Homological properties of balanced Cohen-Macaulay algebras
Homological properties of balanced Cohen-Macaulay algebras
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平衡科恩-麦考莱代数的同调性质
DOI：
10.1090/s0002-9947-02-03166-5
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发表时间：
2002
影响因子：
1.3
通讯作者：
I. Mori
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文献类型：
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作者：
I. Mori
平衡Cohen-Macaulay代数是在Yekutieli（1992）意义上对某整数d和某分级A-A双模ω A具有平衡对偶复数ω A [d]的连通代数A。研究了平衡Cohen-Macaulay代数的一些同调性质。特别地，我们将证明以下定理：定理0.1。设A为Noetherian平衡Cohen-Macaulay代数，M为非零有限生成的分级左A模。: 1。M的有限分辨率为0→ω a （-l mj）→…ω A （1j）→H→M→0，其中H是有限生成的极大Cohen-Macaulay分级左A模。2. M具有有限内射维当且仅当M具有0→ω a （-l mj）→…ωA （-l 1j）→ωA （-l 0j）→M→0。作为推论，我们将得到As Gorenstein代数和As正则代数的下列特征：设A是一个noether平衡Cohen-Macaulay代数。1. A是AS Gorenstein当且仅当ω A作为分级左A模具有有限投影维数。2. 当且仅当每个有限生成的极大Cohen-Macaulay分级左A模都是自由模时，A是AS正则模。
A balanced Cohen-Macaulay algebra is a connected algebra A having a balanced dualizing complex ω A [d] in the sense of Yekutieli (1992) for some integer d and some graded A-A bimodule ω A . We study some homological properties of a balanced Cohen-Macaulay algebra. In particular, we will prove the following theorem: Theorem 0.1. Let A be a Noetherian balanced Cohen-Macaulay algebra, and M a nonzero finitely generated graded left A-module. Then: 1. M has a finite resolution of the form 0 →ω A (-l mj )→...ω A (l 1j )→H→M→0, where H is a finitely generated maximal Cohen-Macaulay graded left A-module. 2. M has finite injective dimension if and only if M has a finite resolution of the form 0 → ω A (-l mj ) →...→ ωA (-l 1j ) → ω A (-l 0j ) → M → 0. As a corollary, we will have the following characterizations of AS Gorenstein algebras and AS regular algebras: Corollary 0.2. Let A be a Noetherian balanced Cohen-Macaulay algebra. 1. A is AS Gorenstein if and only if ω A has finite projective dimension as a graded left A-module. 2. A is AS regular if and only if every finitely generated maximal Cohen-Macaulay graded left A-module is free.
田边和美、村上健二、出云明久、前川清、加藤真由美、三木敦、高田月久：（1989）
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