In mixed characteristic and in equal characteristic \(p\) we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic \(K\)-theory by motivic cohomology. Its graded pieces are related in mixed characteristic to the complex \(A\Omega\) constructed in our previous work, and in equal characteristic \(p\) to crystalline cohomology. Our construction of the filtration on \(\mathrm{THH}\) is via flat descent to semiperfectoid rings.

As one application, we refine the construction of the \(A\Omega \)-complex by giving a cohomological construction of Breuil–Kisin modules for proper smooth formal schemes over \(\mathcal {O}_{K}\), where \(K\) is a discretely valued extension of \(\mathbf {Q}_{p}\) with perfect residue field. As another application, we define syntomic sheaves \(\mathbf {Z}_{p}(n)\) for all \(n\geq 0\) on a large class of \(\mathbf {Z}_{p}\)-algebras, and identify them in terms of \(p\)-adic nearby cycles in mixed characteristic, and in terms of logarithmic de Rham-Witt sheaves in equal characteristic \(p\).

在混合特征和相等特征\（p \）中，我们定义了对拓扑Hochschild同源性及其变体的过滤。该过滤是通过动机同调对代数\（K \）-理论进行过滤的类似物。其分级片混合特性相关的复杂\（A \欧米茄\）在我们以前的工作构造，并且以相等的特性\（P \）到结晶同调。我们在\（\ mathrm {THH} \）上进行过滤的构造是通过平坦下降到半完美环。

作为一个应用，我们细化的结构的\（A \欧米茄\）通过给出的布勒伊切尔Kisin模块同调结构进行适当的平滑正规计划-配合物在\（\ mathcal {ö} _ {K} \） ，其中\（K \）是\（\ mathbf {Q} _ {p} \）的离散值扩展，具有完善的残差字段。作为另一种应用中，我们定义syntomic滑轮\（\ mathbf {Z} _ {P}（N）\）对所有\（N \ GEQ 0 \）上的一大类\（\ mathbf {Z} _ {P} \）-代数，并根据混合特征中的\（p \）- adic附近周期和等价特征\（p \）中的对数de Rham-Witt滑轮来识别它们。