The Stacks project

Lemma 28.20.3. Let $X$ be a reduced scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then the equivalent conditions of Lemma 28.20.2 are also equivalent to

  1. $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite type and the function

    \[ \rho _\mathcal {F} : X \to \mathbf{Z}, \quad x \longmapsto \dim _{\kappa (x)} \mathcal{F}_ x \otimes _{\mathcal{O}_{X, x}} \kappa (x) \]

    is locally constant in the Zariski topology on $X$.

Proof. This lemma immediately reduces to the affine case. In this case the lemma is a reformulation of Algebra, Lemma 10.78.3. $\square$

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