The Stacks project

Lemma 10.126.2. Let $R \to S$ be a ring map. Let $R \to R'$ be a faithfully flat ring map. Set $S' = R'\otimes _ R S$. Then $R \to S$ is of finite presentation if and only if $R' \to S'$ is of finite presentation.

Proof. It is clear that if $R \to S$ is of finite presentation then $R' \to S'$ is of finite presentation. Assume that $R' \to S'$ is of finite presentation. By Lemma 10.126.1 we see that $R \to S$ is of finite type. Write $S = R[x_1, \ldots , x_ n]/I$. By flatness $S' = R'[x_1, \ldots , x_ n]/R'\otimes I$. Say $g_1, \ldots , g_ m$ generate $R'\otimes I$ over $R'[x_1, \ldots , x_ n]$. Write $g_ j = \sum _ i a_{ij} \otimes f_{ji}$ for some $a_{ij} \in R'$ and $f_{ji} \in I$. Let $J \subset I$ be the ideal generated by the $f_{ij}$. By flatness we have $R' \otimes _ R J \subset R'\otimes _ R I$, and both are ideals over $R'[x_1, \ldots , x_ n]$. By construction $g_ j \in R' \otimes _ R J$. Hence $R' \otimes _ R J = R'\otimes _ R I$. By faithful flatness $J = I$. $\square$

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