短局部环的剩余域的线性亏损
Linearity defect of the residue field of short local rings
摘要：
设$(R,\m,k)$是具有极大理想$\m$和剩余域$k$的Noether局部环.表示为$\ld_R(M)$的可编程生成的$R$ -模块$M$的线性缺陷是$M$距离具有线性分辨率有多远的数值度量。研究了剩余域的线性缺陷。对于Herzog和Iyengar提出的问题，即当$\m^4=0$时，$\ld_R(k)<\infty$是否蕴涵$\ld_R(k)=0$，我们给出了肯定的回答.
英文摘要：
Let $(R,\m,k)$ be a Noetherian local ring with maximal ideal $\m$ and residue field $k$. The linearity defect of a finitely generated $R$-module $M$, which is denoted $\ld_R(M)$, is a numerical measure of how far $M$ is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether $\ld_R(k)<\infty$ implies $\ld_R(k)=0$, in the case when $\m^4=0$.
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