Approximating Artinian Rings by Gorenstein Rings and 3-Standardness of the Maximal Ideal

Abstract

We study two different problems in this dissertation. In the first part, we wish to understand how one can approximate an Artinian local ring by a Gorenstein Artin local ring. We make this notion precise in Chapter 2, by introducing a number associated to an Artin local ring, called its Gorenstein colength . We study the basic properties and give bounds on this number in this chapter. We extend results due to W. Teter, C.Huneke and A. Vraciu by studying the relation of Gorenstein colength with self-dual ideals. In particular, we also answer the question as to when the Gorenstein colength is at most two. In Chapter 3, we show that there is a natural upper bound for Gorenstein colength of some special rings. We compute the Gorenstein colengths of these rings by constructing some Gorenstein Artin rings. We further show that the Gorenstein colength of Artinian quotients of two-dimensional regular local rings are also bounded above by the same upper bound by using a formula due to Hoskin and Deligne. Given two Gorenstein Artin local rings, L. Avramov and W. F. Moore construct another Gorenstein Artin local ring called a connected sum. We use this to improve a result of C. Huneke and A. Vraciu in Chapter 4. We also define the notion of a connected sum more generally and apply it to give bounds on the Gorenstein colengths of fibre products of Artinian local rings. In the second part of the thesis, we study a notion called n-standardness of ideals primary to the maximal ideal in a Cohen-Macaulay local ring. We first prove the equivalence of n-standardness to the vanishing of a certain Koszul homology module up to a certain degree. We go over the properties of Koszul complexes and homology needed for this purpose in Chapter 5. In Chapter 6, we study conditions under which the maximal ideal is 3-standard. We first prove results when the residue field is of prime characteristic and use the method of reduction to prime characteristic to extend the results to the characteristic zero case. As an application, we see that this helps us extend a result due to T. Puthenpurakal in which he shows that a certain length associated to a minimal reduction of the maximal ideal does not depend on the minimal reduction chosen.