Lemma 82.4.1. 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
\text{length}_{\mathcal{O}_{X, \overline{x}}} \mathcal{F}_{\overline{x}} = d
for some étale 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
for any étale 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 66.22.1 and 66.29.4. 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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