Geometric regularity of powers of two-dimensional squarefree monomial ideals
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Geometric regularity of powers of two-dimensional squarefree monomial ideals Dancheng Lu1 Received: 30 August 2018 / Accepted: 18 February 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let I be a two-dimensional squarefree monomial ideal of a polynomial ring S. We evaluate the geometric regularity, ai -invariants of S/I n for i ≥ 2. It turns out that they are all linear functions in n from n = 2. Also, it is shown that g-reg(S/I n ) = reg(S/I (n) ) for all n ≥ 1. Keywords Two-dimensional squarefree monomial ideal · Local cohomology · Geometric regularity Mathematics Subject Classification Primary 13D45; Secondary 13C99
Introduction Let S := K [x1 , . . . , xr ] be the polynomial ring in variables x1 , . . . , xr over a field K and m the maximal homogeneous ideal of S. Let M be a finitely generated graded S-module. For each 0 ≤ i ≤ dim M, the ai -invariant of M is defined by i (M)t = 0}, ai (M) := max{t : Hm i (M) is the ith local cohomology module of M with support in m, and we where Hm understand max ∅ = −∞. The regularity of M is defined by
reg(M) := max{ai (M) + i : 0 ≤ i ≤ dim M}. Let I be a homogeneous ideal of the polynomial ring S. It was proved that reg(S/I n ) is a linear function in n for n 0; see [6,14,22]. In other words, there exist integers d, e and n 0 such that reg(S/I n ) = dn + e for all n ≥ n 0 . Based on this result,
B 1
Dancheng Lu [email protected] School of Mathematical Sciences, Soochow University, 215006 Suzhou, People’s Republic of China
123
Journal of Algebraic Combinatorics
many authors have studied properties of the regularity of powers of homogeneous ideals. Roughly speaking, their researches fall into two classes. The one is devoted to understanding the nature of integers d, e and n 0 for some special or general ideals I ; see e.g., [4,7,9]. The other is to computing explicitly or to bounding the regularity function reg(S/I n ) for some special classes of ideals I ; see e.g., [1,2,13]. Let be a simplicial complex on [r ] := {1, 2, . . . , r }. The Stanley–Reisner ideal of is defined to be the ideal of S I := (x F : F is a minimal non-face of ), where x F is the squarefree monomial i∈F xi . Every two-dimensional squarefree monomial ideal containing no variables is the Stanley–Reisner ideal of a simplicial complex of dimension one. Note that a simplicial complex of dimension one can be regarded as a simple graph that may contains isolated vertices. Two-dimensional squarefree monomial ideals attract many authors’ interests. For example, the Buchsbaum property of symbolic powers and ordinary powers of these ideals was studied in [15] and [16], respectively, and the Cohen–Macaulayness of symbolic powers and ordinary powers of such ideals was characterized in terms of the properties of their associated graphs in [17]. Recently, the regularity of symbolic powers of such ideals was computed explicitly in [12]. Inspired by the regularity for sheaves on projective spaces, M.E. Rossi et al. introduced the following weaker but natural
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