A new version of the notes is available here. (It, and all older versions, are available at the usual place.) For the first time, I would say that the content and editing is “potentially done”. All content is potentially polished. (Not done: beautification issues e.g. fighting with latex over line breaks. And the index is very rough.) I am thus very interested in any suggestions and corrections you may have, including things you told me before. (I think I’ve implemented all the edits from my to-do list from comments you have made through the years, by email or here on this site.)

In more detail: This is a fairly substantial revision. I also taught from the notes in the last two quarters, and the excellent comments of the excellent people in the class helped tremendously.    I may not do much more before I declare this project “done”.

New arguments added recently (perhaps in the last few revisions):

  • a scheme with no closed points, and a little more about coproducts
  • fixed proof of Serre duality (long in coming)
  • a glimpse of the Koszul complex, and proof of the Hilbert Syzygy Theorem (I learned how to think about this from Michael Kemeny)
  • improved exposition of proof of formal function theorem (unlike other cases, really the same proof)
  • all the important flatness facts are now done much more easily
  • and many more things I can’t remember right now.

I’m definitely looking for any small remaining issues.  And also any mathematical mistakes or omissions.   

A random cool thing I wanted to share: I have always wanted a way to order mathematical notes, in perhaps a semi-public way or a private way, that would be nonlinear and easy and robust. Wikis are good but have some imperfections. The “back end” of the stacks project (gerby) is fantastic for public presentation of huge amounts of interconnected material, but less suited to person note-taking because it is somewhat fragile, and has significant start-up time to use properly. (For more on gerby, see: https://gerby-project.github.io/ .) I stumbled on Jon Sterling’s “forest”, and I can do nothing better than just point you here: https://forest.jonmsterling.com/index.xml and recommend that you take a look and explore. He has done a lot of thinking on “tools for mathematical thought”, which is precisely the the sort of thing I was wanting to think through more myself. It is something akin to a manifesto. (As is unfortunately common as many of you know, I owe him an email.) I would like to explore this further (despite extreme lack of time), and I wanted to advertise this, in case others would like to try it out too!