Homotopical stuff

April 14, 2011

I’ve added a skeleton of a new chapter on homotopical stuff. In particular, the hope is that this will cover the basic theory of model categories and simplicial sets. The motivation is to discuss applications of homotopical methods in commutative algebra, in particular the cotangent complex.

So far the material is largely taken from these notes on algebraic topology.

Puiseux’s theorem

March 22, 2011

In Local Fields, Serre uses ramification groups to prove that the algebraic closure of the power series field $K((T))$ over an algebraically closed field $K$ of characteristic zero is the colimit of $K((T^{1/n}))$ for $n > 0$; this is, as he calls it, a formal analog of the usual Puiseux theorem. If I am not being silly, there is an easier way to see this. For any finite extension, the unique valuation on the power series field extends uniquely (since these are all complete); the residue class field extension is trivial, since that of the power series field is already $K$. Thus the extension is tamely ramified, and now by general facts, any totally and tamely ramified extension of a local field is obtained by adjoining a root of a uniformizer. (Cf. Lang’s Algebraic Number Theory; this follows from Hensel’s lemma.)

I added this to the chapter on completions. Also, as promised in the comments to this post, there is now some material on finite presentation in a chapter currently loosely marked various. It is far from complete, though.

Krull-Akizuki, the Amitsur complex

March 19, 2011

These are some of the few recent additions. Krull-Akizuki is nice because it implies that (I didn’t know this before) the integral closure of a Dedekind domain in any finite extension, not necessarily separable, is Dedekind. One of the applications that should eventually appear is that a noetherian local domain can always be dominated by a DVR (as opposed to just a plain valuation ring).

The Amitsur complex is what Tamme calls it; others just write  it out. This is the sequence that you get for a faithfully flat extension of rings, whose exactness is what leads to all sorts of descent theorems, which will someday appear here. (There is a MathOverflow question on the history of it.)

A finite presentation chapter?

March 15, 2011

There are a whole slew of results in commutative algebra and algebraic geometry that are essentially elaborations on a standard set of tricks for finitely presented objects. For instance, one has the following fact: if $\{ A_\alpha\}$ is an inductive system of rings, then any finitely presented module over the colimit descends to one of the $A_\alpha$. Moreover, the category of f.p. modules over the colimit is the “colimit category” of the categories of f.p. modules over the $A_\alpha$. Similarly, any f.p. algebra over the colimit descends to one of the $A_\alpha$. This, together with fpqc descent, is behind Grothendieck’s extremely awesome proof of Chevalley’s theorem that a quasi-finite morphism is quasi-affine; this trick, in EGA IV-3, is what lets him reduce to the case where the target scheme is the Spec of some local ring.  So I think it would be fun to have a whole bunch of these sorts of results.

On the other hand, I’m not sure whether it would be pedantic to devote an entire chapter to them. There are probably more important things in commutative algebra proper, and the above results are really cleaner if we can use the language of schemes a bit (then we can talk about quasi-coherent sheaves on projective limits, and even derive ZMT!), though it is an open question exactly how much we should delve into algebraic geometry.

Thoughts?

Flatness, henselianization

March 13, 2011

I’ve been working on the chapter on flatness in an attempt to make it look something like a portion of a book. This means, for instance, a reorganization of material like $\mathrm{Tor}$ and greater material on faithful flatness. I added the canonical example of a faithfully flat algebra over a ring $R$ (the product $\prod R_{f_i}$ where the $f_i$ generate the unit ideal); one of the nice things is that the Amitsur complex of this faithfully flat algebra (with respect to some $R$-module $N$) is the Cech complex of the associated quasi-coherent sheaf with respect to the covering $\{ D(f_i) \}$. In particular, the acyclicity of the Amitsur complex (which should be added soon!) lets you get Serre’s cohomological vanishing on an affine in a somewhat quicker way than Koszul complexes.

I think it would be great to include material on faithfully flat descent, though it will perhaps be hard to motivate without the language of schemes (though right now we are getting fairly close to introducing the language of schemes already!). Perhaps things like Hilbert’s Theorem 90 and Galois descent would be good items to include.

I’ve also been fleshing out the material on henselianization, following Raynaud’s Anneaux locaux henseliens; this is pretty fun stuff too.

Yes, we still exist

March 5, 2011

I realize nothing has been posted on this blog in approximately a month now, but this project is still being updated. The academic year makes it difficult to get much work done on outside projects. The CRing project is still different from what it was a month ago, though no sweeping changes have occurred.

Since we have a lot of submitted material now that has yet to be edited into the project, I plan to work on that a bit during spring break. Other people should feel free to be involved in this!

This is cool (Freyd’s initial object theorem)

February 7, 2011

I blogged about it here, and added it to the CRing project here. Basically, the point is that complete categories are prone to having initial objects (of course, cocomplete categories are!) if they have a “weak” initial object: one that homs into every other object, but not necessarily uniquely.

I don’t think we’ll ever use it in the project, but it is neat. Covering a bit of basic category theory for its own sake seems like a reasonable thing to do, moreover.

Changes

February 5, 2011

We have received a new set of donations, again from a live-TeXed course. One thing that I hadn’t seen before is a theorem of McCoy that gives a criterion (in the “density” of certain ideals generated by minors) of when a matrix mapping of free modules is injective.

In addition, various edits keep being made, slowly. There have been several articles in the Monthly that give short semi-expository pieces of mathematics; a brief account of some of them have appeared in the book. Here is one, giving a short proof of the following lemma: a field extension which is finitely generated as an algebra is algebraic.

To see this, one can use the following argument of McCabe. Let $k'/k$ be an extension finitely generated as an algebra. There is a subring $k[x_1, \dots, x_n]$ over which $k’$ is algebraic. The finite generation hypothesis implies that we can find a localization $k[x_1, \dots, x_n]_y$ over which $k’$ is even integral. But if one has an integral extension and the top ring is a field, then so is the bottom one; this implies $k[x_1, \dots,x_n]_y$ is a field. So $y \in k[x_1, \dots, x_n]$ must lie in every maximal ideal, meaning it is zero, contradiction.

Euclidean domains

January 23, 2011

We all learn in intro abstract algebra that a euclidean domain is a PID. It turns out that the converse is almost true. Namely, if one relaxes the definition of a euclidean norm (instead of a euclidean algorithm, you have something a bit weaker) you get something entirely equivalent to being a PID. This is apparently due to Greene in the Monthly, 1997 (and has a quick proof). Now, this material is in ch. 1.

Lang’s Algebra (as well as some of his other books, too) has a lot of these kinds of isolated references to scattered results in the literature. Some of these are quite interesting; it is probably worth adding more of these. Doing so will also make the book less “canonical”!

It happens, coincidentally, that we also got a donation on euclidean domains, which has been partially merged in.

Split injections of free modules over local rings

January 17, 2011

The main latest change is the addition of the following lemma: Suppose given two free modules $F, F'$ over a local ring, of finite rank, and a morphism $\phi$ between them. Then $\phi$ is a split injection iff the base-change $F \otimes k \to F' \otimes k$ to the residue field is an injection. This is not too difficult to prove, but I realized today that Hartshorne uses it at a key point in proving that a nonsingular subvariety of a nonsingular variety is a local complete intersection. It is kind of glossed over there,  probably for good reasons, but this lemma is now explained in our book.

The makefile is also fixed so that running “make” actually resolves cross-references. Apparently, you run pdflatex twice after invoking bibtex, and not the other way around. That makes sense.