Lemma 81.10.3 (0AES)—The Stacks project

Table of contents
Part 4: Algebraic Spaces
Chapter 81: Pushouts of Algebraic Spaces
Section 81.10: Formal glueing of quasi-coherent modules
Lemma 81.10.3 (cite)

numberstags

Lemma 81.10.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism. Let $\overline{x}$ be a geometric point of $X$ and let $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}) \to X$ be the canonical morphism. For a quasi-coherent module $\mathcal{G}$ on $Y$ we have

\[ f_*\mathcal{G}_{\overline{x}} = \Gamma (Y \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}), p^*\mathcal{F}) \]

where $p : Y \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}) \to Y$ is the projection.

Proof. Observe that $f_*\mathcal{G}_{\overline{x}} = \Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}), h^*f_*\mathcal{G})$ where $h : \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}) \to X$. Hence the result is true because $h$ is flat so that Cohomology of Spaces, Lemma 69.11.2 applies. $\square$

Comments (0)

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).

Comments (0)

Post a comment