Comparison Isomorphisms for Smooth Formal Schemes

TL;DR

This paper establishes a comparison isomorphism linking etale cohomology of the generic fiber with crystalline cohomology of the special fiber for smooth proper schemes over unramified DVRs, bridging two cohomological theories.

Contribution

It proves a new comparison isomorphism connecting etale and crystalline cohomology in the setting of smooth formal schemes over unramified DVRs.

Findings

Established a comparison isomorphism for smooth proper schemes over unramified DVRs.

Bridged etale cohomology of the generic fiber with crystalline cohomology of the special fiber.

Extended the understanding of cohomological relations in mixed characteristic settings.

Abstract

For a smooth proper scheme or formal scheme over an unramified, complete DVR of mixed characteristics we prove a comparison isomorphism relating etale cohomology of the generic fiber with values in a crystalline etale sheaf to the crystalline cohomology of its special fiber with values in the associated F-isocrystal.