I’m just going to start MaBloWriMo here. Anyway, as Qiaochu observes in the comments, National Blog Writing Month (as opposed to National *Novel *Writing Month) was in October. Before I start, I think it might be appropriate to post my half-complete notes on commutative algebra from a course I’m taking. Since I’m not going to be self-contained, they might be a helpful reference for readers.

For the next few posts, we shall always assume that **all rings are commutative and noetherian.** Commutativity is always needed. Noetherianness won’t be needed for now, but it will soon become indispensable. So let be a ring, and an -module. We want to talk about the definition of a **regular sequence** on . This is going to be a sequence of elements of that act “independently” on in some sense. We are going to make this precise below when we interpret a consequence of this condition via associated graded modules. This definition is basically the groundwork for everything that follows, as you need it for the definitions of notions such as Cohen-Macaulayness and regularity. Let us start by stating the definition.

Definition 1A sequence is-regular(or is an-sequenceif for each , is a nonzerodivisor on the -module and also .

So is a nonzerodivisor on , by the first part. That is, the homothety is injective. The last condition is also going to turn out to be necessary for us.

Example 1The basic example one is supposed to keep in mind is the polynomial ring and . Then the sequence is regular in .

Suppose now that is *finitely generated*. Things become the nicest in this context. Then it is a basic fact of commutative algebra that if is any -module, the guys in that are zerodivisors on are precisely those that lie in the union of the associated primes of . This fact will become crucial to us as we attempt to construct regular sequences. The property of being a regular sequence is inherently an inductive one. Note that is a regular sequence on if and only if is a zerodivisor on and is an -sequence.

** 0.1. Basic properties **

The first observation to make is that regular sequences are **not** preserved by permutation. This is one nice characteristic that we would like but is not satisfied. Nonetheless,

Proposition 2Let be a noetherian local ring and a finite -module. Then if is a -sequence contained in the maximal ideal, so is any permutation .

*Proof:* It is clearly enough to check this for a transposition. Namely, if we have an -sequence

we would like to check that so is

It is here that we use the inductive nature. Namely, all we need to do is check that

is regular on , since the first part of the sequence will automatically be regular. Now will automatically be regular on . So all we need to show is that is regular on . The moral of the story is that we have reduced to the following lemma. **Let be a finite -module and an -sequence contained in the maximal ideal. Then so is .** We can prove this as follows. First, will be a nonzerodivisor on . Indeed, if not then we can write

for some with . But is a nonzerodivisor on , which means that implies . Say . So . As is a nonzerodivisor on , we see that . Thus , contradiction. This part has not used the fact that is local. Now I claim that is a nonzerodivisor on . Suppose and . Since is a nonzerodivisor on , we have that , say . Thus

The fact that is injective implies that . So we can do the same and get , , and so on. It follows that is a multiple of , and hence in for each where is the maximal ideal. The Krull intersection theorem now implies that . Together, these arguments imply that is an -sequence, proving the lemma.

One might wonder what goes wrong; after all, oftentimes we can reduce results to their analogs for local rings. Yet the fact that regularity is preserved by permutations for local rings does not extend to arbitrary rings. The problem is that regular sequences do *not* localize. Well, they almost do, but the final condition that doesn’t get preserved. We can state:

Proposition 3Suppose is an -sequence. Let be a flat -module. Then if , then is an -sequence.

*Proof:* This is actually very easy now. The fact that is injective is preserved when is replaced by because the functor is exact.

In particular, it follows that if we have a good reason for supposing that , then we’ll already be done. For instance, if is the localization of at a prime ideal containing the . Then we see that automatically is an -sequence.

Next time, we’ll continue talking about the fundamentals of what this property really means.

October 30, 2010 at 9:49 pm

How long did it take you to type up those notes? Did you live-tex them?

October 31, 2010 at 8:46 am

Yeah, I live-texed them.

October 31, 2010 at 7:13 pm

It’s great that you live-texed those notes, but I should also recommend you to write some of these notes down with your own hand. The reason for this is that while live-texing is a good way of storing your notes electronically (probably the best way to do this, in fact), writing is in some sense a more involved process. At least for me, I don’t know about other people, it helps me to think more about the material, and imprints it in my mind. Live-texing does this too, but I’ve found that when I write something with my own hand and think 1 year later, I actually remember what my thought process was when I wrote it, but when I type it, I actually forget what I was thinking when I typed it. I’m not sure how you think and what approach suits you best, but it may be something that’s worth considering just to compare the two methods of learning. In some sense, that is both the advantage and disadvantage of the computer; it’s more efficient and effective, but efficiency and effectiveness often come at a cost.

October 31, 2010 at 9:09 pm

Thanks! Yes, I know what you mean — I like to write out material when I am working out an argument. Unfortunately, it so happens that (from experience) I’m really terrible at taking hand-written notes and keeping them organized; I usually end up with much shorter, messier notes near-useless except (barely) for myself. I do manually note-take my other classes, though, to rest my hands. Commutative algebra is the class where the subject (e.g. not too many pictures) and the lecturing style (emphasizing rigor and complete proofs) are such that the notes are most likely to be useful for others. (It’s also the subject that I’m most familiar with out of my three courses.)

October 31, 2010 at 9:57 pm

That’s true. Harvard’s algebra is certainly strong, but if you are reading Hartshorne, you’ll most likely be familiar with most of the commutative algebra anyway.

Note-taking is certainly a messy business. I, for one, like to try out the following (it’s been a long time since I’ve really had to take notes frequently at lectures, but this is what I used to do): I listen to the lecturer throughout the lecture, and after the lecture, try to see what I can recall out of the most important concepts taught in the lecture, and transcribe that. This gives me plenty of time to think about the material without having to keep up with the lectures. The one drawback is that you might not remember everything in the lecture, especially if the lectures are fast-paced (math 55 is an extreme example of this, but math 55 is, in any case, one of the fastest courses at Harvard; I’m sure the comm. algebra course isn’t at the same pace, though it’s obviously still fast). However, you can still take down some things in the lecture, and save the main things till after the lecture. I don’t know how this works for most people; as I said, it’s been a long, long time since I’ve done this, and it’s something that takes time getting used to. But it worked for me at the time.

In any case, I think you’ve acquired a good skill of learning mathematics on your own. Notes are most often useful when the lecturer deviates from the text, as often happens, so it’s better to read the text before the lectures to know what’s most important to write down, but I’m sure you know what you’re doing anyway. 🙂

In any case, it would be a nightmare to live-tex notes from an (advanced) algebraic topology class. I know some people who can, but it’s definitely not something I’d try if I was still taking classes. 🙂

October 31, 2010 at 10:44 pm

That’s a good idea. I should try that (at least if I get the time). Commutative algebra started from the beginning (as in, the definition of a ring), but went very quickly, and we have covered a fair bit of material (I think). The curriculum for Math 55 was standardized some time back, so I don’t think it goes as quickly as it used to. (Not too long ago I was told that they were still on linear algebra.)

I have tried live-TeXing notes from my introductory algebraic topology class, actually, though I recently stopped. They got reasonably far, but it was quite difficult to do it (I can do diagrams with xymatrix all right, but not figures; some people claim an ability to live-Tikz things).

One of the things that writing notes does is to keep me focused — otherwise, I’m more likely to let my mind wander. Livetexing has also made me many friends (you’d be amazed at how many people have asked me for copies by now). It has caused occasional problems when I get distracted by technical problems (fixing TeX errors, etc.), but not as much anymore as it did when I started.