Topological Hochschild homology and integral $p$-adic Hodge theory

TL;DR

This paper introduces a filtration on topological Hochschild homology (THH) in mixed and equal characteristic settings, relating it to motivic cohomology, crystalline cohomology, and $A\, extOmega$-complexes, with applications to Breuil--Kisin modules and syntomic sheaves.

Contribution

It constructs a new filtration on THH that generalizes motivic cohomology filtrations and connects to various cohomological theories, providing new tools for $p$-adic Hodge theory.

Findings

Established a filtration on THH related to motivic cohomology.

Connected the graded pieces of the filtration to $A\, extOmega$ and crystalline cohomology.

Defined syntomic sheaves and identified them with $p$-adic nearby cycles and logarithmic de Rham-Witt sheaves.

Abstract

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 $\mathbb Q_p$ with perfect residue field. As another application, we define syntomic sheaves $\mathbb Z_p(n)$ for all $n\geq 0$ on a large class of $\mathbb Z_p$-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 $p$.