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.

In this paper we define the notion of ampleness for two-sided tilting complexes over finite dimensional algebras and prove its basic properties.We call a finite dimensional k-algebra A of finite global dimension Fano if (A * [−d]) −1 is ample for some d ≥ 0. For example geometric algebras in the sense of Bondal-Polishchuk are Fano. We give a characterization of representation type of a quiver from a noncommutative algebro-geometric view point, that is, a finite acyclic quiver has finite representation type if and only if its path algebra is fractional Calabi-Yau, and a finite acyclic quiver has infinite representation type if and only if its path algebra is Fano. IntroductionLet X be a nonsingular projective variety over a field k and let ω X be its canonical bundle. Then the functor. By this fact, from a noncommutative (or categorical) algebro-geometric view point, one thinks of a triangulated category T as the derived category of coherent sheaves of some "space" X and of the Serre functor S T of T (if exists) as the derived tensor product of " dim X"-shifted "canonical bundle" ω X . From this view point, the notion of Calabi-Yau algebra ( and Calabi-Yau category ) is defined and studied extensively by many researchers.In this paper we introduce the notion of ampleness for two-sided tilting complexes over finite dimensional k-algebras. Let A be a finite dimensional k-algebra of finite global dimension. Definition 0.1 (Definition 2.6). A two-sided tilting complex σ over A is called very ample if Hi (σ) = 0 for i ≥ 1 and σ n is pure for n 0. σ is called ample if σ n is pure for n 0.In Section 2, we justify this definition by using the theory of noncommutative projective schemes due to Artin-Zhang [AZ] and Polishchuk [Po]. In the theory of noncommutative projective schemes , for a graded coherent ring R over k we attach an imaginary geometric object proj R = ( cohproj R, R, (1) ) . An abelian category cohproj R is considered as the category of coherent sheaves on proj R. (See Section 1.) In Section 2 we show that the following facts hold. If σ is a very ample tilting complex over A, then the tensor algebra T := T A (H 0 (σ)) of H 0 (σ) over A is a graded connected coherent ring over A and there is a t-structure D σ defined by σ in Perf A and its heart H σ is equivalent to cohproj T . Moreover the following Theorem holds. where T := T A (H 0 (σ)) is the tensor algebra of H 0 (σ) over A.

Weighted projective lines, introduced by Geigle and Lenzing in 1987, are important objects in representation theory. They have tilting bundles, whose endomorphism algebras are the canonical algebras introduced by Ringel. The aim of this paper is to study their higher dimensional analogs. First, we introduce a certain class of commutative Gorenstein rings R graded by abelian groups L of rank 1, which we call Geigle-Lenzing complete intersections. We study their Cohen-Macaulay representations and hence the stable category CM L R, which coincides with the singularity category D L sg (R). We show that CM L R is triangle equivalent to D b (modA CM ) for a finite dimensional algebra A CM , which we call the CM-canonical algebra. As an application, we classify the (R, L) that are Cohen-Macaulay finite. We also give sufficient conditions for (R, L) to be d-Cohen-Macaulay finite in the sense of higher Auslander-Reiten theory. Secondly, we study a new class of non-commutative projective schemes in the sense of Artin-Zhang, i.e. the category cohX = mod L R/mod L 0 R of coherent sheaves on the Geigle-Lenzing projective space X, which is the quotient stackfor a finite dimensional algebra A ca , which we call the d-canonical algebra. We study when X is d-vector bundle finite, and when X is derived equivalent to a d-representation infinite algebra in the sense of higher Auslander-Reiten theory. Our d-canonical algebras provide a rich source of d-Fano and d-anti-Fano algebras in non-commutative algebraic geometry. We also observe Orlov-type semiorthogonal decompositions of D L sg (R) and D b (cohX).

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We prove a noncommutative version of the John-Nirenberg theorem for nontracial filtrations of von Neumann algebras. As an application, we obtain an analogue of the classical large deviation inequality for elements of the associated BM O space.