Betti Sequences over Standard Graded Commutative Algebras with Two Relations
Betti Sequences over Standard Graded Commutative Algebras with Two Relations
复制标题
具有两个关系的标准分级交换代数上的 Betti 序列
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Zheng Yang
中科院分区：
文献类型：
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作者：
L. Avramov;Zheng Yang
众所周知，如果 Q 是域上 d 个变量的标准分级多项式环，并且 R 是 Q 与 (mathop{mathrm{proj,dim}}
olimits _{Q}R leq 1)，则每个 R 模块 N 的 Betti 序列在 d 步后为常数。我们证明 if (mathop{mathrm{proj,dim}}
olimits _{Q}R = 2)，则 N 的 Betti 序列由已知长度的某个初始段显式确定，可以从 N 的表示中读出。存在三种情况，具体取决于 R 的定义关系的程度以及它们是否具有公约数。
It is well known that if Q is a standard graded polynomial ring in d variables over a field, and R is a quotient of Q with (mathop{mathrm{proj,dim}}
olimits _{Q}R leq 1), then the Betti sequence of every R-module N is constant after d steps. We prove that if (mathop{mathrm{proj,dim}}
olimits _{Q}R = 2), then the Betti sequence of N is determined explicitly by some initial segment of known length, which can be read off a presentation of N. There are three cases, depending on the degrees of the defining relations of R and on whether these have a common divisor.