Update

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Since the last update of June 2013 we have added the following material:

  1. Algebraicity of stack of coherent sheaves Tag 09DS
  2. Epp’s Theorem Tag 09F9 and Tag 09II
  3. Reduced fibre theorem Tag 09IL
  4. New chapter on fields Tag 09FA
  5. Perfect generator for DQCoh(X) for schemes Tag 09IS
  6. Perfect generator for DQCoh(X) for algebraic spaces Tag 09IY
  7. New chapter on differential graded algebra Tag 09JD
  8. Section on Noetherian schemes of dimension 1 Tag 09N7
  9. Finite set of codimension 1 points on separated scheme are in an affine Tag 09NN
  10. Section on curves Tag 0A22
  11. Versions of Avramov’s result Tag 09Q7 and Tag 09QF
  12. Obtaining triangulated categories Tag 09P5 written by John Yu and Yifei Zhao
  13. Rickard’s theorem Tag 09S5
  14. Section on compact objects Tag 09SM
  15. Section on generators in triangulated categories Tag 09SI
  16. DQCoh = D(QCoh) for Noetherian schemes Tag 09TI
  17. DQCoh = D(QCoh) for Noetherian spaces Tag 09TH
  18. Final exam commutative algebra Fall 2013 Tag 09TV
  19. Lots of stuff about pseudo-coherence
  20. Cohomology of locally compact spaces Tag 09V0
  21. Proper base change in topology 09V4
  22. New chapter on simplicial spaces Tag 09VI
  23. Cohomological descent for proper hypercoverings in topology Tag 09XA
  24. Section on henselian pairs Tag 09XD
  25. Finite cover by a scheme Tag 0ACX
  26. Partitions and stratifications Tag 09XY
  27. Bunch of stuff about limits of sites and spaces and cohomology
  28. Gabber’s result on Henselian pairs Tag 09Z8
  29. Theorem of formal functiosns via derived completion Tag 0A0H
  30. Lots of edits to the chapter on etale cohomology culminating in a proof of the proper base change theorem Tag 095S
  31. Cohomological dim <= Krull dim for spectral spaces Tag 0A3C
  32. Equivalence beteen torsion and derived complete modules Tag 0A6V
  33. Local duality a la Grothendieck Tag 0A81
  34. Brown representability for triangulated categories Tag 0A8E
  35. Final exam algebraic geometry Spring 2014 Tag 0AAL
  36. Bunch of stuff about twisted inverse image, dualizing complexes, upper shriek functors
  37. New mostly empty chapter on resolution of surfaces Tag 0ADW
  38. Algebraic spaces are schemes in codimension 1 Tag 0ADD
  39. Extremely careful discussion of points on fibres of morphisms of algebraic fibres and when this implies the morphism is quasi-finite at the points Tag 0AC0
  40. Finite groupoids Tag 0AB8
  41. fppf descent data for spaces over spaces are effective Tag 0ADV
  42. New chapter on pushouts of algebraic spaces Tag 0AHT including a discussion of something called formal glueing which should really have another name
  43. Regular local rings are UFDs Tag 0AFW
  44. Approximation for henselian pairs Tag 0AH4

This brings us up to July 1 of this year. At this point I decided to work out what happens with dilatations in Artin’s paper “Formal Moduli II”. In fact, I wrote a blog post trying to figure out what could be true. Anyway, it turns out that Artin’s theorem on dilatations holds without assuming the base Noetherian algebraic space is excellent and in the proof we don’t need to use Artin’s criteria. See Theorem Tag 0ARB and combined this with Lemma Tag 0ARX to obtain Artin’s theorem — I hope to discuss what these results mean in more detail in a later blog post. Here are some related and not so related changes to the Stacks project.

  1. A chapter on formal algebraic spaces. Our approach is somehow a combination of all approaches in the literature: EGA, McQuillan, Yasuda, Beilinson-Drinfeld, Abbes, Fujiwara-Kato, and Knutson. Namely, we define a formal algebraic space as an fppf sheaf on (Sch) which \’etale locally looks like an Ind-scheme whose transition morphisms are thickenings. For the Stacks project this approach turns out to be exactly the right one and many of the earlier results on algebraic spaces can be brought to bear to get ahead fairly quickly.
  2. A followup chapter discussing the notion of rig-etale (homo)morphisms ending with a proof of the algebraization theorem mentioned above. These rig-etale maps are used by Artin in his paper mentioned above to describe formal modifications and these (as well as the similarly defined rig-smooth ring maps) are the ring maps studied in Elkik’s famous paper.
  3. Adjoint functor theorem Tag 0AHM
  4. Remarkable result by Yasuda that every qcqs formal algebraic space is a filtered colimit of algebraic spaces along thickenings Tag 0AJD
  5. Non flat completions Tag 0AL8
  6. QCoh not abelian for formal algebraic spaces Tag 0ALF
  7. Ofer Gabber answered completely this question see Tag 0APW
  8. Algebraic spaces are fpqc sheaves by Ofer Gabber Tag 03W8

Enjoy!