Proposition 13.14.8 (05SE)—The Stacks project

Proposition 13.14.8. Assumptions and notation as in Situation 13.14.1.

The full subcategory $\mathcal{E}$ of $\mathcal{D}$ consisting of objects at which $RF$ is defined is a strictly full triangulated subcategory of $\mathcal{D}$ .
We obtain an exact functor $RF : \mathcal{E} \longrightarrow \mathcal{D}'$ of triangulated categories.
Elements of $S$ with either source or target in $\mathcal{E}$ are morphisms of $\mathcal{E}$ .
Any element of $S_\mathcal {E} = \text{Arrows}(\mathcal{E}) \cap S$ is mapped to an isomorphism by $RF$ .
The set $S_\mathcal {E}$ is a saturated multiplicative system in $\mathcal{E}$ compatible with the triangulated structure.
The functor $S_\mathcal {E}^{-1}\mathcal{E} \to S^{-1}\mathcal{D}$ is a fully faithful exact functor of triangulated categories.
We obtain an exact functor

$RF : S_\mathcal {E}^{-1}\mathcal{E} \longrightarrow \mathcal{D}'.$
If $\mathcal{D}'$ is Karoubian, then $\mathcal{E}$ is a saturated triangulated subcategory of $\mathcal{D}$ .

A similar result holds for $LF$ .

Proof. Since $S$ is saturated it contains all isomorphisms (see remark following Categories, Definition 4.27.20). Hence (1) follows from Lemmas 13.14.4, 13.14.6, and 13.14.5. We get (2) from Lemmas 13.14.3, 13.14.5, and 13.14.6. We get (3) from Lemma 13.14.4. Part (4) follows from Lemma 13.14.4. Part (5) follows from the definitions and part (3). The fully faithfulness in (6) follows from (3) and the definitions. The fact that $S_\mathcal {E}^{-1}\mathcal{E} \to S^{-1}\mathcal{D}$ is exact follows from the fact that a triangle in $S_\mathcal {E}^{-1}\mathcal{E}$ is distinguished if and only if it is isomorphic to the image of a distinguished triangle in $\mathcal{E}$ , see proof of Proposition 13.5.6. The factorization of $RF : \mathcal{E} \to \mathcal{D}'$ through an exact functor $S_\mathcal {E}^{-1}\mathcal{E} \to \mathcal{D}'$ follows from Lemma 13.5.7. Finally, part (8) follows from Lemma 13.14.7. $\square$

Comments (2)

Comment #8386 by Elías Guisado on May 09, 2023 at 08:54

To the statement of (3) maybe one could add at the beginning " $S_\mathcal{E}$ is a saturated multiplicative system in $\mathcal{E}$ compatible with the triangulated structure". All axioms MS $n$ are trivial except maybe MS2, which follows from (3). Also maybe one could move the definition of $S_\mathcal{E}$ from (5) to (4), where it is firstly mentioned.

Comment #8998 by Stacks project on April 17, 2024 at 11:40

Thanks and fixed here.

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