Topologies and descent, Fal 2018

Professor A.J. de Jong, Columbia university, Department of Mathematics.

Introduction. The category of schemes has many different geometrically interesting topologies. For different problems different topologies are suitable. Although traditionally we algebraic geometers shied away from non Noetherian gadgets, lately we've found several instances where limit constructions produced nice non Noetherian gadgets. To descend results from these big ring/schemes back to the original Noetherian situation one uses descent. This seminar discusses different types of descent, culminating in some talks discussing more recent results.

Organizational:

1. This semester the students will give the lectures.
2. Please email me if you want to be on the associated mailing list.
3. Time and place: Room 407, Fridays 10:30 -- 12:00 AM.
4. First short organizational meeting: Friday, September 7 at 10:30 AM in Room 407. Everybody interested please attend.

Schedule: (things will still move around, etc)

• 9/14 Dmitrii Pirozhkov, on topic 1
• 9/21 AGNES no lecture
• 9/28 Shizhang Li, on topic 4
• 10/5 Johan de Jong, on topics 2, 3
• 10/12 Carl Lian and Noah Olander, on topic 5, 6, 7
• 10/19 Carl Lian and Noah Olander, on topic 5, 6, 7
• 10/26 Alex Perry, on topc 13
• 11/2 Raymond Cheng, on topic 10
• 11/9 Bhargav Bhatt, on topic 14
• 11/16 NO LECTURE
• 11/23 NO LECTURE
• 11/30 Johan de Jong, on topic 21

Lectures: The emphasis will be on carefully stating the results and discussing the key steps of proofs (skipping as much as possible abstract generalities) or even giving proofs only in (interesting) special cases.

1. Fpqc descent of quasi-coherent modules, exactness of the Amitsur complex, cohomology of quasi-coherent modules is the same in Zar, fppf, etale, syntomic, smooth topologies, etc. You can also present descent for quasi-coherent modules along universally injective ring maps if you like. Links: 023F, 03DR, and 08WE
2. Equalities of topologies:
   1. The fpqc topology is generated by Zariski coverings and faithfully flat morphisms of affines. See 022H.
   2. The etale topology is the smooth topology, see 055V.
   3. The fppf equals the fpqf topology. See 0572.
   4. The fppf topology is generated by Zariski coverings and surjective finite locally free morphisms of affines. See 05WN
   Try to give only the key algebra arguments.
3. Comparison V-topology with h-topology, see David Rydh, Section 3. In the Stacks project we define the
   1. ph topology, see 0DBC. This topology is generated by Zariski coverings and proper surjective morphisms.
   2. V topology, see 0ETA. This topology is generated by Zariski coverings and morphisms of affines satisfying a lifting property for valuation rings.
   3. h topology, see 0ETQ. This topology is generated by Zariski coverings and morphisms of finite presentation between affines which are V covers or equivalently ph covers.
   4. V = pro-V = pro-h = pro-ph, see 0EVM. This also shows that limit preserving sheaves for the h topology are sheaves for the V topology.
4. Pro-etale versus weakly etale. See Bhatt-Scholze, Theorem 1.3 or see 097Y.
5. Comparing cohomology:
   1. 09WY and especially this result. This fun result tells us that for a Hausdorff locally compact space X the usual cohomology H^i(X, Z) agrees with the cohomology of the constant sheaf Z computed in the "big" site of all locally compact Hausdorff spaces over X endowed with a topology akin to the fpqc topology.
   2. 0DDK Similar to the above but comparing etale and fppf cohomology of an etale sheaf on a scheme.
   3. 0DDV same as above but comparing etale and ph cohomology of an etale sheaf on a scheme.
   4. 0EW7 same as about but comparing etale and h cohomology of an etale sheaf on a scheme.
   5. same as above but comparing etale and v cohomology for etale sheaves.
   6. Add your favorite comparison theorem on cohomology here (but not cases where the topologies are the same!).
   The Stacks project uses the same proof scheme to prove 1, 2, and 3. However the exposition is clumsy and there is an opportunity here to improve the material while giving the talk! Another opportunity is to try and find a clever proof of 4 avoiding 1, 2, or 3. These results form the basis for proving the results on proper hypercoverings below.
6. Computation of cohomology and cohomological descent in the case of a hypercovering of an object given by a simplicial object of a site, see 09X8. This is a fundamental fact about how cohomology (of bounded below complexes) works in a site. Outline of the talk: (a) define the notion of a hypercovering, see 01ZF, (b) prove the vanishing in 01GE, (c) prove the isomorphism in 0D8G, and (d) deduce the results in section 09X8. Avoid any confusion and misunderstandings by working in a site having a final object X and all fibre products and working with a hypercovering of X given by a simplicial object of the site thereby avoiding semi-representable objects alltogether.
7. Proper hypercoverings compute cohomology
   1. Topological case 09XA,
   2. The case of schemes and algebraic spaces 0DHI.
   The proof of the main result here is a straightforward combination of lectures 5 and 6 but it takes some time to setup notation. A good thing here would be to give examples.
8. Glueing complexes (BBD glueing lemma, see the book by Beilinson, Berstein, Deligne, and Gabber, Theorem 3.2.4), see
   1. 0D65 for topological spaces, and
   2. 0DC8 on sites.
   You can directly do the proof as it is presented in BBD. Outline of alternative version of talk: (a) introduce simplicial systems in the derived category, see 0D9G, (b) prove such a cartesian simplicial system arises from an actual object of the simplicial derived category, see 0D9L, and (c) prove the actual glueing lemma 0DCB. If time permits you can discuss unbounded versions of this result where one has some bounded cohomological dimension assumptions as in Laszlo-Olsson, section 2.3. This is also discussed in 0DCC which itself relies on the discussion in 0DC1. Another unbounded glueing result that is useful is a slightly different version for complexes of modules with quasi-coherent cohomology sheaves, see 0DLA.
9. Apply the BBD glueing lemma to construct relative dualizing complexes for schemes locally of finite type over a base endowed with a dualizing complex, see 0AU5. Explain how the BBD glueing lemma helps in proving that Max Lieblich's stack of perfect complexes is a stack. Add more applications of BBD glueing here (for example discuss perverse sheaves and you can also discuss coherent perverse sheaves!).
10. Descent of vectorbundles on perfect schemes, see Bhatt-Scholze, Section 4. A related interesting paper is this paper, Section 4 by Paolo Cascini and Hiromu Tanaka. We could have two talks here: (1) explain the argument in Bhatt-Scholze, and (2) discuss the results of the other paper in the same context.
11. An algebraic space satisfies the sheaf property for the fpqc topology (Gabber), see 03W8. This is an good example of using descent methods as in EGA (but in an entirely new way).
12. Almost math. In Bhargav's Eilenberg lectures, he will need to use the descent of almost finite etale covers along faithfully flat maps. Gabber-Ramero's book gives a systematic discussion of this in section 3.4 of their book, see Gabber-Ramero. In fact, with the work they've done in the previous sections, the actual descent statement is rather straightforward (see the very shortsubsection 3.4.1). Since Gabber-Ramero is a bit dry otherwise, the following seems like it might make for a reasonable seminar talk: explain basics of almost mathematics including definitions of almost finite etale covers (say as in section 4 of Scholze's perfectoid spaces paper), cover Theorems 2.2 and 2.3 in Faltings' p-adic Hodge theory paper (here) giving a direct "hands on" proof of topological invariance of this notion, and then explain the descent statement in Gabber-Ramero 3.4.1. [Unfortunately, there's no concrete payoff about non-almost math.]
13. The quasisyntomic topology. Bhargav in his Eilenberg lectures will need at some point to use the quasisyntomic topology. It seems it could make a fun talk to do the following (replacing talk 18 below and possibly including 15 below as well): explain what the quasisyntomic site is (stick to characteristic p), explain what the A_{crys}(-) functor on this category is and why it's a sheaf, and prove that the cohomology of this sheaf on a smooth scheme is crystalline cohomology. The reference here is sections 4 and 8 of 1802.03261, but please just talk to me or Bhargav. [The fun thing about this talk is that it gives a way of defining crystalline cohomology without talking about the crystalline site and fancy topos theory.]
14. Rigid etale cohomology. At some very late point (probably the last lecture) in Bhargav's Eilenberg lectures, he'll need to use some form of rigid GAGA. The statement needed can be formulated in terms of arc topology business in paper by Bhatt and Matthew. See Corollary 6.18 of the paper. Someone proficient with rigid geometry could also then explain why this implies algebraic and analytic etale cohomology agree for proper schemes over nonarch fields.
15. The cotangent complex satisfies descent in flat topology, see Bhatt's paper, Remark 2.8. Optional 1: explain as an "application" why HP(R/F_p) is a 2-periodic version of de Rham cohomology of R/F_p (for a regular F_p-algebra R). Optional 2: explain why this means certain stacks have a cotangent complex.
16. Artin's theorem. Explain how one can replace flat maps by smooth or etale maps sometimes. See: Artin, Versal deformations and algebraic stacks, Invent. Math. 27, Section 6. This argument is the one for the proof of 06DC but we strongly suggest reading the much simpler argument in Artin. Application: [S/G] is an algebraic stack if G → S is flat and locally of finite presentation. Fun additional descent result: show that conversely if [S/G] is an algebraic stack, then G → S has to be flat and locally of finite presentation. Here one encounters a different kind of descent problem.
17. Algebraic de Rham cohomology is a ph-sheaf in characteristic 0 (Deligne's theorem). See also exposition by Ben Lee and by Huber-Jorder.
18. Syntomic descent for crystalline cohomology, and the following consequence on representing crystalline cohomology of smooth algebras by canonical complexes: if R is a smooth (or just regular) F_p-algebra, then applying the A_{crys}(-) functor to the Cech nerve of R → R_{perf} gives a canonical complex computing the crystalline cohomology of R relative to Z_p.
19. Given a smooth group scheme G over a base S we have H^1_{fppf}(S, G) = H^1_{etale}(S, G) if G is nonabelian where H^1 is the pointed set of isomorphism classes of torsors and we have H^i_{fppf}(S, G) = H^i_{etale}(S, G) for all i if G is abelian. See SGA ??. Extra credit: give an example of a torsor (for fpqc topology and some group scheme) which is not an fppf torsor.
20. Discuss fpqc descent of "being a locally projective module", see 05JF. Discuss the open problem of whether or not we have descent for "being locally free" for modules in the fppf topology, see 05VF. What about the V topology?
21. This paper by Amnon Neeman or perhaps rather this paper by Krause. Please find a write-up of the corresponding mathematics here.
22. Add more here.