Comparison between rigid syntomic and crystalline syntomic cohomology for strictly semistable log schemes with boundary

TL;DR

This paper introduces a new rigid syntomic cohomology theory for strictly semistable log schemes and compares it to existing crystalline syntomic cohomology, extending the Hyodo-Kato theory for broader classes of schemes.

Contribution

It develops a rigid syntomic cohomology framework for strictly semistable log schemes and generalizes Hyodo-Kato theory, enabling comparison with crystalline syntomic cohomology.

Findings

Established a comparison between rigid and crystalline syntomic cohomology.

Extended Hyodo-Kato theory to schemes with boundary.

Provided tools for cohomology calculations in mixed characteristic.

Abstract

We introduce rigid syntomic cohomology for strictly semistable log schemes over a complete discrete valuation ring of mixed characteristic (0,p). In case a good compactification exists, we compare this cohomology theory to Nekov\'a\v{r}-Nizio{\l}'s crystalline syntomic cohomology of the generic fibre. The main ingredients are a modification of Gro{\ss}e-Kl\"onne's rigid Hyodo-Kato theory and a generalisation of it for strictly semistable log schemes with boundary.