论文标题
准注射尺寸
Quasi-injective dimension
论文作者
论文摘要
在我们先前关于准标记维度的工作之后，在本文中，我们引入了准注射尺寸作为注射尺寸的概括。我们在准注射尺寸的背景下（例如以下内容）恢复了关于注射式和戈伦斯坦注射尺寸的几个知名结果。 （a）如果本地环$ r $上有限生成的模块$ m $的准注射尺寸是有限的，则等于$ r $的深度。 （b）如果存在有限生成的有限的准文化尺寸和最大krull尺寸的模块，则$ r $是Cohen-Macaulay。 （c）如果存在一个有限的投影尺寸和有限的准注射尺寸的非零有限生成的模块，则$ r $是Gorenstein。 （d）在Gorenstein局部环上，有限生成的模块的准注射尺寸是有限的，并且仅当其准标准尺寸是有限的。
Following our previous work about quasi-projective dimension, in this paper, we introduce quasi-injective dimension as a generalization of injective dimension. We recover several well-known results about injective and Gorenstein-injective dimensions in the context of quasi-injective dimension such as the following. (a) If the quasi-injective dimension of a finitely generated module $M$ over a local ring $R$ is finite, then it is equal to the depth of $R$. (b) If there exists a finitely generated module of finite quasi-injective dimension and maximal Krull dimension, then $R$ is Cohen-Macaulay. (c) If there exists a nonzero finitely generated module with finite projective dimension and finite quasi-injective dimension, then $R$ is Gorenstein. (d) Over a Gorenstein local ring, the quasi-injective dimension of a finitely generated module is finite if and only if its quasi-projective dimension is finite.
