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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