Timeline for Would full resolution of singularities have cohomological implications beyond the alteration theory?
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| Aug 5, 2020 at 18:38 | comment | added | Mikhail Bondarko | I wonder whether Gabber-type $l'$-alterations for all(!) primes $l$ would be sufficient for all cohomological applications. | |
| Aug 3, 2020 at 7:33 | comment | added | D.-C. Cisinski | In the setting of motivic sheaves à la Voevodsky, in positive characteristic $p$, the technology of de Jong alteration only gives results for cohomology modulo $p$-torsion. Full resolution of singularities would allow genuine integral coefficients. This has consequences for our understanding of algebraic cycles in positive characteristics, for instance. We could also hope to get a better understanding of motivic sheaves of arithmetic schemes with integral coefficients (letting $p$ vary a little bit). | |
| Aug 2, 2020 at 23:32 | comment | added | Jason Starr | Yes it would. You might look at the papers of Esnault et al. on congruences for numbers of rational points over finite fields of specializations of smooth projective varieties with a (rational) decomposition of the diagonal. Most of these theorems assume that the flat, proper model over the Witt vectors is regular, precisely because we do not know resolution. Esnault does have results using only alterations, but this only gives existence of a rational point, not a congruence for the number of rational points. | |
| Aug 2, 2020 at 21:04 | history | asked | user158636 | CC BY-SA 4.0 |