ABSTRACT
CRAIG HUNEKE Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA E-mail: huneke@math.ku.edu
DANIEL KATZ Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA E-mail: dlk@math.ku.edu
WOLMER V. VASCONCELOS Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA E-mail: vasconce@math.rutgers.edu
1 Introduction Let (R,m) be a local Noetherian ring. Given an R-ideal I of grade g, a closely related object to I is its integral closure I. This is the set (ideal, to be precise) of all elements in R that satisfy an equation of the form
Xm +b1Xm−1 +b2Xm−2 + · · ·+bm−1X +bm = 0,
with b j ∈ I j and m a non-negative integer. Clearly one has that I ⊆ I ⊆ √
I, where
√ I is the radical of I and consists instead of the elements of R that satisfy
an equation of the form Xq−b = 0 for some b ∈ I and q a non-negative integer. While [EHV] already provides direct methods for the computation of
√ I, the
nature of I is complex. Even the issue of validating the equality I = I is quite hard and relatively few methods are known [CHV]. In general, computing the integral closure of an ideal is a fundamental problem in commutative algebra. Although it is theoretically possible to compute integral closures, practical computations at present remain largely out-of-reach, except for some special ideals, such as monomial ideals in polynomial rings over a field. One known computational approach is through the theory of Rees algebras: It requires the computation of the integral closure of the Rees algebra R of I in R[t]. However, this method is potentially wasteful since the integral closures of all the powers of I are being computed at the same time. On the other hand, this method has the advantage that for the integral closure A of an affine algebra A there are readily available conductors: given A in terms of generators and relations (at least in characteristic zero) the Jacobian ideal Jac of A has the property that Jac ·A⊆ A, in other words, A ⊆ A : Jac. This fact is the cornerstone of most current algorithms to build A [deJ, V].
