### Journals Information

**
Mathematics and Statistics Vol. 9(5), pp. 639 - 647 DOI: 10.13189/ms.2021.090503 Reprint (PDF) (505Kb) **

## The Theory of Pure Algebraic (Co)Homology

**Alaa Hassan Noreldeen ^{1}^{,*}, Wageeda M. M. ^{1}, O. H. Fathy ^{2}**

^{1}Department of Mathematics, Faculty of Science, Aswan University, Aswan, Egypt

^{2}Department of Basic and Applied Sciences, Arab Academy for Science, Technology & Maritime Transport, Aswan, Egypt

**ABSTRACT**

Polynomial: algebra is essential in commutative algebra since it can serve as a fundamental model for differentiation. For module differentials and Loday's differential commutative graded algebra, simplified homology for polynomial algebra was defined. In this article, the definitions of the simplicial, the cyclic, and the dihedral homology of pure algebra are presented. The definition of the simplicial and the cyclic homology is presented in the Algebra of Polynomials and Laurent's Polynomials. The long exact sequence of both cyclic homology and simplicial homology is presented. The Morita invariance property of cyclic homology was submitted. The relationship was introduced, representing the relationship between dihedral and cyclic (co)homology in polynomial algebra. Besides, a relationship , was examined, defining the relationship between dihedral and cyclic (co)homology of Laurent polynomials algebra. Furthermore, the Morita invariance property of dihedral homology in polynomial algebra was investigated. Also, the Morita property of dihedral homology in Laurent polynomials was studied. For the dihedral homology, the long exact sequence was obtained of the short sequence . The long exact sequence of the short sequence was obtained from the reflexive (co)homology of polynomial algebra. Studying polynomial algebra helps calculate COVID-19 vaccines.

**KEYWORDS**

Homology Theory, Pure Algebras, Exact Sequence, Polynomial Algebra, Dihedral Homology

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Alaa Hassan Noreldeen , Wageeda M. M. , O. H. Fathy , "The Theory of Pure Algebraic (Co)Homology," Mathematics and Statistics, Vol. 9, No. 5, pp. 639 - 647, 2021. DOI: 10.13189/ms.2021.090503.

(b). APA Format:

Alaa Hassan Noreldeen , Wageeda M. M. , O. H. Fathy (2021). The Theory of Pure Algebraic (Co)Homology. Mathematics and Statistics, 9(5), 639 - 647. DOI: 10.13189/ms.2021.090503.