Proposition 10.9.12 (00CS)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.9: Localization
Proposition 10.9.12 (cite)

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Localization is exact.

Proposition 10.9.12. Let $L\xrightarrow {u} M\xrightarrow {v} N$ be an exact sequence of $R$ -modules. Then $S^{-1}L \to S^{-1}M \to S^{-1}N$ is also exact.

Proof. First it is clear that $S^{-1}L \to S^{-1}M \to S^{-1}N$ is a complex since localization is a functor. Next suppose that $x/s$ maps to zero in $S^{-1}N$ for some $x/s \in S^{-1}M$ . Then by definition there is a $t\in S$ such that $v(xt) = v(x)t = 0$ in $M$ , which means $xt \in \mathop{\mathrm{Ker}}(v)$ . By the exactness of $L \to M \to N$ we have $xt = u(y)$ for some $y$ in $L$ . Then $x/s$ is the image of $y/st$ . This proves the exactness. $\square$

Comments (2)

Comment #842 by Johan Commelin on July 24, 2014 at 18:37

Suggested slogan: Localization of modules is exact

Comment #844 by Pieter Belmans on July 24, 2014 at 19:13

There is a dash missing in the first sentence.

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