Lemma 10.75.7. Let R be a Noetherian ring. Let M, N be finite R-modules. Then \text{Tor}_ p^ R(M, N) is a finite R-module for all p.
Proof. This holds because \text{Tor}_ p^ R(M, N) is computed as the cohomology groups of a complex F_\bullet \otimes _ R N with each F_ n a finite free R-module, see Lemma 10.71.1. \square
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