Applications of the defect of a finitely presented functor

For an abelian category AA, the defect sequence0⟶F0⟶F⟶φ(w(F),_)⟶F1⟶0 of a finitely presented functor is used to establish the CoYoneda Lemma. An application of this result is the fp-dual formula which states that for any covariant finitely presented functor F  , F⁎≅(_,w(F))F⁎≅(_,w(F)). The defect sequence is shown to be isomorphic to both the double dual sequence0⟶Ext1(TrF,Hom)⟶F⟶F⁎⁎⟶Ext2(TrF,Hom)⟶0 and the injective stabilization sequence0⟶F‾⟶F⟶R0F⟶F˜⟶0 establishing the fp-injective stabilization formula F‾≅Ext1(TrF,Hom) for any finitely presented functor F  . The injectives of fp(Mod(R),Ab)fp(Mod(R),Ab) are used to compute the left derived functors Lk(_)⁎Lk(_)⁎. These functors are shown to detect certain short exact sequences in Mod(R)Mod(R).