中文摘要：

让 𝒞；是有 Auslander-Reiten 三角形，和 𝒞 的一个函子地有限的僵硬子范畴的一个 triangulated 范畴；。在那里存在，是众所周知的 Auslander-Reiten 序列在现代派。在这份报纸，我们明确地给在 Auslander-Reiten 翻译之间的关系，序列在现代派并且 Auslander-Reiten 函子，在 𝒞 的三角形；分别地。而且，如果 𝒯；是 𝒞 的一个倾斜簇的子范畴；并且现代派的 𝒯；是一个 Frobenius 范畴，我们也得到 Auslander-Reiten 函子和现代派的 𝒯 的翻译函子；相应于在 𝒞 的；。作为后果，如果，我们得到那倾斜子范畴的簇是的 d-Calabi-Yau triangulated 范畴模的商 Frobenius，那么，它的稳定的范畴是(2d-1 )-Calabi-Yau。这结果被凯勒和 Reiten，然后由 Dugas 首先在一般的格中在情况 d=2 证明，用不同方法。

英文摘要：

Let C be a triangulated category which has Auslander-Reiten triangles, and Ra functorially finite rigid subcategory of C. It is well known that there exist Auslander-Reiten sequences in rood 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 Probenius, 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. 2010 Mathematics Subject Classification： 16G20, 16G70