Idempotent completion of pretriangulated categories. (English) Zbl 1340.18011

Summary: A pretriangulated category is an additive category with left and right triangulations such that these two triangulations are compatible. In this paper, we first show that the idempotent completion of a left triangulated category admits a unique structure of left triangulated category and dually this is true for a right triangulated category. We then prove that the idempotent completion of a pretriangulated category has a natural structure of pretriangulated category. As an application, we show that a torsion pair in a pretriangulated category extends uniquely to a torsion pair in the idempotent completion.


18E30 Derived categories, triangulated categories (MSC2010)
18E40 Torsion theories, radicals
18E05 Preadditive, additive categories
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