The Theory of Pure Algebraic (Co)Homology

Alaa Hassan Noreldeen,Wageeda M M,O H Fathy

doi:10.13189/ms.2021.090503

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 <img src=image/13424161_01.gif> was introduced, representing the relationship between dihedral and cyclic (co)homology in polynomial algebra. Besides, a relationship <img src=image/13424161_02.gif>, <img src=image/13424161_03.gif> 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 <img src=image/13424161_04.gif> was obtained of the short sequence <img src=image/13424161_05.gif>. The long exact sequence of the short sequence <img src=image/13424161_05.gif> was obtained from the reflexive (co)homology of polynomial algebra. Studying polynomial algebra helps calculate COVID-19 vaccines.

Highlights

The homology theory of algebras over a field refers to the Hochschildhomology in the mathematical field
Certainhomology theories for associative algebras in disciplines of mathematics and non-commutative geometry that generalise de Rham homology and cohomology of manifolds are referred to as cyclichomology
The dihedral homology and cohomology of the group can be defined as ααHDDnn(PP) = TTTTrrnnkk[EE]oooo(kkDD, EEPPαDαD), ααHDDnn(PP) = EExxttknkn[EE](EEPPαDαD, kkDD)

Summary

Introduction

The homology theory of algebras over a field refers to the Hochschild (co)homology in the mathematical field. Algebra's reflexive and dihedral (co)homology was investigated, and the remaining (co)homology group was studied in 1989. The dihedral cohomology groups of certain operator algebras were analysed [6]. Calculating operator algebras' group symmetry, bisymmetry, and Weil (co)homology have stalled, but the cohomology module KK -module was investigated [13]. The discovery of an isomorphism between the bounded derived categories of graded modules DDgbgbgg(R), and DDgbgbgg(AA ) , where polynomial algebra on nn + 1 variables is R, was a crucial step in classifying them. In 1991, if F is polynomial and KK is a ring with unital, the cyclic homology of algebra AA = KK[XX]/(F) is calculated. The concept of cyclic homology will be expanded, defining both reflexive and dihedral algebraic homology

Homology Theory of Algebras

Homology of Polynomial Algebras

Main Results