  If $A$ is an algebra with finite right global dimension, then for any
automorphism $\alpha$ and $\alpha$-derivation $\delta$ the right global
dimension of $A[t; \alpha, \delta]$ satisfies \[
  \text{rgld} \, A \le \text{rgld} \, A[t; \alpha, \delta] \le \text{rgld} \, A
+ 1. \] We extend this result to the case of holomorphic Ore extensions and
smooth crossed products by $\mathbb{Z}$ of $\hat{\otimes}$-algebras.
