Kato's Ramification filtration via de Rham-Witt complex and applications

TL;DR

This paper introduces a new filtration on the de Rham-Witt complex for schemes with normal crossing divisors, linking it to Kato's ramification filtration and applying it to refine duality theorems and Lefschetz results in positive characteristic.

Contribution

It defines a filtration on the de Rham-Witt complex that explicitly describes Kato's ramification filtration, extending classical results and applying to duality and Lefschetz theorems in positive characteristic.

Findings

Explicit description of Kato's ramification filtration in terms of de Rham-Witt complex

Refinements of duality theorems for schemes over finite fields and henselian DVRs

Lefschetz theorems for ramification filtrations and Brauer groups

Abstract

Given an $F$-finite regular scheme $X$ of positive characteristic and a simple normal crossing divisor $E$ on $X$, we introduce a filtration on the de Rham-Witt complex $W_m\Omega^\bullet_{X\setminus E}$. When $X$ is the spectrum of a henselian discrete valuation ring $A$ with quotient field $K$, this extends the classical filtration on $W_m(K)$ due to Brylinski. We show that Kato's ramification filtration on $H^q_\et(X \setminus E, {\Q}/{\Z}(q-1))$ for $q \ge 1$ admits an explicit description in terms of the above filtration of the de Rham-Witt complex of $X \setminus E$. When $q =1$, this specializes to the results of Kato and Kerz-Saito. As applications, we prove refinements of the duality theorem of Jannsen-Saito-Zhao for smooth projective schemes over finite fields and the duality theorem of Zhao for semi-stable schemes over henselian discrete valuation rings of positive characteristic with finiteresidue fields. We also prove a modulus version of the duality theorem of Ekedahl. As another application, we prove Lefschetz theorems for Kato's ramification filtrations for smooth projective varieties over $F$-finite fields. This extends a result of Kerz-Saito for $H^1$ to higher cohomology. Similar results are proven for the Brauer group.