An explicit approach to residues on and dualizing sheaves of arithmetic surfaces

TL;DR

This paper develops a residue theory for arithmetic surfaces, proving reciprocity laws and explicitly constructing dualizing sheaves, extending known results from surfaces over perfect fields to more general arithmetic contexts.

Contribution

It introduces a residue framework for arithmetic surfaces and explicitly constructs dualizing sheaves, generalizing classical results to arithmetic settings.

Findings

Residue maps satisfy reciprocity laws around points.

Explicit construction of dualizing sheaves for arithmetic surfaces.

In local complete intersection cases, dualizing and canonical sheaves coincide.

Abstract

We develop a theory of residues for arithmetic surfaces, establish the reciprocity law around a point, and use the residue maps to explicitly construct the dualizing sheaf of the surface. These are generalisations of known results for surfaces over a perfect field. In an appendix, explicit local ramification theory is used to recover the fact that in the case of a local complete intersection the dualizing and canonical sheaves coincide.