## Archive for January, 2011

### 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.

### The CHANCE project

January 10, 2011

Whoa. I didn’t realize that there was yet another one of us. It also uses the same license (the GNU FDL) and even has a similar-sounding name.  As the name suggests, it’s about probability.

Apparently, the bandwagon we have jumped on is bigger than I thought.

### 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.