The rigid syntomic ring spectrum

Abstract

The aim of this paper is to show that Besser syntomic cohomology 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 syntomic cohomology such as h-descent and the compatibility of cycle classes with Gysin morphisms. Along the way, we prove that motivic ring spectra induces a complete Bloch-Ogus cohomological formalism and even more. Finally, following a general motivic homotopical philosophy, we exhibit a natural notion of syntomic coefficients.

Final version to appear in the Journal de l'institut des Mathématiques de Jussieu. Many typos have been corrected and the exposition has been improved according to the suggestions of the referees: we thank them a lot!

References in corpus (2)

• Orientation theory in arithmetic geometry
• Karoubi's relative Chern character, the rigid syntomic regulator, and the Bloch-Kato exponential map

Cited by in corpus (9)

• Syntomic cohomology and regulators for varieties over p-adic fields
• Rankin--Eisenstein classes for modular forms
• The Monsky-Washnitzer and the overconvergent realizations
• Finite polynomial cohomology for general varieties
• Bivariant theories in motivic stable homotopy
• The Beilinson regulator is a map of ring spectra
• Karoubi's relative Chern character, the rigid syntomic regulator, and the Bloch-Kato exponential map
• Overconvergent global analytic geometry
• The filtered Ogus realisation of motives