Fundamental gerbes

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 $\hat{I}º$ 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.

43 pages, uses tikz-cd. New section on gerbes of multiplicative type