On the Weil-\'etale cohomology of number fields

TL;DR

This paper provides a detailed description of sheaves on the Weil-étale site of a number ring, introduces a spectral sequence linking Weil-étale and Artin-Verdier étale cohomology, and constructs complexes for computing Weil-étale cohomology.

Contribution

It offers a direct description of sheaves on the Weil-étale site and develops tools to compute Weil-étale cohomology using spectral sequences and complexes.

Findings

Established a spectral sequence relating Weil-étale and Artin-Verdier étale cohomology.

Constructed complexes of étale sheaves for Weil-étale cohomology.

Provided a categorical description of sheaves on the Weil-étale site.

Abstract

We give a direct description of the category of sheaves on Lichtenbaum's Weil-\'etale site of a number ring. Then we apply this result to define a spectral sequence relating Weil-\'etale cohomology to Artin-Verdier \'etale cohomology. Finally we construct complexes of \'etale sheaves computing the expected Weil-\'etale cohomology.