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Syntomic regulators and $$p$$-adic integration. I: Rigid syntomic regulators. (English) Zbl 1001.19003
The author constructs a new version of syntomic cohomology defined for smooth schemes $$X$$ of finite type over the ring of integers $$\mathcal V$$ of a $$p$$-adic field extending previous definitions of M. Gros [Invent. Math. 115, No. 1, 61-79 (1994; Zbl 0799.14010)] and W. Nizioł [Invent. Math. 127, No. 2, 375-400 (1997; Zbl 0928.14014)]. The definition of the syntomic cohomology groups $$H^i_{\text{syn}}(X,n)$$ is more refined than the previous ones – it allows, e.g., log-singularities –, and the cohomology groups turn out to be finite. In addition the author constructs syntomic Chern classes $c_j^p:K_p(X) \rightarrow H^{2j-p}_{\text{syn}}(X,j)$ and Chern characters based on the general framework provided in A. Huber’s “Mixed motives and their realization in derived categories” [Lect. Notes Math. 1604 (1995; Zbl 0938.14008)]. See also the second part of thie paper [A. Besser, Isr. J. Math. 120, Pt. B, 335-359 (2000; Zbl 1001.19004)].

##### MSC:
 19E20 Relations of $$K$$-theory with cohomology theories 14F30 $$p$$-adic cohomology, crystalline cohomology
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