Exercise 8, § 4

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

a) Prove that there exist two spectral sequences of $k$-modules ${}^1E$ and ${}^2E$, converging to the same graded $k$-module, such that

(Consider the double complex $L(M)\otimes_A(N\otimes_B L(P))$.)

b) If $N$ is a flat $B$-module, prove that the spectral sequence ${}^2E$ converges to $\operatorname{Tor}^A(M,N\otimes_B P)$. If $N$ is a flat $A$-module, the spectral sequence ${}^1E$ converges to $\operatorname{Tor}^B(M\otimes_A N,P)$.

c) Let $\rho:A\to B$ be a homomorphism of $k$-algebras. Deduce from b) a spectral sequence $E$ such that

converging to $\operatorname{Tor}^A(M,P)$.