扩张子范畴的幂等完备化

Idempotent Completions of Extension Subcategories

给定三角范畴D的子范畴X,Y,证明了若X,Y是幂等完备的且HomD(X,Y[i])=0,其中i=0,-1,则其扩张子范畴X*Y也是幂等完备的。应用到t-结构上,证明了包含t-结构的心的最小的有厚度子范畴是幂等完备的。应用到右(左)recollement上,证明了两端的三角范畴是幂等完备的充要条件是中间的三角范畴是幂等完备的。

Let D be a triangulated category, we prove that the extension subcategory X,Y is also idempotent completion if X,Y, are idempotent completions and HomD （X,Y [i]）=0, where i=0,-1. When it is applied to the t- structure, we prove that the smallest thick subcategory of triangulated category containing the heart of t-structure is idempotent completion. When it is applied to the right （or left） recollement, we prove that the triangulated categories between the two side of recollement are idempotent completions if and only if the middle term is idempotent completion.