The sixteenth post is the June 27 version in the usual place. This post covers until 25.7 (although 25.8, still a work in progress, is also included). The next post should appear in late July.

Status report.
Most of the progress since the 15th post has been in tweaking, and gradually digesting people’s comments. I have a lot of valuable suggestions still to digest, and am steadily making progress.

We’re nearing the end of the year-long project. I foresee two more regular posts (one in late July one in mid to late August), by which time only a couple of chapters will remain unrevealed. Then I expect a period of disruption, as if all goes well, I’ll become a father again near the start of September (a second son expected). I’ll then be teaching from these notes this coming academic year, and really testing what can be done in a single year. I will also fill out the last couple of chapters, and continue to work through all the wise advice I’ve been given over the year on this site.

For learners.
The fifteenth post was longer than it should have been from the point of view of learning, so please go back and get comfortable with that material.

Here are some flatness exercises. (Section 25.5 is there thanks to Georges.)
25.3.A, 25.3.E, 25.4.A-C, 25.4.H, 25.5.A, 25.5.D, 25.5.E, 25.5.F, 25.6.A, 25.7.A, 25.7.C.

If you would like to work out an explicit example in detail, you may want to try the explicit flat limit at the end of 25.4. If you get stuck at any point, please let me know, and I will try to get you unstuck. (And as always, if you get stuck anywhere in these notes where you think a few words might help you out, please let me know — those few words should probably be added to the text!)

For experts.

I’m not sure what the right definition of “flat of relative dimension n” is. Currently I only discuss it in the locally Noetherian setting, because I want it to be closed under composition (with the obvious additivity of relative dimension), and my method of showing it uses Noetherian techniques. Can someone tell me (or point me to) the right definition (e.g. the location in EGA)?

Advertisements