A study on the core of ideals

Louiza Fouli, Purdue University


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 and depth R/Ij ≥ 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 > rJ(I) - ℓ + g then core(I) = Jn+1 : In = Jn +1 : y∈I (J,y)n for n ≥ max{rJ(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 Jn +1 : In ⊊ core (I) ⊊ Jn+1 : y∈I (J,y)n, and we prove several theoretical results that support the validity of these computations. ^ We then provide new classes of ideals I for which core( I) = Jn+1 : In for 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) = Jn+1 : In for 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 grI (R) of I is Cohen-Macaulay then core( I) = Jn+1 : In for every minimal reduction J of I if and only if n ≥ max{ r(I) - ℓ(I) + g, 0}. ^




Bernd Ulrich, Purdue University.

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