Lemma 28.20.1. 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.
28.20 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.
Proof. Follows from the definitions, see Modules, Definition 17.14.1 and Algebra, Definition 10.78.1. $\square$
We can characterize finite locally free modules in many different ways.
Lemma 28.20.2. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The following are equivalent:
$\mathcal{F}$ is a flat $\mathcal{O}_ X$-module of finite presentation,
$\mathcal{F}$ 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,
$\mathcal{F}$ is a locally free, finite type $\mathcal{O}_ X$-module,
$\mathcal{F}$ is a finite locally free $\mathcal{O}_ X$-module, and
$\mathcal{F}$ 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
is locally constant in the Zariski topology on $X$.
\[ \rho _\mathcal {F} : X \to \mathbf{Z}, \quad x \longmapsto \dim _{\kappa (x)} \mathcal{F}_ x \otimes _{\mathcal{O}_{X, x}} \kappa (x) \]
Proof. This lemma immediately reduces to the affine case. In this case the lemma is a reformulation of Algebra, Lemma 10.78.2. The translation uses Lemmas 28.16.1, 28.16.2, 28.19.1, and 28.20.1. $\square$
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
$\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite type and the function
is locally constant in the Zariski topology on $X$.
\[ \rho _\mathcal {F} : X \to \mathbf{Z}, \quad x \longmapsto \dim _{\kappa (x)} \mathcal{F}_ x \otimes _{\mathcal{O}_{X, x}} \kappa (x) \]
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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