A^1-connectedness in reductive algebraic groups

TL;DR

This paper investigates the A^1-connectedness properties of reductive algebraic groups, revealing conditions under which certain constructions fail to be A^1-local and characterizing A^1-connected groups over fields of characteristic zero.

Contribution

It introduces sheaves of A^1-connected components to analyze the failure of A^1-locality in reductive groups lacking isotropy, and characterizes A^1-connected reductive groups over characteristic zero fields.

Findings

Failure of A^1-locality without isotropy hypotheses

A^1-invariance of torsors fails on smooth affine schemes for certain groups

Characterization of A^1-connected reductive groups over characteristic zero fields

Abstract

Using sheaves of A^1-connected components, we prove that the Morel-Voevodsky singular construction on a reductive algebraic group fails to be A^1-local if the group does not satisfy suitable isotropy hypotheses. As a consequence, we show the failure of A^1-invariance of torsors for such groups on smooth affine schemes over infinite perfect fields. We also characterize A^1-connected reductive algebraic groups over a field of characteristic 0.