As I mentioned earlier I have started to look into Hilbert and Quot schemes/spaces/stacks. This led me to think about the existence of a flattening stratification and then I started thinking about some of the results obtained in the paper [RG] by Raynaud and Gruson. For example, as I first mentioned here, I think that using Theorem 4.1.2 of their paper would be a good way to prove the existence of flattening stratifications (as formulated in this post) in the “correct” level of generality.

Thus it now seems to me that the basic “devissage” result of [RG] would be a worthwhile addition to the stacks project. Consequently, I have started adding material on geometric fibres in families which is needed for one of the key geometric lemmas in [RG].

I think there is a lot more, besides flattening stratifications, to gain from adding this to the stacks project. For example, there are several places in [RG] with directions for the reader to rewrite several parts of EGA using their methods, and I feel that others will show up as we go along.

I’d love to hear about any interesting foundational applications of [RG] you know about!

[RG] Raynaud, Michel; Gruson, Laurent

CritÃ¨res de platitude et de projectivitÃ©. Techniques de “platification” d’un module. (French)

Invent. Math. 13 (1971), 1–89.