Lemma 15.81.15. Let $R$ be a ring. Let $A \to B$ be a map of finite type $R$-algebras. Let $m \in \mathbf{Z}$. Let $K^\bullet$ be a complex of $B$-modules. Assume $A$ is pseudo-coherent relative to $R$. Then the following are equivalent

1. $K^\bullet$ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $A$, and

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

Proof. Choose a surjection $R[x_1, \ldots , x_ n] \to A$. Choose a surjection $A[y_1, \ldots , y_ m] \to B$. Then we get a surjection

$R[x_1, \ldots , x_ n, y_1, \ldots , y_ m] \to A[y_1, \ldots , y_ m]$

which is a flat base change of $R[x_1, \ldots , x_ n] \to A$. By assumption $A$ is a pseudo-coherent module over $R[x_1, \ldots , x_ n]$ hence by Lemma 15.64.13 we see that $A[y_1, \ldots , y_ m]$ is pseudo-coherent over $R[x_1, \ldots , x_ n, y_1, \ldots , y_ m]$. Thus the lemma follows from Lemma 15.64.11 and the definitions. $\square$

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