Untwisting algebras with van den Bergh duality into Calabi-Yau algebras

Jake Goodman and Ulrich Kr"ahmer have recently shown that a twisted Calabi-Yau algebra $A$ with modular automorphism $\sigma$ and dimension $d$ can be “untwisted,” in the sense that the Ore extensions $A[X;\sigma ]$ and $A[X^{\pm 1};\sigma ]$ are Calabi-Yau algebras of dimension $d+1$. In this note we show that this in fact extends more generally to the case where we start with an algebra with van den Bergh duality.