nLab
locally free module

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. 1 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.

Revised on October 8, 2012 18:47:36 by Urs Schreiber (82.169.65.155)