Sur une $q$-déformation locale de la théorie de Hodge non-abélienne en caractéristique positive

Abstract

For $p$ a prime number and $q$ a non trivial $p$th root of 1, we present the main steps of the construction of a local $q$-deformation of the "Simpson correspondence in characteristic $p$" found by Ogus and Vologodsky in 2005. The construction is based on the Morita-equivalence between a ring of $q$-twisted differential operators and its center. We also explain the expected relations between this construction and those recently done by Bhatt and Scholze. For the sake of readability, we limit ourselves to the case of dimension 1.

arXiv commentThis text is an updated version (references, ...) of a note written in Oct. 2017. It now includes (see §6) new results and speculations about the relations with the recent Bhatt-Scholze's work on prismatic cohomology. In French