Recent additions

Here is a list of things we’ve added to the Stacks project since last summer. Newer things are listed first:

  1. Artin’s theorem on contractions, see Tag 0GH7. The exposition follows Artin’s proof very closely. It was added recently, so improvements can be made and suggestions as to how to do so are welcomed.
  2. Rachel Webb pointed out a serious error in the proof of Lemma 0A9Q in the chapter on duality for schemes and the corresponding Lemma 0E58 for algebraic spaces. See Example 0GEU for a counter example to the original formulation.
  3. Noah Olander added some material on detecting boundedness of quasi-coherent complexes using a generator. See Section 0GEI and the material at the end of Section 0CSI. We also have the analogous material for algebraic spaces, see Section 0GFE and Lemma 0GFJ and Lemma 0GFL.
  4. We upgraded some of the discussion in Pushouts of Spaces because it was needed for the proof of Artin’s theorem on contractions.
  5. We discussed various “descent of \’etale sheaves” issues, e.g., if you have a proper surjective morphism f : X —> Y and an \’etale sheaf on X which is constant on the fibres of f, then it comes from an \’etale sheaf on Y. For a precise statement, see discussion in Section 0GEX. There is an analogous section for algebraic spaces somewhere.
  6. Thanks to prompting by Tuomas Tajakka, we added the algebraic spaces case of the discussion of ample invertible modules and cohomology. See Section 0GF9
  7. We added a rather large amount of material on formal algebraic spaces in Chapter 0AM7. In particular, given an adic, finite type morphism f : X —> Y of locally Noetherian formal algebraic spaces, we introduce carefully a number of “rig-properties” of f and prove some initial lemmas on these. A “rig-property” of f is a property of the restriction of f to the “generic fibres” of X and Y, except that the Stacks project doesn’t contain enough theory to make this precise. Anyway, I want to point something out here: the notion “rig-flat” is a rather tricky one! See Section 0GGK for the corresponding algebra discussion.
  8. Yet another application local criterion flatness: Lemma 0GEB is the lemma you always wanted to know about, but you didn’t know it! No, really!
  9. Thanks to discussions with Jarod Alper around his lectures on moduli theory on hikes here in WA, we much improved the discussion of the \’etale local structure of morphisms of schemes in Section 0CAT.
  10. We explicitly formulated Artin’s axioms in the Noetherian setting for algebraic spaces, see Section 0GE6.
  11. We revamped the discussion on algebraization of rig-etale and rig-smooth algebras as discussed in Elkik’s Theorem 7. You can read this in Section 0ALU, Section 0GAU, and Section 0AK5.
  12. We fixed several errors pointed out by 李一笑
  13. We added relative Poincare duality for de Rham cohomology, see Section 0G8F. Thanks to Shizhang Li for helping me with this. Let f : X —> S be a smooth proper morphism of relative dimension n. The key is to prove that the map d : Rnf*Ωn – 1X/S ——> Rnf*ΩnX/S is zero. The hard case is when S is the spectrum of a (nonreduced) Artinian local ring. After trying a *lot* of things that didn’t work, we found a proof using in some sense that the construction of this map is compatible with kunneth and the gysin map for the diagonal of X/S. I would appreciate references to places where relative Poincare is discussed in the literature.
  14. We added some additional Kunneth formulas, see Section 0FLN, Section 0G4A, and Section 0FXX.
  15. Pullbacks of K-flats with flat terms are K-flat with flat terms. Somehow we missed this the first time around. See Lemma 0G7E
  16. We added a bunch of stuff on gysin maps in Hodge cohomology and related lemmas on cohomology with supports of quasi-coherent modules.
  17. Lichtenbaum’s theorem Section 0G5D.
  18. Duality for compactly supported cohomology coherent modules, see Section 0G59.
  19. Bertini a la Jouanolou: just an amazing argument, no idea how you would come up with this. Read the original or see Section 0G4C.

Enjoy!

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