The infinite entry

[sniffnoy]

...by which I mean "dealing with infinite things". I just wanted to point out some neat infinite things I'd seen recently (mostly due to MathOverflow):

EDIT next day: changed some stuff in paragraph 3 to make obvious inferences that I hadn't previously...

1. In an infinite symmetric group, every element is a commutator. I don't just mean the group is perfect, I mean every element is a commutator. Proof due to Ore in 1951: http://www.jstor.org/pss/2032506

2. Over a noetherian ring which does not decompose as a product, every projective module which is *not* finitely generated is free! Proof due to Hyman Bass, 1962. Still reading through this one. Seems a bit of an odd hypothesis, no? It works by making use of Kaplansky's theorem that any projective module can be written as a direct sum of countably generated modules - I looked that up, for such a scary-sounding statement, the proof is surprisingly simple - and the Eilenberg-Mazur swindle. That, and a bunch of terrible manipulations with infinite matrices to handle the countable case...

3. Speaking of that theorem of Kaplansky, here's something I was trying to find the other day. (Note: Everything in the following paragraphs is just stuff I've read, not stuff I actually know.) I recalled reading somewhere a while ago - I thought it was on MathOverflow, but I can't find it there - some guy arguing that we use the wrong definition of "projective module". His argument was as follows: He was commenting on the fact that it is impossible in ZFC to determine the projective dimension of C(x,y,z) over C[x,y,z] (it's 2 if the continuum hypothesis is true, 3 otherwise). Most people would say, wow, that's a surprisingly non-set-theoretic statement to be independent of ZFC. But he said, it is a set-theoretic statement, because our definition of projective is wrong!

Apparently there is a theorem that a module is projective if and only if it is 1) flat, 2) Mittag-Leffler, and 3) a direct sum of countably generated modules. (Don't ask me what it means for a module to be "Mittag-Leffler". I mean, I know the definition on account of having looked it up, but that's it.) He argued, for all the purposes we use projective modules in algebra, merely being flat and Mittag-Leffler is sufficient! (I have to wonder what purposes he had in mind - surely there's some times when you just want to lift some homomorphisms?) And these two conditions are nice algebraic condition. But condition number 3? Being a direct sum of countably generated modules? That's a set-theoretic condition! And one we never need in algebra! Hence our notion of "projective module" is actually hiding a set-theoretic condition inside it, and we should redefine "projective" to mean "flat and Mittag-Leffler", and if we did that, then with that notion of "projective" it would indeed be possible to determine the projective dimension of C(x,y,z) over C[x,y,z]. (I don't remember if he mentioned what it was - at most 2, I assume, but perhaps less for all I know.)

I guess I don't really have an opinion on this, since I don't really know enough to. But I'm wondering if anyone else has seen this, because I can't seem to find it anymore.

EDIT: See comments for a counterargument.

EDIT 16 December 2011: Found the thing I was talking about; it's here (not sure how I missed it). Seems the idea is due to Vladimir Drinfel'd.

-Harry

Comments

[joshuazelinsky.livejournal.com]

Regarding 1, Is slightly surprising but not very much so. I don't have very good intuitions for groups that I can't embed in GL(n,F) for some nice field F. By nice I mean, F is C, or is the algebraic closure of some finite field. So I may just lack the experience to be surprised by this result.

Regarding 3, I don't know much about this, but isn't the standard definition of a projective module just in terms of direct summands? That seems to be purely algebraic. And that definition seems somewhat natural (although this might be connected to the fact that like you I don't really know much about Mittag-Leffler). (Additional disclaimer: I'm just now learning k-theory so anything I say about such issues may be confused.)

[sniffnoy.livejournal.com]

You're right, 1. isn't really that surprising; it wasn't something I had thought about before. It's still neat though.

Regarding 3: Well, I would consider the standard definition to be about lifting homomorphisms, but yeah, same thing. A perfectly sensible algebraic property, which this guy claims being flat and Mittag-Leffler can substitute for in all practical cases. Which I have to say I would find pretty surprising if true. I never said I agreed with the guy, I was just trying to find it. :)

In fact, here's a counterargument: Let's say we're working over a noetherian ring which does not decompose as a product (since noetherian is a reasonable assumption, and if it decomposes as a [finite, since it's noetherian] product, we can decompose the modules). Then by 2) any projective module is either finitely generated - in which case there should not be any set-theoretic weirdness - or free, in which case, well... it's free. OK, I admit the second branch of that disjunction is a little weak (free on what set? set theory could probably still affect things that way), but free shouldn't be very complicated, right? :)

[sniffnoy.livejournal.com]

Also it may be worth noting that it's only when the permutation you're trying to write as a commutator is finite that it requires you to use elements from outside the permutation in writing it as a commutator. (When I speak of an element being "in" a permutation I mean that it's not fixed by it.)