Exercise 16, § 2

Let C be an A-complex, $(F^p C)_{p \in \mathbf{Z}}$ a decreasing sequence of subcomplexes of C. a) For $p, q, r \in \mathbf{Z}$, let $M_r^{pq}$ be the set of elements $x$ of $(F^p C)^{p+q}$ such that $dx \in F^{p+r} C$; for $r \geqslant 0$, put $E_r^{pq} = M_r^{pq}/(d(M_{r-1}^{p-r+1,q+r-2}) + M_r^{p+1,q-1})$ and denote by $d_r : E_r^{pq} \to E_r^{p+r,q-r+1}$ the homomorphism induced by $d$ by passing to quotients. Now prove that there exists for $r \geqslant 0$ an isomorphism $\alpha_r : H(E_r) \to E_{r+1}$, such that the sequence $(E_r, d_r)_{r \geqslant 2}$, endowed with the mappings $\alpha_r$, is a spectral sequence. One has

b) Suppose that $\bigcup_p F^p C = C$ and that for every $n$, there exists an integer $p$ such that $(F^p C)^n = 0$. Now prove that the spectral sequence defined in a) converges (exercise 14) to the graded A-module $H(C)$, endowed with the sequence of graded submodules $F^p H(C) = \operatorname{Im} H(i_p)$ where $i_p : F^p C \to C$ is the canonical injection. c) If $F^0 C = C$, the A-module $E_r^{pq}$ is zero for $p < 0$; if $(F^p C)^n = 0$ for $p > n$, one has $E_r^{pq} = 0$ for $q < 0$. d) Let C’ be a second complex, $(F^p C')_{p \in \mathbf{Z}}$ a decreasing sequence of subcomplexes of C’, E’ the associated spectral sequence. Let $u : C \to C'$ be a morphism of complexes such that $u(F^p C) \subset F^p C'$ for every $p$. Now prove that $u$ induces a morphism of spectral sequences $\tilde{u} : E \to E'$. e) With the notations of d), let $v : C \to C'$ be a second morphism such that $v(F^p C) \subset F^p C'$ for every $p$, and let $h$ be a homotopy relating $u$ to $v$. Let $k$ be an integer $\geqslant 0$ such that $h(F^p C) \subset F^{p-k} C'$ for every $p$. Now prove that $\tilde{u}_r = \tilde{v}_r$ for $r > k$.