Lemma 15.81.16. Let $R \to A$ be a finite type ring map. Let $K^\bullet$ be a complex of $A$-modules. Let $m \in \mathbf{Z}$. Let $f_1, \ldots , f_ r \in A$ generate the unit ideal. The following are equivalent

1. each $K^\bullet \otimes _ A A_{f_ i}$ is $m$-pseudo-coherent relative to $R$, and

2. $K^\bullet$ is $m$-pseudo-coherent relative to $R$.

The same equivalence holds for pseudo-coherence relative to $R$.

Proof. The implication (2) $\Rightarrow$ (1) is in Lemma 15.81.11. Assume (1). Write $1 = \sum f_ ig_ i$ in $A$. Choose a surjection $R[x_1, \ldots , x_ n, y_1, \ldots , y_ r, z_1, \ldots , z_ r] \to A$. such that $y_ i$ maps to $f_ i$ and $z_ i$ maps to $g_ i$. Then we see that there exists a surjection

$P = R[x_1, \ldots , x_ n, y_1, \ldots , y_ r, z_1, \ldots , z_ r]/(\sum y_ iz_ i - 1) \longrightarrow A.$

Note that $P$ is pseudo-coherent as an $R[x_1, \ldots , x_ n, y_1, \ldots , y_ r, z_1, \ldots , z_ r]$-module and that $P[1/y_ i]$ is pseudo-coherent as an $R[x_1, \ldots , x_ n, y_1, \ldots , y_ r, z_1, \ldots , z_ r, 1/y_ i]$-module. Hence by Lemma 15.64.11 we see that $K^\bullet \otimes _ A A_{f_ i}$ is an $m$-pseudo-coherent complex of $P[1/y_ i]$-modules for each $i$. Thus by Lemma 15.64.14 we see that $K^\bullet$ is pseudo-coherent as a complex of $P$-modules, and Lemma 15.64.11 shows that $K^\bullet$ is pseudo-coherent as a complex of $R[x_1, \ldots , x_ n, y_1, \ldots , y_ r, z_1, \ldots , z_ r]$-modules. $\square$

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