## 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.

December 31, 2010

The major change in the past few days is the addition of a section on Galois theory, which was recently donated. As a result, the chapter on fields has expanded somewhat.

The chapter on flatness has expanded slightly (with sections on the local and infinitesimal criteria).

Various snippets of code donated (e.g. problem sets) have been moved into the main document, often as worked examples (such as the computation of the radical of a polynomial ring $R[x]$ in terms of the radical of $R$).

There is now a section discussing a general observation (due to Lam and Reyes) that ideals maximal with respect to a certain property (e.g. being not finitely generated) are often prime.

Importantly for those editing the source, the package ntheorem has now replaced amsthm in the document. Hopefully this will allow for auto-generated theorem lists eventually. In addition, the package cleveref is now used, so that references of the form “Theorem~\ref{name}” can be simplified to “\cref{name}.” This also provides greater versatility if we want to change theorems into propositions or vice versa.

Finally, the book is nearing 300 pages!

## Source files

December 26, 2010

The source files on the main website will no longer contain contributions that have not been merged into the main document. The justification is that otherwise the source files would become very large! (There are some JPEGs in the full source.) Right now, the abbreviated source is a couple of hundred kilobytes. The much larger full source files can, as always, be downloaded from the git repository.

The main website no longer contains .tar.gz files (for the source), but now instead uses bzip2, which typically leads to better compression. (Windows users may find it easier to open the larger .zip archives.)

A few minor changes were made: a typo in the name of the chapter on homological methods, pointed out by Keith Conrad, was corrected, and a bit of material on faithful flatness was added. In fact, there is now a new chapter on “flatness” that is in its infancy. The topic is introduced earlier, but there is much more one can say especially after one has introduced derived functors.

## Major restructuring

December 21, 2010

Several major restructuring changes have happened in the past few days. In chronological order:

1. Much material was merged in from the tobeadded folder. In particle, most of the Math 221 notes (including material on Serre’s criterion and the homological theory) are now in the main document. They need editing!
2. Material was added on Dedekind domains and basic results on how primes split (e.g. in Galois extensions).
3. Material was added on graded rings. Right now, these are based mostly off EGA II and are geared towards that source. In particular, much emphasis is placed on homogeneous localization and the Proj of a graded ring (which seems to me reasonable since this book is intended to prepare for study in algebraic geometry; feel free to disagree). However, there are more topics (e.g. the associated graded of a filtered ring) that should be added.
4. The homological algebra preliminaries have all been moved to the end of the book. (Material such as exactness still appears at the beginning, but derived functors do not.) I think this is reasonable: one doesn’t need to know what a derived functor is to understand flatness, and those constructions are somewhat more advanced than most of the first part of the book. At the same time, it will probably be worthwhile to add an additional chapter at the end when flatness is done more systematically via $\mathrm{Tor}$ (which is necessary for things like the infinitesimal and local criteria, if I remember correctly).

The book is now much longer (about 250 pages).

## 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! Read the rest of this entry »

## First round of changes

December 18, 2010

(N.B. In this post, I shall avoid naming which contributor wrote what — this is visible in the git logs to the curious.)

A bunch of things have happened to the CRing project in the past few days. (As usual, reasonably up-to-date PDFs may be downloaded from the main site.)

First of all, things are actually moving! There have been six contributors so far, and it has been less than a week. While the project started with a (slightly edited) version of lecture notes from a commutative algebra class, they now have — very rough — chapters on category theory, field theory, homological algebra, and formally etale stuff.

The Stacks project has kindly released a section of the stacks code to us, on formally etale and smooth morphisms. This appears as the final chapter of the current project. So far, it is somewhat more advanced than the rest of the project, as well as written in a different style, so we will need to make significant edits.

There is a chapter being written on field theory. Right now, it contains a discussion of transcendental extensions, the characteristic of a field, and a proof of the existence of the algebraic closure of a field.  Since this chapter is being written from “scratch,” it is currently rather short and hopefully will be filled in in the future.
Read the rest of this entry »

## A niche for CRing?

December 14, 2010

Since this project is pretty new, it’s probably worth spending a little time determining its niche. One thing is clear: it’s an undergraduate-level, open-source, collaborative textbook on commutative algebra. But there are a lot of textbooks out there on commutative algebra, even if none (that I’m aware of; please correct me if I’m wrong) are open source.  So we need a raison d’etre to explain exactly why hammering away on this is worth our time, and to determine what we can say in a sales pitch when asking for contributions.

One way we could set a niche is by being really categorical. This is kind of the current approach: the initial chapter (as I write) is on category theory, while the second is on homological algebra. In this sense, the CRing project attempts a broader mission: to educate about general algebra (including abstract nonsense) instead of either simply assuming familiarity with Tor, Ext, and exact sequences and plunging ahead or by sketching details without proofs. As an undergraduate textbook, this level of self-containment is probably a good idea. On the other hand, being really categorical in a proper sense will avoid the “cop-out” of sketching proofs: since this is a collaborative project, we can do a lot.

That’s another sense in which we have a niche. Since there are already several people collaborating (and we hope more will join), editing can proceed at a much faster pace than it could if only one person were working. The use of the git server has made it extremely efficient to keep track of versions and to allow us to work on it simultaneously. As a result, we should probably aim for significant length.

But these are just a few initial, rambling thoughts. Does anyone want to offer any better ideas?