On some finiteness results in real \'etale cohomology

TL;DR

This paper establishes finiteness and constructibility results in real étale cohomology for schemes, showing preservation under direct images and computing related Grothendieck groups, with implications for motivic homotopy theory.

Contribution

It introduces a natural topological notion of constructibility in real étale cohomology and proves key properties like base change and preservation under direct images.

Findings

Constructibility in real étale cohomology aligns with topological notions.

Derived direct image functor preserves constructibility under certain conditions.

Computed the Grothendieck group of the constructible rational stable motivic homotopy category.

Abstract

We show that for quasi-compact quasi-separated schemes of finite dimension, the constructibility condition in real \'etale cohomology agrees with a notion of constructibility arising naturally from topology. As application we prove that the derived direct image functor preserves constructibility under some assumptions, and compute the Grothendieck group of the constructible rational stable motivic homotopy category for reasonable schemes. We prove the generic base change property for constructible real \'etale sheaves, and deduce the same property for rational motivic spectra and $b$-sheaves.