The Exact Structure Dervied From A Cotortion Pair

Let(A,ε)be an exact category,and let χ(?)A be a subcategory.Let εχ ={0→A→B→C→0(*)exact | for any X ∈ χ,Hom(X,-)preserves the exactness of(*)}.In Chapter 2,we show that εχ is an exact substructure of(A,ε).Let εχ-proj={X | X is a εχ-projective object},we prove that Add(χ)(?)εχ-proj.In Chapter 3,let(F,C)be a cotorsion pair.We prove that εF-proj=F;Moreover,if(F,C)is complete,then εF has enough projective objects.Let ΦF={φ|φ is a εF-phantom morphism},and let ΨF={φ|φ is a εF-cophantom morphism}.We show that both ΦF and ΨF are ideals in A.If A has enough injective objects,and([F,C is a complete cotorsion pair,we prove that ΨF:is a special preenveloping ideal.