The Stacks project

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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