Cycle classes and the syntomic regulator

Abstract

Let $\mathcal{V}=Spec\left(R\right)$ and $R$ be a complete discrete valuation ring of mixed characteristic $\left(0,p\right)$. For any flat $R$-scheme $\mathcal{X}$, 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 ${reg}_{syn}:{CH}^i\left(\mathcal{X}∕\mathcal{V},2i-n\right)\to H_{syn}^n\left(\mathcal{X},i\right)$ when $\mathcal{X}$ is smooth over $R$ with values in 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.