Timeline for Finiteness of etale cohomology for arithmetic schemes
Current License: CC BY-SA 3.0
21 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Nov 4, 2017 at 14:14 | history | bounty ended | CommunityBot | ||
| S Nov 4, 2017 at 14:14 | history | notice removed | CommunityBot | ||
| Nov 2, 2017 at 13:58 | history | edited | Daniel Loughran | CC BY-SA 3.0 |
deleted 9 characters in body
|
| Oct 30, 2017 at 15:21 | comment | added | Daniel Loughran | @MartinBright: Thanks for the references! I have had a look at the paper of Sato, but it is written in the language of derived categories which are not my forte.... Could you please be more specific which result is most relevant? He seems to have many different purity results in his paper. | |
| Oct 30, 2017 at 14:52 | comment | added | Martin Bright | A couple of references: the purity result of Gabber referred to is here arxiv.org/abs/1207.3648 Exposé XVI, and indeed assumes the torsion order is invertible. For p-torsion there is some kind of purity theorem in this article of Sato arxiv.org/abs/math/0610426 but I don't know whether it is any use to you. | |
| Oct 30, 2017 at 13:58 | history | edited | Daniel Loughran | CC BY-SA 3.0 |
added 4 characters in body
|
| Oct 27, 2017 at 12:57 | history | edited | Daniel Loughran | CC BY-SA 3.0 |
added 3 characters in body
|
| S Oct 27, 2017 at 12:57 | history | bounty started | Daniel Loughran | ||
| S Oct 27, 2017 at 12:57 | history | notice added | Daniel Loughran | Authoritative reference needed | |
| Oct 25, 2017 at 15:46 | comment | added | nfdc23 | @JasonStarr: as you know, there's always an excision sequence relating the cohomology of a scheme to that of an open subscheme with the extra term being cohomology with supports along the closed complement (rather than cohomology on that complement), so the Artin-Schreier stuff doesn't directly intervene (but there's likely no purity theorem for this situation with torsion-orders not invertible; with reasonable torsion-orders I think Gabber has proved purity theorems for abstract regular schemes). | |
| Oct 25, 2017 at 15:22 | comment | added | Jason Starr | I have one more comment. For every flat, regular, affine $\mathbb{Z}$-scheme $X$ whose reduction $X_p$ over $\mathbb{Z}/p\mathbb{Z}$ is regular of dimension $\geq 1$, the cohomology $H^1_{\text{et}}(X_p,\mathbb{Z}/p\mathbb{Z})$ is infinite, because of the many different Artin-Schreier covers. If there were a long exact sequence, presumably this would imply that $H^2_{\text{et}}(X,\mathbb{Z}/p\mathbb{Z})$ is also infinite. That might be evidence that there is no long exact sequence, or it might be evidence that the cohomology is infinite. | |
| Oct 25, 2017 at 15:12 | comment | added | Jason Starr | The first paper of Esnault that I know including such a Gysin sequence (the one with the joint appendix with Deligne) is "Deligne's Integrality Theorem in Unequal Characteristic and Rational Points over Finite Fields", Ann. of Math. 164 (2006), pp. 715--730. A second paper, where she uses alterations, is "Coniveau over $\mathfrak{p}$-adic Fields and Points over Finite Fields", C. R. Acad. Sci. Paris, Ser. I 345 (2007), pp. 73--76. | |
| Oct 25, 2017 at 14:24 | comment | added | Daniel Loughran | @Jason Starr: Which paper of Esnault are you referring to? | |
| Oct 25, 2017 at 13:45 | comment | added | Jason Starr | Esnault's uses alterations (in the strong form of a $G$-Galois cover that has a regular modification) and the map from the cohomology of the original scheme to the $G$-invariants of the cohomology of the $G$-cover to deal with the regularity properties. You are completely correct that she needs to invert several integers. | |
| Oct 25, 2017 at 13:19 | comment | added | nfdc23 | @JasonStarr: The purity theorems (such as in Milne's notes that you mention) require the closed complement to satisfy regularity properties (which $X \bmod p$ may not) and more importantly assume torsion-orders for the sheaf are invertible on the scheme being considered (and avoiding the latter assumption is the main thrust of the question here). | |
| Oct 25, 2017 at 13:16 | comment | added | nfdc23 | @JasonStarr: In the papers of Deligne and/or Esnault surely the relevant torsion sheaves have torsion orders invertible on that base (and section 6 of Weil II assumes properness). With "invertible" torsion-orders one can use Deligne's "generic base change" expose from SGA 4.5 to see that the higher direct images on ${\rm{Spec}}(\mathbf{Z})$ are constructible, and it is a general theorem (perhaps due to some combination of Artin, Mazur, Zink?) that the cohomology of any constructible abelian sheaf (no hypothesis on torsion-orders) on the ring of $S$-integers of a number field is finite. | |
| Oct 25, 2017 at 12:18 | history | edited | Ariyan Javanpeykar |
added two tags
|
|
| Oct 25, 2017 at 12:18 | comment | added | Jason Starr | . . . Esnault cites Equation 6 in Section 3.6, p. 389 of "La Conjecture de Weil, II" by Pierre Deligne. Deligne considers a finite type flat scheme over a DVR, but he seems to assume that it has equicharacteristic. However, in the papers of Esnault (including the paper with a joint appendix with Deligne), the exact sequence is applied when the DVR has mixed characteristic . . . | |
| Oct 25, 2017 at 12:06 | comment | added | Jason Starr | The version in Milne is Theorem 16.1 of "Lectures on Etale Cohomology", but he only formulates there the theorem for finite type schemes over a field. The purity theorem over a Dedekind domain base has been used by Esnault in her work on congruences for rational points . . . | |
| Oct 25, 2017 at 12:02 | comment | added | Jason Starr | There is a purity theorem that is really a Gysin sequence relating etale cohomology of $U$, etale cohomology of $X$, and etale cohomology with supports in the closed complement of $U$ (which then gets turned into etale cohomology of the complement via purity). There is a reference in Milne. I will write it in a moment. | |
| Oct 25, 2017 at 11:51 | history | asked | Daniel Loughran | CC BY-SA 3.0 |