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首页> 外文期刊>Journal of algebra and its applications >Almost split morphisms in subcategories of triangulated categories
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Almost split morphisms in subcategories of triangulated categories

机译：Almost split morphisms in subcategories of triangulated categories

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For a suitable triangulated category ? with a Serre functor S and a full precovering subcategory ? closed under summands and extensions, an indecomposable object C in ? is called Ext-projective if Ext1(C,?)=0. Then there is no Auslander–Reiten triangle in ? with end term C. In this paper, we show that if, for such an object C, there is a minimal right almost split morphism β:B→C in ?, then C appears in something very similar to an Auslander–Reiten triangle in ?: an essentially unique triangle in ? of the form Δ=X→ξB→βC→ΣX, where X is an indecomposable not in ? and ξ is a ?-envelope of X. Moreover, under some extra assumptions, we show that removing C from ? and replacing it with X produces a new subcategory of ? closed under extensions. We prove that this process coincides with the classic mutation of ? with respect to the rigid subcategory of ? generated by all the indecomposable Ext-projectives in ? apart from C.When ? is the cluster category of Dynkin type An and ? has the above properties, we give a full description of the triangles in ? of the form Δ and show under which circumstances replacing C by X gives a new extension closed subcategory.

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