Lemma 15.64.7. Let $R$ be a ring. Let $K^\bullet $ be a complex of $R$-modules. Let $m \in \mathbf{Z}$.

If $H^ i(K^\bullet ) = 0$ for all $i \geq m$, then $K^\bullet $ is $m$-pseudo-coherent.

If $H^ i(K^\bullet ) = 0$ for $i > m$ and $H^ m(K^\bullet )$ is a finite $R$-module, then $K^\bullet $ is $m$-pseudo-coherent.

If $H^ i(K^\bullet ) = 0$ for $i > m + 1$, the module $H^{m + 1}(K^\bullet )$ is of finite presentation, and $H^ m(K^\bullet )$ is of finite type, then $K^\bullet $ is $m$-pseudo-coherent.

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