Lemma 10.75.2 (00M0)—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 10.75.2 (cite)

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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. $\square$

Comments (2)

Comment #3058 by Tanya Kaushal Srivastava on January 15, 2018 at 21:20

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

Comment #3162 by Johan on February 02, 2018 at 03:13

OK, thanks. How about this?

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