基于Abel范畴的下有界复形均有内射分解的证明

A proof of an injective resolution with bounded below complexity in the Abel categories

Abel范畴是同调代数中的核心概念,三角范畴中的好三角是Abel范畴中短正合列的替代物。三角范畴成为数学中的重要工具和研究对象,是描述数学与数学物理中许多复杂研究对象的基本语言和分类依据。文章基于范畴理论,首先给出了预三角范畴定义中公理(TR3)的两个等价刻画;其次给出了预三角范畴中好三角的可裂单态射与可裂满态射的几条性质;最后证明了对于Abel范畴中的任意一个下有界复形X均存在拟同构f:X→I,其中是内射复形,说明了有足够多的内射对象的Abel范畴中的下有界复形均有内射分解。

Abel category is the core concept of homological algebra.Distinguished triangle in triangulated category for a short exact sequence in Abel category is necessarily substituted.Triangulated categories are importanttools and research objects of mathematics.These are basic languages and classification basis in describing many complex research objects of mathematics and mathematical physics.Based on category theory,two equivalent ofaxiom(TR3)of pre-triangulated category are shown;Several properties on of splitting monomorphism and split-ting epimorphism on distinguished triangles are got.For any bounded below complex,existing an quasi-isomor-phism,where is an injective complex,so an injective resolution of any bounded below complex in Abel categories with enough injective object is given.