Lemma 33.44.10 (0C6P)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 33: Varieties
Section 33.44: Degrees on curves
Lemma 33.44.10 (cite)

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Lemma 33.44.10. Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is reduced and connected. Let $\kappa = H^0(X, \mathcal{O}_ X)$. Then $\kappa /k$ is a finite extension of fields and $w = [\kappa : k]$ divides

$\deg (\mathcal{E})$ for all locally free $\mathcal{O}_ X$-modules $\mathcal{E}$,
$[\kappa (x) : k]$ for all closed points $x \in X$, and
$\deg (D)$ for all closed subschemes $D \subset X$ of dimension zero.

Proof. See Lemma 33.9.3 for the assertions about $\kappa $. For every quasi-coherent $\mathcal{O}_ X$-module, the $k$-vector spaces $H^ i(X, \mathcal{F})$ are $\kappa $-vector spaces. The divisibilities easily follow from this statement and the definitions. $\square$

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