The Stacks project

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. $\square$

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