Covering theory for linear categories with application to derived categories
Covering theory for linear categories with application to derived categories
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DOI：
10.1016/j.jalgebra.2014.02.016
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发表时间：
2014-05
影响因子：
0.9
通讯作者：
R. Bautista;Shiping Liu
中科院分区：
文献类型：
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作者：
R. Bautista;Shiping Liu
我们将Bongartz-Gabriel关于骨架线性范畴的伽罗瓦覆盖理论推广到一般线性范畴。证明了Krull-Schmidt范畴间的伽罗瓦覆盖保留了不可约态射和几乎分裂序列。在派生范畴方面，我们研究了局部有界线性范畴之间的伽罗瓦覆盖在有限维模的有界派生范畴之间的伽罗瓦覆盖。作为一个应用，我们证明了每一个局部有界且根号为零的线性范畴都存在一个可分级的伽罗瓦覆盖，该伽罗瓦覆盖在有限维模的有界派生范畴之间，以及这些有界派生范畴的Auslander-Reiten振之间。在以后的论文中，这将使我们能够得到有限维模的有界派生范畴的完整描述。
We extend the Galois covering theory introduced by Bongartz–Gabriel for skeletal linear categories to general linear categories. We show that a Galois covering between Krull–Schmidt categories preserves irreducible morphisms and almost splits sequences. Specializing to derived categories, we study when a Galois covering between locally bounded linear categories induces a Galois covering between the bounded derived categories of finite dimensional modules. As an application, we show that each locally bounded linear category with radical squared zero admits a gradable Galois covering, which induces a Galois covering between the bounded derived categories of finite dimensional modules, and a Galois covering between the Auslander–Reiten quivers of these bounded derived categories. In a future paper, this will enable us to obtain a complete description of the bounded derived category of finite dimensional modules over a finite dimensional algebra with radical squared zero.