Krull dimension criterion for rank-one generalized Weyl algebras

For a commutative Noetherian base $D$ of finite dimension $d$, the noncommutative Krull dimension of $D(\sigma ,t)$ equals $d$ unless a height-$d$ maximal ideal is periodic under $\sigma$ or contains infinitely many translates ${\sigma }^n(t)$. This is the commutative-base specialization of Theorem 5.3 in Krull dimension of Generalized Weyl Algebras with non-commutative coefficients.