The Stacks project

Lemma 81.4.1. Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $x \in |X|$. Let $d \in \{ 0, 1, 2, \ldots , \infty \} $. The following are equivalent

  1. $\text{length}_{\mathcal{O}_{X, \overline{x}}} \mathcal{F}_{\overline{x}} = d$

  2. for some ├ętale morphism $U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$ we have $\text{length}_{\mathcal{O}_{U, u}} (\mathcal{F}|_ U)_ u = d$

  3. for any ├ętale morphism $U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$ we have $\text{length}_{\mathcal{O}_{U, u}} (\mathcal{F}|_ U)_ u = d$

Proof. Let $U \to X$ and $u \in U$ be as in (2) or (3). Then we know that $\mathcal{O}_{X, \overline{x}}$ is the strict henselization of $\mathcal{O}_{U, u}$ and that

\[ \mathcal{F}_{\overline{x}} = (\mathcal{F}|_ U)_ u \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} \]

See Properties of Spaces, Lemmas 65.22.1 and 65.29.4. Thus the equality of the lengths follows from Algebra, Lemma 10.52.13 the fact that $\mathcal{O}_{U, u} \to \mathcal{O}_{X, \overline{x}}$ is flat and the fact that $\mathcal{O}_{X, \overline{x}}/\mathfrak m_ u\mathcal{O}_{X, \overline{x}}$ is equal to the residue field of $\mathcal{O}_{X, \overline{x}}$. These facts about strict henselizations can be found in More on Algebra, Lemma 15.45.1. $\square$

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