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

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 $n\ge 0$ 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 $p$.

arXiv comment88 pages, minor updates, final version

References in corpus (9)

• Higher Topos Theory
• The pro-étale topology for schemes
• p-adic derived de Rham cohomology
• Completions and derived de Rham cohomology
• Enhancing the filtered derived category
• The de Rham-Witt complex and p-adic vanishing cycles
• K-theory and topological cyclic homology of henselian pairs
• Frobenius gauges and a new theory of p-torsion sheaves in characteristic p. I
• $p$-adic vanishing cycles as Frobenius-fixed points

Cited by in corpus (12)

• Kaledin's degeneration theorem and topological Hochschild homology
• A Universal HKR Theorem
• $K$-theory of valuation rings
• $p$-adic vanishing cycles as Frobenius-fixed points
• Integral p-adic Hodge theory of formal schemes in low ramification
• On Grothendieck's standard conjectures of type C and D in positive characteristic
• Dieudonné Theory via Cohomology of Classifying Stacks
• Complex orientations for THH of some perfectoid fields
• On a theorem of Scholze-Weinstein
• On the Dundas-Goodwillie-McCarthy theorem
• Syntomic complexes of F-crystals and Tamagawa number conjecture in characteristic p, with Appendix by Arthur Ogus
• Derived $F$-zips