_{2}^{p, q} = 0$ except for finitely many pairs $(p, q)$ entails therefore that there exists an integer $r \geq 2$ such that $E_{\infty}^{p, q} = E_{r}^{p, q}$ for every pair $(p, q)$ ; whence the assertion concerning $χ (E_{\infty}^{(∙)})$ .

Éléments de Géométrie Algébrique — English