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 BreuilKisin 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~RhamWitt sheaves in equal characteristic $p$.
 Publication:

arXiv eprints
 Pub Date:
 February 2018
 arXiv:
 arXiv:1802.03261
 Bibcode:
 2018arXiv180203261B
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  KTheory and Homology;
 Mathematics  Number Theory;
 13D03;
 19D55;
 19D50;
 19F27;
 14F30;
 14F20;
 19E20
 EPrint:
 88 pages, minor updates, final version