Absolute, Relative, and Tate Cohomology of Modules of Finite Gorenstein Dimension
Absolute, Relative, and Tate Cohomology of Modules of Finite Gorenstein Dimension
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DOI：
10.1112/s0024611502013527
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发表时间：
2002-09
影响因子：
1.8
通讯作者：
L. Avramov;A. Martsinkovsky
中科院分区：
文献类型：
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作者：
L. Avramov;A. Martsinkovsky
本文研究了环R上的N-生成模M，两边都是Noether。若M在Auslander和布里杰意义下有有限Gorenstein维数G‐dimRM，则它确定了除绝对上同调函子ExtRn（M，）给出的上同调理论之外的另外两个上同调理论。相对上同调函子ExtGn（M，）定义为所有非负整数n;它们将Gorenstein维数为0的模视为投射模，并且当n > G-dimRM时为零。对所有整数n定义Tate上同调函子Ext^Rn（M，N）;所有群Ext^Rn（M，N）为零，若M或N有有限投射维数。比较态射εGn：ExtGn（M，）→ExtRn（M，）和εRn：ExtRn（M，）→Ext^Rn（M，）连接这些函子。我们给出了有限G维模的自包含处理，建立了相对上同调和Tate上同调的基本性质，并将比较态射嵌入到规范长正合序列0→ ExtG 1（M，）→ExtGn（M，）→ExtRn（M，）→Ext^Rn（M，）→ExtGn+1（M，）→ ExtGn中。这些结果为计算交换局部环上模的新旧数值不变量提供了有效的工具。
We study finitely generated modules M over a ring R, noetherian on both sides. If M has finite Gorenstein dimension G‐dimRM in the sense of Auslander and Bridger, then it determines two other cohomology theories besides the one given by the absolute cohomology functors ExtRn(M, ) . Relative cohomology functors ExtGn(M, ) are defined for all non‐negative integers n; they treat the modules of Gorenstein dimension 0 as projectives and vanish for n > G‐dimRM. Tate cohomology functors Ext^Rn(M, ) are defined for all integers n; all groups Ext^Rn(M,N) vanish if M or N has finite projective dimension. Comparison morphisms εGn:ExtGn(M, )→ExtRn(M, ) and εRn:ExtRn(M, )→Ext^Rn(M, ) link these functors. We give a self‐contained treatment of modules of finite G‐dimension, establish basic properties of relative and Tate cohomology, and embed the comparison morphisms into a canonical long exact sequence 0→ExtG1(M, )→⋯→ExtGn(M, )→ExtRn(M, )→Ext^Rn(M, )→ExtGn+1(M, )→⋯ . We show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings.