A structure theorem for subgroups of 𝐺𝐿_{𝑛} over complete local Noetherian rings with large residual image

Manoharmayum, Jayanta

doi:10.1090/S0002-9939-2015-12306-4

A structure theorem for subgroups of $GL_n$ over complete local Noetherian rings with large residual image
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Proc. Amer. Math. Soc. 143 (2015), 2743-2758 Request permission

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 $\boldsymbol {k}$.

References

Gebhard Böckle, Demuškin groups with group actions and applications to deformations of Galois representations, Compositio Math. 121 (2000), no. 2, 109–154. MR 1757878, DOI 10.1023/A:1001746207573
V. P. Burichenko, On some 2-cohomology of the groups $\textrm {SL}(n,q)$, 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, DOI 10.1007/s10469-008-9034-9
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 399283
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
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}
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
B. Mazur and A. Wiles, On $p$-adic analytic families of Galois representations, Compositio Math. 59 (1986), no. 2, 231–264. MR 860140
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, DOI 10.1007/978-3-540-37889-1
Richard Pink, Classification of pro-$p$ subgroups of $\textrm {SL}_2$ over a $p$-adic ring, where $p$ is an odd prime, Compositio Math. 88 (1993), no. 3, 251–264. MR 1241950
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 315016, DOI 10.2307/1970825
Chih Han Sah, Cohomology of split group extensions, J. Algebra 29 (1974), 255–302. MR 393273, DOI 10.1016/0021-8693(74)90099-4
Chih Han Sah, Cohomology of split group extensions. II, J. Algebra 45 (1977), no. 1, 17–68. MR 463264, DOI 10.1016/0021-8693(77)90360-X
Jean-Pierre Serre, Abelian $l$-adic representations and elliptic curves, W. A. Benjamin, Inc., New York-Amsterdam, 1968. McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute. MR 0263823
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, DOI 10.1016/j.jalgebra.2012.02.028