论文标题
同源转移理想
Homological shift ideals
论文作者
论文摘要
对于单一的理想$ i $，我们认为$ i $ i $ $ i $ $ i $ $ i $ text {hs} _i（i）$表示的$ i $ th同源偏移理想，也就是说，这是$ i $ $ i $的$ i $ i $ th multigraded shift产生的理想。研究了该理想的一些代数特性。结果表明，对于任何单一的理想$ i $和任何单元理想$ p $，$ \ text {hs} _i（i（p））\ subseteq \ subseteq \ text {hs} _i _i（i）（p）$ for All $ i $，其中$ i（p）$是$ i $ i $ i $ $ i $的单一元素。特别是，我们考虑了一些具有线性商的单一理想家族的同源转变理想。对于任何$ \ textbf {c} $ - 有限的主borel理想$ i $，对于任何路径图的边缘理想，证明$ \ text {hs} _i _i（i）$具有所有$ i $的线性字样。作为$ \ textbf {c} $ - 有限的主要骨质理想的示例，考虑了Veronese类型的理想，并且表明这些理想的同源性转移理想是多功能的。这意味着，对于满足强大交换属性的任何多功能理想，$ \ text {hs} _j（i）$再次是所有$ j $的多摩托理想。 此外，对于线性分辨率的任何边缘理想，理想的$ \ text {hs} _j（i）$都是表征的，并且显示出$ \ text {hs} _1（i）$具有线性商。
For a monomial ideal $I$, we consider the $i$th homological shift ideal of $I$, denoted by $\text{HS}_i(I)$, that is, the ideal generated by the $i$th multigraded shifts of $I$. Some algebraic properties of this ideal are studied. It is shown that for any monomial ideal $I$ and any monomial prime ideal $P$, $\text{HS}_i(I(P))\subseteq \text{HS}_i(I)(P)$ for all $i$, where $I(P)$ is the monomial localization of $I$. In particular, we consider the homological shift ideal of some families of monomial ideals with linear quotients. For any $\textbf{c}$-bounded principal Borel ideal $I$ and for the edge ideal of complement of any path graph, it is proved that $\text{HS}_i(I)$ has linear quotients for all $i$. As an example of $\textbf{c}$-bounded principal Borel ideals, Veronese type ideals are considered and it is shown that the homological shift ideal of these ideals are polymatroidal. This implies that for any polymatroidal ideal which satisfies the strong exchange property, $\text{HS}_j(I)$ is again a polymatroidal ideal for all $j$. Moreover, for any edge ideal with linear resolution, the ideal $\text{HS}_j(I)$ is characterized and it is shown that $\text{HS}_1(I)$ has linear quotients.
