Selfinjective Koszul algebras of finite complexity

Abstract

In this paper, we study selfinjective Koszul algebras of finite complexity. We prove that the complexity is a nonnegative integer when it is finite; and that the category Open image in new window of modules with complexity less or equal to t, is resolving and coresolving. We show that for each 0 ≤ l ≤ m there exist a family of modules of complexity l parameterized by G(l,m), the Grassmannian of l-dimensional subspaces of an m-dimensional vector space V, for the exterior algebra of V. Using complexity, we also give a new approach to the representation theory of a tame symmetric algebra with vanishing radical cube over an algebraically closed field of characteristic 0, via skew group algebra of a finite subgroup of SL(2, C) over the exterior algebra of a 2-dimensional vector space.