阿贝尔范畴黏合上的纯投射维数

Pure projective dimensions on recollements of Abelian categories

在阿贝尔范畴中引入了纯投射维数的概念。令R_(ab)(A,B,C)为阿贝尔范畴黏合,讨论了三个阿贝尔范畴之间的纯投射对象和纯投射维数的关系。作为应用,研究了形式三角矩阵环上的纯投射模及纯投射维数。最后,证明了在一定条件下,阿贝尔范畴B的纯投射维数有限当且仅当阿贝尔范畴A与C的纯投射维数有限。

The concepts of pure projective dimensions in Abelian categories are introduced.Let R_(ab)(A,B,C)be a recollement of Abelian categories where A,B and C are Abelian categories,the relations of pure projective objects and pure projective dimensions of objects in three Abelian categories are studied.As applications,the pure projective modules and pure projective dimensions over formal triangular matrix rings are studied.Finally,it is proved that pure projective dimension of B is finite if and only if the pure projective dimensions of A and C are finite under some conditions.