Connected Graded Gorenstein Algebras with Enough Normal Elements
Connected Graded Gorenstein Algebras with Enough Normal Elements
复制标题
DOI：
10.1006/jabr.1996.6885
复制
发表时间：
1997-03
影响因子：
0.9
通讯作者：
James J. Zhang
中科院分区：
文献类型：
--
作者：
James J. Zhang
我们将[12，1.1和1.2]推广到以下情况。定理1.设A是内射维度d的连通分次Noether代数，使得A的每个非单分次素因子环包含一个正次齐次正规元。然后：(1)A是Auslander-Gorenstein和Cohen-Macaulay。(2)A有拟Frobenius商环。(3)极小素理想P是分次的，且GKdim A/P=d。(4)如果A有有限的整体维度，则A是其商除环中的整环和极大阶。为了证明上面的结论，我们需要下面的结果，它是[3，2.46(Ii)]的推广。定理2.设A是内射维为d的连通分次Notherian AS-Gorenstein代数，则：(1)AAA的极小内射分解的末项同构于A*的一个移位。(2)对每个Noether分次A-模M，Extd(M，A)在k上是有限维的。
We generalize [12, 1.1 and 1.2] to the following situation. Theorem 1.Let A be a connected graded noetherian algebra of injective dimension d such that every nonsimple graded prime factor ring of A contains a homogeneous normal element of positive degree. Then: (1) A is Auslander–Gorenstein and Cohen–Macaulay. (2) A has a quasi-Frobenius quotient ring. (3) Every minimal prime ideal P is graded andGKdim A/P = d. (4) If, moreover, A has finite global dimension, then A is a domain and a maximal order in its quotient division ring. To prove the above we need the following result, which is a generalization of [3, 2.46(ii)]. Theorem 2.Let A be a connected graded noetherian AS-Gorenstein algebra of injective dimension d. Then: (1) The last term of the minimal injective resolution of AAis isomorphic to a shift of A*. (2) For every noetherian graded A-module M,Extd(M, A)is finite dimensional over k.