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

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

## Comments (2)

Comment #3058 by Tanya Kaushal Srivastava on

Comment #3162 by Johan on

There are also: