Period and index, symbol lengths, and generic splittings in Galois cohomology

TL;DR

This paper explores the relationship between symbol length bounds in Galois cohomology and minimal splitting field degrees, introducing new constructions of versal classes and generic splitting varieties.

Contribution

It introduces the concept of presentable functors and uses them to connect cohomological symbol bounds with splitting field degrees, advancing understanding of Galois cohomology structures.

Findings

Established a link between symbol length bounds and splitting field degrees

Constructed generic splitting varieties for degree 2 cohomology

Developed new methods based on presentable functors

Abstract

We use constructions of versal cohomology classes based on a new notion of "presentable functors," to describe a relationship between the problems of bounding symbol length in cohomology and of finding the minimal degree of a splitting field. The constructions involved are then also used to describe generic splitting varieties for degree 2 cohomology with coefficients in a commutative algebraic group of multiplicative type.