Lemma 10.75.7 (0AZ4)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.75: Tor groups and flatness
Lemma 10.75.7 (cite)

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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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