an:07268500
Zbl 1453.13078
Cioffi, Francesca
The range of all regularities for polynomial ideals with a given Hilbert function
EN
J. Algebra 566, 435-442 (2021).
00454716
2021
j
13P10 14Q99 68W30 11Y55 13C99
Castelnuovo-Mumford regularity; Hilbert function; minimal function; strongly stable ideal
Let \(A\) be a polynomial ring over a field \(K\) and let \(I\) denote any homogeneous ideal of \(A\). This paper presents a proof of the following statement suggested by Le Tuan Hoa in a private communication,
Theorem 1: Given an Hilbert function \(u\), the set \(\{reg(I) : A/I \text{ has Hilbert function } u \}\) is an interval of integers.
Since for every homogenous ideal \(I\), there exists a strongly stable ideal \(J\) such that \(reg(I)=reg(J)\) and \(reg(J)\) is the highest degree, denoted by \(\nu(J)\), of a minimal generator of \(J\), the theorem 1 is equivalent to the following one,
Theorem 2: Given an Hilbert function \(u\), the set \(\{\nu(J) : J \text{ is strongly stable and }A/J \text{ has Hilbert function } u\}\) is an interval of integers.
The proof is based on constructive arguments presented in the paper [\textit{F. Cioffi} et al., Exp. Math. 24, No. 4, 424--437 (2015; Zbl 1333.13028)], mainly on a construction called \textit{expanded lifting}, that allows, given a saturated homogeneous ideal \(I\) with Hilbert function \(\preceq u\), to create a homogenous ideal \(J\) with Hilbert function \(u\).
Gema Maria Diaz Toca (Murcia)
Zbl 1333.13028