m-Koszul Artin-Schelter regular algebras

Abstract

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 in a tensor power of some vector space V. By definition, D(w,i) is the quotient of the tensor algebra TV by the ideal generated by all i-th order left partial derivatives of w. We 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 $\sigma$ of an $m$-Koszul Artin-Schelter regular algebra D(w,i) is the scalar hdet($\sigma$) given by the formula hdet($\sigma$) w =$\sigma^{\otimes m+i}$(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.