The theory of pure algebraic (CO)homology

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 mathematical equation was introduced, representing the relationship between dihedral and cyclic (co)homology in polynomial algebra. Besides, a relationship mathematical equation 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 mathematical equation was obtained of the short sequence mathematical equation. The long exact sequence of the short sequence mathematical equation was obtained from the reflexive (co)homology of polynomial algebra. Studying polynomial algebra helps calculate COVID-19 vaccines. © 2021 by authors, all rights reserved.


Introduction
The homology theory of algebras over a field refers to the Hochschild (co)homology in the mathematical field. Hochschild introduced simplicial cohomology for algebras [1], and Henri and Samuel expanded it over rings in [2].
Certain (co)homology 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 cyclic (co)homology. Boris Tsygan [3] and Alain Connes [4] pioneered the concepts of homology and cohomology independently. Many older branches of mathematics, such as de Rham's speculation, simplicial (co)homology, group (co)homology, and -theory, have intriguing relationships with these invariants.
The dihedral (co)homology, proposed independently and demonstrated in various algebras, is the hermitian equivalent of a cyclic (co)homology [4]. The dihedral homology of algebras over the field is defined as the algebraic homology of the dihedral group [5]. The dihedral (co)homology is denoted as (co)homology with group symmetry [6]. There are two types of (co)homology theory: in-discrete and discrete. The Hochschild mentioned in reference [1] is related to Hochschild in the discrete field's (co)homology of algebra with . The first nontrivial (co)homology group was introduced by Tsygan [3] and Connes [4]. In 1987, the involutive unital algebra's reflexive and dihedral (co)homology was investigated, and the remaining (co)homology group was studied in 1989. The analogue simplified cohomology of operator algebras was studied [7]. The Banach cyclic (co)homology has been explored [8] [9] [10]. The group Banach dihedral cohomology and its relation to cyclic cohomology were investigated [12]. 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 -module was investigated [13].
Grothendieck made a breakthrough in cohomology with his proof of the Zeta function ( ) of finite scheme over finite ring . He proved that the factor ( ) of ( ) be a polynomial that operates on a -adic cohomology group, namely ( , ℚ ) = lim → ( , ℤ/ ( )).
Deligne proved the "Riemann hypothesis" over using étale cohomology in 1972: the Frobenius eigenvalues on ( , ℚ ) were correct algebraic numbers with absolute value /2 . It was the final step in proving the famous Weil Conjectures, cementing the value of étale cohomology. Bernstein, Gelfand, and Gelfand classified vector bundles on projective space over a field in 1978 using derived categories as graded modules over the exterior algebra on + 1 variables. The discovery of an isomorphism between the bounded derived categories of graded modules (ℛ), and ( ) , where polynomial algebra on + 1 variables is ℛ, was a crucial step in classifying them. This result revealed that ( ) did not evaluate the category , which was unexpected (see [14], [15]).
In 1985 and 1987, cyclic homology of algebra was determined when the characteristic is 0. In 1991, if ℱ is polynomial and is a ring with unital, the cyclic homology of algebra = [ ]/(ℱ) is calculated. The dimensions of simplicial (co)homology of periodic infinite algebras of polynomial growth were defined [11]. For applying this, a nonstandard periodic representation was obtained that is not analysable since the polynomial's infinite algebra does not derive from scalar algebra.
The first section will outline the definitions of Hochschild homology and cyclic homology. The long exact sequences of simplicial homology and cyclic homology will be presented. In the second part, the algebraic definitions of polynomials and Laurentian polynomials will be reviewed. The simplicial and cyclic homology will be identified first, followed by a discussion of their leng exact sequences. In part three, the dihedral homology of polynomial algebra will be examined, discussing the long exact between cyclic, dihedral homology in polynomial algebra and Laurent polynomials algebra. The Morita of dihedral homology property and Morita's invariance for dihedral homology will be examined in polynomial algebra and Laurent polynomials algebra, implying that the trace map is inverse to the inclusion map.
The following section contains definitions for the homology theorem. A Hochschild homology description of pure algebra will be presented. The concept of cyclic homology will be expanded, defining both reflexive and dihedral algebraic homology.

Homology Theory of Algebras
The cyclic and dihedral homologies for pure algebra were introduced to define Hochschild homology. Let be a pure algebra over and ℳ bimodule with involution * : → ; → * ∀ ∈ . First, the Hochschild homology was determined by defining the chain complex then the (co)homology of the upper complex is .
It is defined as -simplicial algebra's homology and is denoted by ℋℋ ( ). Another definition of Hochschild (co)homology was reported [16]. If is a tensor product = ⊗ , it can be defined by ( ) and ( ) as following: Before defining periodic homology, the cyclic operator If the next complex called the subcomplex of ∁ ( ), and ∶= 1 + + ⋯ , then its homology called cyclic homology and given as follows ( [17], [18]): Alternatively, if is a category, the cyclic homology and cohomology are as follows: The reflexive homology was defined by identifying a subcomplex of ∁ ( ) as following: Its (co)homology is given by , and it is called hyperhomology (hypercohomolog) of [19].
) was obtained by using the (1) and (2) on ∁( ), which is a sub-complex of ∁ ( ). If the homology of this complex ∁ * ( ) is taken, the dihedral homology of algebra can be defined. It is given by An alternative perspective on dihedral (co)homology is as follows: In the following section, the basic definitions of a polynomial will be discussed. We present both the Hochschild homology and the cyclic homology definitions of the polynomial. The Morita Invariance is introduced for simplicial and cyclic homology more detail see ( [19], [20], [21]).

Homology of Polynomial Algebras
Define as algebra and [ ] as class of polynomials; where [ ] is a polynomial algebra with the multiplication of coefficient of polynomials. A Laurent polynomial in a field [ , −1 ] with coefficients is defined as follows: If the coefficients of two Laurent polynomials are the same, they are equal, achieving Suppose that [ ] is a polynomial algebra over the ring with an involution * : → ; → * for all ∈ [12]. We define a complex ∁( [ ]) = (∁ ( ), ), since is the boundary operator It is well known that +1 = 0 . The simplicial homology of algebra , denoted by ℋℋ ( ), is the homology of this complex;

Concerning
[ ] polynomial algebra, the cyclic homology of this is given by The cyclic homology of Laurent polynomial algebra The polynomial can be expressed using the matrix if all its coefficients are matrix and called polynomial matrices [19]. It can be mathematically expressed as follows: where ( ) denotes a matrix of constant coefficients. Let be an associative unital polynomial algebra over , and let ( ) be matrices algebra with coefficients in of order .

Theorem 2-1:
Let ′ ⊆ is a polynomial algebra. The long exact sequence is then obtained as follows: Proof: See [14].

Theory 2-3.
The long exact sequences are known as Connes' exact periodicity sequences; Where; ℐ is an inclusion map, ℬ is a boundary map and is the periodicity map.

Theory 2-4:
If is -unital -algebra, then the algebra of matrices ( ) is -untial, where The following section will examine the dihedral homology of polynomial algebra. The relation between dihedral and cyclic homology will be introduced for polynomial algebra and Laurent polynomials algebra. Morita's property of dihedral homology and Morita's invariance for dihedral homology in polynomial algebra and Laurent polynomials algebra will be examined, which states the trace map is inverse to the inclusion map. Finally, the long sequence of the dihedral homology will be studied.

Let
= [ ] be the dihedral submodule of the dihedral module generated by Polynomial 1 , ⋯ , 2 ∈ , with involution (∑ ) * = ∑ * , ∈ . The dihedral homology and cohomology of the group can be defined as If is an involutive Polynomial algebra, then ℋ ( ) = ℋ ( ∁∁ + ( )) is the dihedral homology of , and the skew-dihedral homology of is There is a canonical splitting of cyclic homology, which follows directly from the preceding definition: The direct sum of the following two exact sequences breaks up Connes' periodicity exact sequence naturally: when contains ℚ: If is Laurent polynomial algebra [ , −1 ], with the natural involution: Then we can define the dihedral of Laurent polynomial algebra by a formula;

If
contains ℚ, the last relation given as; will be shown in the following theory.

Theorem 3-1:
If is a polynomial algebra, then there is an isomorphism dihedral and cyclic homology;

Proof:
If is a commutative ring with unital, then the long exact sequence for dihedral and cyclic homology groups is is used, the above exact sequence is broken down into short exact sequences: We can also express the sequence by an infinite commutative diagram of exact columns and rows: implying the first isomorphism, when  (3) and (4), we get implying the second isomorphism. In the following theory, the relation between dihedral and cyclic homology of Laurent polynomial algebras will be discussed.

Proof:
From the definition and we take = ±1 then the following hold: From equations (5) and (6) we get Similarly, from the following we get We know that From equations (7), (10), and (11), we get In the following theorem, we prove the Morita Invariance of polynomial algebra.

Proof:
Assume that the bicomplex Λ( ) = Λ( ) has the group ℤ 2 ⁄ . If natural projection, : ℛ( ( )) → Λ( ( )) , and ℛ( ( )) = ker , then the following short sequence of ℤ 2 ⁄ -complexes is exact: This short sequence's hyperhomology yields the long exact sequence; If is a homomorphism between Applying : ( ) → on the cyclic homology, we get If we apply : ( ) → on the dihedral homology with consideration; In the following theorem, we prove the inclusion map of the Laurent polynomial algebra.

Theorem 3-4:
If is Laurent polynomial algebra over [ , is an isomorphism for all ≥ 1 and ≥ 0.
and ℛ ( ( [ , −1 ])) = kℯr , where natural projection, then the short sequence of ℤ 2 ⁄ -complexes after that is The hyper-homology of the upper short exact sequence gives a long sequence; If is a homomorphism between If we take the cyclic of; In the following theory, we will demonstrate that the trace map of dihedral is inverse of the inclusion map.

Proof:
The following commutative diagram is the extension morphism; where the vertical and horizontal maps are executed. Isomorphism can be seen in the left vertical arrow by extension property of polynomial algebra, and the rows are accurate. If we consider the upper diagram's dihedral homology, then we obtain Consequently, the right vertical arrow is isomorphic, and we get a commutative diagram; . If the right is an isomorphism according to Morita invariance for unital polynomial algebra, then is an isomorphism. Thus, is a right inverse of . The theorem for dihedral (co)homology is introduced and proved in the next theory.

Proof:
For the algebra ′, we define the next short exact sequence; where ′ is the algebra of non-unital involution polynomials over and ′ is ideal in a unital polynomials algebra ′, then a long exact sequence is If be a kernel of the following sequence where is the free algebra of unital polynomials ′, then ′ and are roughly -unital from a long exact sequence for the sequences (12,13).
We have Suppose that the next short sequence is The proof of our theorem comes from the long exact sequence.
The theorem for reflexive (co)homology is introduced and demonstrated in the next theory.

Theorem 3-7:
If there is an exact short sequence 0 → → ′ → ′′ → 0 of polynomials algebras over the field, then we have the next long exact sequence in hyperhomology where ′′ ⊂ ′ ⊂ .
By applying the isomorphism in equation (17) to equation (18), we get the proof of our theory.

Conclusions
We demonstrate the dihedral and reflexive homology of polynomial algebra. The relationship between dihedral and cyclic (co)homology for polynomial algebra was introduced, which is ℋ∁ (  From the sequence 0 → → ′ → ′′ → 0, the exact long sequence of the dihedral homology and hyperhomology of polynomial algebra were obtained as follows: We can apply this result with ( [22], [23]) and give use more generalizations.