Etale cohomology, purity and formality with torsion coefficients

TL;DR

This paper employs Galois group actions on étale cohomology to establish partial formality results for dg-operads and dg-algebras with torsion coefficients, impacting the understanding of algebraic structures related to configuration spaces.

Contribution

It introduces a novel approach using Galois actions to prove degree-dependent partial formality for algebraic structures with torsion coefficients.

Findings

Partial formality up to a certain degree depending on the field's cardinality.

Application to the dg-operad of little disks and the algebra of singular cochains.

Extension of formality results to torsion coefficient settings.

Abstract

We use Galois group actions on \'etale cohomology to prove results of formality for dg-operads and dg-algebras with torsion coefficients. Our theory applies, among other related constructions, to the dg-operad of singular chains on the operad of little disks and to the dg-algebra of singular cochains on the configuration space of points in the complex space. The formality that we obtain is only up to a certain degree, which depends on the cardinality of the field of coefficients.