Foxby duality and Gorenstein injective and projective modules
Foxby duality and Gorenstein injective and projective modules
复制标题
DOI：
10.1090/s0002-9947-96-01624-8
复制
发表时间：
1996
影响因子：
1.3
通讯作者：
E. Enochs;Overtoun M. G. Jenda;Jinzhong Xu
中科院分区：
文献类型：
--
作者：
E. Enochs;Overtoun M. G. Jenda;Jinzhong Xu
1966年，Auslander引入了Cohen-Macaulay Noether环上的G-生成模的G-维数的概念，并发现了这些维数的基本性质。他的结果在允许对偶模的局部Cohen-Macaulay环上是有效的（也参见Auslander和布里杰（Mem. Amer. Math. Soc.，第94卷，1969年））。Enochs和Jenda试图二元论G维的概念。把G维为0的模称为Gorenstein投射模似乎是合适的，所以基本问题是定义Gorenstein内射模。这些在数学Z中定义。220（1995），611-633，并且显示出具有Auslander的结果所预测的性质。我们定义Gorenstein内射模的方法可以对偶化，因此我们可以定义Gorenstein投射模（即G维为0的模），无论这些模是否是非生成的。调查这些模块和Gorenstein平坦模块继续由Enochs，Jenda，徐和Torrecillas。然而，为了获得良好的效果，有必要采取的基础环Gorenstein。H.- B。Foxby在局部Cohen-Macaulay环上的模范畴中引入了两个全子范畴之间的对偶，该环允许一个对偶模。他证明了一类中的n-生成模正是有限G维的模。我们将这一结果扩展到不一定是非线性生成的模块，并证明了对偶结果，即我们刻画了Foxby定义的其他类中的模块。本文的基本结果是，Foxby对偶中涉及的两个类分别对应于具有有限Gorenstein投射维数的模和具有有限Gorenstein内射维数的模的类。我们注意到，这种二元性，然后允许我们将我们的许多结果扩展到原来的Auslander设置。
In 1966, Auslander introduced the notion of the G-dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of G-dimensions. It seemed appropriate to call the modules with G-dimension 0 Gorenstein projective, so the basic problem was to define Gorenstein injective modules. These were defined in Math. Z. 220 (1995), 611–633 and were shown to have properties predicted by Auslander’s results. The way we define Gorenstein injective modules can be dualized, and so we can define Gorenstein projective modules (i.e. modules of G-dimension 0) whether the modules are finitely generated or not. The investigation of these modules and also Gorenstein flat modules was continued by Enochs, Jenda, Xu and Torrecillas. However, to get good results it was necessary to take the base ring Gorenstein. H.-B. Foxby introduced a duality between two full subcategories in the category of modules over a local CohenMacaulay ring admitting a dualizing module. He proved that the finitely generated modules in one category are precisely those of finite G-dimension. We extend this result to modules which are not necessarily finitely generated and also prove the dual result, i.e. we characterize the modules in the other class defined by Foxby. The basic result of this paper is that the two classes involved in Foxby’s duality coincide with the classes of those modules having finite Gorenstein projective and those having finite Gorenstein injective dimensions. We note that this duality then allows us to extend many of our results to the original Auslander setting.