The Stacks project

Lemma 73.5.5. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. The functor $Lf^*$ sends $D_\mathit{QCoh}(\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ Y)$.

Proof. Choose a diagram

\[ \xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ X \ar[r]^ f & Y } \]

where $U$ and $V$ are schemes, the vertical arrows are ├ętale, and $a$ is surjective. Since $a^* \circ Lf^* = Lh^* \circ b^*$ the result follows from Lemma 73.5.2 and the case of schemes which is Derived Categories of Schemes, Lemma 36.3.8. $\square$

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