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 $\mathcal{E}$.

The functor $S_\mathcal {E}^{-1}\mathcal{E} \to S^{-1}\mathcal{D}$ is a fully faithful exact functor of triangulated categories.

Any element of $S_\mathcal {E} = \text{Arrows}(\mathcal{E}) \cap S$ is mapped to an isomorphism by $RF$.

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 $LF$.

## Comments (1)

Comment #8386 by ElĂas Guisado on

There are also: