Exercise 14, § 2

We call spectral sequence of $A$-modules the datum of a sequence of $A$-complexes $(E_r, d_r)_{r \geq 2}$ and of mappings $\alpha_r : H(E_r) \to E_{r+1}$, such that:

(i) The grading of the complex $E_r$ is the total grading associated with a bigrading

one has $d_r(E_r^{pq}) \subset E_r^{p+r, q-r+1}$.

(ii) The mappings $\alpha_r$ are isomorphisms of bigraded $A$-modules. We shall also use the descending bigrading of $E_r$; we shall denote $E_r^{pq} = E_r^{-p, -q}$.

a) We shall denote here by $Z_{r+1}(E_r)$ (resp. $B_{r+1}(E_r)$) the bigraded $A$-module $\mathrm{Ker}\ d_r$ (resp. $\mathrm{Im}\ d_r$). Let

be the mapping of passing to the quotient and $i_r : E_{r+1} \to E_r / B_{r+1}(E_r)$ the mapping composed of $\alpha_r^{-1}$ and the natural injection. For all $r \geq 2$, define by induction on $k$ the bigraded submodules $Z_{r+k}(E_r)$ and $B_{r+k}(E_r)$ of $E_r$ by putting: Prove that one has the inclusions: and that $E_k$ is isomorphic to $Z_k(E_r)/B_k(E_r)$ for $k \geq r+1$.

We put $Z_\infty(E_2) = \bigcap_{r \geq 3} Z_r(E_2)$, $B_\infty(E_2) = \bigcup_{r \geq 3} B_r(E_2)$ and $E_\infty = Z_\infty(E_2)/B_\infty(E_2)$. The spectral sequence is said to be regular if the decreasing sequence $(Z_r(E_2^{pq}))_{r \geq 2}$ is stationary for every pair $(p, q)$.

b) Let $E = (E_r, d_r, \alpha_r)_{r \geq 2}$ and $E' = (E'_r, d'_r, \alpha'_r)_{r \geq 2}$ be two spectral sequences of $A$-modules; we call a morphism from $E$ into $E'$ a sequence of morphisms of $A$-complexes $u_r : E_r \to E'_r$, compatible with the bigradations, such that $\alpha'_r \circ H(u_r) = u_{r+1} \circ \alpha_r$ for all $r \geq 2$. Prove that for all $r \geq 2$ and all $k \geq r+1$, $u_r$ induces a homomorphism from $Z_k(E_r)$ (resp. $B_k(E_r)$) into $Z_k(E'_r)$ (resp. $B_k(E'_r)$). In particular, $u_2$ induces homomorphisms $Z_\infty(u_2) : Z_\infty(E_2) \to Z_\infty(E'_2)$, $B_\infty(u_2) : B_\infty(E_2) \to B_\infty(E'_2)$ and $u_\infty : E_\infty \to E'_\infty$; all these homomorphisms are compatible with the bigradations.

c) Suppose that $u_2$ is an isomorphism. Now prove that $u_r$ is an isomorphism for all $r \geq 2$, as well as $B_\infty(u_2)$. If, moreover, the sequences $E$ and $E'$ are regular, $Z_\infty(u_2)$ and $u_\infty$ are isomorphisms.

d) Let $G = \bigoplus_{n \in \mathbf{Z}} G^n$ be a graded $A$-module, endowed with a decreasing sequence $(F^p G)_{p \in \mathbf{Z}}$ of graded sub-$A$-modules, such that $\bigcap_p F^p G = 0$ and $\bigcup_p F^p G = G$. We say that the spectral sequence $E$ approaches $(G, (F^p G)_{p \in \mathbf{Z}})$ (or simply $G$) if there exist isomorphisms $\beta^{pq} : E_\infty^{pq} \to (F^p G)^{p+q}/(F^{p+1} G)^{p+q}$ for all

We say that the spectral sequence E converges to G if it approaches G and if, moreover: (i) The spectral sequence E is regular; (ii) For every integer n, there exists an integer p such that $(F^p G)^n = 0$.

Suppose that E (resp. E') converges to G (resp. G'). Let $u : E \to E'$ be a morphism of spectral sequences, $v : G \to G'$ a homomorphism of degree 0, such that $v(F^p G) \subset F^p G'$ for all p and that the induced homomorphisms $\bar{v}^{p,n} : (F^p G)^n/(F^{p+1} G)^n \to (F^p G')^n/(F^{p+1} G')^n$ satisfy $\bar{v}^{p,p+q} \circ \beta^{p,q} = {\beta'}^{p,q} \circ u_{\infty}^{p,q}$ for every pair $(p, q)$. Now prove that if $u_2^{pq}$ is an isomorphism for every pair $(p, q)$, then $v$ is an isomorphism.