Lemma 76.52.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. The full subcategory of $D(\mathcal{O}_ X)$ consisting of $Y$-perfect objects is a saturated^{1} triangulated subcategory.

**Proof.**
This follows from Cohomology on Sites, Lemmas 21.45.4, 21.45.6, 21.46.6, and 21.46.8.
$\square$

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