We prove equality of the various rational $p$-adic period morphisms for smooth, not necessarily proper, schemes. We start with showing that the $K$-theoretical uniqueness criterion we had found earlier 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 étale, and motivic constructions. We continue with showing that the $h$-cohomology period morphism agrees with the syntomic and almost étale period morphisms whenever the latter morphisms are defined (and up to a change of Hyodo–Kato cohomology). We do it by lifting the syntomic and almost étale period morphisms to the $h$-site of varieties over a field, where their equality with the $h$-cohomology period morphism can be checked directly using the Beilinson Poincaré lemma and the case of dimension $0$. This also shows that the syntomic and almost étale period morphisms have a natural extension to the Voevodsky triangulated category of motives and enjoy many useful properties (since so does the $h$-cohomology period morphism).
Towards non–abelian 𝑝–adic Hodge theory in the good reduction case