论文标题
koszul多元素代数$ l $ -borel理想
Koszul multi-Rees algebras of principal $L$-Borel Ideals
论文作者
论文摘要
鉴于多项式环和多项式戒指变量的子集$ l $的单一$ m $，$ m $产生的主要$ l $ borel理想是所有单元所产生的理想，可以从$ m $中获得$ m $，通过在$ m $中替换$ m $的$ m $的$ m $，而这些变量$ m $由$ m $由$ $ l $ $ $ $ $和较小的索引。 Given a collection $\mathcal{I}=\{I_1,\ldots,I_r\}$ where $I_i$ is $L_i$-Borel for $i=1,\ldots,r$ (where the subsets $L_1,\ldots,L_r$ may be different for each ideal), we prove in essence that if the bipartite incidence graph among the subsets $ l_1，\ ldots，l_r $是和弦双方，然后是$ \ Mathcal {i} $的多元素代数的定义方程，具有在词汇图下具有方形铅词的四边形基础的基础。因此，这种理想集合的多元代数为Koszul，Cohen-Macaulay和正常。这显着概括了Ohsugi和Hibi在Koszul二分图上的定理。作为推论，我们可以获得一系列主要的Borel理想的多元代数是Koszul。为了证明我们的主要结果，我们在曲线图的内核中使用光纤gröbner基准标准，并引入了sturmfels的排序算法的修改。
Given a monomial $m$ in a polynomial ring and a subset $L$ of the variables of the polynomial ring, the principal $L$-Borel ideal generated by $m$ is the ideal generated by all monomials which can be obtained from $m$ by successively replacing variables of $m$ by those which are in $L$ and have smaller index. Given a collection $\mathcal{I}=\{I_1,\ldots,I_r\}$ where $I_i$ is $L_i$-Borel for $i=1,\ldots,r$ (where the subsets $L_1,\ldots,L_r$ may be different for each ideal), we prove in essence that if the bipartite incidence graph among the subsets $L_1,\ldots,L_r$ is chordal bipartite, then the defining equations of the multi-Rees algebra of $\mathcal{I}$ has a Gröbner basis of quadrics with squarefree lead terms under lexicographic order. Thus the multi-Rees algebra of such a collection of ideals is Koszul, Cohen-Macaulay, and normal. This significantly generalizes a theorem of Ohsugi and Hibi on Koszul bipartite graphs. As a corollary we obtain that the multi-Rees algebra of a collection of principal Borel ideals is Koszul. To prove our main result we use a fiber-wise Gröbner basis criterion for the kernel of a toric map and we introduce a modification of Sturmfels' sorting algorithm.
