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A note on GK dimension of skew polynomial extensions
Authors:James J. Zhang
Affiliation:Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
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.

Keywords:Gelfand-Kirillov dimension   Polynomial extension   automorphism of algebra
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