symmetric monoidal (∞,1)-category of spectra
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.
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 $R_I$.
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 $R_I$.
Let $R$ be a ring and $N \in R$Mod.
The following are equivalent
$N$ is finitely generated and projective,
$N$ is locally free, def. 1 and locally finitely generated.
For instance (Clark, theorem 7.20).
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)
Last revised on October 8, 2012 at 18:47:36. See the history of this page for a list of all contributions to it.