Cycle classes and the syntomic regulator

Abstract

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

A Néron–Ogg–Shafarevich criterion for K3 surfaces

“…We must show that the isomorphism identifies with . Since , we can replace crystalline cohomology by rigid cohomology, and since rigid cohomology and algebraic de Rham cohomology both satisfy étale descent, by and [, Exposé VII; Proposition 4.3], respectively, the result follows by pulling back to an étale hypercover of by smooth ‐schemes and applying [, Corollary 1.5.1].…”

The filtered Ogus realisation of motives

“…Let j FOg : P 2 FOg (X)(1) → End FOg (H 1 FOg (A)) 4 The compatiblity of the crystalline and de Rham cycle class has been generalized to the rigid setting in [11,13].…”

The Rigid Syntomic Ring Spectrum

“…is conditional to the conjecture. The latter result concerns the regulator of a proper and smooth surface S over R. We also note that Point (4) has already been used (in the projective morphism case, although stated for proper maps) in [Lan11, p. 505] but the references given there is a draft of [CCM12] which turns to be different from the published version and does not contain the above statement, neither its proof.…”

“…its Berkovich or adic points). This is enough as explained in [CCM12,§ 3] or [Tam11,§ 3]. From now on we will simply write Gdm instead of Gdm P t(X) with X as above.…”

“…There is a regulator map for this theory and one can also interpret rigid syntomic cohomology as an absolute cohomology [Ban02,CCM12].…”