Cycle classes and the syntomic regulator

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 $r:C{H}^i(X/V,2i-n)\to H_{syn}^n(X,i)$, 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.

23 pages, improved exposition

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• The filtered Ogus realisation of motives
• Hyodo-Kato theory with syntomic coefficients