Exercise 17, § 3

In this exercise, if $(\mathbf{K}, d', d'')$ is a bicomplex (X, p. 174, exercise 11), we shall denote by $\mathbf{K}_{p,*}$ the complex $(\bigoplus_n \mathbf{K}_{p,n}, d'')$. Every complex will be considered as a bicomplex, zero in degree $(p, q)$ for $q \neq 0$.

Let $C$ be a complex of $A$-modules. A projective resolution of $C$ is a pair $(P, p)$, where $P$ is a bicomplex and $p : P \to C$ a morphism of bicomplexes, such that $P_{p,q} = 0$ for $q < 0$ and that the complex $P_{p,*}$ (resp. $Z'_{p,*}(P), B'_{p,*}(P), H'_{p,*}(P)$) defines a projective resolution of $C_p$ (resp. $Z_p(C), B_p(C), H_p(C)$) for every $p \in \mathbf{Z}$. For $(P, p)$ to be a projective resolution of $C$, it is sufficient that the complexes $B'_{p,*}(P)$ and $H'_{p,*}(P)$ define for every $p$ projective resolutions of $B_p(C)$ and $H_p(C)$.

(a) Now prove that every $A$-complex $C$ admits a projective resolution (choose for every $p$ projective resolutions $B_{p,*}$ and $H_{p,*}$ of $B_p(C)$ and $H_p(C)$; using exercise 16, construct projective resolutions $Z_{p,*}$ of $Z_p(C)$ and $P_{p,*}$ of $C_p$ as well as exact sequences:

(b) Let $C'$ be another complex, $(P', p')$ a projective resolution of $C'$, $u : C \to C'$ a morphism of complexes. Now prove that there exists a morphism of bicomplexes $\tilde{u} : P \to P'$ such that $u \circ p = p' \circ \tilde{u}$ (for every $p \in \mathbf{Z}$, choose morphisms $B'_{p,*}(P) \to B'_{p,*}(P')$ and $H'_{p,*}(P) \to H'_{p,*}(P')$; construct $\tilde{u}$ as previously, using exercise 16, b)).

(c) Let $v : C \to C'$ be a morphism of complexes homotopic to $u$, and let $\tilde{v} : P \to P'$ be a morphism of bicomplexes such that $v \circ p = p' \circ \tilde{v}$. Now prove that $\tilde{u}$ and $\tilde{v}$ are homotopic (in the sense of X, p. 174, exercise 11, c); use exercise 16, c)).

d) Let $E$ be a subset of $\mathbf{Z}$, $C$ an $A$-complex such that $C_p = 0$ for $p \in E$. Prove that there exists a projective resolution $(P, p)$ of $C$ such that $P_{p,q} = 0$ for $p \in E$.

e) Suppose that there exists an integer $n$ such that every $A$-module admits a projective resolution of length $\leq n$. Prove that every $A$-complex $C$ admits a projective resolution $(P, p)$ such that $P_{p,q} = 0$ for $q > n$.

f) Let $C$ be an $A$-complex. An injective resolution of $C$ is a pair $(e, 1)$, where $1$ is a bicomplex and $e : C \to 1$ a morphism of bicomplexes, such that $1^{p,q} = 0$ for $q < 0$ and such that the complex $I^{p,*}$ (resp. $'Z^{p,*}(1), 'B^{p,*}(1), 'H^{p,*}(1)$) defines an injective resolution of $C^p$ (resp. $Z^p(C), B^p(C), H^p(C)$) for every $p \in \mathbf{Z}$. State and prove the analogues of results a) to e) for injective resolutions.