The rigid syntomic ring spectrum

TL;DR

This paper demonstrates that Besser syntomic cohomology can be represented by a rational ring spectrum within the motivic homotopy framework, leading to new insights and formal properties for syntomic cohomology.

Contribution

It establishes the representability of Besser syntomic cohomology by a rational ring spectrum and introduces a criterion for such representability, expanding the understanding of syntomic cohomology in motivic homotopy theory.

Findings

Besser syntomic cohomology is representable by a rational ring spectrum.

The paper proves h-descent and compatibility of cycle classes with Gysin morphisms.

Motivic ring spectra induce a complete Bloch-Ogus cohomological formalism.

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.