Archive for the 'announcements' Category

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.

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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.)

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.

 

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.

Injectives

January 12, 2011

So now we have a proof (in chapter 3) that category of modules over  a commutative ring has enough injectives. Actually, two proofs. One is the standard dualization argument that appears in most textbooks. The other is a variant of the “small object argument” in homotopy theory and uses a bit more set-theoretic machinery. The latter has the advantage that it can be used to show that large classes of abelian categories have enough injectives (as Grothendieck does in his Tohoku paper).  In my commutative algebra class, the teacher hinted that one could prove the theorem this way.

The idea is somewhat explained in this blog post, but not very well, and some of the technical points (e.g. filtered ordinals) are obscured there. Thanks to Johan de Jong for pointing this out.

Also, the formatting has changed a little. The chapter and section titles are not simply the defaults.

In which the CRing project’s website expands

January 7, 2011

The main website for the CRing project is now slightly improved. Namely, there’s now a downloads page which allows you to view individual chapters of the book. This idea was shamelessly copied from the analog for the Stacks project, of course. As usual, the website will be updated about once a day (which is slightly less frequently than the project actually gets edited!).

The project itself has been evolving as usual the past few days. I am not sure it makes sense to give a blow-by-blow account of every small edit (that’s what the git repository is for), but the major new addition is a small section on Oka families of ideals in the chapter on the Spec of a ring. This is basically an axiomatization of the familiar observation that an ideal maximal with respect to some property is often prime. We also have some new donations, which will start trickling into the main document soon.

The source files also now contain a bunch of Perl scripts that may be useful. This is entirely irrelevant to compiling the main document (CRing.pdf) but might help in other cases. Let me briefly explain what they do:

  • scripts/makenamelist.perl keeps the list of chapters (in the tmp/ directory) up to date.
  • scripts/script.perl updates the makefile (which should be done after you add a new chapter or remove a chapter) and creates files in the aux/ directory that when compiled will produce precisely one chapter. This only needs to be run after you add a new chapter. However, there is a better way to do this: make update_tmp will run the script as well as the one that updates the name list.
  • Speaking of which, the makefile is now better. “make chflat.pdf” (or more generally “make ch(name).pdf”) will, for instance, produce a PDF file containing the chapter on flatness alone. The xr package is used to get the cross-references with the rest of the document working. “make chapters” will do all the chapters (and, incidentally, the whole book as well).
  • If you want to run a script by itself, this should be done from the main directory.
  • If for whatever reason you don’t have “make” (e.g. you use Windows), you can run “pdflatex aux/ch(name).tex” from the main directory twice (after compiling the book itself, pdflatex CRing.tex) to get the individual chapters.

Not that these are likely to be used too often by contributors — they’re probably most useful for now in getting the website automatically updated. Later we might need them if we want to put a table of contents in each chapter or something like that (and for whatever reason can’t use shorttoc). Also, I don’t know programming, so people should feel free to edit these.