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1. • CommentRowNumber1.
   • CommentAuthorAndrew Stacey
   • CommentTimeNov 7th 2011
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexI'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 $A$ is flat if tensoring **with $A$** 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.

   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 $A$ is flat if tensoring with $A$ 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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 7th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeNov 7th 2011
   • (edited Nov 7th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI corrected it: tensoring with $M$ goes from $A$-modules to $k$-modules, not to $A$-modules, as $M$ is not an $A$-bimodule (it is a $k$-$M$-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).

   I corrected it: tensoring with $M$ goes from $A$-modules to $k$-modules, not to $A$-modules, as $M$ is not an $A$-bimodule (it is a $k$-$M$-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).

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 7th 2011
   • (edited Nov 7th 2011)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexAt [[flat module]], to Andrew's words > The module $M$ 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?

   At flat module, to Andrew’s words

   The module $M$ 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?

5. • CommentRowNumber5.
   • CommentAuthorTom Leinster
   • CommentTimeNov 7th 2011
   • PermaLink
   Author: Tom Leinster
   Format: MarkdownItexI thought Andrew's use of "absurd" in comment 1 was simply because **with** should be **over**. Tensoring an $A$-module with $A$ doesn't do very much :-)

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

6. • CommentRowNumber6.
   • CommentAuthorAndrew Stacey
   • CommentTimeNov 7th 2011
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexTom's right about my use of "absurd". My apologies for not being clearer.

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

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 7th 2011
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexRe #5: oh well, then it was more or less just a typo, wasn't it? Tensoring with the module $M$, not $A$!

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

8. • CommentRowNumber8.
   • CommentAuthorAndrew Stacey
   • CommentTimeNov 7th 2011
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexYes, but a confusing typo! Particularly since $M$ 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.

   Yes, but a confusing typo! Particularly since $M$ 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.

9. • CommentRowNumber9.
   • CommentAuthorzskoda
   • CommentTimeNov 7th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexSurely with $M$, that is why it does not end in $A$-mod but in $k$-mod. Regarding that it was written $A$ before, the codomain ended in a wrong category.

   Surely with $M$, that is why it does not end in $A$-mod but in $k$-mod. Regarding that it was written $A$ before, the codomain ended in a wrong category.

10. • CommentRowNumber10.
    • CommentAuthorAndrew Stacey
    • CommentTimeNov 8th 2011
    • PermaLink
    Author: Andrew Stacey
    Format: MarkdownItexI did the unwinding and spelled out how one gets from the element characterisation to the filtered colimit of free modules.

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

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2012
    • (edited Oct 17th 2012)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI 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 .

    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 .

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2012
    • (edited Oct 17th 2012)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have created a section _[Equivalent characterizations](http://ncatlab.org/nlab/show/flat+module#EquivalentCharacterizations)_ with some indications for why indeed -- as claimed right at the beginning of the entry going back to #1 above -- a module $N$ is flat precisely if every linear combination of 0 in $N$ comes from linear combinations of 0 in $R$. This follows pretty directly from yet another equivalent characterization, namely that it is sufficient to check left exactness of $(-)\otimes_R N$ on ideal inclusions. And this implication I have now written out. However, that left exactness of $(-)\otimes_R N$ can be checked already on just ideal inclusions is a bit more work...

    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 $N$ is flat precisely if every linear combination of 0 in $N$ comes from linear combinations of 0 in $R$.

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

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

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2012
    • (edited Oct 18th 2012)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have edited the flow of the _[Definition section](http://ncatlab.org/nlab/show/flat+module#Definition)_ 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.

    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.

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2013
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded the definition of _[[faithfully flat module]]_

    added the definition of faithfully flat module

15. • CommentRowNumber15.
    • CommentAuthorspitters
    • CommentTimeMar 28th 2018
    • PermaLink
    Author: spitters
    Format: MarkdownItexWhat is the reference to Blass and Wraith ?

    What is the reference to Blass and Wraith ?