The Weil-Etale Topology for Number Rings

TL;DR

This paper introduces a new Grothendieck topology for rings of algebraic integers, linking cohomology groups to special zeta-function values, and demonstrating finite generation and Euler characteristic relations.

Contribution

It constructs a novel topology for number rings and establishes a connection between its cohomology and zeta-function special values at s=0.

Findings

Cohomology of the constant sheaf Z is related to zeta-function behavior at s=0.

The topology ensures finitely generated cohomology groups.

Euler characteristics are connected to zeta-function special values.

Abstract

We would like to construct a new Grothendieck topology for arithmetic schemes, whose cohomology groups associated with motivic complexes of sheaves are finitely generated and whose Euler characteristics are related to special values of zeta-functions. In this paper we construct this topology for rings of algebraic integers and show that the cohomology of the constant sheaf Z is related to the behavior of zeta-functions at s = 0.