On Kato's ramification filtration

TL;DR

This paper provides a cohomological framework for Kato's ramification filtrations on Galois cohomology groups over fields of positive characteristic, using de Rham-Witt sheaves, with applications to duality and Brauer groups.

Contribution

It introduces a cohomological description of Kato's ramification filtrations via de Rham-Witt complexes and proves new duality and Lefschetz theorems in positive characteristic settings.

Findings

Refined duality for finite fields

Duality for smooth projective curves over local fields

Lefschetz theorem for Brauer groups

Abstract

For a Henselian discrete valued field $K$ of characteristic $p>0$, Kato defined a ramification filtration $\{{\rm fil}_nH^q(K,\mathbb Q_p/\mathbb Z_p(q-1))\}_{n \ge 0}$ on $H^q(K,\mathbb Q_p/\mathbb Z_p(q-1))$. One can also define a ramification filtration on $H^q(U,\mathbb Z/p^m(q-1))$ using the local Kato-filtration, where $U$ is the complement of a simple normal crossing divisor in a regular scheme $X$ of characteristic $p>0$. The main objective of this thesis is to provide a cohomological description of these filtrations using de Rham-Witt sheaves and present several applications. To achieve our goal, we study a theory of the filtered de Rham-Witt complex of $F$-finite regular schemes of characteristic $p>0$ and prove several properties which are well known for the classical de Rham-Witt complex of regular schemes. As applications, we prove a refined version of Jannsen-Saito-Zhao's duality over finite fields, and a similar duality for smooth projective curves over local fields. As another application, we prove a Lefschetz theorem for unramified and ramified Brauer group (with modulus) of smooth projective $F$-finite schemes over a field of characteristic $p>0$. Further applications are given in [49] and [50].