Asymptotic Sequences and Quasi-Unmixed Rings

In this chapter we give various characterizations of quasi-unmixed local rings. Most of the results are contained in papers by Ratliff, although with different proofs. Recently it has been recognized that a very useful tool for these characterizations are asymptotic sequences, which are somewhat analogous to regular sequences for the characterization of local Cohen-Macaulay rings. The theory of asymptotic sequences has been developed by Ratliff [10] and independently by Katz [8]. Our treatment follows closely the treatment by Katz [8]. We start by giving some auxiliary results.