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

By B. Bhatt, M. Morrow, Peter Scholze. Publications mathématiques (Bures-sur-Yvette), 2018

264 citations recorded by the source corpus.

In mixed characteristic and in equal characteristic p$p$ we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic K$K$-theory by motivic cohomology. Its graded pieces are related in mixed characteristic to the complex AΩ$A\Omega$ constructed in our previous work, and in equal characteristic p$p$ to crystalline cohomology. Our construction of the filtration on THH$\mathrm{THH}$ is via flat descent to semiperfectoid rings.As one application, we refine the construction of the AΩ$A\Omega $-complex by giving a cohomological construction of Breuil–Kisin modules for proper smooth formal schemes over OK$\mathcal {O}_{K}$, where K$K$ is a discretely valued extension of Qp$\mathbf {Q}_{p}$ with perfect residue field. As another application, we define syntomic sheaves Zp(n)$\mathbf {Z}_{p}(n)$ for all n≥0$n\geq 0$ on a large class of Zp$\mathbf {Z}_{p}$-algebras, and identify them in terms of p$p$-adic nearby cycles in mixed characteristic, and in terms of logarithmic de Rham-Witt sheaves in equal characteristic p$p$.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/610c73a006e463ff8565466b423a91c8b17c47cc

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In Geometry and Topology

  1. Dustin Clausen
  2. A. Mathew
  3. M. Morrow
  4. B. Bhatt
  5. Peter Scholze
  6. Lennart Meier
  7. N. Axel Naumann
  8. J. Noel

In Mathematical Physics

  1. Dustin Clausen
  2. A. Mathew
  3. M. Morrow
  4. B. Bhatt
  5. Peter Scholze
  6. Lennart Meier
  7. N. Axel Naumann
  8. J. Noel

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