Cycle classes and the syntomic regulator

TL;DR

This paper establishes compatibility between de Rham and rigid fundamental classes for schemes over mixed characteristic rings, constructs a syntomic regulator map, and verifies axioms for syntomic cohomology as an absolute cohomology theory.

Contribution

It introduces a compatible syntomic regulator map for schemes over mixed characteristic rings and verifies foundational axioms for syntomic cohomology as an absolute theory.

Findings

Compatibility of de Rham and rigid fundamental classes

Construction of a syntomic regulator map

Verification of Bloch-Ogus axioms for syntomic cohomology

Abstract

Let $V=Spec(R)$ and $R$ be a complete discrete valuation ring of mixed characteristic $(0,p)$. For any flat $R$-scheme $X$ we prove the compatibility of the de Rham fundamental class of the generic fiber and the rigid fundamental class of the special fiber. We use this result to construct a syntomic regulator map $r:CH^i(X/V,2i-n)\to H^n_{syn}(X,i)$, when $X$ is smooth over $V$, with values on the syntomic cohomology defined by A. Besser. Motivated by the previous result we also prove some of the Bloch-Ogus axioms for the syntomic cohomology theory, but viewed as an absolute cohomology theory.