Stability And Covering Relations In Torsion Theory

Stability condition was introduced in algebraic geometry to study moduli spaces under the action of a group.This tool has important applications in different branches of mathematics such as τ-tilting theory and cluster algebra.Torsion theory is one of the popular research directions of algebraic representation theory,and also one of the basic tools of geometry,topology and other disciplines.Category theory is an important tool and researching object of algebra and many other disciplines,and the exact category is a generalization of Abelian categories.We draw many new conclusions under the framework of an exact category.We always assume that(A,E)is an essentially small weakly idempotent complete exact length category.This paper includes the following two parts:The first part studies the construction of torsion pairs.Stability function and φ-stable objects are defined in(A,E),and the result of Schur’s lemma is generalized,and a wide subcategory is constructed by using the stable function and φ-stable objects.Two classes of maximally destabilising admissible objects are constructed in(A,E),and it is proved that they are unique up to isomorphism,so we can construct the torsion class and torsion free class by using the stability function and these two classes of objects,thus obtaining a class of torsion pairs.The second part studies the covering relations in torsion theory,defines the covering of torsion class and the filtering sequence of torsion class subobjects in(A,E),and proves that the filtering F ilt(S)of a class of indecomposable subobjects S is closed under extension,and if S is closed under admissible quotient objects(subobjects),then F ilt(S)is also closed under admissible quotient objects(subobjects).Then,the minimal extending object of torsion class is defined,and the mapping(?) is proved to be a bijection.So we obtain the equivalent conditions of minimal extending object.As its dual,the minimal co-extending object of the torsion free class has similar properties,in the end,the relationship between the two classes of objects is characterized by using the indecomposable object.