Exactness of the Functors $\text{Hom}_{\mathscr{A}}(X,-), \text{Hom}_{\mathscr{A}}(-,X)$, $\text{Hom}_{\text{Comp}(\mathscr{A})}(X,-), \text{Hom}_{\text{Comp}(\mathscr{A})}(-,X)$ and the Homological Functors $\tilde{H}_n(X,-) \text{ and } \tilde{H}_n(-,X)$ (European Journal of Pure and Applied Mathematics, 2025)
This article presents several results concerning the exactness of covariant and contravariant Hom functors and their derived functors in a balanced abelian category \( \mathscr{A} \). In particular:(i) The functors \( Hom_{\mathscr{A}}(X,-) \) and \( Hom_{\mathscr{A}}(-,X) \) are left exact, and become exact if and only if \( X \) is projective (resp. injective).(ii) The functors \( Hom_{Comp(\mathscr{A})}(X,-) \) and \( Hom_{Comp(\mathscr{A})}(-,X) \) on the category of complexes \( Comp(\mathscr{A}) \) preserve this behavior.(iii) The homological functors \( \tilde{H}_n(X,-) \) and \( \tilde{H}_n(-,X) \) are constructed for all \( n \in \mathbb{Z} \).(iv) For projective \( X \), the connecting morphism \( λ_n: \tilde{H}_n(X,-)((T,γ)) → \tilde{H}_{n+1}(X,-)((Y,α)) \) allows \( \tilde{H}_n(X,-) \) to send short exact sequences in \( Comp(\mathscr{A}) \) into long exact sequences in \( Ab \).(v) Similarly, for injective \( X \), the morphism \( δ_n: \tilde{H}_n(-,X)((Y,α)) → \tilde{H}_{n+1}(-,X)((T,γ)) \) shows that \( \tilde{H}_n(-,X) \) also preserves long exact sequences.
Keywords: Abelian category; balanced category; Homological functor; category of abelian groups