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. $\square$

There are also:

• 2 comment(s) on Section 10.75: Tor groups and flatness

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).