论文标题
狄拉克的定理和多面的syzygies
Dirac's theorem and multigraded syzygies
论文作者
论文摘要
令$ g $为简单的有限图。 Dirac的著名定理说，$ G $是和弦，并且仅当$ G $承认完美的消除订单时。弗罗伯格（Fröberg）知道，$ g $的边缘理想$ i（g）$ of $ g $具有线性分辨率，并且仅当互补图$ g^c $ $ g $是和弦。在本文中，我们在边缘理想的同源转移理想理论中讨论了Dirac定理的一些代数后果。回想一下，如果$ i $是单一的理想，则$ \ mbox {hs} _k（i）$是$ k $ th多式偏移$ i $的单一理想。我们证明$ \ mbox {hs} _1（i）$具有线性商，对于任何单一的理想$ i $，带有单个程度生成的线性商。对于带有线性商的理想$ i（g）$，$ \ mbox {hs} _k（i（g））$具有所有$ k \ ge0 $的线性商。另一方面，如果$ g^c $是适当的间隔图或森林，我们证明是这种情况。最后，我们讨论了Bandari，Bayati和Herzog的猜想，该猜想预测，如果$ i $是多肌功能，则$ \ mbox {hs} _k（i）$也是多肌动物，对于所有$ k \ ge0 $。我们能够证明该猜想适用于第二级产生的所有多功能理想。
Let $G$ be a simple finite graph. A famous theorem of Dirac says that $G$ is chordal if and only if $G$ admits a perfect elimination order. It is known by Fröberg that the edge ideal $I(G)$ of $G$ has a linear resolution if and only if the complementary graph $G^c$ of $G$ is chordal. In this article, we discuss some algebraic consequences of Dirac's theorem in the theory of homological shift ideals of edge ideals. Recall that if $I$ is a monomial ideal, $\mbox{HS}_k(I)$ is the monomial ideal generated by the $k$th multigraded shifts of $I$. We prove that $\mbox{HS}_1(I)$ has linear quotients, for any monomial ideal $I$ with linear quotients generated in a single degree. For and edge ideal $I(G)$ with linear quotients, it is not true that $\mbox{HS}_k(I(G))$ has linear quotients for all $k\ge0$. On the other hand, if $G^c$ is a proper interval graph or a forest, we prove that this is the case. Finally, we discuss a conjecture of Bandari, Bayati and Herzog that predicts that if $I$ is polymatroidal, $\mbox{HS}_k(I)$ is polymatroidal too, for all $k\ge0$. We are able to prove that this conjecture holds for all polymatroidal ideals generated in degree two.
