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