# The de Rham-Witt complex and p-adic vanishing cycles

@article{Geisser2003TheDR, title={The de Rham-Witt complex and p-adic vanishing cycles}, author={Thomas H. Geisser and Lars Hesselholt}, journal={Journal of the American Mathematical Society}, year={2003}, volume={19}, pages={1-36} }

We determine the structure modulo p of the de Rham-Witt complex of a smooth scheme X over a discrete valuation ring of mixed characteristic with log-poles along the special fiber Y and show that the sub-sheaf fixed by the Frobenius is isomorphic to the sheaf of p-adic vanishing cycles. We use this together with results of the second author and Madsen to evaluate the $K$-theory with finite coefficients of the quotient field K of the henselian local ring of X at a generic point of Y. The result…

## 31 Citations

Duality for Relative Logarithmic de Rham-Witt Sheaves on Semistable Schemes over F q [ [ t ] ]

- Mathematics
- 2019

We study duality theorems for the relative logarithmic de Rham-Witt sheaves on semi-stable schemes X over a local ring Fq[[t]], where Fq is a finite field. As an application, we obtain a new…

Integral p-adic Hodge theory

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- 2016

We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of C_p. It takes values in a mixed-characteristic analogue of Dieudonne modules, which was previously…

The absolute de Rham-Witt complex

- Mathematics
- 2005

This note is a brief survey of the absolute de Rham-Witt complex. We explain the structure of this complex for a smooth scheme over a complete discrete valuation ring of mixed characteristic (0, p)…

Duality for relative logarithmic de Rham–Witt sheaves and wildly ramified class field theory over finite fields

- MathematicsCompositio Mathematica
- 2018

In order to study $p$ -adic étale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect field of characteristic $p>0$ , we introduce new $p$ -primary torsion sheaves. It…

p-adic deformation of algebraic cycle classes

- Mathematics
- 2012

We study the p-adic deformation properties of algebraic cycle classes modulo rational equivalence. We show that the crystalline Chern character of a vector bundle on the closed fibre lies in a…

Integral p$p$-adic Hodge theory

- MathematicsPublications mathématiques de l'IHÉS
- 2018

We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of Cp$\mathbf {C}_{p}$. It takes values in a mixed-characteristic analogue of Dieudonné modules,…

$p$-adic vanishing cycles as Frobenius-fixed points

- Mathematics
- 2018

Given a smooth formal scheme over the ring of integers of a mixed-characteristic perfectoid field, we study its $p$-adic vanishing cycles via de Rham--Witt and $q$-de Rham complexes.

The big de Rham–Witt complex

- Mathematics
- 2010

This paper gives a new and direct construction of the multi-prime big de Rham–Witt complex, which is defined for every commutative and unital ring; the original construction by Madsen and myself…

On K(1)-local TR

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- 2021

We discuss some general properties of TR and its K(1)-localization. We prove that after K(1)-localization, TR of HZ-algebras is a truncating invariant in the Land–Tamme sense, and deduce h-descent…

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