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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.
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.
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).
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?
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 :-)
Tom’s right about my use of “absurd”. My apologies for not being clearer.
Re #5: oh well, then it was more or less just a typo, wasn’t it? Tensoring with the module $M$, not $A$!
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.
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.
I did the unwinding and spelled out how one gets from the element characterisation to the filtered colimit of free modules.
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 .
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…
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.
added the definition of faithfully flat module
What is the reference to Blass and Wraith ?
Re #21: also proven in PID.
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