Some consequences of Frobenius descent for torsors
Niels Borne, Mohamed Rafik Mammeri
TL;DR
This work reframes torsor descent in positive characteristic through Frobenius descent, linking non-commutative Lie-valued differential forms to torsors under Frobenius kernels. It provides a concrete framework for classifying and understanding torsors via curvature and p-curvature obstructions, and it highlights affine line bundles as a pivotal non-abelian example where explicit Lie-valued data governs the structure. The main contributions include a Frobenius-descent equivalence between G-torsors on X^{(p)} and torsors on X with vanishing curvature, a de Rham interpretation of affine line bundles, and a precise description of G^F-torsors in terms of Maurer–Cartan data, with explicit results in the abelian case and clear limitations in the non-abelian setting. Overall, the paper offers a cohesive, geometric account of Frobenius descent for torsors, with practical classifications tied to Cartier theory, the Cartier operator, and Maurer–Cartan maps, and it underscores the role of Frobenius kernels in non-commutative torsor theory.
Abstract
We show how the formalism of Frobenius descent for torsors enables to study torsors under Frobenius kernels in terms of non-commutative, Lie-valued differential forms. We pay particular attention to affine line bundles trivialized by the Frobenius.