Fundamental gerbes

Abstract

For a class of affine algebraic groups $\mathcal{C}$ over a field $\kappa$, we define the notion of $\mathcal{C}$-fundamental gerbe of a fibered category, generalizing what we did for finite group schemes in a 2015 paper.

We give necessary and sufficient conditions on $\mathcal{C}$ implying that a fibered category $X$ over $\kappa$ satisfying mild hypotheses admits a Nori $\mathcal{C}$-fundamental gerbe. We also give a tannakian interpretation of the gerbe that results by taking as $\mathcal{C}$ the class of virtually unipotent group schemes, under a properness condition on $X$.

Finally, we prove a general duality result, generalizing the duality between group schemes of multiplicative type and Galois modules, that yields a construction of the multiplicative gerbe of multiplicative type which is independent of the previous theory, and requires weaker hypotheses. This gives a conceptual interpretation of the universal torsor of Colliot-Thélène and Sansuc.