Fundamental gerbes

TL;DR

This paper generalizes the concept of fundamental gerbes for affine algebraic groups, providing conditions for their existence and a tannakian interpretation in specific cases, extending prior work on finite group schemes.

Contribution

It introduces the notion of $ ext{C}$-fundamental gerbes for a class of affine algebraic groups and establishes existence criteria for fibered categories, extending previous finite group scheme results.

Findings

Established existence of $ ext{C}$-fundamental gerbes for virtually abelian and virtually unipotent groups.

Provided a tannakian interpretation of the gerbe under properness conditions.

Generalized the fundamental gerbe concept beyond finite group schemes.

Abstract

For a class of affine algebraic groups $\mathcal C$ over a field, we define the notions of $\mathcal C$-fundamental gerbe of a fibered category, generalizing what we had done in arXiv:1204.1260 for finite group schemes. We give sufficient conditions on $\mathcal C$ implying that a fibered category $X$ over $\kappa$ satisfying mild hypotheses admits a Nori $\mathcal C$-fundamental gerbe. We show that these are verified in particular by the classes of virtually abelian and virtually unipotent group schemes. In the second situation, under a properness condition on $X$, we give a tannakian interpretation of the resulting gerbe.