Lemma 13.4.20 (05QZ)—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 13.4.20 (cite)

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Lemma 13.4.20. Let $\mathcal{D}$ be a pre-triangulated category. Let $\mathcal{A}$ be an abelian category. Let $H : \mathcal{D} \to \mathcal{A}$ be a homological functor.

Let $\mathcal{D}'$ be a pre-triangulated category. Let $F : \mathcal{D}' \to \mathcal{D}$ be an exact functor. Then the composition $H \circ F$ is a homological functor as well.
Let $\mathcal{A}'$ be an abelian category. Let $G : \mathcal{A} \to \mathcal{A}'$ be an exact functor. Then $G \circ H$ is a homological functor as well.

Proof. Omitted. $\square$

Comments (4)

Comment #1290 by jpg on February 01, 2015 at 17:00

In (2) "Hence" is repeated instead of "Let", "then"

Comment #1302 by Johan on February 10, 2015 at 01:45

Thanks! Fixed here.

Comment #3048 by Chung Ching on January 02, 2018 at 17:27

In (1), it should be $H\circ F$ instead of $G\circ F$ .

Comment #3153 by Johan on February 02, 2018 at 01:16

Thanks, fixed here.

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