Edged Crystalline Cohomology

This proposed project aims at opening new horizons in Grothendieck and Berthelot's theories of crystalline and rigid cohomology. These are p-adic cohomology theories that are used to study algebraic varieties in positive characteristic. In the last years, the subject has seen an incredible development.

Recent important achievements have been, for example, Kedlaya's new proof of the Riemann Hypothesis in positive characteristic and Abe's construction of a p-adic Langlands correspondence for overconvergent F-isocrystals. On the other hand, there are still some fundamental open questions. The main weakness of the theory of rigid cohomology is the difficulty of performing classical geometric operations.

For example, it is not known whether the direct image functors have all the desirable propreties (Berthelot's conjecture).

This is mainly due to the fact that the definitions rely on differential forms, which need smoothness assumptions to be defined.

The Applicant D'Addezio wants to use the edged crystalline site, a new site that he has recently constructed, to solve this issue.

In particular, he wants to show that the edged crystalline site gives an alternative new definition of rigid cohomology and overconvergent isocrystals and then use this to prove Berthelot's conjecture.

For this second step, he will exploit the fact that the definition of the edged crystalline site is completely algebraic.

Other applications that will be developped include the construction of an integral structure for rigid cohomology and the construction of the category of F-isocrystals with log-decay for smooth varieties of arbitrary dimension (extending the results of Kramer--Miller).