The Stacks project

Lemma 70.11.2. Let $k$ be a field. Let $X$ be an algebraic space over $k$. Let $x \in |X|$. The following are equivalent

  1. $X$ is geometrically reduced at $x$,

  2. for some étale neighbourhood $(U, u) \to (X, x)$ where $U$ is a scheme, $U$ is geometrically reduced at $u$,

  3. for any étale neighbourhood $(U, u) \to (X, x)$ where $U$ is a scheme, $U$ is geometrically reduced at $u$.

Proof. Recall that the local ring $\mathcal{O}_{X, \overline{x}}$ is the strict henselization of $\mathcal{O}_{U, u}$, see Properties of Spaces, Lemma 64.22.1. By Varieties, Lemma 33.6.2 we find that $U$ is geometrically reduced at $u$ if and only if $\mathcal{O}_{U, u}$ is geometrically reduced over $k$. Thus we have to show: if $A$ is a local $k$-algebra, then $A$ is geometrically reduced over $k$ if and only if $A^{sh}$ is geometrically reduced over $k$. We check this using the definition of geometrically reduced algebras (Algebra, Definition 10.42.1). Let $K/k$ be a field extension. Since $A \to A^{sh}$ is faithfully flat (More on Algebra, Lemma 15.44.1) we see that $A \otimes _ k K \to A^{sh} \otimes _ k K$ is faithfully flat (Algebra, Lemma 10.38.7). Hence if $A^{sh} \otimes _ k K$ is reduced, so is $A \otimes _ k K$ by Algebra, Lemma 10.162.2. Conversely, recall that $A^{sh}$ is a colimit of étale $A$-algebra, see Algebra, Lemma 10.154.2. Thus $A^{sh} \otimes _ k K$ is a filtered colimit of étale $A \otimes _ k K$-algebras. We conclude by Algebra, Lemma 10.161.7. $\square$


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