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引用次数: 6
摘要
设$S=K[x_1,\ldots,x_n]$是域上的多项式环,$ a $是标准的分级$S$-代数。根据定义理想$J$ ($A$)的Gröbner基,我们给出了一个条件,称为$x$-条件,它意味着$A$ ($A$)的所有分级分量$A_k$具有线性商,并且在附加假设下是分量线性的。这种代数的典型例子是一个分级理想的Rees环$\mathcal{R}(I)$或一个模$M$的对称代数$\textrm{Sym}(M)$。应用该判据研究了若干对称代数和若干图的顶点覆盖理想的幂。
Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gröbner basis of the defining ideal $J$ of $A$ we give a condition, called the $x$-condition, which implies that all graded components $A_k$ of $A$ have linear quotients and with additional assumptions are componentwise linear. A typical example of such an algebra is the Rees ring $\mathcal{R}(I)$ of a graded ideal or the symmetric algebra $\textrm{Sym}(M)$ of a module $M$. We apply our criterion to study certain symmetric algebras and the powers of vertex cover ideals of certain classes of graphs.
期刊介绍:
Mathematica Scandinavica is a peer-reviewed journal in mathematics that has been published regularly since 1953. Mathematica Scandinavica is run on a non-profit basis by the five mathematical societies in Scandinavia. It is the aim of the journal to publish high quality mathematical articles of moderate length.
Mathematica Scandinavica publishes about 640 pages per year. For 2020, these will be published as one volume consisting of 3 issues (of 160, 240 and 240 pages, respectively), enabling a slight increase in article pages compared to previous years. The journal aims to publish the first issue by the end of March. Subsequent issues will follow at intervals of approximately 4 months.
All back volumes are available in paper and online from 1953. There is free access to online articles more than five years old.