Lars Christensen

It was a long standing conjecture, due to Auslander, that local rings would have the property (ac): For any finitely generated R-module M there is an n(M) > 0 such that the following implication holds for all finitely generated R--modules N:

Ext^i (M,N)=0 for i sufficiently large implies Ext^i (M,N)=0 for i > n(M).

This conjecture was disproved by D.Jorgensen and L. Sega in 2003. The class of rings that have the property (ac) is still poorly understood. I will survey some of the questions that arise in attempts to understand it. In this process, we will see that the Auslander-Reiten conjecture holds over rings that have the (ac) property. The writings of Auslander appear not to address this connection between the two conjectures, which is somewhat surprising.