论文标题
解决用2线性分辨率在边缘环上的猜想的解决方案
Solution to a conjecture on edge rings with 2-linear resolutions
论文作者
论文摘要
对于图$ g =（v，e）$ the edge ring $ k [g] $ is $ k [x_1，\ ldots，x_n]/i（g）$，其中$ n = | v | $和$ i（g）$由$ \ {x_ix_j; \ \ \ \ {i，j \} \ in e \ \ \ \ \} $生成。我们对待的猜想是以下内容。 如果$ k [g] $具有2条线的分辨率，则$ k [g] $，pd $（k [g]）$的投影尺寸等于$ g $的顶点的最大程度。 据我们所知，吉特勒和瓦伦西亚的一篇论文中首先提到了这种猜想，在那里被称为Eliahou-Villarreal猜想。该猜想是在艾哈迈德（Ahmed），马菲（Mafi）和纳米克（Namiq）最近的一份论文中对待的。 Moradi和Kiani已经注意到了反例。通过将$ k [g] $解释为史丹利 - 赖斯纳戒指，我们能够表征猜想所持的那些图形。
For a graph $G=(V,E)$ the edge ring $k[G]$ is $k[x_1,\ldots,x_n]/I(G)$, where $n=|V|$ and $I(G)$ is generated by $\{ x_ix_j;\{ i,j\}\in E\}$. The conjecture we treat is the following. If $k[G]$ has a 2-linear resolution, then the projective dimension of $K[G]$, pd$(k[G])$, equals the maximal degree of a vertex in $G$. As far as we know, this conjecture is first mentioned in a paper by Gitler and Valencia, and there it is called the Eliahou-Villarreal conjecture. The conjecture is treated in a recent paper by Ahmed, Mafi, and Namiq. That there are counterexamples was noted already by Moradi and Kiani. By interpreting $k[G]$ as a Stanley-Reisner ring, we are able to characterize those graphs for which the conjecture holds.
