Regularity and h‐polynomials of monomial ideals
Regularity and h‐polynomials of monomial ideals
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DOI：
10.1002/mana.201700476
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发表时间：
2017-11
影响因子：
1
通讯作者：
T. Hibi;Kazunori Matsuda
中科院分区：
文献类型：
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作者：
T. Hibi;Kazunori Matsuda
设S=K[x1，.，xn]表示域K上的n元多项式环，且每个degxi=1，设I ∈ S是S的齐次理想，且dimS/I=d . S/I的希尔伯特级数的形式为hS/I（λ）/（1−λ）d，其中hS/I（λ）= h 0 +h1λ+h2λ2+ h2 +hsλs，其中hs 0是S/I的h-多项式。已知当S/I是Cohen-Macaulay时，有reg（S/I）=deghS/I（λ），其中reg（S/I）是S/I的（Castelnuovo-Mumford）正则性。本文对给定的任意整数r和s（r≥1，s≥1），构造了S=K[x1，..，xn]（n ≥ 0）中的一个单项理想I，其中reg（S/I）=r，deghS/I（λ）=s.进一步，我们给出了一类满足reg（S/I）=deghS/I（λ）且S/I不是Cohen-Macaulay的Cameron-Walker图的边理想I ∈ S.
Let S=K[x1,…,xn] denote the polynomial ring in n variables over a field K with each degxi=1 and let I⊂S be a homogeneous ideal of S with dimS/I=d . The Hilbert series of S/I is of the form hS/I(λ)/(1−λ)d , where hS/I(λ)=h0+h1λ+h2λ2+⋯+hsλs with hs≠0 is the h‐polynomial of S/I . It is known that, when S/I is Cohen–Macaulay, one has reg(S/I)=deghS/I(λ) , where reg(S/I) is the (Castelnuovo–Mumford) regularity of S/I . In the present paper, given arbitrary integers r and s with r≥1 and s≥1 , a monomial ideal I of S=K[x1,…,xn] with n≫0 for which reg(S/I)=r and deghS/I(λ)=s will be constructed. Furthermore, we give a class of edge ideals I⊂S of Cameron–Walker graphs with reg(S/I)=deghS/I(λ) for which S/I is not Cohen–Macaulay.