Lemma 10.102.4 (00MU)—The Stacks project

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Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.102: What makes a complex exact?
Lemma 10.102.4 (cite)

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Lemma 10.102.4. In Situation 10.102.1. Let $R$ be a Artinian local ring. Suppose that $0 \to R^{n_ e} \to R^{n_{e-1}} \to \ldots \to R^{n_0}$ is exact at $R^{n_ e}, \ldots , R^{n_1}$. Then the complex is isomorphic to a direct sum of trivial complexes.

Proof. This is a special case of Lemma 10.102.3 because an Artinian local ring has depth $0$. $\square$

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