m-Koszul Artin-Schelter regular algebras

This paper studies the homological determinants and Nakayama automorphisms of not-necessarily-noetherian m-Koszul twisted Calabi-Yau or, equivalently, m-Koszul Artin-Schelter regular, algebras. Dubois-Violette showed that such an algebra is isomorphic to a derivation quotient algebra D(w,i) for a unique-up-to-scalar-multiples twisted superpotential w. By definition, D(w,i) is the quotient of the tensor algebra TV, where V=D(w,i)1, by (∂iw), the ideal generated by all i-th-order left partial derivatives of w. The restriction map σ↦σ|V is used to identify the group of graded algebra automorphisms of D(w,i) with a subgroup of GL(V). We show that the homological determinant of a graded algebra automorphism σ of an m-Koszul Artin-Schelter regular algebra D(w,i) is given by the formula hdet(σ)w=σ⊗(m+i)(w). It follows from this that the homological determinant of the Nakayama automorphism of an m-Koszul Artin-Schelter regular algebra is 1. As an application, we prove that the homological determinant and the usual determinant coincide for most quadratic noetherian Artin-Schelter regular algebras of dimension 3.