边理想的普通幂与符号幂在正则性和深度方面的比较
Comparisons between ordinary and symbolic powers of edge ideals with respect to regularity and depth
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中文总结 AI 辅助
本文研究边理想普通幂与符号幂在正则性和深度上的差异，证明单纯图情形下二者正则性一致，并给出深度不等式成立条件及差异模的Krull维数图论公式。
中文摘要 AI 辅助
本文研究了边理想的普通幂与符号幂在Castelnuovo-Mumford正则性和深度上的差异。作为主要定理，我们证明了对于单纯图的边理想，其普通幂与符号幂的正则性是一致的，这是Minh猜想的一个部分结果。关于深度，我们首先证明，对于每个$k\geq2$，不等式$\operatorname{depth} S/I^{(k)}\geq\operatorname{depth} S/I^{k}$对于一般平方自由单项理想并不成立。另一方面，我们证明当$k=2,3$时，该不等式对边理想成立。我们还研究了图$G$的边理想$I(G)$的符号-普通差异模$I(G)^{(k)}/I(G)^{k}$，并给出了其Krull维数的一个图论公式，该公式用$G$的诱导奇圈表示，从而回答了Ha和Minh提出的一个问题和一个问题。
英文摘要
In this paper, we investigate the difference between ordinary and symbolic powers of edge ideals on the Castelnuovo-Mumford regularity and the depth. As a main theorem, we prove that the regularities of ordinary and symbolic powers coincide for edge ideals of simplicial graphs, as a partial result of Minh's conjecture. For the depth, we first show that, for each $k\geq2$, the inequality $\operatorname{depth} S/I^{(k)}\geq\operatorname{depth} S/I^{k}$ does not hold for squarefree monomial ideals in general. On the other hand, we prove that it holds for edge ideals when $k=2,3$. We also study the symbolic-ordinary discrepancy module $I(G)^{(k)}/I(G)^{k}$ of the edge ideal $I(G)$ of a graph $G$ and give a graph-theoretic formula for its Krull dimension in terms of induced odd cycles of $G$, thereby answering a question and a problem posed by Ha and Minh.
发表机构
- Okayama University（冈山大学）
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