Lemma 15.7.4. Let A, A', B, B', C, C', D, D', I, M', M, N, \varphi be as in Lemma 15.7.2. If N finite over D and M' finite over C', then N' = N \times _{\varphi , M} M' is finite over D'.
Proof. Recall that D' \to D \times _ C C' is surjective by Lemma 15.6.5. Observe that N' = N \times _{\varphi , M} M' is a module over D \times _ C C'. We can apply Lemma 15.6.7 to the data C, C', D, D', IC', M', M, N, \varphi to see that N' = N \times _{\varphi , M} M' is finite over D \times _ C C'. Thus it is finite over D'. \square
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