Gorenstein dimension and Gorenstein algebras

我们考虑了非交换滤子环及其上的滤子模。我们的主要目标是计算同调不变量以及同调不变量的一些等式或不等式，如Gorenstein维、等级数。在这种情况下，我们看到相关的分次环和模在处理这种同调不变量方面起着重要的作用。我们得到了滤环上的伴随分次模GRM的Gorenstein维数大于或等于滤子模M的Gorenstein维数。我们还得到在一般条件下，相关分次模的等级数与滤子模的等级数重合。该条件由模块化条件描述。因此，我们可以研究不一定是Gorenstein滤环上的模的阶数，并且可以看到这种滤环上的左和右Cohen-Macaulay模之间存在由Ext-群给出的对偶。在推广Auslander正则环性质的基础上，研究了Gorenstein滤环的性质。例如，我们得到一个模是纯的当且仅当它是几何纯的，并且没有孤立的相关素数。

We consider noncommutative filtered rings and filtered modules over them. Our main objective is to calculate homological invariants and some equalities or inequalities of homological invariants, like Gorenstein dimension, grade. When this is the case, we see that associated graded rings and modules play important role to handle such homological invariants. We get that Gorenstein dimension of an associated graded module grM is greater than or equal to that of a filtered module M over a filtered ring. We also get that, under general condition as before, grade of an associated graded module coincides with that of a filtered module. This condition is described by module-wise condition. Hence we can study grade of modules over not necessarily Gorenstein filtered rings and see that there is a duality given by Ext-group between left and right Cohen-Macaulay modules over such filtered ring. The properties of Gorenstein filtered rings are also studied under the point of view of generalization of properties of Auslander regular ring. We get, for example, that a module is pure if and only if it is geometrically pure and has no isolated associated primes.