REFLEXIVE AND DIHEDRAL ( CO ) HOMOLOGY OF A PRE-ADDITIVE CATEGORY

The group dihedral homology of an algebra over a field with characteristic zero was introduced by Tsygan (1983). The dihedral homology and cohomology of an algebra with involution over commutative ring with identity, associated with the small category, were studied by Krasauskas et al. (1988), Loday (1987), and Lodder (1993). The aim of this work is concerned with dihedral and reflexive (co)homology of small pre-additive category. We also define the free product of involutive algebras associated with this category and study its dihedral homology group. Finally, following Perelygin (1990), we show that a small pre-additive category is Morita equivalence. 2000 Mathematics Subject Classification. 55N91, 55P91, 55Q91. 1. Preliminaries. Suppose that∆ is a small category, with objects, the set {[0],[1], . . . ,[n], . . .}, and the following family of morphisms δn : [n] → [n− 1], 0 ≤ i ≤ n, σ n : [n]→ [n+1], 0≤ i≤n, τn : [n]→ [n], ρn : [n]→ [n] such that δn+1δn = δn+1δ n , i < j, σ nσ n+1 = σi nσ n+1, i≤ j, σ n+1δn =   σi−1 n−2δ j n−1, i≤ j, Id, i= j,j+1, σ i n−2δ i−1 n−1, i > j, τnδn = δi−1 n τn−1, 0≤ i≤n, ( τn )n+1 = 1n, ρ n = 1, τnσ j n = σj−1 n τn+1, 0≤ j ≤n, τnρn = ρnτ−1 n . (1.1) Definition 1.1. The category ∆ is called a dihedral category. Note that the category generated by only the morphisms δn and σ j n is called a simplicial category and is denoted by ∆, the category generated by δn, σ j n, and τn is a cyclic category and is denoted by ∆C (see [6]), and the category generated by the family of morphisms δn, σ n, and ρn is called a reflexive category and is denoted by ∆R. Definition 1.2 (see [3]). Let k be a commutative ring with identity and involution. An algebra over k associated with the category ∆ (∆R) is an algebra with identity generated by the morphisms δn,σ j n,τn, and ρn (δn,σ j n,ρn). 430 YASIEN GH. GOUDA Definition 1.3. For an arbitrary category C, following [3], for case of presentation, a simplicial object in the category C is a functor : ∆op → C (the category ∆op is the inverse of ∆). Definition 1.4. Following [2] (see also [6]), for an arbitrary category C, the dihedral (reflexive) object in C is a functor : ∆Dop → C, ( : ∆ op → C). If we drop the morphism ρn from the group family of morphisms (δn,σ j n,τn,ρn), we get a cyclic object of an arbitrary category C (see [5]). Suppose that ([n]) = Xn, (δn) = dn, (σ n)= s n, (τn)= tn, (ρn)= rn. We write the dihedral (reflexive) object by the family (Xn,dn,s j n,tn,rn) (Xn,dn,s j n,rn). We can easily check that the morphisms dn,s j n,tn, and rn satisfy relations (1.1). Definition 1.5. Let k be a commutative ring with identity and involution and let C be a category of k-modules. The dihedral k-module in C is defined to be the dihedral objects (Xn,dn,s j n,tn,rn) in the category C. 2. The reflexive and dihedral homology of pre-additive category. In this section, we define the dihedral k-module associated with a pre-additive category and study its (co)homology. Definition 2.1. Let k be a commutative ring with identity and involution. Following [2], the k-category A with an involution is defined to be a small pre-additive category with objects the k-modules of set morphisms A(i,j), where, i,j are in A, and the bilinear maps A(i,j)xA(j,k) → A(i,k), as morphisms. Suppose that, for all objects i,j ∈A, there exists a k-linear map ∗ :A(i,j)→A(j,i), such that ∗2 :A(i,j)→A(i,j). Define the family M = {Mn}n≥0 of k-modules and k-morphisms as follows: M0 = ⊕ i0∈|A| A ( i0, i0 ) , . . . , Mn = ⊕ i0,i1,...,in∈|A| A ( i0, i0 )⊗ k A ( i1, i2 )⊗ k ···⊗ k A ( in,i0 ) . (2.1) On the family M = (Mn), define the morphisms dn′s j n′tn′rn as follows: dn :Mn →Mn−1, s n :Mn →Mn−1, rn,tn :Mn →Mn, (2.2) such that dn ( a0⊗···⊗an )= din(a0⊗···⊗an)+(−1)nana0⊗···⊗an−1, dn ( a0⊗···⊗an )= n−1 ∑ k=0 (−1)a0⊗···⊗akak+1⊗···⊗an, s n ( a0⊗···⊗an )= a0⊗a1⊗···⊗ai⊗1⊗ai+1⊗···⊗an, tn ( a0⊗···⊗an )= (−1)n(an⊗a0⊗···⊗an−1), rn ( a0⊗···⊗an )=α(−1)n(n+1)/2a∗0 ⊗an⊗···⊗a1 , α=±1, (2.3) where ai ’s are the image of the elements ai (0 ≤ i ≤ n) under the involution ∗. Clearly, the moduleM = {Mn} under the last morphisms is a dihedral k-module. Now, REFLEXIVE AND DIHEDRAL (CO)HOMOLOGY . . . 431 we define the dihedral homology group. Suppose that M = (Mn)= Cp,q,r , p, q, r > 0, and consider the complex (Cp,q,r ,δi), i= 1,2 (see [3]), where δ : Cp,q,r → Cp−1,q,r , δ : Cp,q,r → Cp,q−1,r , (2.4)

1. Preliminaries.Suppose that ∆Ᏸ is a small category, with objects, the set {[0], [1], ..., [n],...}, and the following family of morphisms δ i n : Id, i = j, j + 1, (1.1) Definition 1.1.The category ∆Ᏸ is called a dihedral category.Note that the category generated by only the morphisms δ i n and σ j n is called a simplicial category and is denoted by ∆, the category generated by δ i n , σ j n , and τ n is a cyclic category and is denoted by ∆C (see [6]), and the category generated by the family of morphisms δ i n , σ j n , and ρ n is called a reflexive category and is denoted by ∆R.Definition 1.2 (see [3]).Let k be a commutative ring with identity and involution.An algebra over k associated with the category ∆Ᏸ(∆R) is an algebra with identity generated by the morphisms δ i n ,σ j n ,τ n , and ρ n (δ i n ,σ j n ,ρ n ).
Definition 1.3.For an arbitrary category C, following [3], for case of presentation, a simplicial object in the category C is a functor Ᏺ : ∆ op → C (the category ∆ op is the inverse of ∆).Definition 1.4.Following [2] (see also [6]), for an arbitrary category C, the dihedral (reflexive) object in C is a functor Ᏺ : ∆D op → C, (Ᏺ : ∆ op → C).If we drop the morphism ρ n from the group family of morphisms (δ i n ,σ j n ,τ n ,ρ n ), we get a cyclic object of an arbitrary category C (see [5]).Suppose that Ᏺ( We can easily check that the morphisms d i n , s j n ,t n , and r n satisfy relations (1.1).Definition 1.5.Let k be a commutative ring with identity and involution and let C be a category of k-modules.The dihedral k-module in C is defined to be the dihedral objects (X n ,d i n , s j n ,t n , r n ) in the category C.

The reflexive and dihedral homology of pre-additive category.
In this section, we define the dihedral k-module associated with a pre-additive category and study its (co)homology.Definition 2.1.Let k be a commutative ring with identity and involution.Following [2], the k-category A with an involution is defined to be a small pre-additive category with objects the k-modules of set morphisms A(i, j), where, i, j are in A, and the bilinear maps A(i, j)xA(j, k) → A(i, k), as morphisms.Suppose that, for all objects i, j ∈ A, there exists a k-linear map * : A(i, j) → A(j, i), such that * 2 : A(i, j) → A(i, j).Define the family M = {M n } n≥0 of k-modules and k-morphisms as follows: On the family M = (M n ), define the morphisms d i n s j n t n r n as follows: such that where a * i 's are the image of the elements a i (0 ≤ i ≤ n) under the involution * .Clearly, the module M = {M n } under the last morphisms is a dihedral k-module.Now, we define the dihedral homology group.Suppose that M = (M n ) = C p,q,r , p, q, r > 0, and consider the complex (C p,q,r ,δ i ), i = 1, 2 (see [3]), where δ 1 : C p,q,r → C p−1,q,r , δ 2 : C p,q,r → C p,q−1,r , ( are defined by (2.5) Clearly, by definition, δ i • δ i = 0, i = 1, 2. The complex (C p,q,r ,δ i ) can be illustrated by Tsygan's bicomplex C(M) (see [5] where the morphisms b, −b : C p,q,r → C p,q−1,r are given by b Following [5], the homology of the bicomplex (2.6) gives the cyclic homology group: ᏴC n (M) = Ᏼ n (C(M)).Following [3], if we act by the group Z/2 on the bicomplex (2.6): on the column 2 ( > 0) by means of the automorphism and on the column 2 + 1 by means of the automorphism (−1) n(n−1)/2+ +1 r n = (−1) R n T n , we get the tricomplex (C p,q,r ,δ i ), i = 1, 2, 3.The differentials δ 1 ,δ 2 are defined in (2.5) and δ 3 : C p,q,r +1 → C p,q,r is defined by (2.8) Following [4], the dihedral homology of the module M( α ᏴᏰ(M)) is the homology of the complex (C p,q,r ,δ i ), i = 1, 2, 3, α = ±1.

Definition 2.2. The dihedral homology of a k-category A with involution is the dihedral homology of the associated dihedral k-module M={M n
Definition 2.3.The reflexive homology of a k-category A with an involution is the reflexive homology of k-module {M refl }: where M refl is the reflexive k-module M = {M n }.Similarly, if we take the cyclic kmodule M cycl , we obtain the cyclic homology of the k-category A (see [6]): ᏴC n (A) = Ᏼ n (M cycl ).Following [3,4], the dihedral (reflexive) homology of the dihedral (reflexive) module M can be considered as derived functor where ) is the algebra associated with the dihedral (reflexive) category.
Note that (see [3]) the dihedral (reflexive) homology is considered as the hyperhomology of the group Z/2 with coefficients in Tsygan bicomplex (simplicial (Hochschild) complex).The relations between the cyclic and the dihedral homology and also the reflexive and the dihedral homology of pre-additive category are given by the following assertions.
Theorem 2.4.Let k be a commutative ring and let A be a k-category with an involution.Then there exist the following exact sequences Proof.The proof follows from [3].
Corollary 2.5.Let 1/2 ∈ k.Then there exists the natural isomorphism (2.12) Note that we can define the reflexive cohomology and the dihedral cohomology of a pre-additive category in the same manner.
Consider also the following diagram of algebras and homomorphisms between them: Following [3], let R A and R B be the free involutive resolution of the algebras A and B over the homomorphisms i 1 and i 2 , respectively.Consequently, we get the following diagram: Consider the diagrams Clearly, they are commutative.If we define the homomorphisms j A 1 and j B 2 as follows: we get the diagram where We define an involution on R A * c R B as follows: Remarks.(i) The differential on R A * c R B is defined by Leibniz formula for differential graded algebras [5].
(ii) The chain complex RA is a free C-biomodule resolution of the C-biomodule Ā.Consider the complex We act on the complex by means of an automorphism γ n as follows: where (3.13) In the following lemma, we explain the existence of the homomorphism µ and prove that it is an isomorphism.pqp − pqp −(−1) deg(p•q)•deg p p pq = (−1) deg(pq)•deg p p p q, p, p ∈ RA , q ∈ RB .
(3.15)In the complex T (n) we have the following: where γn is an automorphism on the graded K-module H • (Z/(n + 1); ) is induced by the automorphism γ n on the complex T (n) (C; RA ⊗ c RB ).From the isomorphism µ, we get the following isomorphism: we have Lemma 3.4.The following isomorphism holds: Proof.This follows from the fact that Note that the last three relations are obtained from the long exact sequence of relative dihedral homology of algebras [5].Following [3] (also, see [4]), the automorphisms t n and γ n give the representation of the dihedral group Ᏸ n+1 on the complex Then if char(k) = 0, we get the following isomorphism: (3.29) Let A be a k-category with an involution, and let Mod A be the category of right A-modules and P (A) be full subcategory in Mod A, consisting of the finite projective modules.Consider the category M rt (k) with objects, the k-categories with involution and morphisms f : A → B are k-factors F : Mod A → Mod B, such that for every X ∈ P (A), f (X) ∈ P (B), f commutes with an involution.We call these morphisms, Moritamorphisms.Evidently, if f is an equivalence, then f is a Morita-morphism.Following [7] (also, see [8]), the cyclic (co)homology of k-category Morita equivalence.Using this fact and considering the deep results of [3], the following fact follows.
Theorem 3.6.The reflexive cohomolgy and the dihedral (co)homology of the kcategory A with involution are invariant under Morita equivalence.

3 .
The dihedral homology of free product algebras.In this section, we study the product of the algebras associated with a pre-additive k-category, where k has characteristic zero.Let A, B, and C be arbitrary involutive algebras.The free product of the algebras A and B with respect to algebra C is denoted by A * c B. Following [1], the algebra A * c B is C-bimodule.For the algebras A, B, and C,

Lemma 3 . 1 .
where H is a hyperhomology of RA .From (3.1), (i), (ii), and (iii), we haveTor c i Ā, Ā = Tor c i Ā, B = Tor c i B, Ā = Tor c i B, B = 0. (3.9)From (3.7) and (3.9), we have A * c B = H • ( RA ⊗ c RB ) (see [1]).Consider the following diagram: The diagram (3.10) is commutative.Proof.The proof follows from the fact that the differential graded algebra R A * c R B is an involutive resolution of the algebra A * c B over the inclusion i 1 * c i 2 .

Lemma 3 . 2 .
A chain complex homomorphism µ is an isomorphism.Proof.Clearly, in R A * c R B , there exists a subcomplex C + RA + RB (but in T (n) (C; RA ⊗ c RB ) there is not), and we can factorize R A * c R B by this subcomplex.The elements in RA ⊗ c RB can be compared by modulo with the commutant of the algebra R A * c R B with elements in RA ⊗ be compared by modulo with the commutant of the algebra R A * c R B with elements in RA ⊗ c RB , at the same time; .18) This gives the required isomorphism.It is easily seen that the differentials in the complexes⊗ ∞ n=0 T (n) (C; RA ⊗ c RB )/ Im(1−t n )+Im(1− γ n ) and (R A * c R B )/C + RA + RB + [R A * c R B ,R A * c R B ] + Im(1 −r) coincide.Using the condition Tor c i ( Ā, B) = 0, i > 0, we find that RA ⊗ c RB is a free C-module resolution of the algebra R ⊗ c R. Then by considering the isomorphism µ, we get

Lemma 3 . 3 .
The right-hand side of relation(3.20) is isomorphic to the group