The Ext-algebra and associated monomial algebra of a finite dimensional K-algebra

Abstract

It is well known that if the ext-algebra E(A) of an algebra A is generated in degrees 0 and 1, A must be a Koszul algebra. We seek to generalize this notion. To do so, we set A=KQ/I where Q is a finite quiver and I is an admissible ideal. We use a construction from [13], [1], and [11] to form a family of elements in KQ which yields a projective resolution of A¯, called the AGS resolution. We denote this family {fij} and consider the case where the AGS resolution is minimal. Then we consider the associated monomial algebra of A, found in [8] and [9], which we denote AMON. We prove that if the AGS resolution is minimal and E(AMON) is finitely generated, then E(A) is finitely generated. Next we look at 2-d-determined algebras, as defined in [11]. If A is a 2-d-determined algebra, we prove that if the AGS resolution is minimal, then E(A) is generated in degrees 0,1, and 2. However, if the AGS resolution is not minimal, we prove that E(A) need not be K2.