Skew polynomial algebras with coefficients in Koszul Artin-Schelter regular algebras

Let A be a Koszul Artin-Schelter regular algebra with Nakayama automorphism ξ. We show that the Yoneda Ext-algebra of the skew polynomial algebra A[z;ξ] is a trivial extension of a Frobenius algebra. Then we prove that A[z;ξ] is Calabi-Yau; and hence each Koszul Artin-Schelter regular algebra is a subalgebra of a Koszul Calabi-Yau algebra. A superpotential wˆ is also constructed so that the Calabi-Yau algebra A[z;ξ] is isomorphic to the derivation quotient of wˆ. The Calabi-Yau property of a skew polynomial algebra with coefficients in a PBW-deformation of a Koszul Artin-Schelter regular algebra is also discussed.