Applications of the Defect of a Finitely Presented Functor

TL;DR

This paper explores the defect sequence of finitely presented functors in abelian categories, establishing key duality and stabilization formulas, and analyzing derived functors to understand exact sequences.

Contribution

It introduces the CoYoneda Lemma, the fp-dual formula, and connects the defect sequence with double dual and stabilization sequences, advancing the theory of finitely presented functors.

Findings

Established the CoYoneda Lemma using the defect sequence.

Proved the fp-dual formula relating a functor to its dual.

Connected the defect sequence with double dual and stabilization sequences.

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.