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Transcendence degree of division algebras

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Abstract

We define a transcendence degree for division algebras by modifying the lower transcendence degree construction of Zhang. We show that this invariant has many of the desirable properties one would expect a noncommutative analogue of the ordinary transcendence degree for fields to have. Using this invariant, we prove the following conjecture of Small. Let k be a field, let A be a finitely generated k-algebra that is an Ore domain, and let D denote the quotient division algebra of A. If A does not satisfy a polynomial identity, then GKdim(K) ≤ GKdim(A) − 1 for every commutative subalgebra K of D.

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Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

    Jason P. Bell

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  1. Jason P. Bell
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Correspondence to Jason P. Bell.

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This work supported by NSERC grant 31-611456.

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Bell, J.P. Transcendence degree of division algebras. Isr. J. Math. 190, 195–211 (2012). https://doi.org/10.1007/s11856-012-0001-8

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  • DOI: https://doi.org/10.1007/s11856-012-0001-8

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