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Dimension Theory in Iterated Local Skew Power Series Rings


Many well-known local rings, including soluble Iwasawa algebras and certain completed quantum algebras, arise naturally as iterated skew power series rings. We calculate their Krull and global dimensions, obtaining lower bounds to complement the upper bounds obtained by Wang. In fact, we show that many common such rings obey a stronger property, which we call triangularity, and which allows us also to calculate their classical Krull dimension (prime length). Finally, we correct an error in the literature regarding the associated graded rings of general iterated skew power series rings, but show that triangularity is enough to recover this result.


  1. Ardakov, K.: Krull Dimension of Iwasawa Algebras. J. Algebra 280, 190–206 (2004)

    MathSciNet  Article  Google Scholar 

  2. Bergen, J., Grzeszczuk, P.: Skew Power Series Rings of Derivation type. Journal of Algebra and its Applications 10(6), 1383–1399 (2011)

    MathSciNet  Article  Google Scholar 

  3. Brown, K.A., Goodearl, K.R.: Lectures on Algebraic Quantum Groups. Advanced courses in mathematics–CRM Barcelona. Birkhäuser (2002)

  4. Brumer, A.: Pseudocompact Algebras, Profinite Groups and Class Formations. J.‘ Algebra 4, 442–470 (1966)

    MathSciNet  Article  Google Scholar 

  5. Cartan, H., Eilenberg, S.: Homological Algebra Princeton University Press (1956)

  6. Chan, D., Wu, Q.-S., Zhang, J.J.: Pre-Balanced Dualizing Complexes. Israel J. Math. 132(1), 285–314 (2002)

    MathSciNet  Article  Google Scholar 

  7. Chase, S.U.: Direct Products of Modules. Trans. Amer. Math. Soc. 97(3), 457–473 (1960)

    MathSciNet  Article  Google Scholar 

  8. Dixon, J.D., du Sautoy, M.P.F., Mann, A., Segal, D.: Analytic Pro-p Groups Cambridge University Press (1999)

  9. Goodearl, K.R., Warfield, R.B. Jr.: An Introduction to Noncommutative Noetherian Rings Cambridge University Press (2004)

  10. Greenfeld, B., Smoktunowicz, A., Ziembowski, M.: On Radicals of Ore Etensions and Related Questions. arXiv:1702.08103v2 (2017)

  11. Horton, K.L.: The Prime and Primitive Spectra of Multiparameter Quantum Sympectic and Euclidean Spaces Comm. Alg. 31(10), 4713–4743 (2003)

    Article  Google Scholar 

  12. Huishi, L., Van Oystaeyen, F.: Zariskian Filtrations. Springer Science+Business Media (1996)

  13. Lazard, M.: Groupes analytiques p-adiques. Publications Mathé,matiques de l’IHÉS 26, 5–219 (1965)

    Article  Google Scholar 

  14. Letzter, E.S.: Prime Ideals of Noetherian skew Power Series Rings. Israel J. Math. 192, 67–81 (2012)

    MathSciNet  Article  Google Scholar 

  15. Letzter, E.S., Wang, L: Prime Ideals of Q-Commutative Power Series Rings. Algebr. Represent. Theor. 14, 1003–1023 (2011)

    MathSciNet  Article  Google Scholar 

  16. Marubayashi, H., Van Oystaeyen, F.: Prime Divisors and Noncommutative Valuation Theory. Lecture Notes in Mathematics, 2059 Springer (2012)

  17. McConnell, J.C., Robson, J.C.: Noncommutative Noetherian Rings American Mathematical Society (2001)

  18. Nǎstǎsescu, C., van Oystaeyen, F.: Graded ring Theory North-Holland Publishing Company (1982)

  19. Rinehart, G.S., Rosenberg, A.: The Global Dimensions of Ore Extensions and Weyl Algebras. Algebra, Topology, and Category Theory, pp. 169–180, 0

  20. Schneider, P., Venjakob, O.: On the Codimension of Modules over skew Power Series Rings with Applications to Iwasawa Algebras. J. Pure Appl. Algebra 204, 349–367 (2005)

    MathSciNet  Article  Google Scholar 

  21. Schneider, P., Venjakob, O.: Localisations and Completions of skew Power Series Rings. Am. J. Math. 132(1), 1–36 (2010)

    Article  Google Scholar 

  22. Venjakob, O.: A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory. J. Reine Angew. Math. 559, 153–191 (2003)

    MathSciNet  MATH  Google Scholar 

  23. Wang, L.: Completions of Quantum Coordinate Rings. Proc. Amer. Math Soc. 137(3), 911–919 (2009)

    MathSciNet  Article  Google Scholar 

  24. Warner, S.: Topological Rings. 178. North-Holland Mathematics Studies (1993)

  25. Woods, W.: On the Structure of Virtually Nilpotent Compact P-adic Analytic Groups. J. Group Theory 21(1), 165–188 (2018)

    MathSciNet  Article  Google Scholar 

  26. Yekutieli, A., Zhang, J.J.: Rings with Auslander Dualizing Complexes. J. Algebra 213, 1–51 (1999)

    MathSciNet  Article  Google Scholar 

  27. Yekutieli, A.: Dualizing Complexes over Noncommutative Graded Algebras. J. Algebra 153, 41–84 (1992)

    MathSciNet  Article  Google Scholar 

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I am very grateful to K. A. Brown for some interesting discussions and his extensive comments on an early draft of this paper. I also gratefully acknowledge a helpful discussion with Adam Jones about soluble Iwasawa algebras; Example 2.16(ii) and Non-example 2.20 are in large part due to him.

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Correspondence to Billy Woods.

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Presented by: Kenneth Goodearl

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Woods, B. Dimension Theory in Iterated Local Skew Power Series Rings. Algebr Represent Theor (2022).

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  • Iwasawa algebras
  • Skew power series rings
  • Local rings
  • Quantum algebras

Mathematics Subject Classification (2010)

  • 16S34
  • 16S36
  • 16S99
  • 16P60