A periodic module of infinite virtual projective dimension
A periodic module of infinite virtual projective dimension
复制标题
无限虚拟射影维数的周期模
DOI：
10.1016/0022-4049(89)90016-9
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发表时间：
1989
期刊：
影响因子：
--
通讯作者：
I. Peeva
中科院分区：
文献类型：
--
作者：
L. Avramov;Vesselin Gasharov;I. Peeva
如果 R 的完成 R̂ 可以以 Q/(x) 的形式表示，其中 x 是局部环 Q 中的规则序列，并且 M̂ 的投影维数（被视为 Q 模块）是有限的，则在局部环 R（具有无限剩余域）上移动的有限生成模块被称为具有有限的虚拟投影维数。众所周知，如果 Mare 的贝蒂数有界，则其虚拟射影维数的有限性意味着其最小自由分辨率最终成为周期 2 的周期性。本注释的目的是构造示例，以证明逆向方程一般不成立。
A finitely generated moduleMover a local ringR(with infinite residue field) is said to have finite virtual projective dimension, if the completionR̂ofRcan be presented in the formQ/(x), wherexis a regular sequence in the local ringQ, and the projective dimension ofM̂, viewed as aQ-module, is finite. It is known that if the Betti numbers ofMare bounded, then the finiteness of its virtual projective dimension implies that its minimal free resolution eventually becomes periodic of period 2. The purpose of this note is to construct examples which show that the converse does not hold in general.