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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on GK dimension
of skew polynomial extensions


Author: James J. Zhang
Journal: Proc. Amer. Math. Soc. 125 (1997), 363-373
MSC (1991): Primary 16P90, 16S36
MathSciNet review: 1350966
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Abstract: Let $A$ be a finitely generated commutative domain over an algebraically closed field $k$, $\sigma $ an algebra endomorphism of $A$, and $\delta $ a $\sigma $-derivation of $A$. Then $ \operatorname {GKdim}(A[x,\sigma ,\delta ])= \operatorname {GKdim}(A)+1$ if and only if $\sigma $ is locally algebraic in the sense that every finite dimensional subspace of $A$ is contained in a finite dimensional $\sigma $-stable subspace.

Similarly, if $F$ is a finitely generated field over $k$, $\sigma $ a $k$-endomorphism of $F$, and $\delta $ a $\sigma $-derivation of $F$, then $ \operatorname {GKdim} (F[x,\sigma ,\delta ])= \operatorname {GKdim}(F)+1$ if and only if $\sigma $ is an automorphism of finite order.


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Additional Information

James J. Zhang
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Email: zhang@math.washington.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03602-2
PII: S 0002-9939(97)03602-2
Keywords: Gelfand-Kirillov dimension, Polynomial extension, automorphism of algebra
Received by editor(s): June 19, 1995
Received by editor(s) in revised form: August 24, 1995
Additional Notes: This research was supported in part by the NSF
Communicated by: Ken Goodearl
Article copyright: © Copyright 1997 American Mathematical Society