Separable algebras and coflasque resolutions

TL;DR

This paper introduces a new invariant for reductive algebraic groups using Galois cohomology, characterizing when separable algebra invariants fail to distinguish algebraic objects, and relates it to coflasque resolutions and the Tate-Shafarevich group.

Contribution

It provides a new cohomological invariant linked to coflasque resolutions that determines the effectiveness of separable algebra invariants for algebraic groups.

Findings

Invariant is trivial for many fields, including some number fields.

Invariant coincides with the Tate-Shafarevich group at real places.

Characterizes failure of separable algebra invariants using coflasque resolutions.

Abstract

Over a non-closed field, it is a common strategy to use separable algebras as invariants to distinguish algebraic and geometric objects. The most famous example is the deep connection between Severi-Brauer varieties and central simple algebras. For more general varieties, one might use endomorphism algebras of line bundles, of indecomposable vector bundles, or of exceptional objects in their derived categories. Using Galois cohomology, we describe a new invariant of reductive algebraic groups that captures precisely when this strategy will fail. Our main result characterizes this invariant in terms of coflasque resolutions of linear algebraic groups introduced by Colliot-Th\'el\`ene. We determine whether or not this invariant is trivial for many fields. For number fields, we show it agrees with the Tate-Shafarevich group of the linear algebraic group, up to behavior at real places.