Lemma 22.23.1 (09L0)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 22: Differential Graded Algebra
Section 22.23: The canonical delta-functor
Lemma 22.23.1 (cite)

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Lemma 22.23.1. Let $(A, \text{d})$ be a differential graded algebra. The functor $\text{Mod}_{(A, \text{d})} \to D(A, \text{d})$ defined has the natural structure of a $\delta$ -functor, with

$\delta _{K \to L \to M} = - p \circ q^{-1}$

with $p$ and $q$ as explained above.

Proof. We have already seen that this choice leads to a distinguished triangle whenever given a short exact sequence of complexes. We have to show functoriality of this construction, see Derived Categories, Definition 13.3.6. This follows from Lemma 22.6.2 with a bit of work. Compare with Derived Categories, Lemma 13.12.1. $\square$

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