Lemma 12.22.4 (011Y)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 12: Homological Algebra
Section 12.22: Spectral sequences: differential objects
Lemma 12.22.4 (cite)

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Lemma 12.22.4. Let $\mathcal{A}$ be an abelian category. Let $0 \to (A, d) \to (B, d) \to (C, d) \to 0$ be a short exact sequence of differential objects. Then we get an exact homology sequence

$\ldots \to H(C, d) \to H(A, d) \to H(B, d) \to H(C, d) \to \ldots$

Proof. Apply Lemma 12.13.12 to the short exact sequence of complexes

$\begin{matrix} 0 & \to & A & \to & B & \to & C & \to & 0 \\ & & \downarrow & & \downarrow & & \downarrow \\ 0 & \to & A & \to & B & \to & C & \to & 0 \\ & & \downarrow & & \downarrow & & \downarrow \\ 0 & \to & A & \to & B & \to & C & \to & 0 \end{matrix}$

where the vertical arrows are $d$ . $\square$

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