Homological structure and representation of Noetherian algebras

我们把非交换Noether代数的一个理论称为交换环论。我们主要考虑作为交换Noether环上的模的有限生成的代数。我们用表亲复形定义了Gorenstein代数，并用Bass数刻画了Gorenstein代数。事实上，关于Bass数的事实与交换环完全相似。我们将R.Y.Sharp的结果推广到我们的代数上，即在一定的假设下，给出了有限内射维的有限模范畴与有限保护维的有限模范畴之间的范畴等价。给出了序或Artin代数上的模是对偶模的一个判据，并将它应用于Cohen-Macaulay孤立奇点。我们研究了瓷砖订货机的全球维度。特别地，我们给出了一个处理具有较大整体维度的平铺顺序的理论。

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.