Nonstandard Etale Cohomology

TL;DR

This paper introduces a generalized cohomology theory extending etale cohomology using category enlargements, enabling more flexible coefficients while preserving key properties and facilitating comparisons with classical cohomologies.

Contribution

It develops a new cohomology framework via category enlargement, broadening coefficient choices and maintaining essential properties of etale cohomology.

Findings

Defines a cohomology theory with flexible coefficients

Establishes a Weil cohomology using *Z/P coefficients

Enables comparison with classical l-adic cohomology

Abstract

A lot of good properties of etale cohomology only hold for torsion coefficients. We use "enlargement of categories" as developed in http://arxiv.org/abs/math.CT/0408177 to define a cohomology theory that inherits the important properties of etale cohomology while allowing greater flexibility with the coefficients. In particular, choosing coefficients *Z/P (for P an infinite prime and *Z the enlargement of Z) gives a Weil cohomology, and choosing *Z/l^h (for l a finite prime and h an infinite number) allows comparison with ordinary l-adic cohomology. More generally, for every N in *Z, we get a category of *Z/N-constructible sheaves with good properties.