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

Theorems & Definitions (93)

• Remark 1.2: Relative cycles
• Definition 1.3
• Remark 1.4
• Theorem 1.6: Cycle class compatibility
• proof
• Corollary 1.7
• proof
• Definition 1.8
• Proposition 1.9: Syntomic cycle class
• proof