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，我们定义了拓扑Hochschild同调及其变体的过滤。这种过滤类似于通过动机上同调对代数\(K\)理论进行过滤。它的分级片段在混合特征上与我们之前工作中构建的复数\(A\Omega\)相关，在平等特征\(p\)上与晶体上同调相关。我们对\(\mathrm{THH}\)的过滤构造是通过平坦下降到半完美环。

作为一种应用，我们通过给出 Breuil-Kisin 模的上同调构造来完善\(A\Omega \)复合体的构造，以在\(\mathcal {O}_{K}\)上实现适当的平滑形式方案，其中\(K\)是具有完美残差场的\(\mathbf {Q}_{p}\)的离散值扩展。作为另一个应用，我们为一大类\ (\mathbf {Z}_{p}上的所有\(n\geq 0\)定义语法滑轮 \ (\mathbf {Z}_{p} (n)\) \) -代数，并根据混合特征中的\(p\) -adic 附近循环以及相等特征\(p\)中的对数 de Rham-Witt 滑轮来识别它们。