Ampleness of two-sided tilting complexes

Abstract

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. 0 Introduction Let X be a nonsingular projective variety over a field k and let ωX be its canonical bundle. Then the functor SX := −⊗X ωX [dimX] : D(cohX) −→ D(cohX) is the Serre functor ,i.e., HomX(G ·,F ·)∗ is functorially isomorphic to HomX(F ·, SX(G ·)) for F ·,G · ∈ D(cohX). 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 ST of T (if exists) as the derived tensor product of ” dimX”-shifted ”canonical bundle” ωX . 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 H(σ) = 0 for i ≥ 1 and σ is pure for n A 0. σ is called ample if σ is pure for n A 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 projR = ( cohprojR, R, (1) ) . An abelian category cohprojR is considered as the category of coherent sheaves on projR. (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 := TA(H (σ)) of H(σ) over A is a graded connected coherent ring over A and there is a t-structure D defined by σ in Perf A and its heart H is equivalent to cohprojT . Moreover the following Theorem holds. Theorem 0.2 (Theorem 2.8). Let A be a finite dimensional k-algebra of finite global dimension and let σ be a very ample two-sided tilting complex. Then there is a natural equivalence of triangulated categories D (mod-A) ∼ −→ D (cohprojT ) . where T := TA(H (σ)) is the tensor algebra of H(σ) over A.