Homological structure and representation of Noetherian algebras

本文将非交换Noether代数的一个理论称为交换环理论。主要考虑交换Noether环上模生成的代数。利用Cousin复形定义了Gorenstein代数，并利用Bass数给出了Gorenstein代数的一个刻画.事实上，关于巴斯数的事实与交换环的事实完全相似。我们推广了R. Y.与我们的代数类似，在一定的假设下，我们给出了有限内射维数的有限模范畴与有限保护维数的有限模范畴之间的范畴等价性.给出了一个判定序代数上的模是对偶模的准则，并将其应用于Cohen-Macaulay孤立奇点。本文研究了一个瓦片有序系统的整体维数。特别地，我们给出了一个理论来处理具有一个大的整体维数的平铺秩序。

We consider a theory of non-commutative Noetherian algebras as commutative ring theory. Mainly, we consider an algebra which is finitely generated as a module over a commutative Noetherian ring. We define a Gorenstein algebra by the Cousin complex and give a characterization of a Gorenstein algebra by the Bass number. Indeed, the fact concerning Bass numbers holds completely similar to that of commutative rings. We extend the results due to R. Y. Sharp to our algebras, namely, under some assumption, we give a category equivalence between the category of finite modules of finite injective dimension and that of finite modules of finite protective dimension. We give a criterion that a module over an order or artin algebra is a dualizing modulen and apply it to a Cohen-Macaulay isolated singularities. We study a global dimension of a tiled oreder. Especially, we give a theory to deal with a tiled order having a large global dimension.