On $p$-adic absolute Hodge cohomology and syntomic coefficients, I

TL;DR

This paper interprets syntomic cohomology as a $p$-adic absolute Hodge cohomology, introduces syntomic coefficients, and explores their properties and applications in $p$-adic realizations of motives and fundamental groups.

Contribution

It provides a new interpretation of syntomic cohomology, constructs syntomic coefficients, and extends $p$-adic realizations of mixed motives and fundamental groups.

Findings

Syntomic cohomology is interpreted as $p$-adic absolute Hodge cohomology.

Syntomic descent spectral sequence is constructed and shown to degenerate.

Syntomic coefficients form a triangulated subcategory in the derived category of Galois representations.

Abstract

We interpret syntomic cohomology of Nekov\'a\v{r}-Nizio{\l} as a $p$-adic absolute Hodge cohomology. This is analogous to the interpretation of Deligne-Beilinson cohomology as an absolute Hodge cohomology by Beilinson and generalizes the results of Bannai and Chiarellotto, Ciccioni, Mazzari in the good reduction case, and of Yamada in the semistable reduction case. This interpretation yields a simple construction of the syntomic descent spectral sequence and its degeneration for proper and smooth varieties. We introduce syntomic coefficients and show that in dimension zero they form a full triangulated subcategory of the derived category of potentially semistable Galois representations. Along the way, we obtain $p$-adic realizations of mixed motives including $p$-adic comparison isomorphisms. We apply this to the motivic fundamental group generalizing results of Olsson and Vologodsky.