# A study on the core of ideals

#### Abstract

Let *R* be a local Gorenstein ring with infinite residue field *k* and let *I* be an *R*-ideal. The core of *I*, core(*I*), is defined to be the intersection of all (minimal) reductions. We will usually assume that * I* satisfies *G*_{ℓ} and depth * R/I ^{j}* ≥ dim

*R/I - j*+ 1 for 1 ≤

*j*≤ ℓ -

*g*, where ℓ = ℓ(

*I*) is the analytic spread of

*I*and

*g*= ht

*I*> 0. Under these conditions Polini and Ulrich show that if char

*k*= 0 or char

*k*>

*r*(

_{J}*I*) - ℓ +

*g*then core(

*I*) =

*J*

^{n}^{+1}:

*I*=

^{n}*J*

^{n}^{ +1}: y∈I (

*J,y*)

*for*

^{n}*n*≥ max{

*r*(

_{J}*I*) - ℓ +

*g*, 0} and any minimal reduction

*J*of

*I*([27]). ^ This formula for the core depends on the characteristic of the residue field

*k*. We propose a conjecture for the core of such an ideal that should hold in any characteristic. We exhibit a series of examples that support this conjecture. Using the computer algebra program Macaulay 2 ([8]) we provide examples where

*J*

^{n}^{ +1}:

*I*⊊ core (

^{n}*I*) ⊊

*J*

^{n}^{+1}: y∈I (

*J,y*)

*, and we prove several theoretical results that support the validity of these computations. ^ We then provide new classes of ideals*

^{n}*I*for which core(

*I*) =

*J*

^{n}^{+1}:

*I*for

^{n}*n*>> 0 and any minimal reduction

*J*of

*I*. Our main theorem states that if in addition

*k*is perfect and the special fiber ring F (

*I*) of

*I*has embedding dimension at most 1 locally at every minimal prime of maximal dimension, then core (

*I*) =

*J*

^{n}^{+1}:

*I*for

^{n}*n*≥ max{

*r*(

_{ J}*I*) - ℓ +

*g*, 0} and every minimal reduction

*J*of

*I*. ^ We give several applications of our main theorem that show different instances where the above formula holds. In addition we obtain a description for the core of a power of the homogeneous maximal ideal of a standard graded Cohen-Macaulay

*k*-algebra. ^ Finally we investigate the connection between the core and the reduction number. We prove that if the associated graded ring gr

*(*

_{I}*R*) of

*I*is Cohen-Macaulay then core(

*I*) =

*J*+1 :

^{n}*I*for every minimal reduction

^{n}*J*of

*I*if and only if

*n*≥ max{

*r*(

*I*) - ℓ(

*I*) +

*g*, 0}. ^

#### Degree

Ph.D.

#### Advisors

Bernd Ulrich, Purdue University.

#### Subject Area

Mathematics

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