The Homological Dimension Of The ME-exact Structure

Let（A,ε）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,ε）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,ε）.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）.