Lemma 10.75.2. Let $R$ be a ring and let $M$ be an $R$-module. Suppose that $0 \to N' \to N \to N'' \to 0$ is a short exact sequence of $R$-modules. There exists a long exact sequence

$\text{Tor}_1^ R(M, N') \to \text{Tor}_1^ R(M, N) \to \text{Tor}_1^ R(M, N'') \to M \otimes _ R N' \to M \otimes _ R N \to M \otimes _ R N'' \to 0$

Proof. The proof of this is the same as the proof of Lemma 10.71.6. $\square$

Comment #3058 by Tanya Kaushal Srivastava on

The two lines of the long exact sequence for Tor should be switched.

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