Comptes Rendus
Algebra/Homological Algebra
Homological properties of noncommutative Iwasawa algebras
Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 15-20.

For any compact p-adic Lie group G, the Iwasawa algebra ΩG is an Artin–Schelter Gorenstein algebra. We obtain the Auslander–Buchsbaum formula, the Bass's theorem and the No-holes theorem for noetherian modules over ΛG and ΩG, and the dual versions for their artinian modules. It is shown that ΩG is Morita self-dual via dualizing complexes. We finally consider the homological invariant “grade” of filtered modules over ΛG and ΩG, when G is a uniform pro-p group with certain properties.

Pour tout groupe de Lie G p-adique compact, l'algèbre d'Iwasawa ΛG et son image épimorphique ΩG sont des algèbres d'Artin–Schelter Gorenstein. Nous montrons la formule d'Auslander–Buchsbaum, le théorème de Bass et le théorème des « non trous » pour des modules noethériens sur ΩG, ainsi que des versions duales pour leur modules artiniens. Il est montré que ΩG est auto-duale au sens de Morita par des complexes dualisants. Finalement, nous considérons les invariants homologiques « grade » des modules filtrés sur ΛG et ΩG, lorsque G est un groupe uniforme pro-p satisfaisant certaines propriétés.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2010.11.030

Feng Wei 1

1 Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, China
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Feng Wei. Homological properties of noncommutative Iwasawa algebras. Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 15-20. doi : 10.1016/j.crma.2010.11.030. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.11.030/

[1] K. Ajitabh; S.P. Smith; J.J. Zhang Auslander–Gorenstein rings, Comm. Algebra, Volume 26 (1998), pp. 2159-2180

[2] K. Ardakov; K.A. Brown Ring-theoretic properties of Iwasawa algebras: a survey, Doc. Math., Volume Extra Vol. Coates (2006), pp. 7-33

[3] K. Ardakov; F. Wei; J.J. Zhang Reflexive ideals in Iwasawa algebras, Adv. Math., Volume 218 (2008), pp. 865-901

[4] K. Ardakov; F. Wei; J.J. Zhang Nonexistence of reflexive ideals in Iwasawa algebras of Chevalley type, J. Algebra, Volume 320 (2008), pp. 259-275

[5] J. Coates Iwasawa algebras and arithmetic, Astérisque, Volume 290 (2003) no. 896, pp. 37-52

[6] J. Coates; P. Schneider; R. Sujatha Modules over Iwasawa algebras, J. Inst. Math. Jussieu, Volume 2 (2003), pp. 73-108

[7] J. Coates; T. Fukaya; K. Kato; R. Sujatha; O. Venjakob The GL2 main conjecture for elliptic curves without complex multiplication, Publ. Math. IHES, Volume 101 (2005), pp. 163-208

[8] R. Hartshorne Residues and Duality, Lecture Notes in Mathematics, vol. 20, Springer-Verlag, Berlin, 1966

[9] A.V. Jategaonkar Morita duality and Noetherian rings, J. Algebra, Volume 69 (1981), pp. 358-371

[10] M. Lazard Groupes analytiques p-adiques, Publ. Math. IHES, Volume 26 (1965), pp. 389-603

[11] T. Levasseur Some properties of noncommutative regular rings, Glasg. Math. J., Volume 34 (1992), pp. 277-300

[12] H. Li; F. Van Oystaeyen Zariskian Filtrations, K-Monograph in Mathematics, vol. 2, Kluwer Academic Publishers, 1996

[13] K. Nishida, Iwasawa algebras, crossed products and filtered rings, in: Proceedings of the 41st Symposium on Ring Theory and Representation Theory, Symp. Ring Theory Represent., Theory Organ. Comm., Tsukuba, 2009, pp. 63–67.

[14] Y. Ochi; O. Venjakob On the structure of Selmer groups over p-adic Lie extensions, J. Algebraic Geom., Volume 11 (2002), pp. 547-580

[15] O. Venjakob, Iwasawa theory of p-adic Lie extensions, PhD thesis, University of Heidelberg, 2000.

[16] O. Venjakob On the structure theory of the Iwasawa algebra of a p-adic Lie group, J. Eur. Math. Soc. (JEMS), Volume 4 (2002), pp. 271-311

[17] O. Venjakob A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory, J. Reine Angew. Math., Volume 559 (2003), pp. 153-191

[18] O. Venjakob On the Iwasawa theory of p-adic Lie extensions, Compos. Math., Volume 138 (2003), pp. 1-54

[19] F. Wei, Homological properties of noncommutative Iwasawa algebras, II, preprint.

[20] Q.-S. Wu; J.J. Zhang Homological identities for noncommutative rings, J. Algebra, Volume 242 (2001), pp. 516-535

[21] W.-M. Xue Rings with Morita Duality, Lecture Notes in Mathematics, vol. 1523, Springer-Verlag, Berlin, 1992

[22] A. Yekutieli Dualizing complexes over noncommutative graded algebras, J. Algebra, Volume 153 (1992), pp. 41-84

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