The First Hochschild Cohomology of Square Algebras With it ’ s Stability

In this paper, we study on a special case of generalized matrix algebra that we call it square algebra. According to that Hochschild cohomology play a significant role in Geometry for example in orbifolds, we study the first Hochschild cohomology of the square algebra the vanishing of its.


Introduction
Let R be a commutative ring (with unit), let A and B be R-algebras and M be a left A-module and right B-module.A triangular algebra T over R is the following matrix Automorphisms, derivations, commuting mappings and Lie derivations on triangular algebras are studied by Cheung (Cheung, 2001) and (Cheung, 2003).Other useful and valuable literature concerning the structure of derivations and Lie derivations is (Ji & Qi, 2011).Basic examples of triangular algebras are upper triangular matrix algebras and nest algebras which derivations of those considered in (Christensen, 1977), (Coelho, & Milies, 1993), (Donsig, Forrest & Marcoux, 1996).
A generalized matrix algebra is a generalization of triangular matrix algebra.In the triangular algebra T , the element lies in the second row and second column is zero.In generalized matrix algebra, we put a right A-module and left B-module N in zero place.We denote the generalized matrix algebra by G. Algebraic studying on derivations, generalized derivations and Lie derivations have been studied in (Du, & Wang, 2012), (Li & Wei, 2012), (Li, & Xiao, 2011).
Throughout this paper R is a commutative ring (with unit), A and B are R-algebras with units 1 A and 1 B , respectively, M is an R-bimodule, left A-module and right B-module (A, B-module) and N is an R-bimodule, right A-module and left B-module (B, A-module).Define bimodule homomorphisms For more details and applications see (Buchweitz, 2003).We define generalized matrix algebra with the usual 2 × 2 matrix-like addition and multiplication , then we denote it by S and we called that a square algebra.
Let R be a commutative ring (with unit), let A be an R-algebra and M be an A-bimodule.For n = 0, 1, 2, . .., let C n (A, M) be the space of all n-linear (as a R-module map) mappings from where n ≥ 1, x ∈ M and a 1 , . . ., a n+1 ∈ A. The above sequence is a complex for A and M. The n-th cohomology group of C(A, E) is said to be n-th Hochschild cohomology group and denoted by H n (A, M), for more details see (Brodmann & Sharp, 1998), (Rotman, 2009) Thus, we have In this paper, we describe H 1 (S , S ) and vanishing of H 1 (S , X), where X is a two sided S -module (bimodule) is investigated.

Structure of H 1 (S , S )
We begin with the following simple properties of derivations on S as follows: Conversely, if d A and d B are derivations on A and B, respectively, and if τ : M −→ M and σ : N −→ N are any R-linear maps satisfy (i), (ii), (iii) and (iv) then the map D ] defines a derivation on S .
Proof.Let D be a derivation.By the following relations and simple calculation we obtain (i)-(vii): Now, let Z(A) be the center of A and Z(B) be the center of B, x ∈ Z(A) and y ∈ Z(B).Then the Rosenblum R-linear map τ x,y M is called a central Rosenblum R-linear map.We denote the set of all central Rosenblum R-linear maps by ZR A,B (M).Also, we have is a derivation.Moreover, d φ,σ is an inner derivation if and only if φ = τ x,y M and σ = τ y,x N , where τ x,y M ∈ ZR A,B (M) and τ y,x N ∈ ZR B,A (N).Proof.The first statement follows immediately from assume that φ = τ x,y M and σ = τ y,x N where x ∈ Z(A) and y ∈ Z(B).Then Hence d φ,σ is inner.Conversely, assume that d φ,σ is inner.Then there exists . Then, N (n).In particular, φ ∈ ZR A,B (M) and σ ∈ ZR B,A (N).We can now state the main result of this section for describing H 1 (S , S ).
Corollary 4 Let A and B be a commutative ring.By hypothesis of the above Theorem, we have H 1 (S , S ) Hom A,B (M) × Hom B,A (N).

Vanishing of the First Cohomology Group
Let X be a unitary S -bimodule, denote For example, when X = S , we have X AA = A, X BB = B, X AB = M and X BA = N.In this section, the relations between the first cohomology of S with coefficients in X and those of A and B with coefficients in X AA and X BB , respectively, whenever X AB = 0, are investigated.
We started by illustrating the structure of derivations from a square algebra into its bimodules.
Let δ : S −→ X be a derivation.Then Conversely, if δ 1 and δ 2 are derivation from A and B into X AA and X BB , respectively, and τ : M −→ X AB and σ : N −→ X BA are any R-linear maps satisfy in (i), (ii), (iii) and (iv), then the map D a derivation from S into X.If X AB = 0 = X BA , then we may assume that τ and σ are zero.Note that, in this case, Now, we have the following: Proof.Suppose that X AB = 0 = X BA and consider the R-linear map ρ : Der(S , X) −→ H 1 (A, X AA ) ⊕ H 1 (B, X BB ) defined by δ → (δ A + Inn(A, X AA ), δ B + Inn(B, X BB )).
If δ 1 ∈ Der(A, X AA ) and δ 2 ∈ Der(B, X BB ), then is a derivation from S into X and The last equation is deduced from the fact that δ Therefore, we have δ − D ∈ Inn(S , X), and so δ ∈ Inn(S , X).
Proof.With X = M (X = N) we have ) and this is zero.

Stability of the First Hochschild Cohomology
Let A and R be Banach algebras such that A is a Banach R-algebra with compatible actions, that is for all a, b ∈ A, α ∈ R. Let X be a Banach A-bimodule and a Banach R-bimodule with compatible actions, that is . A derivation is a linear map D : A −→ M such that D(ab) = aD(b) + D(a)b (a, b ∈ A) and for x ∈ M, we define the map D x : A −→ M by D x (a) = xa − ax.The map D x is a derivation and such derivations called inner derivations.Let Der(A, M) denote all derivations and Inn(A, M) denote all inner derivations.

Proposition 1
Let D : S −→ S be a derivation, then there are derivations d A : A −→ A, d B : B −→ B, R-linear maps τ : M −→ M and σ : N −→ N and elements m D ∈ M and n D ∈ N such that linear map if there exist derivations d A and d B on A and B, respectively, such that τ M satisfies τ(amb) = d A (a)mb + aτ(m)b + amd B (b) for each a ∈ A, b ∈ B and m ∈ M. Similarly, an R-map τ N : N −→ N is called a generalized Rosenblum R-linear map if there exist derivations d A and d B on A and B, respectively, such that τ N satisfies τ(bna) = d B (b)na + bτ N (n)a + bnd A (a) for each a ∈ A, b ∈ B and m ∈ M. Lemma 2 Let φ ∈ Hom A,B (M) and σ ∈ Hom B,A (N).Then the map d φ,σ : S −→ S given by d φ,σ then xa − ax = 0 for each a ∈ A and yb − by = 0 for each b ∈ B. In particular, x ∈ Z(A) and y ∈ Z(B).Moreover, we have φ(m) = xm + zb − az − my and σ(n) = wa + yn − nx − bw.Since φ ∈ Hom A,B (M) and σ ∈ Hom B,A (N), it follows that zb − az = 0 and wa − bw = 0. Hence φ(m) = xm − my = τ x,y M (m) and σ(n) = yn − nx = τ y,x

)
Proof.Define ϕ : Hom A,B (M) × Hom B,A (N) −→ H 1 (S , S ) by ϕ(φ, σ) = d φ,σ , where d φ,σ represents the equivalence class of d φ,σ in H 1 (S , S ).Clearly, ϕ is R-linear.We shall show that ϕ is surjective.Let d : S −→ S be a derivation.Then there are derivations d A , d B , and R-linear maps τ : M −→ M, σ : N −→ N and elements m d ∈ M, n d ∈ N that satisfy in the conditions (i)-(vii) of Proposition 1.Since H 1 (A, A) = H 1 (B, B) = 0, we can find x ∈ A and y ∈ B such that d A = d x and d B = d y .Define d 0 : S −→ S by