Exercise 9, § 4

Let $P$ be a complex of right $A$-modules, $Q$ a complex of left $A$-modules. Suppose either that $P$ and $Q$ are bounded below, or that there exists an integer $d$ such that every left or right $A$-module admits a projective resolution of length $\leq d$.

a) Now prove that there exist two spectral sequences (X, p. 175, exercise 14) $'E$ and $''E$, converging to the same graded $k$-module, such that

(where $\operatorname{Tor}^A_q(P,Q)$ denotes the complex associated with the bicomplex

)

(Consider projective resolutions (X, p. 183, exercise 17) $\mathscr P$ and $\mathscr Q$ of $P$ and $Q$, and the double complex $C$ such that

)

b) If $P$ is flat, now prove that the spectral sequence $''E$ converges to $H(P\otimes_A Q)$. If $H(P)$ is flat, the spectral sequence $'E$ converges to $H(P)\otimes_A H(Q)$.