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

Fundamental gerbes

Angelo Vistoli
2019

Abstract

For a class of affine algebraic groups C over a field κ, we define the notion of 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 C implying that a fibered category X over κ satisfying mild hypotheses admits a Nori C-fundamental gerbe. We also give a tannakian interpretation of the gerbe that results by taking as 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.
2019
Settore MAT/03 - Geometria
File in questo prodotto:
File Dimensione Formato  
ant-v13-n3-p01-p.pdf

Accesso chiuso

Tipologia: Published version
Licenza: Non pubblico
Dimensione 1.69 MB
Formato Adobe PDF
1.69 MB Adobe PDF   Richiedi una copia
1610.07341.pdf

accesso aperto

Tipologia: Accepted version (post-print)
Licenza: Solo Lettura
Dimensione 502.81 kB
Formato Adobe PDF
502.81 kB Adobe PDF

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/78364
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 4
  • ???jsp.display-item.citation.isi??? 4
  • OpenAlex ND
social impact