## A wish list for CRing?

December 18, 2010

So now that we have several contributions to the CRing project, and several people who have agreed to contribute in the future, we need a wish list of things to add. Here are a few things that I’d like to see. Most of these simply reflect my own interests, so people should feel free to suggest things in the comments.

First, there is material contained in the tobeadded folder but not merged into the main document:

1. Serre’s criterion for normality (and the baby case for reducedness)
2. Homological theory (regular local rings, Cohen-Macaulayness, Koszul complexes, etc.)
3. More on discrete valuation rings

Then there is other material that will require writing “from scratch” (or donations of other people’s notes). First, we have the easy stuff:

1. We don’t have a formal chapter on “graduations, filtrations, and topologies,” to quote Bourbaki. Namely, students of algebraic geometry have to know what a homogeneous ideal is, or what homogeneous localization means. It will also be useful to collect definitions of things like the associated graded (which we use during dimension theory). The Artin-Rees lemma can be moved here. This will be fairly easy to write up, and likely would be pretty short.
2. The chapter on field theory right now has many gaps. It’s not clear how much field theory should go in the present book (which is about commutative algebra), but some is definitely appropriate, and we need to fill in the gaps between the definition of characteristic and algebraic closures!
3. Material on free and finitely generated modules should be added to the chapter on foundations. Also, it would be nice if we had the proof that every module can be imbedded in an injective module; it would be very cool to do this using the baby version of the “small object argument” as in Grothendieck’s Tohoku paper. (I just think of the argument with transfinite push-outs as analogous to the small object argument — I don’t know if this is an incorrect analogy.)
4. We should prove that a PID is a UFD. This is standard, general algebra, but it’s probably worth including for completeness.
5. The category theory chapter should be polished and possibly expanded (e.g. to talk about adjoint functors, abelian categories, etc.).
6. Much more examples! We have quite a few, but there are a lot of subsections lacking them.

There are some more advanced topics that I myself be interested in; other can discuss the appropriateness of them in the comments below.

1. Excellent rings. At one point I had to use this idea in my algebraic geometry course for a problem set (I’m almost sure my argument was sub-optimal, but still), and it really made an impression on me that, for most everyday local rings, a property is true for the initial ring iff it is true for the completion, and that this class of “excellent rings” has all the stability properties you could ever want.
2. Etaleness, smoothness, and unramifiedness. Right now, we have a chapter donated by the Stacks project, but it needs editing to match the style we are using. It also should, I think, cover regular etaleness (resp. smoothness, unramifiedness) before the “formal” concepts (i.e. the ones defined by an infinitesimal lifting property), as in provide background for the exercises in Hartshorne on the material.
3. Further discussion of depth?

### 9 Responses to “A wish list for CRing?”

1. Johan Says:

For the existence of enough injectives in the category of modules over a commutative ring, you don’t need Grothendieck’s argument. The standard argument (at least I think it is the standard one) can be found in the stacks project, in the chapter entitled “Injectives”. A nice bonus is that the injective embedding you get is functorial.

• Akhil Mathew Says:

This is the argument I initially learned (and we should probably include it), but I never really felt comfortable with it; I find Grothendieck’s argument more intuitive: you just add all the push-outs a transfinite number of times! (I guess I’m weird.)

2. Johan Says:

I’d love it if you guys added more basic material about fields, because that is missing from the stacks project, and I could “steal” it. How about: Galois theory for example?

3. darij Says:

Here is my personal wish list for this wonderful project:

1) I know that this is a bug of PDF viewers rather than one of PDF files, but as long as no viewer is doing it right, could you please make page 1 of the PDF be page 1 in the text? Because mental arithmetic isn’t as fun anymore as it used to be in preschool…

2) There are many algebra texts in the world, but there are very few *constructive* algebra texts out there (I only know of one, which is in French and far from perfect – what is not surprising given that it is the first systematic treatise of this area since the 19th Century). I really would be delighted to see at least an attempt at proving constructively valid things in a constructive way (of course, there are genuinely non-constructive parts, which I am not talking about), or at least at reducing the nonconstructive methods to a minimum and not making the text a Zornfest.

Now a couple of mistakes I found:
– §7.1 Proposition 1.4: I don’t believe in the proof of this. It seems to work for noetherian \$R\$ only, because why else should the sequence stabilize? I fear there is no good way to avoid the determinant trick here.
– §7.1 Proposition 1.6: You must require that \$M\$ is faithful.
– §7 1.3: You have not defined what an “integral map” is.

Feel free to use anything you wish from my integrality notes ( http://www.cip.ifi.lmu.de/~grinberg/#integrality , now with sourcecode, although it’s ugly machine-generated LaTeX).

I might contribute something later when my to-do list clears up a bit. As for now, unfortunately I don’t have the time, but I wish you good luck and many nice proofs.

darij

• Akhil Mathew Says:

Hi Darij,

1. OK, I’ll look into this.

2. This is interesting, and I would be happy to see constructive methods included, though I don’t know anything about the subject and can’t probably do it myself anytime soon.

Regarding the mistakes:

7.1 Prop 1.4: If a f.g. module over any ring is a union of an ascending chain of submodules, then one of those submodules is equal to the full module, because each of the generators has to be contained in some spot. (This is just the standard argument, though it may help to clarify in the text.)

7.1 Prop 1.6. Yes, you’re right; I’ll fix this.

7 1.3: Fair enough. I’ll fix this as well.

• darij Says:

Ah, thanks, I see now that the proof of Proposition 1.4 is right. I instinctively cried wolf because I have seen a similar and wrong argument in a book.