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Regularity and Koszul property of symbolic powers of monomial ideals
Mathematische Zeitschrift ( IF 1.0 ) Pub Date : 2021-01-02 , DOI: 10.1007/s00209-020-02657-8
Le Xuan Dung , Truong Thi Hien , Hop D. Nguyen , Tran Nam Trung

Let I be a homogeneous ideal in a polynomial ring over a field. Let $$I^{(n)}$$ I ( n ) be the n -th symbolic power of I . Motivated by results about ordinary powers of I , we study the asymptotic behavior of the regularity function $${{\,\mathrm{reg}\,}}(I^{(n)})$$ reg ( I ( n ) ) and the maximal generating degree function $$\omega (I^{(n)})$$ ω ( I ( n ) ) , when I is a monomial ideal. It is known that both functions are eventually quasi-linear. We show that, in addition, the sequences $$\{{{\,\mathrm{reg}\,}}I^{(n)}/n\}_n$$ { reg I ( n ) / n } n and $$\{\omega (I^{(n)})/n\}_n$$ { ω ( I ( n ) ) / n } n converge to the same limit, which can be described combinatorially. We construct an example of an equidimensional, height two squarefree monomial ideal I for which $$\omega (I^{(n)})$$ ω ( I ( n ) ) and $${{\,\mathrm{reg}\,}}(I^{(n)})$$ reg ( I ( n ) ) are not eventually linear functions. For the last goal, we introduce a new method for establishing the componentwise linearity of ideals. This method allows us to identify a new class of monomial ideals whose symbolic powers are componentwise linear.

中文翻译:

单项式理想符号幂的正则性和 Koszul 性质

让 I 成为域上多项式环中的齐次理想。令 $$I^{(n)}$$ I ( n ) 为 I 的第 n 个符号幂。受 I 的普通幂结果的启发,我们研究了正则函数 $${{\,\mathrm{reg}\,}}(I^{(n)})$$ reg ( I ( n ) ) 和最大生成度函数 $$\omega (I^{(n)})$$ ω ( I ( n ) ) ,当 I 是单项式理想时。众所周知,这两个函数最终都是拟线性的。我们表明,此外,序列 $$\{{{\,\mathrm{reg}\,}}I^{(n)}/n\}_n$$ { reg I ( n ) / n } n和 $$\{\omega (I^{(n)})/n\}_n$$ { ω ( I ( n ) ) / n } n 收敛到相同的极限,可以组合描述。我们构造了一个等维、高度为两个平方的自由单项式理想 I 的例子,其中 $$\omega (I^{(n)})$$ ω ( I ( n ) ) 和 $${{\,\mathrm{reg} \, }}(I^{(n)})$$ reg ( I ( n ) ) 最终不是线性函数。对于最后一个目标,我们引入了一种建立理想的分量线性的新方法。这种方法使我们能够识别一类新的单项式理想,其符号幂是分量线性的。
更新日期:2021-01-02
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