maybe that's an idiot question, however I did not found anything related in the classical references. It's know that a finitely generated projective $A$-module $M$ is locally free ,since each localization $A_\mathfrak{p}$ is a local ring (then the result follows by a theorem of Kaplansky). Moreover, if $A$ is Noetherian and $M$ finitely generated, then flatness and locally freeness coincide. However I don't know exactly if these results holds for arbitrary modules. Furthermore, I don't know the relation between flatness and locally freeness for general rings and modules? Are there any results for such general facts? Any references? How one can (partial) order these properties (flatness, locally freeness, "projectiveness") when assuming certain common classes of rings if necessary? Is it possible to (partial) order these properties for general abelian categories (making some restrictions if necessary)?

Thanks in advance.

  • $\begingroup$ The question does not apply to general abelian categories, since there is no notion of "locally free" and no (obvious) notion of "flat". $\endgroup$ – Zhen Lin May 4 '13 at 21:46
  • $\begingroup$ @ZhenLin There is the notion of flat object and the notion of localization in abelian categories. $\endgroup$ – user40276 May 4 '13 at 21:49
  • $\begingroup$ Flatness makes sense in abelian tensor categories. $\endgroup$ – Martin Brandenburg May 4 '13 at 21:51

Flatness can be checked locally and free modules are flat. Hence, locally free modules are flat. The converse does not hold, even in the finitely generated case. However, finitely presented flat modules coincide with locally free modules of finite rank.

Free modules are flat and direct summands of flat modules are flat. It follows that projective modules are flat. The converse does not hold. However, finitely generated projective modules coincide with finitely presented flat modules.

  • $\begingroup$ And what about the locally freeness and "protectiveness" relationship? $\endgroup$ – user40276 May 4 '13 at 22:48
  • $\begingroup$ I don't know any relationship (beyond the already mentioned one, locally free of finite rank = finitely generated projective). $\endgroup$ – Martin Brandenburg May 5 '13 at 8:44

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