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

TL;DR

The paper constructs a Div$_E(X)$-filtered de Rham–Witt filtration on $W_m\Omega^\bullet_{X\setminus E}$ that extends Brylinski’s filtration on $W_m(K)$ and describes Kato’s ramification filtration on $H^q_{\et}(X\setminus E, \mathbb{Q}_p/\mathbb{Z}_p(q-1))$ via this filtration. It then develops global and local Cartier theory, establishes the goodness of the filtration, and uses these tools to prove refined dualities (Ekedahl, Jannsen–Saito–Zhao, Zhao) with modulus, Lefschetz-type results for ramification filtrations and Brauer groups, and a generalized refined Swan conductor for schemes. The work extends Kerz–Saito to higher cohomology, provides a modulus-compatible framework for ramification and duality, and underpins class field theory with modulus in positive characteristic. Overall, the filtration yields a powerful, unifying approach to ramification, duality, and Swan conductors in the setting of log- and modulus-enabled de Rham–Witt theory.

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 $W_mΩ^\bullet_{X\setminus E}$. 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.

Theorems & Definitions (292)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Corollary 1.9
• Remark 1.10