Lemma 10.62.7 (0B51)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.62: Support and dimension of modules
Lemma 10.62.7 (cite)

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Lemma 10.62.7. Let $R$ be a Noetherian ring. Let $0 \to M' \to M \to M'' \to 0$ be a short exact sequence of finite $R$ -modules. Then $\max \{ \dim (\text{Supp}(M')), \dim (\text{Supp}(M''))\} = \dim (\text{Supp}(M))$ .

Proof. If $R$ is local, this follows immediately from Lemmas 10.62.6 and 10.59.10. A more elementary argument, which works also if $R$ is not local, is to use that $\text{Supp}(M')$ , $\text{Supp}(M'')$ , and $\text{Supp}(M)$ are closed (Lemma 10.40.5) and that $\text{Supp}(M) = \text{Supp}(M') \cup \text{Supp}(M'')$ (Lemma 10.40.9). $\square$

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