Abstract
We study the existence of free subalgebras in division algebras, and prove the following general result: if \(A\) is a noetherian domain which is countably generated over an uncountable algebraically closed field \(k\) of characteristic \(0\), then either the quotient division algebra of \(A\) contains a free algebra on two generators, or it is left algebraic over every maximal subfield. As an application, we prove that if \(k\) is an uncountable algebraically closed field and \(A\) is a finitely generated \(k\)-algebra that is a domain of GK-dimension strictly less than \(3\), then either \(A\) satisfies a polynomial identity, or the quotient division algebra of \(A\) contains a free \(k\)-algebra on two generators.
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Acknowledgments
We thank Jia-feng Lü for many helpful comments. We also thank George Bergman, James Zhang, Sue Sierra, Tom Lenagan, Agata Smoktunowicz, Toby Stafford, Zinovy Reichstein, Lance Small, and Jairo Gonçalves for valuable discussions. Finally, we thank the referee for suggesting Schofield’s paper to prove Lemma 3.1.
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The first-named author was supported by NSERC Grant 31-611456.
The second-named author was supported by NSF Grant DMS-0900981.
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Bell, J.P., Rogalski, D. Free subalgebras of division algebras over uncountable fields. Math. Z. 277, 591–609 (2014). https://doi.org/10.1007/s00209-013-1267-1
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DOI: https://doi.org/10.1007/s00209-013-1267-1