Lemma 15.81.5. Let R be a ring. Let A \to B be a finite map of finite type R-algebras. Let m \in \mathbf{Z}. Let K^\bullet be a complex of B-modules. Then K^\bullet is m-pseudo-coherent (resp. pseudo-coherent) relative to R if and only if K^\bullet seen as a complex of A-modules is m-pseudo-coherent (pseudo-coherent) relative to R.
Proof. Choose a surjection R[x_1, \ldots , x_ n] \to A. Choose y_1, \ldots , y_ m \in B which generate B over A. As A \to B is finite each y_ i satisfies a monic equation with coefficients in A. Hence we can find monic polynomials P_ j(T) \in R[x_1, \ldots , x_ n][T] such that P_ j(y_ j) = 0 in B. Then we get a commutative diagram
\xymatrix{ & R[x_1, \ldots , x_ n, y_1, \ldots , y_ m] \ar[d] \\ R[x_1, \ldots , x_ n] \ar[d] \ar[r] & R[x_1, \ldots , x_ n, y_1, \ldots , y_ m]/(P_ j(y_ j)) \ar[d] \\ A \ar[r] & B }
The top horizontal arrow and the top right vertical arrow satisfy the assumptions of Lemma 15.64.11. Hence K^\bullet is m-pseudo-coherent (resp. pseudo-coherent) as a complex of R[x_1, \ldots , x_ n]-modules if and only if K^\bullet is m-pseudo-coherent (resp. pseudo-coherent) as a complex of R[x_1, \ldots , x_ n, y_1, \ldots , y_ m]-modules. \square
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