A note on Ratliff-Rush filtration, reduction number and postulation number of [formula omitted]-primary ideals

Abstract

Let (R,m) be a Cohen-Macaulay local ring of dimension d≥2, and I an m-primary ideal. Let r(I) be the reduction number of I, n(I) the postulation number and ρ(I) the stability index of the Ratliff-Rush filtration with respect to I. We prove that for d=2, if n(I)=ρ(I)−1, then r(I)≤n(I)+2, and if n(I)≠ρ(I)−1, then r(I)≥n(I)+2. For d≥3, assuming I is integrally closed, depthgr(I)=d−2, and n(I)=−(d−3), we prove that r(I)≥n(I)+d. Our main result generalizes a result by Marley on the relation between the Hilbert-Samuel function and the Hilbert-Samuel polynomial by relaxing the condition on the depth of the associated graded ring to the good behavior of the Ratliff-Rush filtration with respect to I mod a superficial sequence. From this result, it follows that for Cohen-Macaulay local rings of dimension d≥2, if PI(k)=HI(k) for some k≥ρ(I), then PI(n)=HI(n) for all n≥k.