Exercise 8, § 5

Let $P, Q$ be two $A$-complexes. Suppose either that $P$ is bounded below and $Q$ bounded above, or that there exists an integer $d$ such that every $A$-module admits a projective resolution of length $\leq d$. a) Now prove that there exist two spectral sequences $'E$ and $"E$, converging to the same graded $k$-module, such that

(where $\mathrm{Ext}_A^q(P, Q)$ denotes the complex associated with the bicomplex $\bigoplus_{r,s} \mathrm{Ext}_A^q(P_r, Q_s^s)$). b) If $P$ is projective or $Q$ injective, the spectral sequence $"E$ converges to $H(\mathrm{Homgr}_A(P, Q))$. If $H(P)$ is projective or $H(Q)$ injective, the sequence $'E$ converges to $\mathrm{Homgr}_A(H(P), H(Q))$.