Two-dimensional Id\`eles with Cycle Module Coefficients

TL;DR

This paper develops a theory of idèles with cycle module coefficients for smooth surfaces over a field, providing a flasque resolution of cycle module sheaves and establishing the Gersten property for cycle modules in certain local rings.

Contribution

It introduces a new framework for idèles with cycle module coefficients, extending higher adèle theory to handle cycle modules instead of quasi-coherent sheaves.

Findings

Established a flasque resolution of cycle module sheaves in the Zariski topology.

Proved the Gersten property for cycle modules on equicharacteristic complete regular local rings.

Extended higher adèle theory to include cycle modules for smooth surfaces.

Abstract

We give a theory of id\`eles with coefficients for smooth surfaces over a field. It is an analogue of Beilinson/Huber's theory of higher ad\`eles, but handling cycle module sheaves instead of quasi-coherent ones. We prove that they give a flasque resolution of the cycle module sheaves in the Zariski topology. As a technical ingredient we show the Gersten property for cycle modules on equicharacteristic complete regular local rings, which might be of independent interest.