The Homological Dimension Of The ME-exact Structure

Let(A,?)be an exact category.In chapter 2,let Ext(C,A)be the set of the equivalence classes of extensions of C by A.We give a proof that Ext(C,A)is an additive group by using pushout and pullback.In chapter 3,let ME be the exact structure of the category of arrows Arr(A)consisting of ME-extensions,and suppose that(A,?)has enough projective objects.We show that(Arr(A);ME)also has enough projective objects,and it has the same global projective dimension with(A,?).Let c:C₀?C₁ and a:A₀?A₁ be objects in Arr(A).We prove that Ext_ME¹(c,a)=0 if and only if each ME-extension of c by a splits using the projective resolution of c,and we show that Ext_ME²(c,a)?Ext²(C₁,A₀).