Lemma 15.64.4. Let $R$ be a ring. Let $M$ be an $R$-module. Then

$M$ is $0$-pseudo-coherent if and only if $M$ is a finite $R$-module,

$M$ is $(-1)$-pseudo-coherent if and only if $M$ is a finitely presented $R$-module,

$M$ is $(-d)$-pseudo-coherent if and only if there exists a resolution

\[ R^{\oplus a_ d} \to R^{\oplus a_{d - 1}} \to \ldots \to R^{\oplus a_0} \to M \to 0 \]of length $d$, and

$M$ is pseudo-coherent if and only if there exists an infinite resolution

\[ \ldots \to R^{\oplus a_1} \to R^{\oplus a_0} \to M \to 0 \]by finite free $R$-modules.

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