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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Galois theory for iterative connections and nonreduced Galois groups

Author: Andreas Maurischat
Journal: Trans. Amer. Math. Soc. 362 (2010), 5411-5453
MSC (2000): Primary 13N99; Secondary 12H20, 12F15
Published electronically: May 25, 2010
MathSciNet review: 2657686
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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 (cf. Berthelot and Ogus, 1978) 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 1975.

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 given by S. Chase in 1976.

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Additional Information

Andreas Maurischat
Affiliation: Interdisciplinary Center for Scientific Computing, Heidelberg University, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany

Keywords: Galois theory, differential Galois theory, inseparable extensions, connections
Received by editor(s): February 22, 2008
Received by editor(s) in revised form: October 31, 2008
Published electronically: May 25, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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