Bounds on Bass Numbers and Their Dual

Let (R,m) be a commutative Noetherian local ring. We establish some bounds for the sequence of Bass numbers and their dual for a finitely generated R-module. Introduction Throughout this paper, (R,m, k) is a non-trivial commutative Noetherian local ring with unique maximal ideal m and residue field k. Several authors have obtained results on the growth of the sequence of Betti numbers {βn(k)} (e.g., see [9] and [1]). In [10] Ramras gives some bounds for the sequence {βn(M)} when M is a finitely generated non-free R-module. In this paper, we seek to give some bounds for the sequence of Bass numbers. For a finitely generated R-module M , let 0 → M → E → E → · · · → E → · · · be a minimal injective resolution of M . Then, μ(M) denotes the number of indecomposable components of E isomorphic to the injective envelope E(k) and is called Bass number of M . This is a dual notion of Betti number. For a prime ideal p, μ(p,M) denotes the number of indecomposable components of E isomorphic to the injective envelope E(R/p). It is known that μ(M) is finite and is equal to the dimension of Ext i R (R/m,M) considered as a vector space over R/m (note that μ(p,M) = μi(Mp)). These numbers play important role in understanding the injective resolution of M , and are the subject of further work. For example, the ring R of dimension d is Gorenstein if and only if R is Cohen-Macaulay and the dth Bass number μ(R) is 1. This was proved by Bass in [2]. Vasconcelos conjectured that one could delete the hypothesis that R be Cohen-Macaulay. This was proved by Paul Roberts in [12]. For a finitely generated R-module M , it turns out that the least i for which μ(M) > 0 is the depth of M , while the largest i with μ(M) > 0 is the injective 2000 Mathematics Subject Classification : 13C11, 13H10.