Universal deformation rings and dihedral defect groups
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- by Frauke M. Bleher PDF
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Abstract:
Let $k$ be an algebraically closed field of characteristic $2$, and let $W$ be the ring of infinite Witt vectors over $k$. Suppose $G$ is a finite group and $B$ is a block of $kG$ with dihedral defect group $D$, which is Morita equivalent to the principal $2$-modular block of a finite simple group. We determine the universal deformation ring $R(G,V)$ for every $kG$-module $V$ which belongs to $B$ and has stable endomorphism ring $k$. It follows that $R(G,V)$ is always isomorphic to a subquotient ring of $WD$. Moreover, we obtain an infinite series of examples of universal deformation rings which are not complete intersections.References
- J. L. Alperin, Local representation theory, Cambridge Studies in Advanced Mathematics, vol. 11, Cambridge University Press, Cambridge, 1986. Modular representations as an introduction to the local representation theory of finite groups. MR 860771, DOI 10.1017/CBO9780511623592
- Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR 1314422, DOI 10.1017/CBO9780511623608
- D. J. Benson, Representations and cohomology. I, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. Basic representation theory of finite groups and associative algebras. MR 1110581
- Frauke M. Bleher, Universal deformation rings and Klein four defect groups, Trans. Amer. Math. Soc. 354 (2002), no. 10, 3893–3906. MR 1926858, DOI 10.1090/S0002-9947-02-03072-6
- Frauke M. Bleher, Deformations and derived equivalences, Proc. Amer. Math. Soc. 134 (2006), no. 9, 2503–2510. MR 2213727, DOI 10.1090/S0002-9939-06-08269-4
- Frauke M. Bleher and Ted Chinburg, Universal deformation rings and cyclic blocks, Math. Ann. 318 (2000), no. 4, 805–836. MR 1802512, DOI 10.1007/s002080000148
- Frauke M. Bleher and Ted Chinburg, Universal deformation rings need not be complete intersections, C. R. Math. Acad. Sci. Paris 342 (2006), no. 4, 229–232 (English, with English and French summaries). MR 2196003, DOI 10.1016/j.crma.2005.12.006
- Frauke M. Bleher and Ted Chinburg, Universal deformation rings need not be complete intersections, Math. Ann. 337 (2007), no. 4, 739–767. MR 2285736, DOI 10.1007/s00208-006-0054-2
- Richard Brauer, On $2$-blocks with dihedral defect groups, Symposia Mathematica, Vol. XIII (Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972) Academic Press, London, 1974, pp. 367–393. MR 0354838
- Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over $\mathbf Q$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939. MR 1839918, DOI 10.1090/S0894-0347-01-00370-8
- Michel Broué, Equivalences of blocks of group algebras, Finite-dimensional algebras and related topics (Ottawa, ON, 1992) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 1–26. MR 1308978, DOI 10.1007/978-94-017-1556-0_{1}
- M. C. R. Butler and Claus Michael Ringel, Auslander-Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra 15 (1987), no. 1-2, 145–179. MR 876976, DOI 10.1080/00927878708823416
- Jakub Byszewski, A universal deformation ring which is not a complete intersection ring, C. R. Math. Acad. Sci. Paris 343 (2006), no. 9, 565–568 (English, with English and French summaries). MR 2269865, DOI 10.1016/j.crma.2006.09.015
- Problems from the Workshop on Automorphisms of Curves, Rend. Sem. Mat. Univ. Padova 113 (2005), 129–177. MR 2168985
- S. B. Conlon, Certain representation algebras, J. Austral. Math. Soc. 5 (1965), 83–99. MR 0185024
- Gary Cornell, Joseph H. Silverman, and Glenn Stevens (eds.), Modular forms and Fermat’s last theorem, Springer-Verlag, New York, 1997. Papers from the Instructional Conference on Number Theory and Arithmetic Geometry held at Boston University, Boston, MA, August 9–18, 1995. MR 1638473, DOI 10.1007/978-1-4612-1974-3
- Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
- Bart de Smit and Hendrik W. Lenstra Jr., Explicit construction of universal deformation rings, Modular forms and Fermat’s last theorem (Boston, MA, 1995) Springer, New York, 1997, pp. 313–326. MR 1638482
- Michael Dettweiler and Stefan Wewers, Variation of local systems and parabolic cohomology, Israel J. Math. 156 (2006), 157–185. MR 2282374, DOI 10.1007/BF02773830
- Karin Erdmann, Blocks of tame representation type and related algebras, Lecture Notes in Mathematics, vol. 1428, Springer-Verlag, Berlin, 1990. MR 1064107, DOI 10.1007/BFb0084003
- Paul Fong, A note on splitting fields of representations of finite groups, Illinois J. Math. 7 (1963), 515–520. MR 153741
- Daniel Gorenstein and John H. Walter, The characterization of finite groups with dihedral Sylow $2$-subgroups. I, J. Algebra 2 (1965), 85–151. MR 177032, DOI 10.1016/0021-8693(65)90027-X
- B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
- Steffen König and Alexander Zimmermann, Derived equivalences for group rings, Lecture Notes in Mathematics, vol. 1685, Springer-Verlag, Berlin, 1998. With contributions by Bernhard Keller, Markus Linckelmann, Jeremy Rickard and Raphaël Rouquier. MR 1649837, DOI 10.1007/BFb0096366
- Henning Krause, Maps between tree and band modules, J. Algebra 137 (1991), no. 1, 186–194. MR 1090218, DOI 10.1016/0021-8693(91)90088-P
- Markus Linckelmann, A derived equivalence for blocks with dihedral defect groups, J. Algebra 164 (1994), no. 1, 244–255. MR 1268334, DOI 10.1006/jabr.1994.1061
- B. Mazur, Deforming Galois representations, Galois groups over $\textbf {Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385–437. MR 1012172, DOI 10.1007/978-1-4613-9649-9_{7}
- Jeremy Rickard, Derived equivalences as derived functors, J. London Math. Soc. (2) 43 (1991), no. 1, 37–48. MR 1099084, DOI 10.1112/jlms/s2-43.1.37
- Jeremy Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. (3) 72 (1996), no. 2, 331–358. MR 1367082, DOI 10.1112/plms/s3-72.2.331
- Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380
- Takehito Shiina, Regular Galois realizations of $\textrm {PSL}_2(p^2)$ over $\Bbb Q(T)$, Galois theory and modular forms, Dev. Math., vol. 11, Kluwer Acad. Publ., Boston, MA, 2004, pp. 125–142. MR 2059760, DOI 10.1007/978-1-4613-0249-0_{6}
- Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR 1333036, DOI 10.2307/2118560
- G. Wiese, On projective linear groups over finite fields as Galois groups over the rational numbers. ArXiv math.NT/0606732.
- Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, DOI 10.2307/2118559
Additional Information
- Frauke M. Bleher
- Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242-1419
- Email: fbleher@math.uiowa.edu
- Received by editor(s): July 26, 2006
- Received by editor(s) in revised form: April 27, 2007
- Published electronically: February 10, 2009
- Additional Notes: The author was supported in part by NSF Grant DMS01-39737 and NSA Grant H98230-06-1-0021.
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 3661-3705
- MSC (2000): Primary 20C20; Secondary 20C15, 16G10
- DOI: https://doi.org/10.1090/S0002-9947-09-04543-7
- MathSciNet review: 2491895