Restricted homological dimensions and Cohen-Macaulayness
Restricted homological dimensions and Cohen-Macaulayness
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DOI：
10.1006/jabr.2001.9115
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发表时间：
2002-05
影响因子：
0.9
通讯作者：
Lars Christensen;H. Foxby;Anders J. Frankild
中科院分区：
文献类型：
--
作者：
Lars Christensen;H. Foxby;Anders J. Frankild
摘要经典的同调维数--投射维数、平坦维数和内射维数--通常用解析度来定义，然后用相应的导函子的消失来证明它们是可计算的。在本文中，我们定义了限制同调维数的消失相同的派生函子，但在类的测试模块，限制，以确保自动有限交换Noether环的有限Krull维数。当环是局部的，我们使用的方法从经典的交换代数和理论的同调维数的混合物表明，消失这些函子揭示了基础环是一个科恩-麦考利环，或至少接近。
Abstract The classical homological dimensions—the projective, flat, and injective ones—are usually defined in terms of resolutions and then proved to be computable in terms of vanishing of appropriate derived functors. In this paper we define restricted homological dimensions in terms of vanishing of the same derived functors but over classes of test modules that are restricted to assure automatic finiteness over commutative Noetherian rings of finite Krull dimension. When the ring is local, we use a mixture of methods from classical commutative algebra and the theory of homological dimensions to show that vanishing of these functors reveals that the underlying ring is a Cohen–Macaulay ring—or at least close to being one.