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A LOCALISABLE CLASS OF PRIMITIVE IDEALS OF UNIFORM NILPOTENT IWASAWA ALGEBRAS

Part of: Chain conditions, growth conditions, and other forms of finiteness Structure and classification of infinite or finite groups Rings and algebras arising under various constructions Local rings and generalizations

Published online by Cambridge University Press:  21 July 2015

JONATHAN NELSON
JONATHAN NELSON*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, UK
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Abstract

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We study the injective hulls of faithful characteristic zero finite dimensional irreducible representations of uniform nilpotent pro-p groups, seen as modules over their corresponding Iwasawa algebras. Using this we prove that the kernels of these representations are classically localisable.

MSC classification

Primary: 16L30: Noncommutative local and semilocal rings, perfect rings
Secondary: 16P50: Localization and Noetherian rings 16S99: None of the above, but in this section 20E18: Limits, profinite groups
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 1 , January 2016 , pp. 1 - 15
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

REFERENCES

1.Ardakov, K., Localisation at augmentation ideals in Iwasawa algebras, Glasgow Math. J. 48 (2) (2006), 197–203.CrossRefGoogle Scholar
2.Ardakov, K., The controller subgroup of one-sided ideals in completed group rings, Contemp. Math. 562 (2012), 11–26.CrossRefGoogle Scholar
3.Ardakov, K. and Brown, K., Ring-theoretic properties of Iwasawa algebras: a survey, Doc. Math. (electronic) Extra Vol. (2006), 7–33.CrossRefGoogle Scholar
4.Ardakov, K. and Wadsley, S., Characteristic elements for p-torsion Iwasawa modules, J. Algebr. Geom. 15 (2) (2006), 339–377.CrossRefGoogle Scholar
5.Coates, J., Fukaya, T., Kato, K., Sujatha, R. and Venjakob, O., GL 2 main conjecture for elliptic curves without complex multiplication, Publ. Math. IHES 101 (1) (2005), 163–208.CrossRefGoogle Scholar
6.Coates, J., Schneider, P. and Sujatha, R., Modules over Iwasawa algebras, J. Inst. Math. Jussieu 2 (1) (2003), 73–108.CrossRefGoogle Scholar
7.Dixon, J., du Sautoy, M., Mann, A. and Segal, D., Analytic pro-p groups, 2nd ed. (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar
8.Goodearl, K. and Warfield, R., An introduction to noncommutative noetherian rings, 2nd ed. (London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1989).Google Scholar
9.Hall, P., Nilpotent groups, The collected works of Philip Hall (The Clarendon Press Oxford University Press, Oxford, 1988), 415–462.Google Scholar
10.Ingleton, A. W., The Hahn-Banach theorem for non-archimedean-valued fields, Math. Proc. Cam. Phil. Soc. 48 (01) (1952), 41–45.CrossRefGoogle Scholar
11.Jategaonkar, A. V., Localization in noetherian rings, London Mathematical Society Lecture. Note Series, no. 98 (Cambridge University Press, Cambridge, 1986).CrossRefGoogle Scholar
12.Kakde, M., Proof of the main conjecture of noncommutative Iwasawa theory for totally real number fields in certain cases, J. Algebr. Geom. 20 (4) (2011), 631–683.CrossRefGoogle Scholar
13.Lazard, M., Groupes analytiques p-adiques, Publ. Math. IHES 26 (1965), 389–603.Google Scholar
14.Musson, I., On the structure of certain injective modules over group algebras of soluble groups of finite rank, J. Algebra 85 (1) (1983), 51–75.CrossRefGoogle Scholar
15.Nelson, J., Cliques and localisability in soluble Iwasawa algebras, in preparation.Google Scholar
16.Northcott, D., Injective envelopes and inverse polynomials, J. London Math. Soc. 8 (2) (1974), 290–296.CrossRefGoogle Scholar
17.Passman, D., The algebraic structure of group rings (Interscience, New York, 1977).Google Scholar
18.Schneider, P. and Venjakob, O., On the codimension of modules over skew power series rings with applications to Iwasawa algebras, J. Pure Appl. Algebra 204 (2) (2006), 349–367.CrossRefGoogle Scholar
19.Venjakob, O., On the structure theory of the Iwasawa algebra of a compact p-adic Lie group, J. Eur. Math. Soc. 4 (3) (2002), 271–311.CrossRefGoogle Scholar
20.Wilson, J. S., Profinite groups, London Math. Soc. Monographs New Series, no. 19 (Oxford Science Publications, Oxford, 1998).CrossRefGoogle Scholar