Applications of the Defect of a Finitely Presented Functor

Abstract

For an abelian category $\mathcal{A}$, the defect sequence $$0⟶F_0⟶F\stackrel{\phi }{⟶}(w(F),\underline{})⟶F_1⟶0$$ of a finitely presented functor is used to establish the CoYoneda Lemma. An application of this result is the $\text{fp}$-dual formula which states that for any covariant finitely presented functor $F$, $F^{\ast }\cong (\underline{},w(F))$. The defect sequence is shown to be isomorphic to both the double dual sequence $$0⟶{\text{Ext}}^1(\text{Tr}F,\text{Hom})⟶F⟶F^{\ast \ast }⟶{\text{Ext}}^2(\text{Tr}F,\text{Hom})⟶0$$ and the injective stabilization sequence $$0⟶\overline{F}⟶F⟶R^{0}F⟶\tilde{F}⟶0$$ establishing the $\text{fp}$-injective stabilization formula $\overline{F}\cong {\text{Ext}}^1(\text{Tr}F,\text{Hom})$ for any finitely presented functor $F$. The injectives of $\text{fp}(\text{Mod}(R),\text{Ab})$ are used to compute the left derived functors $L^k(\underline{}{)}^{\ast }$. These functors are shown to detect certain short exact sequences.