nLab locally free module

Idea

Under the interpretation of modules as generalized vector bundles a module being locally free corresponds to the corresponding bundle being locally trivial bundle, hence a fiber bundle.

Since a trivial bundle corresponds to a free module, a locally free module is such that its localization to any maximal ideal is a free module.

Definition

Definition

An $R$-module $N$ over a Noetherian ring $R$ is called a locally free module if there is a cover by ideals $I \hookrightarrow R$ such that the localization $N_I$ is a free module over the localization $R_I$.

Definition

For $R$ a commutative ring, an $R$-module $N$ is called a stalkwise free module if for every maximal ideal $I \hookrightarrow R$ the localization $N_I$ is a free module over the localization $R_I$.

Properties

Proposition

Let $R$ be a ring and $N \in R$Mod.

The following are equivalent

1. $N$ is finitely generated and projective,

2. $N$ is locally free, def. and locally finitely generated.

For instance (Clark, theorem 7.20).

Proposition

For $R$ a Noetherian ring and $N$ a finitely generated module over $R$, $N$ is a locally free module precisely if it is a flat module.

By Raynaud-Gruson, 3.4.6 (part I)

References

• Pete Clark, Commutative algebra (pdf)

• Michel Raynaud, Laurent Gruson, Critères de platitude et de projectivité, Techniques de “platification” d’un module. Invent. Math. 13 (1971), 1–89.