The Stacks project

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

Lemma 28.20.1 . Let $X = \mathop<\mathrm>(R)$ be an affine scheme. Let $\mathcal = \widetilde$ for some $R$-module $M$. The quasi-coherent sheaf $\mathcal$ is a (finite) locally free $\mathcal_ X$-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 10.78.1. $\square$

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

Lemma 28.20.2 . Let $X$ be a scheme. Let $\mathcal$ be a quasi-coherent $\mathcal_ X$-module. The following are equivalent:

  1. $\mathcal$ is a flat $\mathcal_ X$-module of finite presentation,
  2. $\mathcal$ is $\mathcal_ X$-module of finite presentation and for all $x \in X$ the stalk $\mathcal_ x$ is a free $\mathcal_$-module,
  3. $\mathcal$ is a locally free, finite type $\mathcal_ X$-module,
  4. $\mathcal$ is a finite locally free $\mathcal_ X$-module, and
  5. $\mathcal$ is an $\mathcal_ X$-module of finite type, for every $x \in X$ the stalk $\mathcal_ x$ is a free $\mathcal_$-module, and the function

\[ \rho _\mathcal : X \to \mathbf, \quad x \longmapsto \dim _ <\kappa (x)>\mathcal_ x \otimes _<\mathcal_> \kappa (x) \]

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 $\mathcal$ be a quasi-coherent $\mathcal_ X$-module. Then the equivalent conditions of Lemma 28.20.2 are also equivalent to

    $\mathcal$ is an $\mathcal_ X$-module of finite type and the function

\[ \rho _\mathcal : X \to \mathbf, \quad x \longmapsto \dim _ <\kappa (x)>\mathcal_ x \otimes _<\mathcal_> \kappa (x) \]

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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