Lemma 13.4.14 (05QW)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 13: Derived Categories
Section 13.4: Elementary results on triangulated categories
Lemma 13.4.14 (cite)

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Lemma 13.4.14. Let $\mathcal{D}$ be a pre-triangulated category. If $\mathcal{D}$ has countable products, then $\mathcal{D}$ is Karoubian. If $\mathcal{D}$ has countable coproducts, then $\mathcal{D}$ is Karoubian.

Proof. Assume $\mathcal{D}$ has countable products. By Homology, Lemma 12.4.3 it suffices to check that morphisms which have a right inverse have kernels. Any morphism which has a right inverse is an epimorphism, hence has a kernel by Lemma 13.4.12. The second statement is dual to the first. $\square$

Comments (2)

Comment #3412 by Herman Rohrbach on July 03, 2018 at 04:44

Typo: in the statement of the lemma, the sentence "If $\mathcal{D}$ has countable products, then $\mathcal{D}$ is Karoubian." appears twice.

Comment #3471 by Johan on August 03, 2018 at 16:06

Although it is better not to have the statement repeated, mathematically speaking there is nothing wrong with doing so! Thanks and see fix here.

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