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A structure theorem for subgroups of $ GL_n$ over complete local Noetherian rings with large residual image

Author: Jayanta Manoharmayum
Journal: Proc. Amer. Math. Soc. 143 (2015), 2743-2758
MSC (2010): Primary 20E18, 20E34, 11E57
Published electronically: March 11, 2015
MathSciNet review: 3336600
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Abstract: Given a complete local Noetherian ring $ (A,\mathfrak{m}_A)$ with finite residue field and a subfield $ \boldsymbol {k}$ of $ A/\mathfrak{m}_A$, we show that every closed subgroup $ G$ of $ GL_n(A)$ such that $ G\mod {\mathfrak{m}_A}\supseteq SL_n(\boldsymbol {k})$ contains a conjugate of $ SL_n(W(\boldsymbol {k})_A)$ under some small restrictions on $ \boldsymbol {k}$. Here $ W(\boldsymbol {k})_A$ is the closed subring of $ A$ generated by the Teichmüller lifts of elements of the subfield $ \boldsymbol {k}$.

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  • [1] Gebhard Böckle, Demuškin groups with group actions and applications to deformations of Galois representations, Compositio Math. 121 (2000), no. 2, 109–154. MR 1757878, 10.1023/A:1001746207573
  • [2] V. P. Burichenko, On some 2-cohomology of the groups 𝑆𝐿(𝑛,𝑞), Algebra Logika 47 (2008), no. 6, 687–704, 779 (Russian, with Russian summary); English transl., Algebra Logic 47 (2008), no. 6, 384–394. MR 2508323, 10.1007/s10469-008-9034-9
  • [3] Edward Cline, Brian Parshall, and Leonard Scott, Cohomology of finite groups of Lie type. I, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 169–191. MR 0399283
  • [4] Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
  • [5] B. Mazur, Deforming Galois representations, Galois groups over 𝑄 (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385–437. MR 1012172, 10.1007/978-1-4613-9649-9_7
  • [6] Barry Mazur, An introduction to the deformation theory of Galois representations, Modular forms and Fermat’s last theorem (Boston, MA, 1995) Springer, New York, 1997, pp. 243–311. MR 1638481
  • [7] B. Mazur and A. Wiles, On 𝑝-adic analytic families of Galois representations, Compositio Math. 59 (1986), no. 2, 231–264. MR 860140
  • [8] Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2008. MR 2392026
  • [9] Richard Pink, Classification of pro-𝑝 subgroups of 𝑆𝐿₂ over a 𝑝-adic ring, where 𝑝 is an odd prime, Compositio Math. 88 (1993), no. 3, 251–264. MR 1241950
  • [10] Daniel Quillen, On the cohomology and 𝐾-theory of the general linear groups over a finite field, Ann. of Math. (2) 96 (1972), 552–586. MR 0315016
  • [11] Chih Han Sah, Cohomology of split group extensions, J. Algebra 29 (1974), 255–302. MR 0393273
  • [12] Chih Han Sah, Cohomology of split group extensions. II, J. Algebra 45 (1977), no. 1, 17–68. MR 0463264
  • [13] Jean-Pierre Serre, Abelian 𝑙-adic representations and elliptic curves, McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0263823
  • [14] University of Georgia VIGRE Algebra Group, Second cohomology for finite groups of Lie type, J. Algebra 360 (2012), 21–52. University of Georgia VIGRE Algebra Group: Brian D. Boe, Brian Bonsignore, Theresa Brons, Jon F. Carlson, Leonard Chastkofsky, Christopher M. Drupieski, Niles Johnson, Wenjing Li, Phong Thanh Luu, Tiago Macedo, Daniel K. Nakano, Nham Vo Ngo, Brandon L. Samples, Andrew J. Talian, Lisa Townsley and Benjamin J. Wyser. MR 2914632, 10.1016/j.jalgebra.2012.02.028

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Additional Information

Jayanta Manoharmayum
Affiliation: School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, United Kingdom

Received by editor(s): April 5, 2013
Received by editor(s) in revised form: July 3, 2013
Published electronically: March 11, 2015
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.