Lemma 75.6.4. Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. Then $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ reflects isomorphisms.
Proof. The statement means that a morphism $\alpha : E \to F$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ is an isomorphism if $Rf_*\alpha $ is an isomorphism. We may check this on cohomology sheaves. In particular, the question is étale local on $Y$. Hence we may assume $Y$ and therefore $X$ is affine. In this case the problem reduces to the case of schemes (Derived Categories of Schemes, Lemma 36.5.2) via Lemma 75.4.2 and Remark 75.6.3. $\square$
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Your email address will not be published.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)