On automorphisms of p-torsion G_m-gerbes

2026/01/31 by Noah Olander
#math.AG

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Abstract

Rouquier proved that for a smooth projective variety X, the group scheme Pic⁰_X \rtimes Aut⁰_X is an invariant of the derived category of X. This was generalized to the twisted case by Olsson, who associated a group algebraic space Aut⁰_X to a G_m-gerbe X → X and proved it to be a twisted derived invariant. In characteristic zero, Olsson showed that Aut⁰_X is an extension of Aut⁰_X by a subgroup scheme of Pic_X. It is important in applications to compute Aut⁰_X in positive characteristic as well, which is the problem we consider here. We use deformation theory and representability results of Bragg and Olsson to prove that in many cases of interest, Olsson's description of Aut⁰_X as an extension also holds in characteristic p. As a corollary, we show that twisted derived equivalent abelian varieties are isogenous, simultaneously generalizing results of Honigs and Lane. We also provide an example showing that Olsson's extension description does not hold in general. This uses a result of Illusie on crystalline and flat cohomology and a result of Yang on the Brauer group of an ordinary variety, which we generalize to cohomology in arbitrary degree.

Citations

• Point Objects and Derived Equivalences of Twisted Derived Categories of Abelian Varieties
• Twisted derived categories and Rouquier functors
• Derived invariance of the Albanese relative canonical ring
• Representability of cohomology of finite flat abelian group schemes
• Counterexamples to Hochschild--Kostant--Rosenberg in characteristic p
• Derived invariants arising from the Albanese map
• Derived equivalent surfaces and abelian varieties, and their zeta\n functions
• On the Homology of Completion and Torsion
• Derived invariance of the number of holomorphic 1-forms and vector fields
• Derived categories of coherent sheaves on abelian varieties and equivalences between them

Related

• p-Primary Torsion of the Brauer Group in Characteristic p
• Twisted derived categories and Rouquier functors
• Torsion in the 0-cycle group with modulus
• Derived isogenies and isogenies for abelian surfaces
• Derived invariants and motives, Part II integral derived invariants and some applications