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

$X$ is reduced,

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 $f = 0$.

## Comments (2)

Comment #7801 by 羽山籍真 on

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