#### Vol. 13, No. 3, 2019

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Fundamental gerbes

### Niels Borne and Angelo Vistoli

Vol. 13 (2019), No. 3, 531–576
##### Abstract

For a class of affine algebraic groups $\mathsc{C}$ over a field $\kappa$, we define the notion of $\mathsc{C}$-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 $\mathsc{C}$ implying that a fibered category $X$ over $\kappa$ satisfying mild hypotheses admits a Nori $\mathsc{C}$-fundamental gerbe. We also give a tannakian interpretation of the gerbe that results by taking as $\mathsc{C}$ 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.

##### Keywords
fundamental group scheme, Tannaka theory, algebraic stacks
Primary: 14A20
Secondary: 14H30