Auslander-Reiten Sequences or Triangles Related to Rigid Subcategories
Auslander-Reiten Sequences or Triangles Related to Rigid Subcategories
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DOI：
10.1142/s100538671600002x
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发表时间：
2016-01
影响因子：
0.3
通讯作者：
Ming Lu
中科院分区：
文献类型：
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作者：
Ming Lu
设G是一个具有Auslander-Reiten三角形的三角范畴，G是G的函有限刚性子范畴.众所周知，模中存在Auslander-Reiten序列。本文明确地给出了模空间中的Auslander-Reiten平移、序列与模空间中的Auslander-Reiten函子、三角形之间的关系。进一步，如果模是模的一个簇倾斜子范畴，模是一个Frobenius范畴，我们还得到了模的Auslander-Reiten函子和平移函子对应于模的Auslander-Reiten函子和平移函子.𝒯由此得到，如果d-Calabi-Yau三角范畴模簇倾斜子范畴的商是Frobenius，则其稳定范畴是（2d-1）-Calabi-Yau.这个结果首先由Keller和Reiten在d=2的情况下证明，然后由Dugas在一般情况下使用不同的方法证明。
Let 𝒞 be a triangulated category which has Auslander-Reiten triangles, and ℛ a functorially finite rigid subcategory of 𝒞. It is well known that there exist Auslander-Reiten sequences in mod ℛ. In this paper, we give explicitly the relations between the Auslander-Reiten translations, sequences in mod ℛ and the Auslander-Reiten functors, triangles in 𝒞, respectively. Furthermore, if 𝒯 is a cluster-tilting subcategory of 𝒞 and mod 𝒯 is a Frobenius category, we also get the Auslander-Reiten functor and the translation functor of mod 𝒯 corresponding to the ones in 𝒞. As a consequence, we get that if the quotient of a d-Calabi-Yau triangulated category modulo a cluster tilting subcategory is Frobenius, then its stable category is (2d-1)-Calabi-Yau. This result was first proved by Keller and Reiten in the case d=2, and then by Dugas in the general case, using different methods.