Lemma 13.40.4 (0FXC)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 13: Derived Categories
Section 13.40: Admissible subcategories
Lemma 13.40.4 (cite)

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Lemma 13.40.4. Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{A} \subset \mathcal{D}$ be a full subcategory invariant under all shifts. Then both the right orthogonal $\mathcal{A}^\perp$ and the left orthogonal ${}^\perp \mathcal{A}$ of $\mathcal{A}$ are strictly full, saturated ¹, triangulated subcagories of $\mathcal{D}$ .

Proof. It is immediate from the definitions that the orthogonals are preserved under taking shifts, direct sums, and direct summands. Consider a distinguished triangle

$X \to Y \to Z \to X[1]$

of $\mathcal{D}$ . By Lemma 13.4.16 it suffices to show that if $X$ and $Y$ are in $\mathcal{A}^\perp$ , then $Z$ is in $\mathcal{A}^\perp$ . This is immediate from Lemma 13.40.2. $\square$

[1] Definition 13.6.1.

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