Lemma 22.5.4 (09JR)—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 22.5.4 (cite)

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Lemma 22.5.4. Let $(A, \text{d})$ be a differential graded algebra. The homotopy category $K(\text{Mod}_{(A, \text{d})})$ has direct sums and products.

Proof. Omitted. Hint: Just use the direct sums and products as in Lemma 22.4.2. This works because we saw that these functors commute with the forgetful functor to the category of graded $A$ -modules and because $\prod$ is an exact functor on the category of families of abelian groups. $\square$

Comments (2)

Comment #8480 by Et on June 11, 2023 at 09:49

This proposition isn't hard to prove directly, but I wonder where you use : "and because ∏ is an exact functor on the category of families of abelian groups."?

Comment #9096 by Stacks project on May 10, 2024 at 09:21

To prove that Homs from $A$ into a product is the product of the homs (in the homotopy category).

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