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).

我们证明了各种有理的 $ p $ -adic周期射态是相等的，以实现平稳，不一定正确的方案。我们首先表明，我们先前发现的用于适当平滑方案的 $ K $- 理论唯一性准则在良好约简情况下扩展到了适当的有限简单方案，在半稳定约简情况下扩展到了具有紧致支持的同调性。它产生了同态的周期态射影相等，并使用了语法的，几乎是虚幻的和动机的结构来定义紧凑的支持。我们继续显示 $ h $ 只要定义了后同态（直到Hyodo-Kato同态的变化），-同调周期态就与语法的和几乎成故事的周期态态相吻合。我们通过在一个字段上将符号学上近乎童话时代的态态提升到品种的 $ h $- 位点上，在这里可以使用BeilinsonPoincaré引理直接检验它们与 $ h $-同 调态态态的相等性。尺寸 $ 0 $ 。这也表明，有序和近乎童话时代的态射态自然地扩展到了Voevodsky三角动机类别，并具有许多有用的特性（因为 $ h $-同 调子时期态射态也是如此）。