Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-2010-04966-9
Published electronically: May 25, 2010
MathSciNet review: 2657686
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [AM05] Amano, K.. Masuoka, A.: Picard-Vessiot extensions of Artinian simple module algebras. Journal of Algebra, 285:743-767 (2005) MR 2125463 (2005m:16046)
  • [Ama07] Amano, K.: On a discrepancy among Picard-Vessiot theories in positive characteristics. eprint arXiv:math/0612683v2 (2007)
  • [And01] André, Y.: Différentielles non commutatives et théorie de Galois différentielle ou aux différences. Ann. Scient. Éc. Norm. Sup., (4), 34:685-739 (2001) MR 1862024 (2002k:12013)
  • [BO78] Berthelot, P.. Ogus, A.: Notes on crystalline cohomology. Princeton University Press (1978) MR 0491705 (58:10908)
  • [Cha76] Chase, S.U.: Infinitesimal group scheme actions on finite field extensions. Am. J. Math., (2) 98:441-480 (1976) MR 0424773 (54:12731)
  • [Del90] Deligne, P.: Catégories Tannakiennes, in Grothendieck Festschrift, Vol II. Birkhäuser Boston, pp. 111-195 (1990) MR 1106898 (92d:14002)
  • [DG70] Demazure, M.. Gabriel, P.: Groupes Algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs. North-Holland Pub. Comp., Amsterdam (1970) MR 0302656 (46:1800)
  • [DM89] Deligne, P.. Milne, J.: Tannakian Categories, in Hodge Cycles, Motives and Shimura Varieties. Springer Lecture Notes 900:101-228 (1989)
  • [Dyc08] Dyckerhoff, T.: The Inverse Problem of Differential Galois Theory over the Field $ \mathbb{R}(t)$. eprint arXiv:0802.2897v1 (2008)
  • [Gro64] Grothendieck, A.: Éléments de géométrie algébrique IV. Publ. Math. de l'I. H. É. S. 20 (1964)
  • [Eis95] Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics 150, Springer (1995) MR 1322960 (97a:13001)
  • [Gie75] Gieseker, D.: Flat vector bundles and the fundamental group in non-zero characteristics. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), (2)1:1-31 (1975) MR 0382271 (52:3156)
  • [Hard08] Hardouin, C.: Iterative q-difference Galois Theory. IWR-Preprint (2008) (available at http://www.ub.uni-heidelberg.de/archiv/8278/ )
  • [Hart77] Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics 52, Springer (1977) MR 0463157 (57:3116)
  • [HS37] Hasse, H.. Schmidt, F.K.: Noch eine Begründung der Theorie des höheren Differentialquotienten in einem algebraischen Funktionenkörper in einer Unbestimmten. J. Reine Angew. Math. 177:215-237 (1937)
  • [Hei07] Heiderich, F.: Picard-Vessiot-Theorie für lineare partielle Differentialgleichungen. Heidelberg University Library, Diplom thesis (2007)
  • [Jac64] Jacobson, N.: Lectures in Abstract Algebra, Vol III: Theory of Fields and Galois Theory. Van Nostrand, Princeton, NJ (1964); Springer-Verlag reprint (1975) MR 0172871 (30:3087)
  • [Jan03] Jantzen, J.C.: Representations of algebraic groups. Math. Surveys and Monographs, 107, Am. Math. Soc. (2003) MR 2015057 (2004h:20061)
  • [Kat70] Katz, N.: Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin. Publ. Math. de l'I. H. É. S. 39:175-232 (1970) MR 0291177 (45:271)
  • [Kat87] Katz, N.: On the calculation of some differential Galois groups. Invent. Math. 87:13-61 (1987) MR 862711 (88c:12010)
  • [Mat01] Matzat, B.H.: Differential Galois Theory in Positive Characteristic, notes written by J. Hartmann. IWR-Preprint 2001-35 (2001) (available at http://www.iwr.uni-heidelberg.de/$ \sim$Heinrich.Matzat/publications.html)
  • [Mats89] Matsumura, H.: Commutative ring theory. Cambridge Studies in Adv. Math. 8 (1989) MR 1011461 (90i:13001)
  • [MvdP03] Matzat, B.H.. van der Put, M.: Iterative differential equations and the Abhyankar conjecture. J. Reine Angew. Math. 557:1-52 (2003) MR 1978401 (2004d:12011)
  • [Pap08] Papanikolas, M.: Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms. Invent. Math. 171:123-174 (2008) MR 2358057
  • [Rös07] Röscheisen, A.: Iterative Connections and Abhyankar's Conjecture. Heidelberg University Library, Ph.D. thesis (2007) (available at http://www.ub.uni-heidelberg.de/archiv/7179/)
  • [San07] Dos Santos, J.P.: Fundamental group schemes for stratified sheaves. Journal of Algebra 317:691-713 (2007) MR 2362937 (2008h:14045)
  • [Tak89] Takeuchi, M.: A Hopf Algebraic Approach to the Picard-Vessiot Theory. Journal of Algebra 122:481-509 (1989) MR 999088 (90j:12016)
  • [vdPS03] van der Put, M.. Singer, M.F.: Galois Theory of Linear Differential Equations. Grundlehren Math. Wiss. 328, Springer (2003) MR 1960772 (2004c:12010)
  • [Voj07] Vojta, P.: Jets via Hasse-Schmidt Derivations. In Diophantine Geometry, Proceedings, pp. 335-361. Edizioni della Normale, Pisa (2007) MR 2349665 (2008i:13043)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13N99, 12H20, 12F15

Retrieve articles in all journals with MSC (2000): 13N99, 12H20, 12F15


Additional Information

Andreas Maurischat
Affiliation: Interdisciplinary Center for Scientific Computing, Heidelberg University, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany
Email: andreas.maurischat@iwr.uni-heidelberg.de

DOI: https://doi.org/10.1090/S0002-9947-2010-04966-9
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.

American Mathematical Society