论文标题
边缘理想几乎具有最大有限指数及其力量
Edge ideals with almost maximal finite index and their powers
论文作者
论文摘要
一个分级的理想$ i $ in $ \ mathbb {k} [x_1，\ ldots，x_n] $，其中$ \ mathbb {k} $是一个字段，如果其最小的自由分辨率是线性的，则几乎具有最大有限索引$ \ mathrm {pd}（i）-1 $，其中$ \ mathrm {pd}（i）$表示$ i $的投影尺寸。在本文中，我们将边缘理想具有此属性的图表进行分类。这尤其表明，对于边缘理想，具有几乎最大有限索引的属性并不取决于$ \ mathbb {k} $的特征。我们还计算了这些理想的非线性betti数字。最后，我们表明，对于边缘理想的$ i $，图$ g $带有几乎最大有限索引，理想的$ i^s $在$ s \ geq 2 $时具有线性分辨率，并且仅当互补图$ \ bar {g} $不包含诱发的长度周期的长度$ 4 $。
A graded ideal $I$ in $\mathbb{K}[x_1,\ldots,x_n]$, where $\mathbb{K}$ is a field, is said to have almost maximal finite index if its minimal free resolution is linear up to the homological degree $\mathrm{pd}(I)-2$, while it is not linear at the homological degree $\mathrm{pd}(I)-1$, where $\mathrm{pd}(I)$ denotes the projective dimension of $I$. In this paper we classify the graphs whose edge ideals have this property. This in particular shows that for edge ideals the property of having almost maximal finite index does not depend on the characteristic of $\mathbb{K}$. We also compute the non-linear Betti numbers of these ideals. Finally, we show that for the edge ideal $I$ of a graph $G$ with almost maximal finite index, the ideal $I^s$ has a linear resolution for $s\geq 2$ if and only if the complementary graph $\bar{G}$ does not contain induced cycles of length $4$.
