Ampleness of two-sided tilting complexes

From the view point of noncommutative algebraic geometry (NCAG), a two-sided tilting complex is an analog of a line bundle. In this paper, we define the notion of ampleness for two-sided tilting complexes over finite-dimensional algebras of finite global dimension, and prove its basic properties, which justify the name “ampleness”. From the view point of NCAG, Serre functors are considered to be shifted canonical bundles. A finite-dimensional algebra A of finite global dimension is called Fano if the shifted Serre functor A*[−d] is anti-ample. Some classes of algebras studied before are Fano. We show by examples that the property of A*[−d] from the view point of NCAG captures a representation theoretic property of the algebra A.