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Dimension Theory in Iterated Local Skew Power Series Rings

Abstract

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.

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Acknowledgements

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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The author did not receive funding from any organisation for the submitted work, and has no relevant financial or non-financial interests to disclose. Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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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). https://doi.org/10.1007/s10468-022-10144-3

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  • DOI: https://doi.org/10.1007/s10468-022-10144-3

Keywords

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

Mathematics Subject Classification (2010)

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