## 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).