The Stacks project

27.20 Locally free modules

On any ringed space we know what it means for an $\mathcal{O}_ X$-module to be (finite) locally free. On an affine scheme this matches the notion defined in the algebra chapter.

Lemma 27.20.1. Let $X = \mathop{\mathrm{Spec}}(R)$ be an affine scheme. Let $\mathcal{F} = \widetilde{M}$ for some $R$-module $M$. The quasi-coherent sheaf $\mathcal{F}$ is a (finite) locally free $\mathcal{O}_ X$-module of if and only if $M$ is a (finite) locally free $R$-module.

Proof. Follows from the definitions, see Modules, Definition 17.14.1 and Algebra, Definition 10.77.1. $\square$

We can characterize finite locally free modules in many different ways.

Lemma 27.20.2. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The following are equivalent:

  1. $\mathcal{F}$ is a flat $\mathcal{O}_ X$-module of finite presentation,

  2. $\mathcal{F}$ is $\mathcal{O}_ X$-module of finite presentation and for all $x \in X$ the stalk $\mathcal{F}_ x$ is a free $\mathcal{O}_{X, x}$-module,

  3. $\mathcal{F}$ is a locally free, finite type $\mathcal{O}_ X$-module,

  4. $\mathcal{F}$ is a finite locally free $\mathcal{O}_ X$-module, and

  5. $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite type, for every $x \in X$ the stalk $\mathcal{F}_ x$ is a free $\mathcal{O}_{X, x}$-module, 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 10.77.2. The translation uses Lemmas 27.16.1, 27.16.2, 27.19.1, and 27.20.1. $\square$

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