Exercise 2, § 6

Let $A, B$ be two $k$-algebras (associative and with unit); let $M$ be a left $A$-module, $P$ a right $B$-module, $N$ an $(A, B)$-bimodule. Consider the homomorphism

such that $\sigma(u \otimes m)(v) = u \circ v(m)$ for $u \in \operatorname{Hom}_B(N, P), v \in \operatorname{Hom}_A(M, N), m \in M$ (II, p. 190, exercise 6). (a) Replacing $M$ by a projective resolution, define with the aid of $\sigma$ a $k$-homomorphism (b) Suppose henceforth that $A$ is left noetherian and that the $A$-module $M$ is of finite type. Now prove that if the $B$-module $P$ is injective, $\tau$ is an isomorphism. (c) Now prove that there exist two spectral sequences $E$ and $'E$ (X, pp. 175-177, exercises 14 to 17), converging to the same graded $k$-module, such that