The Direct Summ and Conjecture and Behavior of Codimension in Graded Extensions.

This thesis deals with the following question on regular Noetherian rings: If R is a regular Noetherian ring and S (R-HOOK) R is module-finite over R (i.e. S is an R-algebra containing s(,1),(.)(.)(.),s(,k) such that S = Rs(,1) + (.)(.)(.) + Rs(,k)), then is R (L-HOOK) S a direct summ and of S as an R-module? An affirmative answer to the question for the cases of degree p extensions and certain degree p('2) extensions, in the case where p is not a unit in R, is given. Some new examples of normal non-Cohen-Macaulay rings are given which are particularly interesting because they are not Cohen-Macaulay purely because p is not a unit in R. A part of this work deals with a general question of very great interest: Given an ideal I of a Noetherian ring S, what is the supremum of the height (codimension) of IT as T runs over all finitely generated S-algebras? A small class of algebras where the same supremum can be obtained is given but examples show that there are significant obstructions to sharpening the result to answer the first question. One of the main tools is Zariski's Main Theorem--but it is necessary to prove a kind of generalization for Z('N)-graded rings. A discussion of questions related to the first question is also included.Ph.D.MathematicsUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/159583/1/8324220.pd