Bounding Hilbert coefficients of parameter ideals

Abstract

Abstract. Let (R, m) be a Noetherian local ring of dimension d > 0 and depth R ≥ d − 1. Let Q be a parameter ideal of R. In this paper, we derive uniform lower and upper bounds for the Hilbert coefficient ei(Q) under certain assumptions on the depth of associated graded ringFurther, we obtain a necessary condition for the vanishing of the last coefficient e d (Q). As a consequence, we characterize the vanishing of e2(Q

Castelnuovo-Mumford Regularity and Hilbert Coefficients of Parameter Ideals

“…This implies that if d = 4, depth(A) ≥ 3 and σ(Q) ≥ 2, then e i (Q) ≤ 0 for i = 1, 2, 3, 4. From Proposition 3.5, it follows an early result of Saikia-Saloni [20] and Linh-Trung [13]. As a consequence, it follows that if d ≥ 3 and depth(A) ≥ d − 1, then e i (Q) ≤ 0 for i = 1, 2, 3.…”

“…However, if depth(A) ≥ d − 1, McCune [16] showed that e 2 (Q) ≤ 0 for every parameter ideal Q. In [20], Saikia and Saloni proved that if depth(A) ≥ d − 1, then e 3 (Q) ≤ 0 for every parameter ideal Q. In [16], McCune also proved that if Q is a parameter ideal such that depth(G(Q)) ≥ d − 1, then e i (Q) ≤ 0 for i = 1, .…”

“…. , d. Recently, Saikia-Saloni [20] and Linh-Trung [13] proved that if depth(A) ≥ d − 1 and Q is a parameter ideal such that depth(G(Q)) ≥ d − 2, then e i (Q) ≤ 0 for i = 1, . .…”

On the Hilbert coefficients and reduction numbers

“…showed that 2 ( ) ≤ 0 and Saikia-Saloni [3] proved that 3 ( ) ≤ 0 for every parameter ideal . Recently, Linh-Trung [4] proved that if ℎ( ) ≥ − 1 and is a parameter ideal such that ℎ( ( )) ≥ − 2 , then ( ) ≤ 0 for = 1, … , .…”

On Hilbert coefficients of parameter ideals and Cohen-Macaulayness

“…We obtain that e 3 (I) ≤ 0 if e 2 (I) ≤ 2 and e 3 (I) ≤ 2 if e 2 (I) = 3. Discussions on the signature and vanishing of Hilbert coefficients can be found in [2], [3], [6], [7], [14], [17], [19], [28] etc. Suppose depth G(I) ≥ d − 1.…”