Exercise 7, § 5

Let $C$ be an $A$-complex, $M$ an $A$-module. The extension module of $M$ by $C$ (resp. of $C$ by $M$) is called the graded $k$-module

If $f : C \to C'$ is a morphism of complexes and $g : M \to M'$ a homomorphism of $A$-modules, one sets $\mathscr{E}xt_A(f, g) = H(\mathrm{Homgr}(f, I(g)))$, $\mathscr{E}xt_A(g, f) = H(\mathrm{Homgr}(L(g), f))$. a) If $f : C \to C'$ is an homologism, the $k$-homomorphisms $\mathscr{E}xt_A(f, 1)$ and $\mathscr{E}xt_A(1 f)$ are bijective. If $C$ is projective and bounded on the right (resp. injective and bounded on the left), the graded $k$-module $\mathscr{E}xt_A(C, M)$ (resp. $\mathscr{E}xt_A(M, C)$) is isomorphic to $H(\mathrm{Homgr}_A(C, M))$ (resp. $H(\mathrm{Homgr}_A(M, C))$). b) Let $0 \to C' \to C \to C'' \to 0$ be an exact sequence of $A$-complexes and $0 \to M' \to M \to M'' \to 0$ an exact sequence of $A$-modules. Prove that there exist exact sequences c) Prove that there exists a spectral sequence $"E$ (X, pp. 175-177, exercises 14 to 17) converging to $\mathscr{E}xt_A(C, M)$ such that $"E_2^{pq} = H^q(\mathscr{E}xt_A^p(C, M))$. If $C$ is bounded on the right, prove that there exists a spectral sequence $'E$, converging to $\mathscr{E}xt_A(C, M)$, such that $'E_2^{pq} = \mathrm{Ext}_A^p(H_q(C), M)$ (consider the bicomplexe $\mathrm{Homgr}_A(C, I(M))$).