CO-POINT MODULES OVER KOSZUL ALGEBRAS

Abstract

Let $A$ be a graded algebra finitely generated in degree 1 over a field $k$. Point modules over $A$ introduced by Artin, Tate and Van den Bergh play an important role in studying $A$ in noncommutative algebraic geometry. In this paper, we define a dual notion of point module in terms of Koszul duality, which we call a co-point module. Using co-point modules, we will construct counter-examples to the following condition due to Auslander: for every finitely generated right module $\pi$ over a ring $R$, there is a natural number $n_M\in {\mathbb N}$ such that, for any finitely generated right module $N$ over $R$, ${\rm Ext}^i_R(M, N)=0$ for all $i\gg 0$ implies ${\rm Ext}^i_R(M, N)=0$ for all $i>n_M$.