In this paper, we define a notion of AS-Gorenstein algebra for N-graded algebras, and show that symmetric AS-regular algebras of Gorenstein parameter 1 are exactly preprojective algebras of quasi-Fano algebras. This result can be compared with the fact that symmetric graded Frobenius algebras of Gorenstein parameter −1 are exactly trivial extensions of finite dimensional algebras. The results of this paper suggest that there is a strong interaction between classification problems in noncommutative algebraic geometry and those in representation theory of finite dimensional 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. By definition, D(w, i) is the quotient of the tensor algebra T V , where V = D(w, i) 1 , by (∂ i w), 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.

In this paper, we introduce a notion of ampleness of a group action G on a right noetherian graded algebra A, and show that it is strongly related to the notion of A G to be a graded isolated singularity introduced by the second author of this paper. Moreover, if S is a noetherian AS-regular algebra and G is a finite ample group acting on S, then we will show that D b (tails S G ) ∼ = D b (mod ∇S * G) where ∇S is the Beilinson algebra of S. We will also explicitly calculate a quiver QS,G such that D b (tails S G ) ∼ = D b (mod kQS,G) when S is of dimension 2.

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We will prove a Riemann-Roch like theorem for triangulated categories satisfying Serre duality. As an application, we will prove Riemann-Roch Theorem and Adjunction Formula for noncommutative Cohen-Macaulay surfaces in terms of sheaf cohomology. We will also show that the formulas hold for stable categories over Koszul connected graded algebras in terms of Tate-Vogel cohomology by extending the BGG correspondence.