The \'etale cohomology ring of the ring of integers of a number field

TL;DR

This paper computes the étale cohomology ring of the spectrum of the ring of integers in a number field and applies the results to derive a non-vanishing formula for a Kim invariant.

Contribution

It provides the first explicit computation of the étale cohomology ring for the ring of integers of a number field and connects it to Kim's invariant.

Findings

Explicit description of the étale cohomology ring for $ ext{Spec } ext{O}_K$

A non-vanishing formula for Minhyong Kim's invariant

New tools for studying arithmetic properties via cohomology

Abstract

We compute the \'etale cohomology ring $H^*(\text{Spec } \mathcal{O}_K,\mathbb{Z}/n\mathbb{Z})$ where $\mathcal{O}_K$ is the ring of integers of a number field $K.$ As an application, we give a non-vanishing formula for an invariant defined by Minhyong Kim.