Exercise 11, § 2

One calls a bicomplex of A-modules a triplet $(C, d', d'')$, where C is an A-module graded of type $\mathbf{Z} \times \mathbf{Z}$, $d' : C \to C$ an A-graded endomorphism of degree $(-1, 0)$ and $d'' : C \to C$ an A-graded endomorphism of degree $(0, -1)$, satisfying $d' \circ d' = d'' \circ d'' = d' \circ d'' + d'' \circ d' = 0$. We shall also use the ascending grading defined by $C^{p,q} = C_{-p,-q}$ for $p, q \in \mathbf{Z}$.

We denote by $Z'(C), Z''(C), B'(C), B''(C)$ the sub-bicomplexes $\mathrm{Ker}\,(d'), \mathrm{Ker}\,(d''), \mathrm{Im}\,(d'), \mathrm{Im}\,(d'')$, and by $H'(C)$ (resp. $H''(C)$) the bicomplex $Z'(C)/B'(C)$ (resp. $Z''(C)/B''(C)$).

a) Put $d = d' + d''$. Show that when C is provided with the total grading associated with the given bigrading (defined by $C_n = \bigoplus_{p+q=n} C_{p,q}$, cf. II, p. 164), the pair $(C, d)$ is a complex. It is said that this is the associated complex to the bicomplex $(C, d', d'')$; its homology is denoted by $H(C)$.

b) Let $(C, d', d'')$ and $(K, \delta', \delta'')$ be two bicomplexes. A morphism of $(C, d', d'')$ into $(K, \delta', \delta'')$ is called an A-graded homomorphism $u$ of degree $(0, 0)$ from C into K satisfying: $\delta' \circ u = u \circ d'$ and $\delta'' \circ u = u \circ d''$. Show that $u$ induces a morphism of the associated complexes.

c) Let $u, v$ be two morphisms of bicomplexes from C into K. A homotopy relating $u$ to $v$ is called a pair $(s', s'')$ of A-homomorphisms from C into K, graded of degrees $(1, 0)$ and $(0, 1)$ respectively, such that

Show that $s' + s''$ is a homotopy relating the morphisms of associated complexes induced by $u$ and $v$.