Almost split sequences in subcategories
Almost split sequences in subcategories
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DOI：
10.1016/0021-8693(81)90214-3
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发表时间：
1981-04
影响因子：
0.9
通讯作者：
M. Auslander;S. Smalø
中科院分区：
文献类型：
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作者：
M. Auslander;S. Smalø
设R是交换Artin环，I是R-代数，它是有限生成的R-模。在文[5]中，Auslander和Reiten引入了模A中几乎可裂序列的概念，建立了模A上有限生成模的范畴，并证明了这种短正合列的存在唯一性。几乎分裂序列理论得到了进一步发展，Bautista和Martinez[8]以及Roggenkamp[9]证明了mod/i的某些子范畴的类似结果。本文的主要目的是建立具有几乎可裂序列的mod/i的子范畴的一个更一般的理论，它以前面的例子为特例。与几乎分裂序列的概念密切相关的是极小左、右几乎分裂态射的概念。这些概念是在[61]中为mod/i引入的，并在[7]中进一步发展为mod II的子范畴。本文是基于[7]的，因此我们使用与那里使用的相同的符号和约定。我们现在给出一些定义，然后开始逐节描述论文的内容。设C是模II的一个子范畴，则称C中的态射g：B+C是C中的右几乎可分态射，如果(I)g不是可分的满同态，且(Ii)当C中有不可分的满同态h：C‘+C时，存在h’：C‘-+B使得gh’=h.C中的态射A-f B称为左几乎分裂的，如果(I)f不是可分的单态，(Ii)当存在不可分的单态h：A+A‘时，存在h’：B+A‘使得h’f=h.我们称C有右几乎分裂的态射，如果对所有不可分的对象
Let R be a commutative artin ring and let/i be an R-algebra which is a finitely generated R-module. In [5] Auslander and Reiten introduced the notion of an almost split sequence in mod A, the category of finitely generated modules over A, and the existence and uniqueness of such short exact sequences were established there. The theory of almost split sequences developed further, and similar results for certain subcategories of mod/i were proved by Bautista and Martinez [8] and Roggenkamp [9]. The main purpose of this paper is to develop a more general theory for subcategories of mod/i having almost split sequences, which has the previous examples as special cases. Notions closely related to that of almost split sequences are those of minimal left and right almost split morphisms. These notions were introduced in [61 for mod/i and further developed in [7] for subcategories of mod II. This paper is based upon [7] and we therefore use the same notations and conventions as used there. We now give some definitions and then proceed to describe the content of the paper section by section. Let C be a subcategory of mod II. Then a morphism g: B+ C in C is said to be a right almost split morphism in C if (i) g is not a splittable epimorphism and (ii) whenever there is a nonsplittable epimorphism h: C’+ C in C, there exists an h’: C’-+ B such that gh’= h. Dually, a morphismfi A--f B in C is said to be left almost split if (i) f is not a splittable monomorphism and (ii) whenever there is a nonsplittable monomorphism h: A+ A’there exists an h’: B+ A’such that h’f= h. We say that C has right almost split morphisms if for all indecomposable objects