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Primeness, semiprimeness and localisation in Iwasawa algebras

Author(s): Konstantin Ardakov; Kenneth A. Brown
Journal: Trans. Amer. Math. Soc. 359 (2007), 1499-1515.
MSC (2000): Primary 16P40, 16L30, 11R23, 20C07
Posted: November 3, 2006
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Abstract: Necessary and sufficient conditions are given for the completed group algebras of a compact $ p$-adic analytic group with coefficient ring the $ p$-adic integers or the field of $ p$ elements to be prime, semiprime and a domain. Necessary and sufficient conditions are found for the localisation at semiprime ideals related to the augmentation ideals of closed normal subgroups. Some information is obtained about the Krull and global dimensions of the localisations. The results extend and complete work of A. Neumann and J. Coates et al.

References:

1.
K. Ardakov, Krull dimension of Iwasawa Algebras, J. Algebra 280 (2004), 190-206. MR 2081928 (2005e:16036)

2.
K. Ardakov, Localisation at augmentation ideals in Iwasawa algebras, Glasgow Math J. 48 (2006), 251-267.

3.
A. Bell and R. Farnsteiner, On the theory of Frobenius extensions and its application to Lie superalgebras, Trans. Amer. Math. Soc. 335 (1993), 407-424. MR 1097163 (93c:17049)

4.
A. Brumer, Pseudocompact algebras, profinite groups and class formations, J. Algebra 4 (1966), 442-470. MR 0202790 (34:2650)

5.
K.A. Brown, C.R. Hajarnavis and A.E. MacEacharn, Rings of finite global dimension integral over their centres, Comm. in Algebra 11 (1983), 67-93. MR 0687406 (84b:16029)

6.
J. Coates, T. Fukaya, K. Kato, R. Sujatha, O. Venjakob, The $ GL_2$ main conjecture for elliptic curves without complex multiplication, preprint, arXiv math NT/0404297.

7.
J.D. Dixon, M.P.F. Du Sautoy, A. Mann, D. Segal, Analytic pro$ -p$ groups, 2nd edition, CUP (1999). MR 1720368 (2000m:20039)

8.
L. Huishi and F. van Oystaeyen, Zariskian filtrations, Kluwer Academic Publishers, K-monographs in Mathematics, vol. 2 (1996). MR 1420862 (97m:16083)

9.
A. V. Jategaonkar, Localization in Noetherian Rings, London Math. Soc. Lecture Notes vol. 98, CUP, 1986. MR 0839644 (88c:16005)

10.
M. Lazard, Groupes analytiques $ p$-adiques, Publ. Math. IHES 26 (1965), 389-603. MR 0209286 (35:188)

11.
J.C. McConnell, Localisation in enveloping rings, J. London Math. Soc. 43 (1968), 421-428; Erratum and Addendum, ibid. (2) 3, 409-410. MR 0228532 (37:4112); MR 0313341 (47:1896)

12.
J.C. McConnell, J.C. Robson, Noncommutative Noetherian rings, AMS Graduate Studies in Mathematics, vol. 30. MR 1811901 (2001i:16039)

13.
A. Neumann, Completed group algebras without zero-divisors, Arch. Math. Basel 51 (1988), 496-499. MR 0973723 (90a:20014)

14.
D. Passman, Infinite Crossed Products, Pure and Applied Mathematics vol. 135, Academic Press, San Diego (1989). MR 0979094 (90g:16002)

15.
D. Passman, The Algebraic Structure of Group Rings, Wiley-Interscience, New York (1977). MR 0470211 (81d:16001)

16.
P. F. Smith, Localisation and the AR property, Proc. Lond. Math. Soc. 2 (1971), 39-68. MR 0294383 (45:3453)

17.
J.-P. Serre, Sur la dimension homologique des groupes profinis, Topology 3 (1965), 413-420. MR 0180619 (31:4853)

18.
O. Venjakob, Characteristic elements in non-commutative Iwasawa theory, J. Reine Angew. Math. 583 (2005), 193-236. MR 2146857 (2006d:11133)

19.
O. Venjakob, On the structure theory of the Iwasawa algebra of a compact $ p$-adic Lie group, J. Eur. Math. Soc. (JEMS) 4 (2002), no. 3, 271-311. MR 1924402 (2004h:16029)

20.
R. Walker, Local rings and normalizing sets of elements, Proc. Lond. Math. Soc. 24 (1972), 27-45. MR 0294399 (45:3469)

21.
J. S. Wilson, Profinite Groups, Lond. Math. Soc. Monographs 19, Oxford University Press, Oxford, 1998. MR 1691054 (2000j:20048)

22.
Y. Zhong, Homological dimension of crossed products, J. Algebra 164 (1994), 101-123. MR 1268329 (95a:16039)

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Additional Information:

Konstantin Ardakov
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, United Kingdom
Email: K.Ardakov@dpmms.cam.ac.uk

Kenneth A. Brown
Affiliation: Department of Mathematics, University of Glasgow, Glasgow G12 8QW, United Kingdom
Email: kab@maths.gla.ac.uk

DOI: 10.1090/S0002-9947-06-04153-5
PII: S 0002-9947(06)04153-5
Keywords: Noncommutative Iwasawa theory, pro-$p$ group, completed group algebra, complete noetherian local ring, prime ring, semiprime ring, localisable ideal
Received by editor(s): January 5, 2005
Posted: November 3, 2006
Additional Notes: The first author thanks Christ's College, Cambridge, for financial support.
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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