Open source commutative algebra
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.
In Local Fields, Serre uses ramification groups to prove that the algebraic closure of the power series field 
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.
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.)
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 
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?
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 
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.
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!
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.
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 
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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.
The main latest change is the addition of the following lemma: Suppose given two free modules 
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.
The CRing Project 