hypothesis of the lemma (cf. the Cartan–Eilenberg resolution.) Now, the double complex gives rise to two spectral sequences, horizontal and vertical, which we are now going to examine. On the one hand, by definition, , which is always zero unless q = 0 since is G-acyclic by hypothesis. Hence, and . On the other hand, by the definition and the lemma, Since is an injective resolution of (it is a resolution since its cohomology is trivial), Since and have the same limiting term, the proof is complete. Notes References Computational Examples Sharpe, Eric (2003). Lectures on D-branes and Sheaves (pages 18–19), Spectral sequences