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2009•Journal of Fujian Normal UniversityRequires access

A Note on an Abelian Triangulated Category

Xin Lin

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Abstract

Show that if(C,T,Δ) is a triangulated category,then C is an Abelian category if and only if the collection of diagramsXuYvZwT(X)in C which are isomorphic to the diagrams of the form UV〔0 0 0 1〕WV〔0 0 1 0〕T(U)W〔1 0 0 0〕T(U)T(V) are triangulations.So if C is an Abelian category and T is an additive functor which is an automorphism of the category C.Then have only a way to make(C,T) a triangulated category.Moreover also inverstigate the conditions for the localization C to be an Abelian triangulated category via a Serre class on the Abelian category C.

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Show that if(C,T,Δ) is a triangulated category,then C is an Abelian category if and only if the collection of diagramsXuYvZwT(X)in C which are isomorphic to the diagrams of the form UV〔0 0 0 1〕WV〔0 0 1 0〕T(U)W〔1 0 0 0〕T(U)T(V) are triangulations.So if C is an Abelian category and T is an additive functor which is an automorphism of the category C.Then have only a way to make(C,T) a triangulated category.Moreover also inverstigate the conditions for the localization C to be an Abelian triangulated category via a Serre class on the Abelian category C.

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Available abstract

Show that if(C,T,Δ) is a triangulated category,then C is an Abelian category if and only if the collection of diagramsXuYvZwT(X)in C which are isomorphic to the diagrams of the form UV〔0 0 0 1〕WV〔0 0 1 0〕T(U)W〔1 0 0 0〕T(U)T(V) are triangulations.So if C is an Abelian category and T is an additive functor which is an automorphism of the category C.Then have only a way to make(C,T) a triangulated category.Moreover also inverstigate the conditions for the localization C to be an Abelian triangulated category via a Serre class on the Abelian category C.

Key concepts: Abelian category, Triangulated category, Mathematics, Abelian group, Derived category, Closed category, Functor, Category of groups

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