### Journals Information

**
Mathematics and Statistics Vol. 11(6), pp. 874 - 882 DOI: 10.13189/ms.2023.110602 Reprint (PDF) (745Kb) **

## Some Properties of Cyclic and Dihedral Homology for Schemes

**Samar A. A. Quota ^{1}, Faten. R. Kara ^{1}, O. H. Fathy ^{2}^{,*}, W. M. Mahmoud ^{1}**

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

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

**ABSTRACT**

A scheme is a type of mathematical construction that extends the concept of algebraic variety in a number of ways, including accounting for multiplicities and being defined over any commutative ring. In this article, we study some properties of the cyclic and dihedral homology theory in schemes. We study the long exact sequence of cyclic homology of scheme and prove some results. So, we introduce and study Morita-equivalence in cyclic homology of schemes and proof the main relation between trace map and inclusion map. Our goal is to explain product structures on cyclic homology groups . Especially, we show of algebra. We give the relations between dihedral homology and cyclic homology of schemes, therefore: . We explain the trace map and inclusion map of cyclic homology for scheme algebra which takes form: and . For the shuffle map , we obtain the long exact sequence of cyclic homology for scheme: . We give the long exact sequence of dihedral homology for scheme: . For any three and algebra, we write the next long exact sequence as a commutative diagram: . For all and schemes, we give the long exact sequence of dihedral homology as: .

**KEYWORDS**

Cyclic Homology, Mayer-Vietortis, Morita Invariance, Scheme

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

(a). IEEE Format:

[1] Samar A. A. Quota , Faten. R. Kara , O. H. Fathy , W. M. Mahmoud , "Some Properties of Cyclic and Dihedral Homology for Schemes," Mathematics and Statistics, Vol. 11, No. 6, pp. 874 - 882, 2023. DOI: 10.13189/ms.2023.110602.

(b). APA Format:

Samar A. A. Quota , Faten. R. Kara , O. H. Fathy , W. M. Mahmoud (2023). Some Properties of Cyclic and Dihedral Homology for Schemes. Mathematics and Statistics, 11(6), 874 - 882. DOI: 10.13189/ms.2023.110602.