A Simpson correspondence in positive characteristic

Michel Gros (1) , Bernard Le Stum (1) , Adolfo Quirós (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Departamento de Matemáticas

We define the $p^m$-curvature map on the sheaf of differential operators of level $m$ on a scheme of positive characteristic $p$ as dual to some divided power map on infinitesimal neighborhhods. This leads to the notion of $p^m$-curvature on differential modules of level $m$. We use this construction to recover Kaneda's description of a semi-linear Azumaya splitting of the sheaf of differential operators of level $m$. Then, using a lifting modulo $p^2$ of Frobenius, we are able to define a Frobenius map on differential operators of level $m$ as dual to some divided Frobenius on infinitesimal neighborhhods. We use this map to build a true Azumaya splitting of the completed sheaf of differential operators of level $m$ (up to an automorphism of the center). From this, we derive the fact that Frobenius pull back gives, when restricted to quasi-nilpotent objects, an equivalence between Higgs-modules and differential modules of level $m$. We end by explaining the relation with related work of Ogus-Vologodski and van der Put in level zero as well as Berthelot's Frobenius descent.

Domaines: Géométrie algébrique [math.AG]

Michel Gros, Bernard Le Stum, Adolfo Quirós. A Simpson correspondence in positive characteristic. Publications of the Research Institute for Mathematical Sciences, 2010, 46 (1), pp.1-35. ⟨hal-00337672⟩