Lemma 66.22.4 (04N9)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 66: Properties of Algebraic Spaces
Section 66.22: Stalks of the structure sheaf
Lemma 66.22.4 (cite)

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Lemma 66.22.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ . Let $x \in |X|$ be a point. Let $d \in \{ 0, 1, 2, \ldots , \infty \}$ . The following are equivalent

the dimension of the local ring of $X$ at $x$ (Definition 66.10.2) is $d$ ,
$\dim (\mathcal{O}_{X, \overline{x}}) = d$ for some geometric point $\overline{x}$ lying over $x$ , and
$\dim (\mathcal{O}_{X, \overline{x}}) = d$ for any geometric point $\overline{x}$ lying over $x$ .

Proof. The equivalence of (2) and (3) follows from the fact that the isomorphism type of $\mathcal{O}_{X, \overline{x}}$ only depends on $x \in |X|$ , see Remark 66.19.11. Using Lemma 66.22.1 the equivalence of (1) and (2) $+$ (3) comes down to the following statement: Given any local ring $R$ we have $\dim (R) = \dim (R^{sh})$ . This is More on Algebra, Lemma 15.45.7. $\square$

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