The Stacks project

Lemma 70.4.5. Let $S$ be a scheme. Let $X$ be a normal integral algebraic space over $S$. For every $x \in |X|$ there exists a normal integral affine scheme $U$ and an ├ętale morphism $U \to X$ such that $x$ is in the image.

Proof. Choose an affine scheme $U$ and an ├ętale morphism $U \to X$ such that $x$ is in the image. Let $u_ i$, $i \in I$ be the generic points of irreducible components of $U$. Then each $u_ i$ maps to the generic point of $X$ (Decent Spaces, Lemma 66.20.1). By our definition of a decent space (Decent Spaces, Definition 66.6.1), we see that $I$ is finite. Hence $U = \mathop{\mathrm{Spec}}(A)$ where $A$ is a normal ring with finitely many minimal primes. Thus $A = \prod _{i \in I} A_ i$ is a product of normal domains by Algebra, Lemma 10.36.16. Then $U = \coprod U_ i$ with $U_ i = \mathop{\mathrm{Spec}}(A_ i)$ and $x$ is in the image of $U_ i \to X$ for some $i$. This proves the lemma. $\square$


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