Lemma 10.7.2. Let R \to S be a finite ring map. Let M be an S-module. Then M is finite as an R-module if and only if M is finite as an S-module.
Proof. One of the implications follows from Lemma 10.5.5. To see the other assume that M is finite as an S-module. Pick x_1, \ldots , x_ n \in S which generate S as an R-module. Pick y_1, \ldots , y_ m \in M which generate M as an S-module. Then x_ i y_ j generate M as an R-module. \square
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