The Stacks project

Lemma 15.80.5. Let $R \to A$ be a finite type ring map. Let $M$ be an $A$-module finitely presented relative to $R$. For any ring map $R \to R'$ the $A \otimes _ R R'$-module

\[ M \otimes _ A A' = M \otimes _ R R' \]

is finitely presented relative to $R'$.

Proof. Choose a surjection $R[x_1, \ldots , x_ n] \to A$. Choose a presentation

\[ R[x_1, \ldots , x_ n]^{\oplus s} \to R[x_1, \ldots , x_ n]^{\oplus t} \to M \to 0 \]

Then

\[ R'[x_1, \ldots , x_ n]^{\oplus s} \to R'[x_1, \ldots , x_ n]^{\oplus t} \to M \otimes _ R R' \to 0 \]

is a presentation of the base change and we win. $\square$

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