Arithmetic Cohomology Groups

TL;DR

This paper introduces global arithmetic cohomology groups for quasi-coherent sheaves on arithmetic varieties using an adelic approach, establishing their fundamental properties and topological structures, including self-duality of adelic spaces on arithmetic surfaces.

Contribution

It develops a new framework for arithmetic cohomology groups on arithmetic varieties, proving duality and topological properties, and analyzing ind-pro topologies on adelic spaces.

Findings

Arithmetic cohomology groups are topologically self-dual.

Fundamental properties like duality and exact sequences are established.

Adelic spaces on arithmetic surfaces have well-defined ind-pro topologies.

Abstract

We first introduce global arithmetic cohomology groups for quasi-coherent sheaves on arithmetic varieties, adopting an adelic approach. Then, we establish fundamental properties, such as topological duality and inductive long exact sequences, for these cohomology groups on arithmetic surfaces. Finally, we expose basic structures for ind-pro topologies on adelic spaces of arithmetic surfaces. In particular, we show that these adelic spaces are topologically self-dual.