Preprojective modules over artin algebras
Preprojective modules over artin algebras
复制标题
DOI：
10.1016/0021-8693(80)90113-1
复制
发表时间：
1980-09
影响因子：
0.9
通讯作者：
M. Auslander;S. Smalø
中科院分区：
文献类型：
--
作者：
M. Auslander;S. Smalø
在研究有限维张量代数的表示理论时，Dlab和Ringet描述了一些他们称之为预射影模和预内射模的模。Platzeck在她关于稳定等价于遗传artin代数[14]的artin代数的表示理论的工作中，发现并研究了与Diab和ring引入的预射影和预注入模明显类似的模。本文的目的是发展任意artin代数上的预射影和预注入模的一般理论。正如预期的那样，一般理论所描述的预投影和预内射模块与Dlah、Ringel和Platzeck在各自的情况下所考虑的模块相吻合。鉴于这些评论，也许令人惊讶的是，我们工作的最初动力不是来自遗传的代数或那些稳定等效遗传的代数理论。相反，它来自于解释Gabriel和Roiter[12,151]关于有限表示型的martin代数的更古老的结果的努力，该结果是根据Auslander和Reiten在几乎分裂序列和不可约态射[6,71]方面发展的技术和思想得出的。我们对Gabriel-Roiter结果的推广如下。
In their study of the representation theory of finite-dimensional tensor algebras Dlab and Ringet [Ill described certain modules which they called preprojective and preinjective modules. Platzeck in her work on the representation theory of artin algebras stably equivalent to hereditary artin algebras [14], which includes among other things the tensor, hereditary, and square radical zero artin algebras, found and studied modules which are clearly analogs of the preprojective and preinjective modules introduced by Diab and Ring& Our purpose in this paper is to develop a general theory of preprojective and preinject& modules over arbitrary artin algebras. As would be expected, the preprojective and preinjective modules described by the general theory coincide with the modules considered earlier by Dlah and Ringel and Platzeck in their respective situations.In view of these remarks it is perhaps surprising that the original impetus for our work did not come from the theory of hereditary artin algebras or those stably equivalent to hereditary artin algebras. Rather it came from an effort to explain a much older result of Gabriel and Roiter [12, 151 concerning artin algebras of finite representation type in terms of the technics and ideas developed by Auslander and Reiten in connection with almost split sequcnccs and irreducible morphisms [6, 71. Our generaiization of the Gabriel-Roiter result is as follows.