Exercise 6, § 5

Let $A, B$ be two $k$-algebras (associative and with a unit), $M$ a left $A$-module, $P$ a left $B$-module, $N$ a $(B, A)$-bimodule. (a) Prove that there exist two spectral sequences (X, pp. 175-177, exercises 14 to 17) of $k$-modules $^1E$ and $^2E$, converging to the same graded $k$-module, such that

(b) By composition of the homomorphisms $\gamma$ and $\delta$ of exercise 15, p. 176, define a $k$-homomorphism If $P$ is injective, prove that $\mu$ is an isomorphism. (c) If $N$ is a flat $A$-module, show that the sequence $^1E$ converges to $\mathrm{Ext}_B(N \otimes_A M, P)$; if $N$ is a projective $B$-module, the sequence $^2E$ converges to $\mathrm{Ext}_A(M, \mathrm{Hom}_B(N, P))$. (d) Let $\rho : A \to B$ be a homomorphism of $k$-algebras. Show that there exists a spectral sequence $E$, such that $E_2^{pq} = \mathrm{Ext}_B^p(\mathrm{Tor}_q^A(B, M), P)$, converging to $\mathrm{Ext}_A(M, P)$, and a spectral sequence $'E$, such that converging to $\mathrm{Ext}_A(P, M)$.