The Stacks project

Remark 66.19.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. We claim that for any pair of geometric points $\overline{x}$ and $\overline{x}'$ lying over $x$ the stalk functors are isomorphic. By definition of $|X|$ we can find a third geometric point $\overline{x}''$ so that there exists a commutative diagram

\[ \xymatrix{ \overline{x}'' \ar[r] \ar[d] \ar[rd]^{\overline{x}''} & \overline{x}' \ar[d]^{\overline{x}'} \\ \overline{x} \ar[r]^{\overline{x}} & X. } \]

Since the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ is given by pullback along the morphism $\overline{x}$ (and similarly for the others) we conclude by functoriality of pullbacks.


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