
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 over a field, we define the notions of -fundamental gerbe of a fibered category, generalizing what we had done in arXiv:1204.1260 for finite group schemes. We give sufficient conditions on implying that a fibered category over satisfying mild hypotheses admits a Nori -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 , we give a tannakian interpretation of the resulting gerbe.
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