Abstract
Let A be a right noetherian algebra over a field k. If the base field extension A ⊗kK remains right noetherian for all extension fields K of k, then A is called stably right noetherian over k. We develop an inductive method to show that certain algebras of finite Gelfand-Kirillov dimension are stably noetherian, using critical composition series. We use this to characterize which algebras satisfying a polynomial identity are stably noetherian. The method also applies to many \(\mathbb {N}\)-graded rings of finite global dimension; in particular, we see that a noetherian Artin-Schelter regular algebra must be stably noetherian. In addition, we study more general variations of the stably noetherian property where the field extensions are restricted to those of a certain type, for instance purely transcendental extensions.
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Acknowledgments
We thank John Farina and Lance Small for many interesting and helpful conversations.
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Presented by: Kenneth Goodearl
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The author was partially supported by the NSA grant H98230-15-1-0317.
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Rogalski, D. Stably Noetherian Algebras of Polynomial Growth. Algebr Represent Theor 24, 519–540 (2021). https://doi.org/10.1007/s10468-020-09958-w
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DOI: https://doi.org/10.1007/s10468-020-09958-w
Keywords
- Stably noetherian ring
- Base field extension
- Tensor product
- Gelfand-Kirillov dimension
- Polynomial identity ring
- Graded ring
- Hilbert series
- Homological smoothness