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

DOI:
https://doi.org/10.1090/S0002-9939-97-03602-2

MathSciNet review:
1350966

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a finitely generated commutative domain over an algebraically closed field , an algebra endomorphism of , and a -derivation of . Then if and only if is locally algebraic in the sense that every finite dimensional subspace of is contained in a finite dimensional -stable subspace.

Similarly, if is a finitely generated field over , a -endomorphism of , and a -derivation of , then if and only if is an automorphism of finite order.

**[Ha]**R. Hartshorne,*Algebraic Geometry*, Springer-Verlag, New York, 1977. MR**57:3116****[KL]**G. Krause and T. H. Lenagan,*Growth of algebras and Gelfand-Kirillov dimension*, Research Notes in Mathematics, Pitman Adv. Publ. Program**116**(1985). MR**86g:16001****[LMO]**A. Leroy, J. Matczuk and J. Okninski,*On the Gelfand-Kirillov dimension of normal localizations and twisted polynomial rings*, Perspectives in Ring Theory (F. van Oystaeyen and L. Le Bruyn, eds.), Kluwer Academic Publishers, 1988, pp. 205-214. MR**91c:16020****[Ma]**H. Matsumura,*Commutative Ring Theory*, (Translated by M. Reid), Cambridge University Press, 1986. MR**88h:13001****[MR]**J. C. McConnell and J. C. Robson,*Non-Commutative Noetherian Rings*, Wiley-interscience, Chichester, 1987. MR**89j:16023****[Mu]**I. Musson,*Gelfand-Kirillov dimension of twisted Laurent extensions*, Comm. Alg., vol. 17 (11), 1989, pp. 2853-2856. MR**91a:16018****[Zh]**J. J. Zhang,*On Gelfand-Kirillov transcendence degree*, Trans. Amer. Math. Soc.**348**(1996), 2867-2899.

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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:
https://doi.org/10.1090/S0002-9939-97-03602-2

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