De Jong's result on alterations allows one to show the potential semistability of certain Galois representations arising from cohomology of varieties (among other things). If we knew the existence of resolution of singularities in positive characteristic (and whatever is expected in mixed characteristic) would this have consequences for the study of cohomology of varieties beyond what can be extracted from de Jong?
Would full resolution of singularities have cohomological implications beyond the alteration theory?
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5$\begingroup$ 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. $\endgroup$– Jason StarrCommented Aug 2, 2020 at 23:32
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5$\begingroup$ 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). $\endgroup$– D.-C. CisinskiCommented Aug 3, 2020 at 7:33
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$\begingroup$ I wonder whether Gabber-type $l'$-alterations for all(!) primes $l$ would be sufficient for all cohomological applications. $\endgroup$– Mikhail BondarkoCommented Aug 5, 2020 at 18:38
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