A Study on Auslander Bounds

Abstract: The Auslander bound of a module can be thought of as a generalization of projective dimension. We say that the Auslander bound of $M$ is finite if for all finitely generated modules $N$ such that whenever $\text{Ext}{R}^{n}(M,N)=0$ for $n\gg 0$, there exists an integer $b$ that only depends on $M$ so that $\text{Ext}{R}^{n}(M,N)=0$ for $n>b$. In this paper we give new results on the Auslander bound by weakening the AC condition, generalizing many theorems in the the literature. We then define an Auslander bound for complexes and extend the module results to complexes.