Corollary 2

Let $C$ be a complex, Ann (C) its annihilator. Then $q + \mathrm{Ann}(C)$ annihilates $H^A(u, C)$ and $H_A^*(u, C)$.

For every $\lambda \in q$, the homothety $\lambda_{K(u)}$ is homotopic to zero by the proposition, hence so also are $1_C \otimes \lambda_{K(u)}$ and Homgr ($\lambda_{K(u)}, 1_C$) by X, p. 64, prop. 3 and X, p. 83, prop. 3; it follows that $\lambda$ annihilates $H(u, C)$ and $H^*(u, C)$. If $\lambda \in \mathrm{Ann}(C)$, then $1_{K(u)} \otimes \lambda_C$ and Homgr ($1_{K(u)}, \lambda_C$) are zero.

Suppose $L$ projective (resp. $K(u)$ acyclic in degrees > 0). Then the complex $\Lambda(L)$ is projective by III, p. 87, cor. 2 (resp. is a resolution of $A/q$); by X, p. 102 (resp. p. 100), one therefore has, for every $A$-module $M$, homomorphisms

(2) $H_r^A(u, M) \to \mathrm{Tor}_r^A(A/q, M)$, $\mathrm{Ext}_A^r(A/q, M) \to H_r^A(u, M)$ resp. (3) $\mathrm{Tor}_r^A(A/q, M) \to H_r^A(u, M)$, $H_r(u, M) \to \mathrm{Ext}_A^r(A/q, M)$.

If $L$ is projective and $K(u)$ acyclic in degrees > 0, the homomorphisms (2) and (3) above are bijective and reciprocal to one another (X, p. 102, prop. 1).