arXiv 1107.38992011-07-20math.AC

余维数为3的Artin水平代数

Artinian level algebras of codimension 3

Jeaman Ahn, Young Su Shin

备注：: 15 pages

摘要：

本文通过研究余维为3的Artinian水平代数满足条件$β_{2,d+2}(I^{\rm lex})=β_{1,d+1}(I^{\rm lex})$，继续研究哪些$h$ -向量$\H=(1,3,..., h_{d-1}, h_d, h_{d+1})$可以是水平代数的Hilbert函数，其中$I^{\rm lex}$是与理想$I$相关联的lex-segment理想.我们的方法是采用一种称为{ \it消去原理}的同调方法：通过消去同一移位的相邻项，从$I^{\rm lex}$的最小自由分辨率得到$I$的最小自由分辨率。证明了当$β_{1,d+2}(I^{\rm lex})=β_{2,d+2}(I^{\rm lex})$，$R/I$是Artinian水平$k$ -代数时，只有当$h_{d-1}<h_d<h_{d+1}$或$h_{d-1}=h_d=h_{d+1}=d+1$成立.我们还应用我们的结果表明，对于$\H=(1,3,..., h_{d-1}, h_d, h_{d+1})$，一个余维为3的Artinian代数的Hilbert函数满足条件$h_{d-1}=h_d<h_{d+1}$，（a）如果$h_d\leq 3d+2$，那么$h$ -向量$\H$不能是水平的，（b）如果$h_d\geq 3d+3$，那么对于$h_{d+1}$的某个值，存在一个具有Hilbert函数$\H$的水平代数。

英文摘要：

In this paper, we continue the study of which $h$-vectors $\H=(1,3,..., h_{d-1}, h_d, h_{d+1})$ can be the Hilbert function of a level algebra by investigating Artinian level algebras of codimension 3 with the condition $β_{2,d+2}(I^{\rm lex})=β_{1,d+1}(I^{\rm lex})$, where $I^{\rm lex}$ is the lex-segment ideal associated with an ideal $I$. Our approach is to adopt an homological method called {\it Cancellation Principle}: the minimal free resolution of $I$ is obtained from that of $I^{\rm lex}$ by canceling some adjacent terms of the same shift. We prove that when $β_{1,d+2}(I^{\rm lex})=β_{2,d+2}(I^{\rm lex})$, $R/I$ can be an Artinian level $k$-algebra only if either $h_{d-1}<h_d<h_{d+1}$ or $h_{d-1}=h_d=h_{d+1}=d+1$ holds. We also apply our results to show that for $\H=(1,3,..., h_{d-1}, h_d, h_{d+1})$, the Hilbert function of an Artinian algebra of codimension 3 with the condition $h_{d-1}=h_d<h_{d+1}$, (a) if $h_d\leq 3d+2$, then $h$-vector $\H$ cannot be level, and (b) if $h_d\geq 3d+3$, then there is a level algebra with Hilbert function $\H$ for some value of $h_{d+1}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文