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arXiv 2109.09268math.ACmath.CO
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边理想的小幂的整闭包及其正则性

Integral closure of small powers of edge ideals and their regularity

Nguyen Cong Minh, Thanh Vu

备注：
22 pages. arXiv admin note: text overlap with arXiv:2109.06396, arXiv:2109.05242. Add a section about integral closure of powers of Stanley-Reisner ideals of one-dimensional simplicial complexes
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摘要：

设$I(G)$是域k上单图$G$的边理想.证明了对所有的$s \le 4$，都有$${\rm reg}(\overline {I(G)^s}) = {\rm reg}(I(G)^s),$$.此外，我们提供了一个图$G$的例子，使得对于所有$s \ge 1.$，$${\rm reg} I(G)^s = {\rm reg} \overline{I(G)^s} = {\rm reg} I(G)^{(s)} = \begin{cases} 5 + 2s & \text{ if char k} = 2 \\ 4 + 2s & \text{ if char k} \neq 2, \end{cases}$$

英文摘要：

Let $I(G)$ be the edge ideal of a simple graph $G$ over a field k. We prove that $${\rm reg}(\overline {I(G)^s}) = {\rm reg}(I(G)^s),$$ for all $s \le 4$. Furthermore, we provide an example of a graph $G$ such that $${\rm reg} I(G)^s = {\rm reg} \overline{I(G)^s} = {\rm reg} I(G)^{(s)} = \begin{cases} 5 + 2s & \text{ if char k} = 2 \\ 4 + 2s & \text{ if char k} \neq 2, \end{cases}$$ for all $s \ge 1.$

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