Exercise 17, § 2

Let C be a bicomplex (X, p. 174, exercise 11).

a) We define two decreasing sequences $(\,^{\prime}F^pC\,)_{p\in\mathbf Z}$ and $(\,^{\prime\prime}F^qC\,)_{q\in\mathbf Z}$ of subcomplexes of the complex associated to C by putting

Let $\,^{\prime}E$ and $\,^{\prime\prime}E$ be the corresponding spectral sequences (exercise 16, a)). Now prove that for every pair $(p,q)$, the $A$-module $\,^{\prime}E^{p q}_2$ (resp. $\,^{\prime\prime}E^{p q}_2$) is isomorphic to $\,^{\prime}H^p(H^q(C))$ (resp. $\,^{\prime\prime}H^q(H^p(C))$).

b) Suppose that there exists an integer $n$ such that $C^{pq}=0$ for $p>n$ or for $q\leq n$. Now prove that the spectral sequence $\,^{\prime}E$ converges to $H(C)$. If likewise there exists an integer $m$ such that $C^{pq}=0$ for $p<m$ or for $q\geq m$, the spectral sequence $\,^{\prime\prime}E$ converges to $H(C)$.

c) Suppose that $C^{pq}=0$ for $p<0$ or $q<0$. Now prove that the homomorphism $\,^{\prime}E^{p0}_2\rightarrow H^p(C)$ (exercise 15) is obtained by passing to homology from the canonical injection of complexes $\,^{\prime}Z^0(C)\rightarrow C$. Analogously the homomorphism $\,^{\prime\prime}E^{0q}_2\rightarrow H^q(C)$ is obtained from the injection $\,^{\prime\prime}Z^0(C)\rightarrow C$.

d) Let $\widetilde C$ be a second bicomplex, $\,^{\prime}\widetilde E$ and $\,^{\prime\prime}\widetilde E$ the two associated spectral sequences. Now prove that every morphism of bicomplexes $u:C\rightarrow\widetilde C$ induces morphisms of spectral sequences $\,^{\prime}u:^{\prime}E\rightarrow\,^{\prime}\widetilde E$ and $\,^{\prime\prime}u:^{\prime\prime}E\rightarrow\,^{\prime\prime}\widetilde E$. Now prove that if two morphisms $u,v$ from $C$ to $\widetilde C$ are homotopic, the associated morphisms $\,^{\prime}u$ and $\,^{\prime}v$ (resp. $\,^{\prime\prime}u$ and $\,^{\prime\prime}v$) are equal (use exercise 16, e)).