The Stacks project

Lemma 66.29.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $x \in |X|$ be a point and let $\overline{x}$ be a geometric point lying over $x$. Finally, let $\varphi : (U, \overline{u}) \to (X, \overline{x})$ be an ├ętale neighbourhood where $U$ is a scheme. Then

\[ (\varphi ^*\mathcal{F})_ u \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} = \mathcal{F}_{\overline{x}} \]

where $u \in U$ is the image of $\overline{u}$.

Proof. Note that $\mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, u}^{sh}$ by Lemma 66.22.1 hence the tensor product makes sense. Moreover, from Definition 66.19.6 it is clear that

\[ \mathcal{F}_{\overline{u}} = \mathop{\mathrm{colim}}\nolimits (\varphi ^*\mathcal{F})_ u \]

where the colimit is over $\varphi : (U, \overline{u}) \to (X, \overline{x})$ as in the lemma. Hence there is a canonical map from left to right in the statement of the lemma. We have a similar colimit description for $\mathcal{O}_{X, \overline{x}}$ and by Lemma 66.29.3 we have

\[ ((\varphi ')^*\mathcal{F})_{u'} = (\varphi ^*\mathcal{F})_ u \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{U', u'} \]

whenever $(U', \overline{u}') \to (U, \overline{u})$ is a morphism of ├ętale neighbourhoods. To complete the proof we use that $\otimes $ commutes with colimits. $\square$


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