Lemma 22.5.2 (09JP)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 22: Differential Graded Algebra
Section 22.5: The homotopy category
Lemma 22.5.2 (cite)

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Lemma 22.5.2. Let $(A, \text{d})$ be a differential graded algebra. Let $f, g : L \to M$ be homomorphisms of differential graded $A$ -modules. Suppose given further homomorphisms $a : K \to L$ , and $c : M \to N$ . If $h : L \to M$ is an $A$ -module map which defines a homotopy between $f$ and $g$ , then $c \circ h \circ a$ defines a homotopy between $c \circ f \circ a$ and $c \circ g \circ a$ .

Proof. Immediate from Homology, Lemma 12.13.7. $\square$

Comments (1)

Comment #289 by arp on September 06, 2013 at 07:58

Typo: In the statement of the lemma, the homotopy $h$ should be a map $L \rightarrow M$ not $M \rightarrow N$ .

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