When is a reductive group scheme linear?

TL;DR

This paper characterizes when a reductive group scheme over a base scheme admits a faithful linear representation, linking it to the isotriviality of its radical torus and its splitting after a finite étale cover.

Contribution

It establishes a precise criterion for the linearity of reductive group schemes based on the isotriviality of their radical torus.

Findings

Reductive group scheme admits a faithful linear representation iff its radical torus is isotrivial.

Radical torus splits after a finite étale cover if and only if the group scheme is linear.

Provides a necessary and sufficient condition for linearity of reductive group schemes.

Abstract

We show that a reductive group scheme over a base scheme S admits a faithful linear representation if and only if its radical torus is isotrivial, that is, it splits after a finite {\'e}tale cover.