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nLab > Latest Changes: flat module

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  1.  
    • CommentRowNumber1.
    • CommentAuthorAndrew Stacey
    • CommentTimeNov 7th 2011

    I’ve made a few changes at flat module since I wanted to know what one was and the nLab page simply confused me further. It seemed to be saying that a module over a ring/algebra AS^1/\mathbb{Z[A is flat if tensoring with AA is a flat functor. That seemed absurd so I changed it. The observation “everything happens for a reason” was a little curt so once I’d worked out what it meant I expanded it a bit and put in the analogy to bases. It wasn’t clear from the way that it was phrased whether this condition was due to Wraith and Blass or the fact that it can be put in a more general context. Lastly, the last sentence was originally in the same paragraph as the penultimate sentence where it didn’t seem to fit (or was at best ambiguous) and it also claimed that the module could be non-unital which seemed a little odd.

    If an expert could kindly check that I’ve done no lasting damage to the page, I’d be grateful.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 7th 2011

    I have tried to add some structure.

    That seemed absurd

    Do you really mean “absurd”? For one it explains the term “flat functor”. But of course it should be expanded on.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeNov 7th 2011
    • (edited Nov 7th 2011)

    I corrected it: tensoring with MM goes from AA-modules to kk-modules, not to AA-modules, as MM is not an AA-bimodule (it is a kk-MM-bimodule though). I agree with Urs that there is nothing absurd about it. Historically, maybe the case of modules was first to be characterized by the property of the tensoring (sends SES to SES) which we a posteriori call flat functor. But the definition is still that property, which is basic and categorical, so in a sense it is traditional, though the name flat is more generally and more primarily associated to the functor then to the way the functor is defined (via module).

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 7th 2011
    • (edited Nov 7th 2011)

    At flat module, to Andrew’s words

    The module MM being flat is equivalent to being able always to do this.

    I added

    After some unwinding, it means that a module is flat if and only if it is a filtered colimit of free modules.

    which is a bridge to the more general meanings of flat functor (filtered colimit of representables).

    Edit: By the way, what is this cryptic reference to Wraith and Blass?

    • CommentRowNumber5.
    • CommentAuthorTom Leinster
    • CommentTimeNov 7th 2011

    I thought Andrew’s use of “absurd” in comment 1 was simply because with should be over. Tensoring an AA-module with AA doesn’t do very much :-)

    • CommentRowNumber6.
    • CommentAuthorAndrew Stacey
    • CommentTimeNov 7th 2011

    Tom’s right about my use of “absurd”. My apologies for not being clearer.

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 7th 2011

    Re #5: oh well, then it was more or less just a typo, wasn’t it? Tensoring with the module MM, not AA!

    • CommentRowNumber8.
    • CommentAuthorAndrew Stacey
    • CommentTimeNov 7th 2011

    Yes, but a confusing typo! Particularly since MM wasn’t mentioned by name in the first paragraph. Since I’m not an expert in modules, flat or otherwise, I wanted to highlight the change so that the rest of you could check that it now made sense.

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeNov 7th 2011

    Surely with MM, that is why it does not end in AA-mod but in kk-mod. Regarding that it was written AA before, the codomain ended in a wrong category.

    • CommentRowNumber10.
    • CommentAuthorAndrew Stacey
    • CommentTimeNov 8th 2011

    I did the unwinding and spelled out how one gets from the element characterisation to the filtered colimit of free modules.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2012
    • (edited Oct 17th 2012)

    I have added various material to flat module. There is now characterization in terms of Tor-functors, a pointer to the source for Lazard’s criterion, discussion of the relation to projective and (locally) free modules and a discussion of the example of flat abelian groups .

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2012
    • (edited Oct 17th 2012)

    I have created a section Equivalent characterizations with some indications for why indeed – as claimed right at the beginning of the entry going back to #1 above – a module NN is flat precisely if every linear combination of 0 in NN comes from linear combinations of 0 in RR.

    This follows pretty directly from yet another equivalent characterization, namely that it is sufficient to check left exactness of (−)⊗ RN(-)\otimes_R N on ideal inclusions. And this implication I have now written out.

    However, that left exactness of (−)⊗ RN(-)\otimes_R N can be checked already on just ideal inclusions is a bit more work…

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2012
    • (edited Oct 18th 2012)

    I have edited the flow of the Definition section a bit more. Notably I made the immediate equivalent reformulations a list and pointed to the discussion of the not-so-immediate equivalent reformulations further below. Also I moved the remark that much of the definition works also over non-commutative and non-unital rings to further below.

    Those who care please check if they can live with the way it is now.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2013

    added the definition of faithfully flat module

    • CommentRowNumber15.
    • CommentAuthorspitters
    • CommentTimeMar 28th 2018

    What is the reference to Blass and Wraith ?

  16. Add an example

    Damien Lejay

    diff, v31, current

  17. Add context as to why flat modules are “flat”.

    Damien Lejay

    diff, v32, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJul 16th 2023

    Great. I have given this remark a Remark-environment (now here)

    diff, v33, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJul 16th 2023

    have touched (here) wording of the lead-in paragraphs

    and expanded the typesetting of the first formula

    and added more hyperlinks to technical terms, such as PID and Tor

    diff, v33, current

  20. Create a new section on finitely generated flat modules

    Damien Lejay

    diff, v34, current

  21. Add the theorem flat = torsion-free over a PID

    Damien Lejay

    diff, v34, current

  22. Add a counter-example

    Damien Lejay

    diff, v36, current

  23. Add the proof of the proposition on finitely generated flat modules over an integral domain.

    Damien Lejay

    diff, v38, current

  24. Add the classification theorem of all rings on which finitely generated flat modules are projective.

    Damien Lejay

    diff, v39, current

  25. Add a reduction lemma to cyclic modules.

    Damien Lejay

    diff, v40, current

  26. Add a reduction lemma to cyclic modules.

    Damien Lejay

    diff, v40, current

  27. Add more information on the example RX

    Damien Lejay

    diff, v41, current

  28. Fuse the cyclic reduction with the equivalent charactirisation already present and make it a subsection.

    Damien Lejay

    diff, v41, current

  29. Restructure

    Damien Lejay

    diff, v42, current

  30. Update the examples and add Q to the list.

    Damien Lejay

    diff, v42, current

  31. Fix \varinjlim

    Damien Lejay

    diff, v43, current

  32. Add the example of the field of fractions.

    Damien Lejay

    diff, v43, current

  33. Add a new section on general flat modules. Start populating it with the characterisation of absolutely flat rings.

    Damien Lejay

    diff, v44, current

  34. Add a new section on general flat modules. Start populating it with the characterisation of absolutely flat rings.

    Damien Lejay

    diff, v44, current

  35. Add the example of a flat cyclic module

    Damien Lejay

    diff, v45, current

  36. End the proof of the characterisation of absolutely flat rings.

    Damien Lejay

    diff, v45, current

    • CommentRowNumber37.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 19th 2023

    Re #21: also proven in PID.

  38. Add Prüfer domains as the rings on which flat = torsion-free

    Damien Lejay

    diff, v46, current

  39. Add Prüfer domains as the rings on which flat = torsion-free

    Damien Lejay

    diff, v46, current

  40. Add Prüfer domains as the rings on which flat = torsion-free

    Damien Lejay

    diff, v46, current

    • CommentRowNumber41.
    • CommentAuthorRodMcGuire
    • CommentTimeJul 20th 2023

    linked GAGA

    diff, v47, current

  42. Fix: the torsion-free → flat property holds on semi-hereditary rings and not only on Prüfer domains

    Damien Lejay

    diff, v48, current

  43. Add that R[S^-1] is flat over R.

    Damien Lejay

    diff, v49, current

  44. Add flatness as a local property.

    Damien Lejay

    diff, v50, current

  45. Remove the theorem for PID, since they are only a special case of semi-hereditary ring.

    Damien Lejay

    diff, v50, current

  46. Add subsections

    Damien Lejay

    diff, v50, current

  47. Add characterisation of absolutely flat rings in terms of their local rings.

    Damien Lejay

    diff, v51, current

  48. Add a proposition on the product of flat modules.

    Damien Lejay

    diff, v52, current

  49. Add the proof of the characterisation theorem of semi-hereditary rings

    Damien Lejay

    diff, v54, current

  50. Add a section on countably presented flat modules.

    Damien Lejay

    diff, v55, current

    • CommentRowNumber51.
    • CommentAuthordmjc
    • CommentTimeMar 25th 2024
    I was trying to understand flat modules, but noticed something that seemed wrong on this page, involving the characterisation via identities I hope I fixed it correctly - or someone can tell me I am an idiot !

1 to 51 of 51

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