Kato's residue homomorphisms and reciprocity laws on arithmetic surfaces

TL;DR

This paper investigates Kato's residue homomorphisms in Milnor K-theory, establishing reciprocity laws on arithmetic surfaces and linking these concepts to Contou-Carrère symbols.

Contribution

It provides explicit analysis of Kato's residue homomorphisms and introduces new reciprocity laws for K-groups on arithmetic surfaces.

Findings

Established several reciprocity laws for K-groups on arithmetic surfaces

Connected Kato's residue homomorphisms to Contou-Carrère symbols

Enhanced understanding of local maps in Milnor K-theory

Abstract

We explicitly study Kato's residue homomorphisms in Milnor $K$-theory, which are closely related to Contou-Carr\`ere symbols. As applications we establish several reciprocity laws for certain locally defined maps on $K$-groups that are associated to arithmetic surfaces.