Lemma 10.126.1. 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 type if and only if R' \to S' is of finite type.
Proof. It is clear that if R \to S is of finite type then R' \to S' is of finite type. Assume that R' \to S' is of finite type. Say y_1, \ldots , y_ m generate S' over R'. Write y_ j = \sum _ i a_{ij} \otimes x_{ji} for some a_{ij} \in R' and x_{ji} \in S. Let A \subset S be the R-subalgebra generated by the x_{ij}. By flatness we have A' := R' \otimes _ R A \subset S', and by construction y_ j \in A'. Hence A' = S'. By faithful flatness A = S. \square
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