Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [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.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 16P90, 16S36

Retrieve articles in all journals with MSC (1991): 16P90, 16S36

Additional Information

James J. Zhang
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195

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

American Mathematical Society