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.

The chapter on category theory came out of notes in a few lectures in an algebraic topology class. They are quite rough — much rougher than most of the book, I think — but they are being revised. The present book has somewhat more of a categorical flavor than might be common: some of the exercises in the “foundations” chapter ask the reader to prove representability for certain functors defined on the category of rings. (In reality, however, such exercises amount to getting a feel for some of the universal properties.)

The chapter on dimension theory has been edited somewhat. The basic structure is not changed: Hilbert polynomials are used to define the dimension, and it is proved with some effort that it satisfies the usual properties and coincides with the Krull dimension. However, some things have been moved around, and sections — as opposed to simply subsections — have appeared.

The Nullstellensatz has been added, although it is not edited yet. This was covered in the course notes, but it was only recently moved from the “tobeadded” folder into the main document.

The formatting has now changed significantly. The title page looks much nicer, and includes a list of current contributors as well as a description that it is “an open source textbook on commutative algebra.”  The headers on pages have now been actually formatted (as opposed my having hacked something together at the beginning). Also, \DeclareMathOperator has replaced \mathrm for mathematics operators, which may improve the appearance of the book.



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