Minimal injective resolution of an order over a local dedekind domain
Minimal injective resolution of an order over a local dedekind domain
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局部 dedekind 域上的阶次的最小单射解析
DOI：
10.1080/00927879808826140
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发表时间：
1998
影响因子：
0.7
通讯作者：
H. Fujita
中科院分区：
文献类型：
--
作者：
H. Fujita
设R是具有商域K和唯一极大理想Ar的局部Dedekind域，A是有限维半单Li-代数A=AK中的半完全R-序。对于右A-格L(即R上FG自由的右A-模)，我们可以计算L的一个极小A-格内射分解为L的R-对偶的极小投射分解的R-对偶。(如果右A-格在右A-格范畴内射是A-格内射的，则它是A-格内射的。)本文从L的极小A-格内射分解出发，构造了它的一个极小内射A-模分解。利用这种构造，我们利用A的极小A-格内射归结和A/Sa的极小内射Alaa-模归结给出了R-序A在Auslander[2，$31]意义下是k-Gorenstein的准则。在附录中，我们注意到有限内射维的FBN环的极小内射分解的最后一项有一个本质基座。我们可以假设A是基本的。设EL，…，En是A的正交本原幂等元的完备集，J是A的Jacobson根，则Ela，…E，A是不可分解的投射右A-模，而Ela/el J，…，e，A/e，J是单右A-模。我们用()*表示函子HornR(，R)。则()*诱导左A-格和右A-格之间的对偶。因此，(Ae*)‘，…，(Ae，)’是不可分解的右A-格内射。
Let R be a local Dedekind domain with a quotient field K and a unique maximal ideal aR, and let A be a semiperfect R-order in a finite dimensional, semisimple li-algebra A= AK. For a right A-lattice L (ie, a right A-module which is fg free over R), we can compute a minimal A-lattice injective resolution of L as the R-dual of a minimal projective resolution of the R-dual of L.(A right A-lattice is A-lattice injective if it is injective in the category of right A-lattices.) In this note we construct a minimal injective A-module resolution of L from its minimal A-lattice injective resolution. Using this construction, we give criteria for an R-order A to be k-Gorenstein in the sense of Auslander [2, $31 by a minimal A-lattice injective resolution of A, and by a minimal injective AlaA-module resolution of A/sA. In the appendix, we note that the last term of a minimal injective resolution of an FBN ring of finite injective dimension has an essential socle. We may assume that A is basic. Let el,..., en be a complete set of orthogonal primitive idempotents of A and J the Jacobson radical of A. Then elA,.... e, A are the indecomposable, projective right A-modules, and elA/el J,..., e, A/e, J are the simple right A-modules. We denote the functor HornR (, R) by ()*. Then ()* induces a duality between left A-lattices and right A-lattices. So (Ae*)',...,(Ae,)'are the indecomposable, right A-lattice injectives.