Representations of non-splittable extension algebras
Representations of non-splittable extension algebras
复制标题
不可分可拓代数的表示
DOI：
10.1016/0021-8693(88)90281-5
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发表时间：
1988
影响因子：
0.9
通讯作者：
K. Yamagata
中科院分区：
文献类型：
--
作者：
K. Yamagata
设A是域K上的代数，D是自对偶D：mod A 2 mod AoP，其中AoP是A的相反代数，mod A是有限生成左A-模的范畴。通过一个简单的扩张，我们理解了A上的一个扩张代数T，其核DA是Cartan-Eilenberg [3]意义下的：0-+ DA+ T-+ p A--+ 0，其中p是一个代数满态.在A是遗传的情况下，借助于Heller函数QAXDA，平凡扩张A Kda的Auslander-Reiten rAwDA完全由rA [lo]确定。此外，对于任何扩张T，Tr同构于rAwDA。这一事实似乎表明T上的某些模范畴与A 1x DA上的某些模范畴之间存在更密切的联系，尽管mod T一般不等价于mod A Kda。本文研究了不可分裂扩张与平凡扩张之间的范畴关系，并对遗传代数所属的一类代数建立了它们之间的关系。本文证明了以下定理，其中mod，A表示无投射和的n-生成左A-模的范畴.设A是域上的基本代数，其普通环不含定向圈。设D是A的任意自对偶，T是A上具有核DA的扩张，则mod，T等价于mod，AkDA，且因子代数T/sotT同构于平凡扩张Ak（DA/sotDA）.此外，Tr与rAWoA同构。
Let A be an algebra over a field K and D a self-duality D: mod A 2 mod AoP, where AoP is the opposite algebra of A and mod A the category of finitely generated left A-modules. By an extension for short we understand an extension algebra T over A with kernel DA in the sense of Cartan-Eilenberg [3]: 0-+ DA+ T-+ p A--+ 0, where p is an algebra epimorphism. In the case where A is hereditary, by means of the Heller function QAXDA, the Auslander-Reiten quiver rAwDA of the trivial extension A Kda is completely determined by rA [lo]. Moreover, for any extension T, Tr is isomorphic to rAwDA. It seems that this fact suggests the existence of a closer connection between some categories of modules over T and over A 1x DA, though mod T is not in general equivalent to mod A Kda. In this paper we are concerned with categorical relations between non-splittable extensions and the trivial extensions, and we shall establish some relation between them for some class of algebras to which hereditary algebras belong. We prove the following, where mod, A denotes the category of finitely generated left A-modules without projective summands.THEOREM. Let A be a basic algebra over a field whose ordinary quiver contains no oriented cycles. Let D be an arbitrary self-duality of A, and let T be an extension over A with kernel DA. Then mod, T is equivalent to mod, Ak DA, and the factor algebra T/sot T is isomorphic to the trivial extension Ak (DA/sot DA). Further, Tr is isomorphic to rAWoA.