Homological Functor H̃ n: Comp(𝒜) ⟶ Ab Where 𝒜 Is an Abelian Category and 𝓃 Is an Integer in ℤ

Ablaye Diallo

doi:doi:10.11648/j.ajam.20251306.13

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Homological Functor H̃ _n: Comp(𝒜) ⟶ Ab Where 𝒜 Is an Abelian Category and 𝓃 Is an Integer in ℤ

Ablaye Diallo^*

Published in American Journal of Applied Mathematics (Volume 13, Issue 6)

Received: 10 October 2025 Accepted: 27 October 2025 Published: 6 December 2025

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Abstract

The study of the homological functor has been carried out in particular cases of abelian categories, notably in the category of left A-modules (resp. right A-modules) and Categories of graded A-modules where A is a ring. We call category of complexes of an abelian category 𝒜 the category denoted by Comp(𝒜) whose objects are the complexe sequence of 𝒜 and morphisms are the chain maps of 𝒜. In this paper, we consider the functor H̃_n define by the composition of functors H_n ∘ Hom_Comp(𝒜)(X,-), where Hom_Comp(𝒜)(X,-) is define by Comp(𝒜) → Comp(Ab), the functor H_n is define by Comp(Ab) → Ab and X is a projective object of 𝒜. In this article, we study how the homological functor H̃_n define by Comp(𝒜) ⟶ Ab where 𝒜 is an abelian category and 𝓃 is an integer in ℤ transforms a short exact sequence into a long exact sequence. The main results of this paper are: the construction of the connecting morphism ϕ_n associated to the covariant functor H̃_n, and show that H̃_n transforms all short exact sequence of morphisms in Comp(𝒜) into a long exact sequence of morphisms in Ab, where X is an projectif object in 𝒜 and 𝒜 is an abelian category.

Published in	American Journal of Applied Mathematics (Volume 13, Issue 6)
DOI	10.11648/j.ajam.20251306.13
Page(s)	412-418
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Abelian Category, Homological Functor, Category of Abelian Groups

References

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[2]	Charles A Weibel. An Introduction to HomologicalAlgebra, Number 38. Cambridge university press, 1994.
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[4]	Joseph J Rotman. Notes on homological algebras,university of Illinois, Urbana, 1968.
[5]	Joseph J Rotman. An Introduction to HomologicalAlgebra, Springer, 2nd edition, 2009.
[6]	Aurelien Djament and Antoine Touze. Finitudehomologique des foncteurs sur une categorie additive etapplications, Transactions of the American MathematicalSociety 376(02): 1113-1154, 2023.
[7]	Arij Benkhadra, Two categorical studies: relativehomological algebra of R-modules, fixpoint theorems forQ-categories, Ph.D. Thesis, Universite du Littoral Cˆoted’Opale; Universite Mohammed V (Rabat). Faculte,2023.
[8]	Alex Heller, Homological algebra in abellian categories,Annals of Mathematics 68(3): 484-525, 1958.
[9]	Carol Peercy Walker. Relative homological algebraand abelian groups. Illinois Journal of Mathematics,10(2):186-209, 1966.
[10]	Manfred Hartl and Bruno Loiseau. Internal objectactions in homological categories. arXiv preprint arXiv:1003.0096, 2010.
[11]	Deren Luo. Homological algebra in n-abelian categories,Proceedings-Mathematical Sciences, 127(4): 625-656,2017.
[12]	Lidia Angeleri H¨ugel. Henning Krause: Homologicaltheory of representations, cambridge university press,2021, 375 pp., 2022.
[13]	Sebastian Posur. On free abelian categories for theoremproving, Journal of Pure and Applied Algebra 226(7):106994, 2022.
[14]	Aurelien Djament and Antoine Touze. The homology ofadditive functors in prime characteristic. arXiv preprintarXiv:2407.10522, 2024.
[15]	Xi Tang and Zhao Yong Huang. Homological Transferbetween additive categories and higher differentialadditive categories, Acta Mathematica Sinica, EnglishSeries 40(5): 1325-1344, 2024.

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Diallo, A. (2025). Homological Functor H̃ n: Comp(𝒜) ⟶ Ab Where 𝒜 Is an Abelian Category and 𝓃 Is an Integer in ℤ. American Journal of Applied Mathematics, 13(6), 412-418. https://doi.org/10.11648/j.ajam.20251306.13

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Diallo, A. Homological Functor H̃ n: Comp(𝒜) ⟶ Ab Where 𝒜 Is an Abelian Category and 𝓃 Is an Integer in ℤ. Am. J. Appl. Math. 2025, 13(6), 412-418. doi: 10.11648/j.ajam.20251306.13

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AMA Style

Diallo A. Homological Functor H̃ n: Comp(𝒜) ⟶ Ab Where 𝒜 Is an Abelian Category and 𝓃 Is an Integer in ℤ. Am J Appl Math. 2025;13(6):412-418. doi: 10.11648/j.ajam.20251306.13

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@article{10.11648/j.ajam.20251306.13,
  author = {Ablaye Diallo},
  title = {Homological Functor H̃ n: Comp(𝒜) ⟶ Ab Where 𝒜 Is an Abelian Category and 𝓃 Is an Integer in ℤ
},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {6},
  pages = {412-418},
  doi = {10.11648/j.ajam.20251306.13},
  url = {https://doi.org/10.11648/j.ajam.20251306.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251306.13},
  abstract = {The study of the homological functor has been carried out in particular cases of abelian categories, notably in the category of left A-modules (resp. right A-modules) and Categories of graded A-modules where A is a ring. We call category of complexes of an abelian category 𝒜 the category denoted by Comp(𝒜) whose objects are the complexe sequence of 𝒜 and morphisms are the chain maps of 𝒜. In this paper, we consider the functor H̃n define by the composition of functors Hn ∘ HomComp(𝒜)(X,-), where HomComp(𝒜)(X,-) is define by Comp(𝒜) → Comp(Ab), the functor Hn is define by Comp(Ab) → Ab and X is a projective object of 𝒜. In this article, we study how the homological functor H̃n define by Comp(𝒜) ⟶ Ab where 𝒜 is an abelian category and 𝓃 is an integer in ℤ transforms a short exact sequence into a long exact sequence. The main results of this paper are: the construction of the connecting morphism ϕn associated to the covariant functor H̃n, and show that H̃n transforms all short exact sequence of morphisms in Comp(𝒜) into a long exact sequence of morphisms in Ab, where X is an projectif object in 𝒜 and 𝒜 is an abelian category.},
 year = {2025}
}

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TY  - JOUR
T1  - Homological Functor H̃ n: Comp(𝒜) ⟶ Ab Where 𝒜 Is an Abelian Category and 𝓃 Is an Integer in ℤ

AU  - Ablaye Diallo
Y1  - 2025/12/06
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251306.13
DO  - 10.11648/j.ajam.20251306.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 412
EP  - 418
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251306.13
AB  - The study of the homological functor has been carried out in particular cases of abelian categories, notably in the category of left A-modules (resp. right A-modules) and Categories of graded A-modules where A is a ring. We call category of complexes of an abelian category 𝒜 the category denoted by Comp(𝒜) whose objects are the complexe sequence of 𝒜 and morphisms are the chain maps of 𝒜. In this paper, we consider the functor H̃n define by the composition of functors Hn ∘ HomComp(𝒜)(X,-), where HomComp(𝒜)(X,-) is define by Comp(𝒜) → Comp(Ab), the functor Hn is define by Comp(Ab) → Ab and X is a projective object of 𝒜. In this article, we study how the homological functor H̃n define by Comp(𝒜) ⟶ Ab where 𝒜 is an abelian category and 𝓃 is an integer in ℤ transforms a short exact sequence into a long exact sequence. The main results of this paper are: the construction of the connecting morphism ϕn associated to the covariant functor H̃n, and show that H̃n transforms all short exact sequence of morphisms in Comp(𝒜) into a long exact sequence of morphisms in Ab, where X is an projectif object in 𝒜 and 𝒜 is an abelian category.
VL  - 13
IS  - 6
ER  -

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Author Information

Ablaye Diallo

Department ofApplied Mathematics, Gaston Berger University, Saint-Louis, Senegal

Contact Email

http://orcid.org/0009-0002-4186-2558

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APA Style

Diallo, A. (2025). Homological Functor H̃ n: Comp(𝒜) ⟶ Ab Where 𝒜 Is an Abelian Category and 𝓃 Is an Integer in ℤ. American Journal of Applied Mathematics, 13(6), 412-418. https://doi.org/10.11648/j.ajam.20251306.13

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ACS Style

Diallo, A. Homological Functor H̃ n: Comp(𝒜) ⟶ Ab Where 𝒜 Is an Abelian Category and 𝓃 Is an Integer in ℤ. Am. J. Appl. Math. 2025, 13(6), 412-418. doi: 10.11648/j.ajam.20251306.13

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AMA Style

Diallo A. Homological Functor H̃ n: Comp(𝒜) ⟶ Ab Where 𝒜 Is an Abelian Category and 𝓃 Is an Integer in ℤ. Am J Appl Math. 2025;13(6):412-418. doi: 10.11648/j.ajam.20251306.13

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@article{10.11648/j.ajam.20251306.13,
  author = {Ablaye Diallo},
  title = {Homological Functor H̃ n: Comp(𝒜) ⟶ Ab Where 𝒜 Is an Abelian Category and 𝓃 Is an Integer in ℤ
},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {6},
  pages = {412-418},
  doi = {10.11648/j.ajam.20251306.13},
  url = {https://doi.org/10.11648/j.ajam.20251306.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251306.13},
  abstract = {The study of the homological functor has been carried out in particular cases of abelian categories, notably in the category of left A-modules (resp. right A-modules) and Categories of graded A-modules where A is a ring. We call category of complexes of an abelian category 𝒜 the category denoted by Comp(𝒜) whose objects are the complexe sequence of 𝒜 and morphisms are the chain maps of 𝒜. In this paper, we consider the functor H̃n define by the composition of functors Hn ∘ HomComp(𝒜)(X,-), where HomComp(𝒜)(X,-) is define by Comp(𝒜) → Comp(Ab), the functor Hn is define by Comp(Ab) → Ab and X is a projective object of 𝒜. In this article, we study how the homological functor H̃n define by Comp(𝒜) ⟶ Ab where 𝒜 is an abelian category and 𝓃 is an integer in ℤ transforms a short exact sequence into a long exact sequence. The main results of this paper are: the construction of the connecting morphism ϕn associated to the covariant functor H̃n, and show that H̃n transforms all short exact sequence of morphisms in Comp(𝒜) into a long exact sequence of morphisms in Ab, where X is an projectif object in 𝒜 and 𝒜 is an abelian category.},
 year = {2025}
}

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TY  - JOUR
T1  - Homological Functor H̃ n: Comp(𝒜) ⟶ Ab Where 𝒜 Is an Abelian Category and 𝓃 Is an Integer in ℤ

AU  - Ablaye Diallo
Y1  - 2025/12/06
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251306.13
DO  - 10.11648/j.ajam.20251306.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 412
EP  - 418
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251306.13
AB  - The study of the homological functor has been carried out in particular cases of abelian categories, notably in the category of left A-modules (resp. right A-modules) and Categories of graded A-modules where A is a ring. We call category of complexes of an abelian category 𝒜 the category denoted by Comp(𝒜) whose objects are the complexe sequence of 𝒜 and morphisms are the chain maps of 𝒜. In this paper, we consider the functor H̃n define by the composition of functors Hn ∘ HomComp(𝒜)(X,-), where HomComp(𝒜)(X,-) is define by Comp(𝒜) → Comp(Ab), the functor Hn is define by Comp(Ab) → Ab and X is a projective object of 𝒜. In this article, we study how the homological functor H̃n define by Comp(𝒜) ⟶ Ab where 𝒜 is an abelian category and 𝓃 is an integer in ℤ transforms a short exact sequence into a long exact sequence. The main results of this paper are: the construction of the connecting morphism ϕn associated to the covariant functor H̃n, and show that H̃n transforms all short exact sequence of morphisms in Comp(𝒜) into a long exact sequence of morphisms in Ab, where X is an projectif object in 𝒜 and 𝒜 is an abelian category.
VL  - 13
IS  - 6
ER  -

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