Lemma 75.5.5. 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 étale, and $a$ is surjective. Since $a^* \circ Lf^* = Lh^* \circ b^*$ the result follows from Lemma 75.5.2 and the case of schemes which is Derived Categories of Schemes, Lemma 36.3.8. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)