Corollary 2

Let $A$ be a local ring, $C$ a complex of $A$-modules bounded to the left, $p$ an integer. Suppose that for $m \geq p$, $H_m(C)$ is of finite length and $C_m$ of depth $> m - p$. Then $H_n(C) = 0$ for $n \geq p$.

The $A$-module $\bigoplus_{m \geq p} H_m(C)$ is of finite length. Let $J$ denote its annihilator; by A, VIII, § 1, No. 3, corollary, the ring $A/J$ is artinian, hence $J$ contains a power of the maximal ideal of $A$ (A, VIII, § 10, No. 1, th. 1). It follows that $\operatorname{prof}_A(J; C_m) \geq \operatorname{prof}(C_m) > m - p$ for $m \geq p$ (No. 1, cor. 1 of prop. 2), so that one may apply cor. 1.