Symbolic Powers and other Contractions of Ideals in Noetherian Rings

The results in this thesis are motivated by the following four questions:;1. (Eisenbud-Mazur conjecture): Given a regular local ring ( R, m ) containing a field of characteristic zero, and an unmixed ideal I in R, I(2) ⊆ mI .;2. (Integral closedness of mI ) Given a regular local ring (R, m ) and a radical ideal I ⊂ R, when is mI integrally closed?;3. (Uniform bounds on symbolic powers) Given a complete local domain R, is there a constant k such that for any prime ideal P ⊂ R, P (kn) ⊆ Pn?;4. (General contractions of powers of ideals) Given an extension of Noetherian rings R ⊆ S and an ideal J in S what can be said about the behavior of In := Jn ∩ R?;It is shown that if I is an ideal generated by a single binomial and several monomials in a polynomial ring over a field where m is the homogeneous maximal ideal, mI is integrally closed. One of the main results is to show that in a Noetherian local ring (R, m ), if I = (a1,..., ad)R is integrally closed and mIi ⊆ Ii for 1 ≤ i ≤ d where Ii = (a 1,...,âi,...,a d)R, then, mI is integrally closed. The Eisenbud-Mazur conjecture is shown to hold for the case of certain prime ideals in certain subrings of a formal power series ring over a field. Some computational results using Macaulay2 are discussed. For a Noetherian complete local domain (R, m ), it is shown that there exists a function beta : Z>0→Z >0 such that for any prime ideal P in R we have, P(beta(n )) ⊆ mn . Suppose R ⊆ S is a module-finite extension of domains and R is normal while S is regular, equicharacteristic, then, under mild conditions on R, S, it is shown that there exists a positive integer c such that for any prime ideal P in R, P(cn) ⊆ Pn. It is shown that if R is a principal ideal domain and I an ideal in R[ x], then, In ∩ R = (I ∩ R)n (which in turn implies that the ring ⊕infinityi=1 (In ∩ R) is Noetherian). The corresponding statement is shown to be false in general if R is polynomial ring over a field in more than 1 indeterminate. The rings ⊕infinityi=1 (In ∩ R) are shown to be Noetherian for certain family of ideals I generated by one binomial and several monomials in polynomial rings R in several indeterminates over a field.