The Stacks project

Lemma 66.21.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent

  1. $X$ is reduced,

  2. for every $x \in |X|$ the local ring of $X$ at $x$ is reduced (Remark 66.7.6).

In this case $\Gamma (X, \mathcal{O}_ X)$ is a reduced ring and if $f \in \Gamma (X, \mathcal{O}_ X)$ has $X = V(f)$, then $f = 0$.

Proof. The equivalence of (1) and (2) follows from Properties, Lemma 28.3.2 applied to affine schemes étale over $X$. The final statements follow the cited lemma and fact that $\Gamma (X, \mathcal{O}_ X)$ is a subring of $\Gamma (U, \mathcal{O}_ U)$ for some reduced scheme $U$ étale over $X$. $\square$

Comments (2)

Comment #7801 by 羽山籍真 on

I would like to ask where the notion "local rings" is defined for general algebraic space, since (2) used this (maybe it would be nice if we recall it here). I only know that for decent algebraic space we have Henselian local rings and for geometric points on general algebraic space we have strict Henselian local rings...

Comment #7803 by on

You use Remark 66.7.6 which in tern uses the preceding Definition 66.7.5 to define what this means. So it makes sense even if the local ring does not make sense. OK?

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