  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.

Rings and Algebras;General Mathematics