ar X iv : 1 50 2 . 06 01 5 v 1 [ m at h . R A ] 2 0 Fe b 20 15 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 (∂w), the ideal generated by all ith-order left partial derivatives of w. The restriction map σ 7→ 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.