A note on GK dimension of skew polynomial extensions

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.