This morning I finished incorporating the material from sections 1 through 4 of the paper by Raynaud and Gruson into the stacks project. Most of it isĀ in the chapter entitled More on Flatness. There is a lot of very interesting stuff contained in this chapter and I will discuss some of those results in the following blog posts. Note that I previously blogged about this paper here, here, here, here, here, here, and here.

It turns out that it was kind of a mistake to do this, as the payoff wasn’t as great as I had hoped for. Moreover, I don’t think you are going to find the chapter easy to read. So the benefit of having done this is mainly that I now understand this material very well, but I’m not sure if it is going to help any one else. Maybe the lesson is that I should stick to the strategy I have used in the past: only prove those statements that are actually needed to build foundations for algebraic stacks. This will sometimes require us to go back and generalize previous results but (1) we can do this as the stacks project is a “live” book, and (2) it is probably a good idea to rewrite earlier parts in order to improve them anyway.

The long(ish) term plan for what I want work on for the stacks project now is the following: I will first add a discussion of Hilbert schemes/spaces/stacks parameterizing finite closed subscheme/space/stacks. I will prove just enough so I can prove this theorem of Artin: A stack which has a flat and finitely presented cover by a scheme is an algebraic stack. A preview for the argument is a write-up of Bhargav Bhatt you can find here.

Curiously, Artin’s result for algebraic spaces is already in the stacks project: It is Theorem TAG 04S6. It was proved by a completely different method, namely using a Keel-Mori type argument whose punch line is explained on the blog here.

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