nLab
locally free module

Context

Algebra

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Geometry on formal duals of algebras

Theorems

Contents

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 RR-module NN over a Noetherian ring RR is called a locally free module if there is a cover by ideals IRI \hookrightarrow R such that the localization N IN_I is a free module over the localization R IR_I.

Definition

For RR a commutative ring, an RR-module NN is called a stalkwise free module if for every maximal ideal IRI \hookrightarrow R the localization N IN_I is a free module over the localization R IR_I.

Properties

Proposition

Let RR be a ring and NRN \in RMod.

The following are equivalent

  1. NN is finitely generated and projective,

  2. NN is locally free, def. and locally finitely generated.

For instance (Clark, theorem 7.20).

Proposition

For RR a Noetherian ring and NN a finitely generated module over RR, NN 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.

Last revised on October 8, 2012 at 18:47:36. See the history of this page for a list of all contributions to it.