Just a short update. The semester is in full swing here at Columbia University and there are a lot of things to do (including writing letters of recommendation), so I have had less time to work on the stacks project. I hope/expect to get back to it soon.

Currently, I am still working through the details of the paper by Raynaud and Gruson. I found a (repairable) error in the proof of the main geometric result (existence of devissage; last sentence of the proof of Proposition 1.2.3). It is a small error, but it really is an error and you have to slightly change the set-up in order to fix it. Of course I may be wrong, but I do not think so (for those of you who are taking a look at the paper: try to imagine what it would mean to replace the sentence mentioned above by a fully written out argument, checking all the details). In addition to this, I’m having trouble finding simplifications for almost any of the arguments, as each of the later results in the paper uses the earlier results, in other words, I haven’t been able to split off some parts as independent from the rest.

I am going to finish writing it all up, as soon as I have more time. But for the moment this experience is teaching me a lesson. Namely, I started working through the details of Raynaud-Gruson as I wanted to have a very general result on flattening stratifications. I was eager to do this, as I wanted to discuss Hilbert schemes/spaces/stacks in the “correct” generality. And this in turn I wanted to do because I want to explain the proof of Artin’s result that a stack X in groupoids over (Sch) whose diagonal is representable by algebraic spaces such that there exists a surjective, flat, finitely presented morphism U —> X where U is a scheme is an algebraic stack. Looking back what I should have done is write a chapter on Hilbert schemes/spaces parameterizing finite closed sub schemes/spaces/stacks (maybe even restricting the discussion to the representable separated case). This is much easier, is quite interesting in its own right, and is sufficient for the application in the proof of Artin’s theorem.

On the upside, I have learned a lot more about flatness in the effort to get this material written out fully!

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