Integral $p$-adic Hodge theory - announcement

TL;DR

This paper introduces a new cohomology theory bridging crystalline and étale cohomology for proper smooth schemes over p-adic integers, establishing torsion-freeness transfer under certain conditions.

Contribution

It constructs a novel cohomology framework that interpolates between crystalline and étale cohomology, enabling new torsion-freeness results in p-adic Hodge theory.

Findings

Proves torsion-freeness transfer from crystalline to étale cohomology.

Sketches the construction of a new interpolating cohomology theory.

Prepares for a full comparison isomorphism in forthcoming work.

Abstract

Given a proper, smooth (formal) scheme over the ring of integers of $\mathbb C_p$, we prove that if the crystalline cohomology of its special fibre is torsion-free then the $p$-adic \'etale cohomology of its generic fibre is also torsion-free. In this announcement we sketch the proof, which relies on the construction of a new cohomology theory interpolating between crystalline and \'etale cohomology. Further details and results, including a comparison isomorphism, will be presented in the full forthcoming article.