23 pages, improved exposition. Accepted for publication in Algebra and Number TheoryInternational audienceLet $V=Spec(R)$ 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

The aim of this paper is to show that rigid syntomic cohomology -defined by Besser -is representable by a rational ring spectrum in the motivic homotopical sense. In fact, extending previous constructions, we exhibit a simple representability criterion and we apply it to several cohomologies in order to get our central result. This theorem gives new results for rigid syntomic cohomology such as h-descent and the compatibility of cycle classes with Gysin morphisms. Along the way, we prove that motivic ring spectra induce a complete Bloch-Ogus cohomological formalism and even more. Finally, following a general motivic homotopical philosophy, we exhibit a natural notion of rigid syntomic coefficients. MSC: 14F42; 14F30.

Abstract. According to a well-known theorem of Serre and Tate, the in nitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the in nitesimal deformation theory of its Barsotti-Tate group. We extend this result to -motives.

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