Arithmetic surfaces and adelic quotient groups

TL;DR

This paper explicitly computes an arithmetic adelic quotient group for a sheaf on an arithmetic surface, linking it to cohomology groups and real tori, providing a detailed algebraic and topological analysis.

Contribution

It presents a new explicit calculation of an adelic quotient group for sheaves on arithmetic surfaces, incorporating infinite fiber contributions and connecting to cohomology and real tori.

Findings

Explicit description of the adelic quotient group via a short exact sequence

Identification of the last term with a projective limit of real tori groups

Connection established between the quotient group and first cohomology groups

Abstract

We explicitly calculate an arithmetic adelic quotient group for a locally free sheaf on an arithmetic surface when the fiber over the infinite point of the base is taken into account. The calculations are presented via a short exact sequence. We relate the last term of this short exact sequence with the projective limit of groups which are finite direct products of copies of one-dimensional real torus and are connected with first cohomology groups of locally free sheaves on the arithmetic surface.