Applications of the Defect of a Finitely Presented Functor

Abstract

For an abelian category $\mathcal{A}$, the defect sequence $$0\longrightarrow F_0\longrightarrow F\overset{\varphi}{\longrightarrow} \big(w(F),\hspace{0.05cm}\underline{\ \ }\hspace{0.1cm} \big)\longrightarrow F_1\longrightarrow 0$$ of a finitely presented functor is used to establish the CoYoneda Lemma. An application of this result is the $\textsf{fp}$-dual formula which states that for any covariant finitely presented functor $F$, $F^*\cong \big(\hspace{0.05cm}\underline{\ \ }\hspace{0.1cm} , w(F)\big)$. The defect sequence is shown to be isomorphic to both the double dual sequence $$0\longrightarrow \textsf{Ext}^1(\textsf{Tr} F,\textsf{Hom})\longrightarrow F\longrightarrow F^{**}\longrightarrow \textsf{Ext}^2(\textsf{Tr} F,\textsf{Hom})\longrightarrow 0$$ and the injective stabilization sequence $$0\longrightarrow \overline{F}\longrightarrow F\longrightarrow R^0F\longrightarrow \tilde F\longrightarrow 0$$ establishing the $\textsf{fp}$-injective stabilization formula $\overline{F}\cong \textsf{Ext}^1(\textsf{Tr} F,\textsf{Hom})$ for any finitely presented functor $F$. The injectives of $\textsf{fp}(\textsf{Mod}(R),\textsf{Ab})$ are used to compute the left derived functors $L^k(\hspace{0.05cm}\underline{\ \ }\hspace{0.1cm} )^*$. These functors are shown to detect certain short exact sequences.