On the depth of certain graded rings associated to an ideal

Anna Guerrieri, Purdue University

Abstract

Let (R,m) be a Cohen-Macaulay local ring of dimension $d > 0$ and let $I \subseteq R$ be an ideal. In this thesis we individuate conditions on the ideal I that allow us to obtain information on depth gr$\sb{I}(R),$ depth$R\lbrack It\rbrack,$ depthS(I) and depth$S(I/I\sp2).$ In Chapter II we analyze a class of m-primary ideals such that the depth of gr$\sb{I}(R)$ is at least $d-2.$ In particular we prove the following results. Let I be an m-primary ideal of R and $J \subseteq I$ a minimal reduction of I minimally generated by d elements such that $$\sum\sb{k\ge2} \lambda(I\sp{k}\cap J/I\sp{k-1}J) = 1,$$where $\lambda$ denotes the length function. Then depth gr$\sb{I}(R) = d - 1.$ Moreover if $J\subseteq I$ is a minimal reduction of I minimally generated by d elements such that $\lambda(I\sp2\ \cap J/IJ) = 2$ and $I\sp{n}\ \cap J = I\sp{n-1} J$ for all $n\ge3,$ then depth gr$\sb{I}(R) \ge d - 2.$ In Chapter III we approach the problem with homological techniques. We show that the module $${\bigoplus\limits\sb{k\ge2}}{I\sp{k}\ \cap J\over I\sp{k-1}J}$$is isomorphic to the first homology of a particular Koszul complex. We generalize a result of Lichtenbaum and we give a new proof of one of the previous results. In Chapter IV we find conditions on I that allow us to individuate simple relations among depth $R\lbrack It\rbrack,$ depth gr$\sb{I}(R),$ depth S(I) and depth $S(I/I\sp2).$ These relations are useful also from a practical point of view since it is quite difficult to evaluate depth gr$\sb{I}(R)$ and depth $S(I/I\sp2)$ even with the help of a computer. Furthermore we study the class of ideals that satisfy one of the required conditions and we show that ideals generated by quadratic sequences are in this class.

Degree

Ph.D.

Advisors

Huneke, Purdue University.

Subject Area

Mathematics

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