Reduction numbers and balanced ideals

Summary

Let R be a Noetherian local ring and let I be an ideal in R. The ideal I is called balanced if the colon ideal J : I is independent of the choice of the minimal reduction J of I . Under suitable assumptions, Ulrich showed that I is balanced if and only if the reduction number, r (I ), of I is at most the “expected” one, namely ℓ(I )−ht I +1, where ℓ(I ) is the analytic spread of I . In this article we propose a generalization of balanced. We prove under suitable assumptions that if either R is one-dimensional or the associated graded ring of I is Cohen–Macaulay, then J ⁿ⁺¹ : I ⁿ is independent of the choice of the minimal reduction J of I if and only if r (I ) ≤ ℓ(I )−ht I +n.

Let R be a Noetherian ring and let I be an ideal in R. The Rees algebra (I ) and the associated graded ring gr_I (R) of I are

(I ) = R[I t] =I i ti and grI (R) = R[I t]/I R[I t] =I i/I i+1. i≥0 i≥0

The projective spectrums of (I ) and grI (R) are the blowup of Spec(R) along V(I ) and the normal cone of I , respectively. When studying various algebraic properties of these blowups a natural question to consider is which properties of the ring R are transferred to these graded algebras. When R is a local Cohen–Macaulay ring and I an ideal of positive height then if (I ) is Cohen– Macaulay then so is grI (R) [Huneke 1982]. The converse does not hold true in general. A celebrated theorem of Goto and Shimoda illustrates the intricate relationship between the Cohen–Macaulay property of these blowup algebras and the reduction number of I . It states that when (R,m) is a local Cohen–Macaulay ring, with infinite residue field, dimension d > 0, and I an m-primary ideal, then (I ) is Cohen–Macaulay if and only if grI (R) is Cohen–Macaulay and the reduction number of I is at most d − 1 [Goto and Shimoda 1982]. This theorem has inspired the work of many researchers and many generalizations of it appeared in the literature in the late 1980s and early 1990s; see, for example, [Grothe et al. 1984; Huckaba and Huneke 1992; 1993; Goto and Huckaba 1994; Johnston and Katz 1995; Aberbach et al. 1995; Simis et al. 1995].