Ramification theory for reciprocity sheaves, III, Abbes-Saito formula

TL;DR

This paper introduces a new geometric approach to understanding the ramification filtration of reciprocity sheaves, leading to characteristic forms and applications in singularity theory and motivic cohomology.

Contribution

It provides a novel geometric characterization of the motivic ramification filtration for reciprocity sheaves, extending Abbes-Saito methods and defining characteristic forms.

Findings

New geometric characterization of ramification filtration

Definition of characteristic forms for reciprocity sheaves

Applications to pseudo-rational singularities and cohomology representability

Abstract

We give a new geometric characterization of the motivic ramification filtration of reciprocity sheaves, by imitating a method used by Abbes and (Takeshi) Saito to study the ramification of torsors under finite \'etale groups. This new characterization is used to define characteristic forms for reciprocity sheaves. We obtain applications on pseudo-rational singularities and on questions regarding the representability of certain cohomology groups of reciprocity sheaves in the triangulated category of motives with modulus introduced by Kahn-Miyazaki-Saito-Yamazaki.