Transactions of the American Mathematical Society

Author(s): Frauke M. Bleher
Journal: Trans. Amer. Math. Soc. 361 (2009), 3661-3705.
MSC (2000): Primary 20C20; Secondary 20C15, 16G10
Posted: February 10, 2009
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Abstract: Let

be an algebraically closed field of characteristic

, and let

be the ring of infinite Witt vectors over

. Suppose

is a finite group and

is a block of

with dihedral defect group

, which is Morita equivalent to the principal

-modular block of a finite simple group. We determine the universal deformation ring

for every

-module

which belongs to

and has stable endomorphism ring

. It follows that

is always isomorphic to a subquotient ring of

. Moreover, we obtain an infinite series of examples of universal deformation rings which are not complete intersections.

1.: J. L. Alperin, Local representation theory. Modular representations as an introduction to the local representation theory of finite groups. Cambridge Studies in Advanced Mathematics, vol. 11, Cambridge University Press, Cambridge, 1986. MR 860771 (87i:20002)
2.: M. Auslander, I. Reiten and S. Smalø, Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR 1314422 (96c:16015)
3.: D. J. Benson, Representations and cohomology. I: Basic representation theory of finite groups and associative algebras. Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. MR 1110581 (92m:20005)
4.: F. M. Bleher, Universal deformation rings and Klein four defect groups. Trans. Amer. Math. Soc. 354 (2002), 3893-3906. MR 1926858 (2004a:20014)
5.: F. M. Bleher, Deformations and derived equivalences. Proc. Amer. Math. Soc. 134 (2006), 2503-2510. MR 2213727 (2007e:20007)
6.: F. M. Bleher and T. Chinburg, Universal deformation rings and cyclic blocks. Math. Ann. 318 (2000), 805-836. MR 1802512 (2001m:20013)
7.: 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)
8.: F. M. Bleher and T. Chinburg, Universal deformation rings need not be complete intersections. Math. Ann. 337 (2007), 739-767. MR 2285736
9.: R. Brauer, On -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 (50:7315)
10.: C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the modularity of elliptic curves over $\mathbb{Q}$ : Wild -adic exercises. J. Amer. Math. Soc. 14 (2001), 843-939. MR 1839918 (2002d:11058)
11.: M. Broué, Equivalences of blocks of group algebras. In: 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 (97c:20004)
12.: M. C. R. Butler and C. M. Ringel, Auslander-Reiten sequences with few middle terms and applications to string algebras. Comm. Algebra 15 (1987), 145-179. MR 876976 (88a:16055)
13.: 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)
14.: 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, G. Cornelissen, C. Gasbarri, D. Glass, C. Lehr, M. Matignon, F. Oort, R. Pries and S. Wewers. Rend. Sem. Mat. Univ. Padova 113 (2005), 129-177. MR 2168985 (2006d:14027)
15.: S. B. Conlon, Certain representation algebras. J. Austral. Math. Soc. 5 (1965), 83-99. MR 0185024 (32:2494)
16.: G. Cornell, J. H. Silverman and G. Stevens (eds.), Modular Forms and Fermat's Last Theorem (Boston, 1995), Springer-Verlag, Berlin-Heidelberg-New York, 1997. MR 1638473 (99k:11004)
17.: C. W. Curtis and I. Reiner, Methods of representation theory. Vol. I. With applications to finite groups and orders. John Wiley & Sons, Inc., New York, 1981. MR 632548 (82i:20001)
18.: B. de Smit and H. W. Lenstra, Explicit construction of universal deformation rings. In: Modular Forms and Fermat's Last Theorem (Boston, 1995), Springer-Verlag, Berlin-Heidelberg-New York, 1997, pp. 313-326. MR 1638482
19.: M. Dettweiler and S. Wewers, Variation of local systems and parabolic cohomology. Israel J. Math. 156 (2006), 157-185. MR 2282374 (2007k:14013)
20.: K. Erdmann, Blocks of Tame Representation Type and Related Algebras. Lecture Notes in Mathematics, vol. 1428, Springer-Verlag, Berlin-Heidelberg-New York, 1990. MR 1064107 (91c:20016)
21.: P. Fong, A note on splitting fields of representations of finite groups. Illinois J. Math. 7 (1963) 515-520. MR 0153741 (27:3702)
22.: D. Gorenstein and J. H. Walter, The characterization of finite groups with dihedral Sylow -subgroups. I, II, III. J. Algebra 2 (1965) 85-151, 218-270, 354-393. MR 0177032 (31:1297a)
23.: B. Huppert, Endliche Gruppen. I. Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967. MR 0224703 (37:302)
24.: S. König and A. Zimmermann, Derived equivalences for group rings. With contributions by Bernhard Keller, Markus Linckelmann, Jeremy Rickard and Raphaël Rouquier. Lecture Notes in Mathematics, vol. 1685. Springer-Verlag, Berlin, 1998. MR 1649837 (2000g:16018)
25.: H. Krause, Maps between tree and band modules. J. Algebra 137 (1991), 186-194. MR 1090218 (92j:16010)
26.: M. Linckelmann, A derived equivalence for blocks with dihedral defect groups. J. Algebra 164 (1994), 244-255. MR 1268334 (94m:20030)
27.: B. Mazur, Deforming Galois representations. In: Galois groups over $\mathbb{Q}$ (Berkeley, 1987), Springer-Verlag, Berlin-Heidelberg-New York, 1989, pp. 385-437. MR 1012172 (90k:11057)
28.: J. Rickard, Derived equivalences as derived functors. J. London Math. Soc. 43 (1991), 37-48. MR 1099084 (92b:16043)
29.: J. Rickard, Splendid equivalences: Derived categories and permutation modules. Proc. London Math. Soc. 72 (1996), 331-358. MR 1367082 (97b:20011)
30.: J. P. Serre, Linear representations of finite groups. Translated from the second French edition by Leonard L. Scott. Graduate Texts in Mathematics, vol. 42, Springer-Verlag, New York-Heidelberg, 1977. MR 0450380 (56:8675)
31.: T. Shiina, Regular Galois realizations of $\mathrm{PSL}_2(p^2)$ over $\mathbb{Q}(T)$ . In: Galois theory and modular forms (Tokyo, 2001), Dev. Math., 11, Kluwer Acad. Publ., Boston, 2004, pp. 125-142. MR 2059760 (2005b:12011)
32.: R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. of Math. 141 (1995), 553-572. MR 1333036 (96d:11072)
33.: G. Wiese, On projective linear groups over finite fields as Galois groups over the rational numbers. ArXiv math.NT/0606732.
34.: A. Wiles, Modular elliptic curves and Fermat's last theorem. Ann. of Math. 141 (1995), 443-551. MR 1333035 (96d:11071)

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20C20, 20C15, 16G10

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

DOI: 10.1090/S0002-9947-09-04543-7
PII: S 0002-9947(09)04543-7
Keywords: Universal deformation rings, dihedral defect groups, special biserial algebras, stable endomorphism rings
Received by editor(s): July 26, 2006
Received by editor(s) in revised form: April 27, 2007
Posted: February 10, 2009
Additional Notes: The author was supported in part by NSF Grant DMS01-39737 and NSA Grant H98230-06-1-0021.
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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