On Kato's ramification filtration

TL;DR

This thesis provides a cohomological description of Kato's ramification filtrations for fields of characteristic $p>0$ by developing a robust theory of filtered de Rham–Witt and log de Rham–Witt complexes. It constructs two-term complexes $W_m{\mathcal F}^{q,\bullet}_{D}$ that realize ramification filtrations both locally (Kato filtrations) and globally (filtration with modulus) and proves that these filtrations correspond to explicit cohomological objects. The work establishes refined dualities for Hodge–Witt cohomology with modulus over finite fields and extends to curves over local fields, together with Lefschetz-type results for unramified and ramified Brauer groups. These results bridge ramification theory, $p$-adic étale cohomology, and Brauer group phenomena in positive characteristic, yielding new tools for higher-dimensional class field theory and questions involving divisors with modulus. The constructions also pave the way for future applications in log-crystalline contexts and in the broader program of ramification theory in characteristic $p$.

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].

Theorems & Definitions (312)

• Theorem 1
• Definition 2
• Corollary 3
• Theorem 4
• Theorem 5
• Proposition 6
• Definition 7
• Theorem 8
• Lemma 9
• Theorem 10