Galois theory for iterative connections and nonreduced Galois groups

TL;DR

This paper develops a Galois theory for modules with iterative connections in arbitrary characteristic, establishing a Tannakian framework and relating it to stratifications, flat bundles, and nonreduced Galois groups, extending classical theories.

Contribution

It introduces a general theory of modules with iterative connection in arbitrary characteristic and establishes a Galois correspondence including nonreduced group schemes.

Findings

Modules with iterative connection form a Tannakian category under certain conditions.

Equivalence between stratifications and integrable iterative connections over smooth rings.

Galois correspondence includes nonreduced subgroup schemes and inseparable extensions.

Abstract

This article presents a theory of modules with iterative connection. This theory is a generalisation of the theory of modules with connection in characteristic zero to modules over rings of arbitrary characteristic. We show that these modules with iterative connection (and also the modules with integrable iterative connection) form a Tannakian category, assuming some nice properties for the underlying ring, and we show how this generalises to modules over schemes. We also relate these notions to stratifications on modules, as introduced by A. Grothendieck in order to extend integrable (ordinary) connections to finite characteristic. Over smooth rings, we obtain an equivalence of stratifications and integrable iterative connections. Furthermore, over a regular ring in positive characteristic, we show that the category of modules with integrable iterative connection is also equivalent to the category of flat bundles as defined by D. Gieseker. In the second part of this article, we set up a Picard-Vessiot theory for fields of solutions. For such a Picard-Vessiot extension, we obtain a Galois correspondence, which takes into account even nonreduced closed subgroup schemes of the Galois group scheme on one hand and inseparable intermediate extensions of the Picard-Vessiot extension on the other hand. Finally, we compare our Galois theory with the Galois theory for purely inseparable field extensions.