On uniqueness of $p$-adic period morphisms, II

TL;DR

This paper proves the equality of various $p$-adic period morphisms for smooth schemes, extending previous results to broader cases and unifying different constructions through motive theory.

Contribution

It extends the $K$-theoretical uniqueness criterion to broader classes of schemes and shows the equivalence of multiple $p$-adic period morphisms using motive theory.

Findings

Equality of $p$-adic period morphisms for smooth schemes established.

Extension of uniqueness criteria to finite simplicial schemes and cohomology with compact support.

Unification of different period morphisms via motives and Poincaré Lemma.

Abstract

We prove equality of the various $p$-adic period morphisms for smooth, not necessarily proper, schemes. We start with showing that the $K$-theoretical uniqueness criterium we had found for proper smooth schemes extends to proper finite simplicial schemes in the good reduction case and to cohomology with compact support in the semistable reduction case. It yields the equality of the period morphisms for cohomology with compact support defined using the syntomic, almost \'etale, and motivic constructions. We continue with showing that the $h$-cohomology period morphism agrees with the syntomic and almost \'etale period morphisms whenever the latter morphisms are defined. We do it by lifting the syntomic and almost \'etale period morphisms to Voyevodsky triangulated category of motives, where their equality with the $h$-cohomology period morphism can be checked directly using the Beilinson Poincar\'e Lemma.