Excision theory in the dihedral and reflexive (co)homology of algebras

Abstract In this paper, we study an excision theorem of the dihedral and reflexive (co)homology theory of associative algebras. That is, for such an extension, we obtain a six-term exact sequence in the dihedral cohomology. Also, we present and prove the relation between cyclic and dihedral cohomology of algebras and some examples.

The dihedral (co)homology is referred to as (co)homology with group symmetry by Gouda and Alaa (2009). First, two groups of (co)homology theory are considered to exist: discreet and in-discrete. The Hochschild (co)homology of algebra with id in the discrete field is related to Hochschild (1945). Connes (1985) and Tsygan (1983) introduce the first nontrivial (co)homology group. In 1987, the dihedral and reflexive (co)homology of involutive algebra has been studied and in 1989 the remaining (co)homology groups have been studied. Johnson (1972) studied the analog-simplified cohomology of operator algebras. Gouda (2011), Helemskii (1992), and Helemskii (1991) studied the Banach cyclic (co)homology. Gouda (1997) studied the group Banach dihedral cohomology and relationship with cyclic cohomology. Gouda and Alaa (2009) studied the dihedral cohomology groups of some operator algebras. There is no progress calculating operator algebras group symmetry, bisymmetry, and Weil (co)homology, but the cohomology module k-module is studied by Gouda (2011).
The first to apply the excision theorem was Penner (2020). It is important for the excision property to study the simplicial triviality properties for pure algebra and operator algebra. Lykova and Michael (1998) studied the excision property in simplicial cohomology H n A; A � ð Þ and homology H n A; A ð Þ for short exact sequence 0 ! I ! A ! A=I ! 0.
The notion of H-unitality for algebras has been introduced in 1989, and they conducted it to the short exact sequence 0 ! I ! A ! b ! 0 for the excision in cyclic homology. The bivariant cyclic theory succeeded in the excision of nilpotent extensions because of the theorem by Goodwillie (1985).
The excision theory was developed in 2001 to include Z=2-graded cyclic homology theories based on free extension, but it achieved the Wodzicki's approach. They have studied the Wodzicki's excision theory of simplicial homology and proven it for pure algebra with unital homology in category and they calculated them as an application to the continuous simplicial and cyclic (co) homology by Cortiñas and Valqui (2003).
In our paper, we introduce and study the excision theorem of the reflexive and dihedral (co) homology group of pure algebras. And, we introduce some new proven theorem in the excision theorem of a cyclic homology.
Our work consists of three sections as follows: In Section 2, we introduce a mathematical review on the definition of Hochschild, cyclic, reflexive, and dihedral (co)homology of algebras.
In Section 3, we discuss some results on Hochschild and cyclic homology achievement of the excision property of H-unital algebras, excision of periodic cyclic homology, and excision of cyclic homology.
In Section 4, we provide proven excision theorem of the dihedral (co)homology for short exact sequence So, we prove the relations: The results of the excision theorem of the dihedral cohomology equipped with the results of Intissar (2020) and Kostikov and Romanenkov (2020). Also, our results can introduce this application in a new form.
Suppose that A is an associative unital algebra over K ring and M is bimodule overA with an Þ, since C n ðAÞ ¼ A �ðnþ1Þ ; b n : C n ðAÞ ! C nÀ 1 ðAÞ n � 0 is the boundary operator: b n a 0 ; a 1 ; . . . ; a n ð Þ ¼ ∑ nÀ 1 i¼0 À 1 ð Þi a 0 ; . . . ; aia iþ1 ; . . . ; a n ð Þ þ À 1 ð Þ n a n a 0 ; a 1 ; . . . ; a nÀ 1 ð Þ: It is well known that b n b nþ1 ¼ 0, and henceIm b nþ1 ð Þ � ker b n ð Þ. Consider the following complex, called the Hochschild complex, and the Hochschild boundary: b : M � A �n ! M � A �nÀ 1 is the K-linear map given by the formula: b m; a 1 ; � � � ; a n ð Þ, ¼ ma 1 ; a 2 ; � � � ; a n ð Þ þ ∑ n i¼1 À 1 ð Þ i m; a 1 ; � � � ; a i a iþ1 ; � � � ; a n ð Þ þ À 1 ð Þ n a n m; a 1 ; � � � ; a nÀ 1 ð Þ: The following group is called the Hochschild homology of algebra A: H n ðAÞ ¼ ðHÞ n ðC * ðAÞÞ ¼ kerðb n Þ Imðb nþ1 Þ ; and denoted by HH n A ð Þ. The enveloping of algebra in A is the tensor product A e ¼ A � A op of A with its opposite algebra. In the work by Krasauskas et al. (1988), the simplicial (co)homology of A with coefficients in M in terms of the functors (TorÞ and Ext ð Þ is defined by: We act on the complex C A ð Þ by the cyclic order group n þ 1 ð Þ through the cyclic operator t n : t n a 0 ; . . . ; a nÀ 1 ; a n ð Þ ¼ À 1 ð Þ n a n ; a 0 ; . . . ; a nÀ 1 ð Þ: (1) Þ is a sub-complex of C n A ð Þ. From the work by Helemskii (1991), the homology of the complex CC � A ð Þ is called the cyclic homology of algebra A, and denoted by We act on a complex C A ð Þ by the reflexive group Z=2 ¼ À 1; þ1 f g of order 2 by means of the reflexive operator r n : C n A ð Þ ! C n A ð Þ where, r n a 0 ; . . . ; ; a nÀ 1 ; a n ð where a ¼ �1; a 2 ¼ 1; r n ð Þ 2 ¼ 1 and ai � ¼ Im ai ð Þ under the involution � . If Λ is a category, then another definition of cyclic (co)homology is: where each K-algebra A the cyclic K-module.
From the work by Alaa (2019), the homology of the complex α CR n A ð Þ is called the reflexive homology of algebra A, and denoted by: If we use Equations (1) and (2) together on C A ð Þ, we have the complex From the work by Tsygan (1986), the homology of a complex α CD � A ð Þ is called dihedral homology of algebra A, and denoted by: Another definition of dihedral (co)homology (J-L. Loday, 1998) is Definition (2-1): Let A be K-algebra and I -ideal where A ! A=I is K-split, then there exists the map of the relative homology ((co)homology) for A modulo I : ε : HH n I ð Þ ! HH n A; I ð Þ; ε : HH n I ð Þ ! HH n A; I ð Þ: The ideal I is said to be excision of simplicial homology (cohomology) if a map is an isomorphism (Cartan & Eilenberg, 1956). Then the sequence: is exact.

Definition (2-2):
For K-split sequence A ! A=I where A be K-algebra and I-ideal, map of relative homology (cohomology) for A modulo I respect is: j : CC n ðI Þ ! CC n ðA=I Þ; j : CC n ðI Þ ! CC n ðA=I Þ: The excision of the cyclic homology (cohomology) is the ideal I if the map is an isomorphism (Cartan & Eilenberg, 1956). The sequences are exact.

Theorem (2-3):
The periodicity exact sequence of the cyclic module C � is where the map I is inserted, the simplicial complex for C � becomes bicomplex C �� : If C n ¼ A �n , the periodicity exact sequence of the cyclic sequence takes the form (see J-L. Loday, 1998).
There is a natural long exact sequence for any algebra A over the ring K which contains Q There are long exact sequences, called exact periodicity sequences of Connes: (Noreldeen, 2019).

Theorem (2-6): (Connes' Periodicity Exact Sequence)
As the theorem above, we get the long exact sequences
In the following section, we will show previous studies of excision thermos in the Hochschild and cyclic (co)homologies of associative algebras. We will also explain some results and examples related to previous studies.

Excision in simplicial and cyclic (co)homology
In this part, we introduce some properties and theorems of the Hochschild and cyclic (co)homologies of associative algebras by Buchholtz and Rijke (2019), Quillen (1972), Ralf (2010), Thiel (2006), and Wodzicki (1989). We discuss and study some special theories of excision theorem of simplicial and cyclic (co)homology theory in pure algebras.

Definition (3-1):
Let A is C-algebra and M is right A-module. A and M are unital homologically, if the chain The same definition is for the left modules. By definition, A is unital homology algebra if and only if it is unital homology (Krasauskas et al., 1988). Therefore, M is unital homologically if Let abelian category C be with extensions, then a chain complex is true if its homology vanishes. In this case, M is unital homologically if HH � A; M � V ð Þ ¼ 0: In general, H-unital is unrelated to the vanishing of HH � A; M � V ð Þ (for more details, see Gouda & Alaa, 2009).

Lemma (3-2):
Suppose I ! E ! Q is algebra and M is unital homology J -module. Then we find that the E-module structure is only a structure extended from I -module structure.

Proof:
For p 2 N, let F p be a complex With M � V with zero degree, since I ! E ! Q is pure, and then M � I �K � E �p � V ! M � I �KÀ 1 � E �pþ1 � V is inflation "K; p � 0. Hence, from Tsygan (1983), the canonical map F p ! F pþ1 is inflation "p. Its cokernel is the chain complex where p þ 1 ½ � denotes translation by p þ 1. This chain complex is exact because M is homologically unital as a right I -module. Since F p ! F pþ1 ! F pþ1 =F p is conflation, then the map F p ! F pþ1 is quasiisomorphism by Lemma (3-2) and J. Loday (2013). Thus the inclusion:F 0 ! F p "p 2 N. For p ¼ 0, we get In any fixed degree n, we have ðF p Þ n ¼ HH n ðE; MÞ � V"p � n: Hence, the canonical map HH n I ; M ð Þ ! HH n E; M ð Þ is a pure quasi-isomorphism.

Corollary (3-4):
Consider the pure algebra conflation I ! E ! Q and unital homology I, then

Theorem (3-6):
Consider the pure algebra conflation I ! E ! Q and I is unital homology and M I ! M E ! M Q is pure conflation of E-bimodules. Taking the structure of E-bimodule on M Q come down to the structure of Q-bimodule and M I is unital homology asI -module. Then: If we have short exact sequence of algebras with bijective homomorphism in unital homology, we get the long exact sequence in the Hochschild homology theory obtained in the following theorem.

Theorem (3-7):
Suppose that I ! E ! Q is pure of C-algebras and I is unital homology. Then,

Proof:
Let F p "p � 0 be the co-chain complex where b � is the co-boundary map which uses the right E-module Q � E ! Q structure andM-bimodule structure. For the pure algebra conflation and injective M of C, we get where M is injective and the exact complex in Equation 7 ð Þ. From Theorem (3-2) and Gouda and Alaa (2009), we find that � F p ! � F pþ1 is quasi-isomorphism; then, � F 0 ! � F p "p 2 N. This yields the assertion because Consider the monoidal category C provided with split extensions class. For Q-bimoduleM, split extensions I ! E ! Q in C and I �n ; b 0 ð Þ is exact, then are injective in C .

Theorem (3-11):
Let 0 ! I ! A ! A=I ! 0 be an extension of K-algebra, if I is H-unital. Then we get There is a well-defined functional map for the Hochschild homologyρ : HC n I ð Þ ! HC n A; I ð Þ: In the other hand, it is immediate from the construction of HC n I ð Þ that in the framework of non-unital algebras there is a long exact sequence of Connes. From the work by Guram and Manuel (2014), we consider the exact rows commutative diagram: We know that HH n I ð Þ ! HH n A; I ð Þ is an isomorphism when I is H-untial which implies that HC n I ð Þ ! HC n A; I ð Þ is an isomorphism "n; n 2 Z.

Theorem (3-12):
For A algebra over K and containing Q , the following map is an isomorphism

Proof:
We can define the homotopy as which maps from C n A ð Þ to itself (Cartan & Eilenberg, 1956). One verifies that This satisfies that H 0 ¼ C λ n A ð Þ and the homology of C �� A ð Þ is canonically isomorphic toC λ n A ð Þ. In the next part, we will show the very important idea of H-unita algebra put by Wodzicki.

Theorem (3-14):
Suppose that 0 ! I ! A ! B ! 0 is a pure extension of K-algebras, M on A-bimodule and K-moduleV. For H-unital and I -bimoduleM, then the canonical inclusion i : Proof: see Lykova and Michael (1998).

Corollary (3-15):
Suppose that 0 ! I ! A ! B ! 0 is an extension of pure K-algebra and the k-module V, for H-unital I , we have ω :

Proof:
1 ð Þ ) 2 ð Þ: Take 0 ! I ! A ! B ! 0 is a pure extension of K-algebras, V and K-module and Taking the diagram following for short exact commutation sequences: By corollary (3-15), ω 1 is a quasi-isomorphism and also J . Using theorem (3-14), we complete the proof. sequence is a long, exact sequence not only for differential forms cohomology de Rham but also for integral coefficients cohomology).
3 ð Þ ) 1 ð Þ: let V and A be as in 2 ð Þ ) 1 ð Þ. � ω : C �� A ð Þ ! C �� V ð Þ, the projection canonical and is the sub-complex of ker � ω ð Þ generated by a 0 � � � � � a n ; a 0 0 � � � � � a 0 nÀ 1 ð Þ with some a i and some a 0 n inV. Sine ker � ω ð Þ ¼ C �� I ð Þ � and I satisfies Hochschild homology excision, is exact. Let I be not H-unital. Taking x 2 V � I �n is cycle for b 0 n that is not a boundary. Of note that 0; N x ð Þ ð Þ is cycle for n þ 1 degree in that is un-boundary; it is a contradiction with the exactness of .
In the next section, we will give the main results of this paper. We prove the relations between the cyclic and dihedral (co)homology in algebra, from which we will prove the excision theorems of reflexive and dihedral (co)homology theory as a new result.

Excision in the dihedral and reflexive cohomology of algebras
In this part, we introduce the main result in our paper. We prove the relation between the cyclic and dihedral homology of algebras and the relation between the reflexive and dihedral cohomology of algebras in theorems (4-8) and (4-9). We prove the excision property of the dihedral and reflexive cohomology of pure algebras in theorems (4-11) and (4-12). We use references Buchholtz and Rijke (2019), Cortiñas & Valqui (2003), Penner (2020), Quillen (1972) to study the property of excision theorem.