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The Stacks project

Lemma 12.9.5. Let \mathcal{A} be an abelian category. Let 0 \to A_1 \to A_2 \to A_3 \to 0 be a short exact sequence of \mathcal{A}. Then A_2 is Noetherian if and only if A_1 and A_3 are Noetherian.

Proof. Omitted. \square

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