The Stacks project

Lemma 37.57.11. Let $A \to B$ be a flat ring map of finite presentation. Let $I \subset B$ be an ideal. Then $A \to B/I$ is pseudo-coherent if and only if $I$ is pseudo-coherent as a $B$-module.

Proof. Choose a presentation $B = A[x_1, \ldots , x_ n]/J$. Note that $B$ is pseudo-coherent as an $A[x_1, \ldots , x_ n]$-module because $A \to B$ is a pseudo-coherent ring map by Lemma 37.57.6. Note that $A \to B/I$ is pseudo-coherent if and only if $B/I$ is pseudo-coherent as an $A[x_1, \ldots , x_ n]$-module. By More on Algebra, Lemma 15.64.11 we see this is equivalent to the condition that $B/I$ is pseudo-coherent as an $B$-module. This proves the lemma as the short exact sequence $0 \to I \to B \to B/I \to 0$ shows that $I$ is pseudo-coherent if and only if $B/I$ is (see More on Algebra, Lemma 15.64.6). $\square$


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