Galois theory for iterative connections and nonreduced Galois groups

Details Galois theory for iterative connections and nonreduced Galois groups Galois theory for iterative connections and nonreduced Galois groups

2007-12-21

arxiv.org/abs/0712.3748v4

Transactions of the AMS, 362 (2010), no. 10, pp. 5411-5453

Mathematics

Rings and Algebras

37 pages; v2->v3: more cross references to other papers are added in this version, the introduction is more detailed v3->v4: p

Scientific paper

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.

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