Extension groups between atoms and objects in locally noetherian Grothendieck category

JOURNAL OF ALGEBRA

卷:422

Article

Kanda, Ryo

关键词: Bass number;    Atom spectrum;    Grothendieck category;    E-stable subcategory;    Noetherian algebra;   

DOI  :  10.1016/j.jalgebra.2014.09.009

来源: Elsevier

【 摘 要 】

We define the extension group between an atom and an object in a locally noetherian Grothendieck category as a module over a skew field. We show that the dimension of the i-th extension group between an atom and an object coincides with the i-th Bass number of the object with respect to the atom. As an application, we give a bijection between the E-stable subcategories closed under arbitrary direct sums and direct summands and the subsets of the atom spectrum and show that such subcategories are also closed under extensions, kernels of epimorphisms, and cokernels of monomorphisms. We show some relationships to the theory of prime ideals in the case of noetherian algebras. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free