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Inverse problems for deformation rings


Authors: Frauke M. Bleher, Ted Chinburg and Bart de Smit
Journal: Trans. Amer. Math. Soc. 365 (2013), 6149-6165
MSC (2010): Primary 11F80; Secondary 11R32, 20C20
DOI: https://doi.org/10.1090/S0002-9947-2013-05848-5
Published electronically: May 14, 2013
MathSciNet review: 3091278
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Abstract: Let $ W$ be a complete Noetherian local commutative ring with residue field $ k$ of positive characteristic $ p$. We study the inverse problem for the universal deformation rings $ R_{W}(\Gamma ,V)$ relative to $ W$ of finite dimensional representations $ V$ of a profinite group $ \Gamma $ over $ k$. We show that for all $ p$ and $ n \ge 1$, the ring $ W[[t]]/(p^n t,t^2)$ arises as a universal deformation ring. This ring is not a complete intersection if $ p^nW\neq \{0\}$, so we obtain an answer to a question of M. Flach in all characteristics. We also study the `inverse inverse problem' for the ring $ W[[t]]/(p^n t,t^2)$; this is to determine all pairs $ (\Gamma , V)$ such that $ R_{W}(\Gamma ,V)$ is isomorphic to this ring.


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

  • 1. E. Artin and J. Tate, Class Field Theory. W.A. Benjamin, 1967. MR 0223335 (36:6383)
  • 2. F. M. Bleher and T. Chinburg, Universal deformation rings and cyclic blocks. Math. Ann. 318 (2000), 805-836. MR 1802512 (2001m:20013)
  • 3. F. M. Bleher and T. Chinburg, Universal deformation rings need not be complete intersections. C. R. Math. Acad. Sci. Paris 342 (2006), 229-232. MR 2196003 (2007b:20053)
  • 4. F. M. Bleher and T. Chinburg, Universal deformation rings need not be complete intersections. Math. Ann. 337 (2007), 739-767. MR 2285736 (2008g:11093)
  • 5. F. M. Bleher, T. Chinburg and B. de Smit, Deformation rings which are not local complete intersections, March 2010. arXiv:1003.3143
  • 6. G. Böckle, Presentations of universal deformation rings. In: $ L$-functions and Galois representations, 24-58, London Math. Soc. Lecture Note Ser., 320, Cambridge Univ. Press, Cambridge, 2007. MR 2392352 (2009e:11102)
  • 7. J. Byszewski, A universal deformation ring which is not a complete intersection ring. C. R. Math. Acad. Sci. Paris 343 (2006), 565-568. MR 2269865 (2007i:20051)
  • 8. T. Chinburg, Can deformation rings of group representations not be local complete intersections? In: Problems from the Workshop on Automorphisms of Curves. Edited by Gunther Cornelissen and Frans Oort, with contributions by I. Bouw, T. Chinburg, Cornelissen, C. Gasbarri, D. Glass, C. Lehr, M. Matignon, Oort, R. Pries and S. Wewers. Rend. Sem. Mat. Univ. Padova 113 (2005), 129-177. MR 2168985 (2006d:14027)
  • 9. H. Darmon, F. Diamond and R. Taylor, Fermat's Last Theorem. In : R. Bott, A. Jaffe and S. T. Yau (eds), Current developments in mathematics, 1995, International Press, Cambridge, MA., 1995, pp. 1-107. MR 1474977 (99d:11067a)
  • 10. B. de Smit and H. W. Lenstra, Explicit construction of universal deformation rings. In: G. Cornell, J. H. Silverman and G. Stevens (eds), Modular Forms and Fermat's Last Theorem (Boston, MA, 1995), Springer-Verlag, Berlin-Heidelberg-New York, 1997, pp. 313-326. MR 1638482
  • 11. D. S. Dummit and R.M. Foote, Abstract Algebra. Third edition. John Wiley & Sons, 2004. MR 2286236 (2007h:00003)
  • 12. A. Grothendieck, Éléments de géométrie algébrique, Chapitre IV, Quatriéme Partie. Publ. Math. IHES 32 (1967), 5-361. MR 0238860 (39:220)
  • 13. H. Matsumura, Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, Vol. 8, Cambridge University Press, Cambridge, 1989. MR 1011461 (90i:13001)
  • 14. B. Mazur, Deforming Galois representations. In: Galois groups over $ \mathbb{Q}$ (Berkeley, CA, 1987), Springer-Verlag, Berlin-Heidelberg-New York, 1989, pp. 385-437. MR 1012172 (90k:11057)
  • 15. B. Mazur, An introduction to the deformation theory of Galois representations. In: G. Cornell, J. H. Silverman and G. Stevens (eds), Modular Forms and Fermat's Last Theorem (Boston, MA, 1995), Springer-Verlag, Berlin-Heidelberg-New York, 1997, pp. 243-311. MR 1638481
  • 16. R. Rainone, On the inverse problem for deformation rings of representations. Master's thesis, Universiteit Leiden, Thesis Advisor: Bart de Smit, June 2010. http://www.math.leidenuniv.nl/en/theses/205/
  • 17. M. Schlessinger, Functors of Artin Rings. Trans. of the AMS 130 (1968), 208-222. MR 0217093 (36:184)
  • 18. J. P. Serre, Corps Locaux. Hermann, Paris, 1968. MR 0354618 (50:7096)
  • 19. J. P. Serre, Linear Representations of Finite Groups. Springer-Verlag, New York, 1977. MR 0450380 (56:8675)
  • 20. C. Weibel, An Introduction to Homological Algebra, Cambridge University Press, 1994. MR 1269324 (95f:18001)
  • 21. A. Wiles, Modular elliptic curves and Fermat's last theorem. Ann. of Math. 141 (1995), 443-551. MR 1333035 (96d:11071)

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Additional Information

Frauke M. Bleher
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242-1419
Email: frauke-bleher@uiowa.edu

Ted Chinburg
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Email: ted@math.upenn.edu

Bart de Smit
Affiliation: Mathematisch Instituut, University of Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands
Email: desmit@math.leidenuniv.nl

DOI: https://doi.org/10.1090/S0002-9947-2013-05848-5
Keywords: Universal deformation rings, complete intersections, inverse problems
Received by editor(s): February 24, 2012
Received by editor(s) in revised form: April 5, 2012
Published electronically: May 14, 2013
Additional Notes: The first author was supported in part by NSF Grant DMS0651332 and NSA Grant H98230-11-1-0131. The second author was supported in part by NSF Grants DMS0801030 and DMS1100355. The third author was funded in part by the European Commission under contract MRTN-CT-2006-035495.
Article copyright: © Copyright 2013 Frauke M. Bleher, Ted Chinburg, and Bart de Smit

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