Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Euler characteristics, Akashi series and compact $ p$-adic Lie groups

Author: Simon Wadsley
Journal: Proc. Amer. Math. Soc. 138 (2010), 3455-3465
MSC (2010): Primary 11R23, 16U20
Published electronically: May 5, 2010
MathSciNet review: 2661546
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We discuss Euler characteristics for finitely generated modules over Iwasawa algebras. We show that the Euler characteristic of a module is well-defined whenever the 0th homology group is finite if and only if the relevant compact $ p$-adic Lie group is finite-by-nilpotent and that in this case all pseudo-null modules have trivial Euler characteristic. We also prove some other results relating to the triviality of Euler characteristics for pseudo-null modules as well as some analogous results for the Akashi series of Coates et al.

References [Enhancements On Off] (What's this?)

  • 1. Konstantin Ardakov.
    Localisation at augmentation ideals in Iwasawa algebras.
    Glasg. Math. J., 48(2):251-267, 2006. MR 2256976 (2007h:16030)
  • 2. K. Ardakov and K. A. Brown.
    Ring-theoretic properties of Iwasawa algebras: a survey.
    Doc. Math., Extra Vol.:7-33, 2006. MR 2290583 (2007k:11185)
  • 3. Konstantin Ardakov and Kenneth A. Brown.
    Primeness, semiprimeness and localisation in Iwasawa algebras.
    Trans. Amer. Math. Soc., 359(4):1499-1515 (electronic), 2007. MR 2272136 (2007j:16031)
  • 4. Konstantin Ardakov and Simon Wadsley.
    Characteristic elements for $ p$-torsion Iwasawa modules.
    J. Algebraic Geom., 15(2):339-377, 2006. MR 2199061 (2006m:11152)
  • 5. Konstantin Ardakov and Simon Wadsley.
    $ K\sb 0$ and the dimension filtration for $ p$-torsion Iwasawa modules.
    Proc. Lond. Math. Soc. (3), 97(1):31-59, 2008. MR 2434090 (2010b:11147)
  • 6. John Coates.
    Fragments of the $ \operatorname{GL}\sb 2$ Iwasawa theory of elliptic curves without complex multiplication.
    In Arithmetic theory of elliptic curves (Cetraro, 1997), volume 1716 of Lecture Notes in Math., pages 1-50. Springer, Berlin, 1999. MR 1754685 (2002c:11060)
  • 7. John Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha, and Otmar Venjakob.
    The $ \operatorname{GL}\sb 2$ main conjecture for elliptic curves without complex multiplication.
    Publ. Math. Inst. Hautes Études Sci., 101:163-208, 2005. MR 2217048 (2007b:11172)
  • 8. John Coates, Peter Schneider, and Ramdorai Sujatha.
    Links between cyclotomic and $ \operatorname{GL}\sb 2$ Iwasawa theory.
    Doc. Math., Extra Vol.
    (Kazuya Kato's fiftieth birthday): 187-215 (electronic), 2003. MR 2046599 (2005c:11134)
  • 9. John Coates and Ramdorai Sujatha.
    Euler-Poincaré characteristics of abelian varieties.
    C. R. Acad. Sci. Paris Sér. I Math., 329(4):309-313, 1999. MR 1713337 (2001g:11090)
  • 10. Thierry Levasseur.
    Some properties of noncommutative regular graded rings.
    Glasg. Math. J., 34(3):277-300, 1992. MR 1181768 (93k:16045)
  • 11. Luis Ribes and Pavel Zalesskii.
    Profinite groups, volume 40 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics.
    Springer-Verlag, Berlin, 2000. MR 1775104 (2001k:20060)
  • 12. Joseph J. Rotman.
    An introduction to homological algebra, volume 85 of Pure and Applied Mathematics.
    Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1979. MR 538169 (80k:18001)
  • 13. Jean-Pierre Serre.
    La distribution d'Euler-Poincaré d'un groupe profini.
    In Galois representations in arithmetic algebraic geometry (Durham, 1996), volume 254 of London Math. Soc. Lecture Note Ser., pages 461-493. Cambridge University Press, Cambridge, 1998. MR 1696505 (2000g:22017)
  • 14. Burt Totaro.
    Euler characteristics for $ p$-adic Lie groups.
    Inst. Hautes Études Sci. Publ. Math., 90:169-225, 1999. MR 1813226 (2002f:22032)
  • 15. Charles A. Weibel.
    An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics.
    Cambridge University Press, Cambridge, 1994. MR 1269324 (95f:18001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11R23, 16U20

Retrieve articles in all journals with MSC (2010): 11R23, 16U20

Additional Information

Simon Wadsley
Affiliation: Homerton College, University of Cambridge, Cambridge, CB2 8PQ, United Kingdom

Keywords: Iwasawa algebra, Euler characteristic, Akashi series
Received by editor(s): December 3, 2009
Received by editor(s) in revised form: January 5, 2010
Published electronically: May 5, 2010
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society