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 $\tilde{H}_n(X,-) \text{ and } \tilde{H}_n(-,X)$

Authors

  • Ablaye Diallo Gaston Berger University
  • Mohamed Ben Faraj Ben Maaouia Gaston Berger University
  • Mamadou Sanghare Cheikh Anta Diop, University

DOI:

https://doi.org/10.29020/nybg.ejpam.v18i4.6640

Keywords:

Abelian category , balanced category , Homological functor, category of abelian groups

Abstract

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 \( \mathrm{Hom}_{\mathscr{A}}(X,-) \) and \( \mathrm{Hom}_{\mathscr{A}}(-,X) \) are left exact, and become exact if and only if \( X \) is projective (resp. injective).

(ii) The functors \( \mathrm{Hom}_{\mathrm{Comp}(\mathscr{A})}(X,-) \) and \( \mathrm{Hom}_{\mathrm{Comp}(\mathscr{A})}(-,X) \) on the category of complexes \( \mathrm{Comp}(\mathscr{A}) \) 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 \( \lambda_n: \tilde{H}_n(X,-)((T,\gamma)) \to \tilde{H}_{n+1}(X,-)((Y,\alpha)) \) allows \( \tilde{H}_n(X,-) \) to send short exact sequences in \( \mathrm{Comp}(\mathscr{A}) \) into long exact sequences in \( \mathrm{Ab} \).

(v) Similarly, for injective \( X \), the morphism \( \delta_n: \tilde{H}_n(-,X)((Y,\alpha)) \to \tilde{H}_{n+1}(-,X)((T,\gamma)) \) shows that \( \tilde{H}_n(-,X) \) also preserves long exact sequences.

Author Biographies

  • Mohamed Ben Faraj Ben Maaouia, Gaston Berger University

    Full Professor at Gaston Berger University 

  • Mamadou Sanghare, Cheikh Anta Diop, University

    Full Professor at Cheikh Anta Diop University 

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Published

2025-11-05

Issue

Vol. 18 No. 4: (October 2025)

Section

Algebra

License

Copyright (c) 2025 Ablaye Diallo; Mohamed Ben Faraj Ben Maaouia; Mamadou Sanghare

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How to Cite

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 $\tilde{H}_n(X,-) \text{ and } \tilde{H}_n(-,X)$. (2025). European Journal of Pure and Applied Mathematics, 18(4), 6640. https://doi.org/10.29020/nybg.ejpam.v18i4.6640