Publications mathématiques de l'IHÉS

, Volume 129, Issue 1, pp 199–310 | Cite as

Topological Hochschild homology and integral \(p\)-adic Hodge theory

  • Bhargav BhattEmail author
  • Matthew Morrow
  • Peter Scholze


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 \(p\).


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Copyright information

© IHES and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of MichiganAnn ArborUSA
  2. 2.CNRS and IMJ-PRGSorbonne UniversityParisFrance
  3. 3.Mathematisches InstitutUniversität BonnBonnGermany

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