The Stacks project

Lemma 15.81.7. Let $R \to A$ be a finite type ring map. Let $M$ be an $A$-module. Then

1. $M$ is $0$-pseudo-coherent relative to $R$ if and only if $M$ is a finite type $A$-module,

2. $M$ is $(-1)$-pseudo-coherent relative to $R$ if and only if $M$ is a finitely presented relative to $R$,

3. $M$ is $(-d)$-pseudo-coherent relative to $R$ if and only if for every surjection $R[x_1, \ldots , x_ n] \to A$ there exists a resolution

   \[ R[x_1, \ldots , x_ n]^{\oplus a_ d} \to R[x_1, \ldots , x_ n]^{\oplus a_{d - 1}} \to \ldots \to R[x_1, \ldots , x_ n]^{\oplus a_0} \to M \to 0 \]

   of length $d$, and

4. $M$ is pseudo-coherent relative to $R$ if and only if for every presentation $R[x_1, \ldots , x_ n] \to A$ there exists an infinite resolution

   \[ \ldots \to R[x_1, \ldots , x_ n]^{\oplus a_1} \to R[x_1, \ldots , x_ n]^{\oplus a_0} \to M \to 0 \]

   by finite free $R[x_1, \ldots , x_ n]$-modules.