EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 2, 2021, 480-492 ISSN 1307-5543 – ejpam.com Published by New York Business Global On the E-infinity algebras Alaa Hassan Noreldeen Mohamed1,∗, Samar A. Abo Quota1 1 Department of Mathematics, Faculty of Science, Aswan University, Egypt Abstract. In this paper we study an elementary use of E-infinity modules and E-infinity algebras as together they have a use in terms of describing triangulated categories. Also, we show an interpretation of E-infinity algebras where the modules are fibrant objects within the categories of differential graded co-algebras and co-modules. 2020 Mathematics Subject Classifications: 55Q05, 57Q10 Key Words and Phrases: Coalgebra, DG-algebras, E-infinity algebra, Modules 1. Introduction An operad homology theory appeared in Mays investigation of iterated loop spaces in [11]. The operad model has some properties of the operations there in, for example, com- mutativity and associativity encoded in every operad as realized by the algebra involved. The classification of the DG-modules over ring k is defined by dg-mod. Examples of al- gebra over the operads in dg-mod include A∞-algebras and E∞-algebras generalizing the concept of associativity and commutativity.An E∞-algebra is a DG-module with multipli- cation present which is the associativity and commutativity up to all higher homotopies. An example of E∞-algebras corresponding to n = ∞ has been given in [1].We present the essential explanations and meaningsof E∞-algebras, E∞-modulesand their fundamen- tal properties in this study. This is in addition to the presentation of aderived category, finishing up with the depiction of triangulated classes using E∞-algebras. We present an understanding of E∞-algebras and E∞-modules as the fibrant objects in the category of the model of certain DG-co-algebras (co-modules) following [9]. From the idea of [9], we provide conceptual construction of the E∞-functor categories and use it to construct the canonical bi-algebra structure of the cobar-construction of the simplicial complex of an associative algebra. In the second section, we will present the definition of E∞-algebra with some examples and study some theories and properties of it. In the third part: we explain the idea of co-algebras and the bar-cobar construction. This ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i2.3905 Email addresses: ala2222000@yahoo.com (A. Noreldeen), scientist samar@yahoo.com (S. Abo Quota) http://www.ejpam.com 480 c© 2021 EJPAM All rights reserved. A. Noreldeen, S. Abo Quota / Eur. J. Pure Appl. Math, 14 (2) (2021), 480-492 481 is followed by some relationships and examples. The fourth section is concerned with the study of morphisms and relationships in E∞- algebra. In the fifth section, we will discuss some important theories in E∞-algebra with its proof, and we will also present examples as an application. 2. Mathematical background We start by briefly recalling the fundamental definitions of E∞-algebras and E∞- modules. This is to build up a picture of the different relations between them. We‘ll provide and concentrate on the interpretation of the fibrant objects of the E∞-algebras as in the model category of the differential graded co-algebras. For a gentler introduction, see [3], [2] and [10]. For the associated absolute field, we have used F . Since W is denoted as the graded vector space, i.e. W = ⊗ p∈Z Wp,we define SW or W[1] as the graded space with (SW)p = Wp+1 for each p ∈ Z. SW is the shift of W. Definition 1. [12] An operad = is comprised of a symmetric monoidal category as part of a collection =(j)j≥0. Each =(j) is enriched by the activity of the symmetric group, Σj and that of the morphisms. γs,r1,r2,··· ,rs : =(s)⊗=(r1)⊗ · · · ⊗ =(rs) −→ =(r1 + · · · rs) (1) for every decision of the components; s, r1, · · · , rs ≥ 0, to the associativity and unit axioms, are fulfilled. An E∞ F-algebra A is a graded space with a map, θj : =(j) ⊗ (SA)(j) −→ (SA) and unit, η : x −→ A to such an extent that the clear associativity, commutativity and the unit diagrams are commutative. Example 1. [2] A graded space A = M[ε]/(ε2) with the trivial A-infinity structure given by map m2 by multiplication ofM, since the maps mn = 0 for each value n 6= 2, whereM is the ordinary algebra for N ≥ 1 and ε be uncertain of degree (2 −N). We characterize the linear map f :M⊗N −→M and the deformed multiplication, m ′ n = { mn n 6= N mN + εf n = N } A endowed with m ′ n is A-infinity algebra if and only if f is Hochschild cocycle for M. Definition 2. A weak E∞-F-algebra A is the graded space with map, θ0 : =(j) ⊗ x −→ (SA), θj , j ≥ 0. The morphisms, f : A −→ B of E∞-algebras are the maps θn : =(j) ⊗ (SA)(n) −→ (SA) homogeneous of the degree zero with the ultimate objective that, ∀ n ≥ 1 we get; ∑ i+j+l=n fi+1+l ◦ (1⊗i ⊗ bj ⊗ 1⊗l) = ∑ i1+···+is=n bs ◦ (fi1 ⊗ · · · ⊗ fis) (2) A. Noreldeen, S. Abo Quota / Eur. J. Pure Appl. Math, 14 (2) (2021), 480-492 482 For any two morphisms (f, g) as (f ◦ g) is given as, (f ◦ g)j = ∑ i1+···+is=j fs ◦ (gi1 ⊗ · · · ⊗ gis) (3) Proposition 1. [10] For each E-infinity algebra A, there is the universal E-infinity algebra morphism φ : U(A) −→ A to a differential graded algebra U(A). Moreover, the morphism φ is an E-infinity quasi isomorphism. Proposition 2. If A be E-infinity algebra and f1 : A −→ V is a quasi-isomorphism (semi- isomorphism) of complexes since V is the complex. Then the complex V admits a structure of E-infinity algebra s.h. f1 extends to an E-infinity quasi-isomorphism f1 : A −→ V. Theorem 1. If A is an E-infinity algebra, then H∗(A) admits an E-infinity algebra structure since: (1) b1 = 0, b2 is induced from bA2 , (2) The identity element in the homology is induced by the E-infinity quasi-isomorphism A −→ H∗(A). Note that E∞-quasi-isomorphism is trivial. If, b1 = 0 then E∞-algebra is minimal. The minimal model of an E-infinity algebra A is the space H∗(A) endowed by the structure provided by the theorem. Definition 3. The Yoneda product is characterized between Ext-groups over general rings. However, for algebras over fields, the presentation can be simplified using canonical resolu- tions. For any associative algebra B with a unital, there is a projective resolution P −→M and a right B-module M. Let the DG-endomorphism algebra A = HomB(P, P ) of P with the nth part of A comprise of the graded object morphisms of degree n where its differential is the super commutator with a differential of P . Thus, A is specifically an E-infinity alge- bra with a minimal model. The homology HEn ∗ (A) is isomorphic for m2 to Ext∗B(M,M), which is the Yoneda algebra. Definition 4. Let A be the strict unit for an E-infinity algebra. It is an element, 1 ∈ E0, which is the unit of m2 and such that, for n 6= 2, the map bn takes the value 0 when one of its contentions rises to 1. HEn ∗ (A) is the homological unit for associative algebra A with the multiplication induced by m2. Consequently, the Homological unitality is saved under E-infinity quasi-isomorphism. Proposition 3. Each homologically unital E-infinity algebra is E-infinity quasi-isomorphic to a strictly unital E-infinity algebra. An E-infinity module over A is a spectrum M with maps, λj : =(j)× (Aj−1 ∧M) −→M (4) that are λj, namely suitably, unital, associative, and equivalent. An E-infinity module M is an F-module with the unital, equivariant, and associative systems within the action maps λj : =(j)⊗Aj−1 ⊗ SM −→ SM (5) A. Noreldeen, S. Abo Quota / Eur. J. Pure Appl. Math, 14 (2) (2021), 480-492 483 That homogeneous of degree 1 is such that the identity of the definition 1 holds for j ≥ 1. We define an ∞-algebra A as a module over itself. The map of A-modules is semi-isomorphic if there is an actuating isomorphism on the homology. Definition 5. A derived category is characterized as the stable homotopic category of spectra and signified by h̄=. If the map of spectra induces an isomorphism in the homotopy groups, then it is weak equivalence and h̄= is constructed from a homotopy category of the spectra by formally altering the weak equivalences. The derived category of A-modules DA = h̄MA is constructed from the homotopy category of A-modules by formally altering a semi-isomorphisms. The exact triangle sequence, M f−→ N −→ Cf −→ ∑ M prompts a triangulation of the derived category, for a map f :M−→ N . Definition 6. [9] The derived category D∞(A) is the localization of the category E-infinity modules with degree 0 morphisms regarding a class of a quasi-isomorphisms. Note that, the objects of the derived category D∞(A) are E-infinity modules, and its mor- phisms are obtained from the morphisms of E-infinity modules by formally rearranging every single semi isomorphisms and, D(ModA) −→ D∞A. Theorem 2. [11] The category of F-linear algebraiclly triangulated T with the split idem- potent and the generator G. Then for m1 = 0, the structure of E-infinity algebra is as follows: A = ⊗ n∈ZHomT (G,G[n]) (6) m2 is given as composition and that the functor; T −→ Grmod(A,m2), U 7−→ ⊗ n∈ZHomT (G,G[n]) wgich lifts to the triangle equivalence, T −→ per(A). Definition 7. [8] A cyclic fibration (co-fibration) is the map with fibration (co-fibration) and weak equiva- lence. A cofibrant object is a one of a kind morphism (φ −→ X ) from the underlying item that is a co-fibration. The fibrant object is the special morphism (X −→ ∗) concerning the terminal object which is a fibration. Definition 8. The classes are supposed to satisfy the Quillen axioms as following: (1) C Have limits and co-limits which is finite. (2) If µ and γ are composable in C , and for any two of µ, γ and µ, γ are weak equivalences, at that point also is the third. (3) A The draw in the morphism’s category of C of a weak equivalence, fibration, or A. Noreldeen, S. Abo Quota / Eur. J. Pure Appl. Math, 14 (2) (2021), 480-492 484 cofibration is individually a weak equivalence, fibration, or cofibration. (4) The following commuting arrow diagram A −→ X i ↓ l↗ ↓ p B −→ Y (7) with a cofibration i and a fibration p, if i or p is weak equivalence, then the lifting l exists making both triangles commute. Every morphism can be considered as (1) a cyclic cofibration taken after by a fibration, and as (2) a cofibration took after by a cyclic fibration. Definition 9. [12] The morphisms (W ∩ fib) of W are called trivial fibrations. The morphisms in (W ∩ C) of W called trivial cofibrations. Fibration involves the morphisms with privilege lifting property for any trivial cofibrations and complex C of the morphisms with the left lifting property concerning all trivial fibration. A left (right) legitimate model class is a one where the weak equivalences are steady under push forward along cofibrations. Example 2. [11] The class on the form C = C+(ModA) of the left bounded complex · · · −→ 0 −→ · · · −→ Xp −→ Xp+1 −→ · · · of right modules over the ring A. For an arbitrary W be a class of a semi-isomorphisms C, the set of morphism i : X −→ Y , ∀ in, n ∈ Z, is injective and fib is the set of morphisms p : X −→ Y with morphism pn, n ∈ Z, is surjective. From [10], we get C is a model classification. Since C is the underlying and the terminal protest, consequently the morphism 0 −→ X is dependably a cofibration, since the morphism X −→ 0 is fibration iff the all components X n, n ∈ Z are injective X −→ 0 ∼↘ ↗ I (8) For a self-assertive class C, an object X is fibrant if the morphism X −→ ∗ is a fibration. Correspondingly, if the morphism φ −→ Y is a cofibration, then the object Y is cofibrant. All complexes X are cofibrant and a complex Y is fibrant if and only if it has injective parts. In the following section, we study the co-algebras With examples illustrated. 3. Co-algebras and the cobar construction In this part, we recall the idea of co-algebras and the bar-cobar construction. This is followed by some relations and examples. The fundamental references are [13],[5] and [14]. A. Noreldeen, S. Abo Quota / Eur. J. Pure Appl. Math, 14 (2) (2021), 480-492 485 Definition 10. An algebraic operad comprises a gathering of the chain complexes, O(n), n ≥ 0, an accumulation of a chain maps γ : O(k)⊗O(j1)⊗ · · ·O(jK) −→ O(j1 + · · ·+ jk) (9) An O-coalgebra is a chain complex C together with chain maps θ : O(j)⊗ C −→ Cj Where, O is an operad, fulfilling the conditions; (i) Associativity: For ∑k s=1 js = j, then the diagram; O(k)⊗O(j1)⊗ · · ·O(jK)⊗ C γ⊗id −→ O(j)⊗ C ↓ θ id⊗ θ ↓ Cj ↑ θk O(j1)⊗ · · ·O(jK)⊗ Ck −→ shuffle O(j1)⊗ · · ·O(jK)⊗ C (10) is commutes. (ii) Unity: The accompanying diagram commutes: R⊗ C ∼= −→ C γ ⊗ id ↓ ↗ θ O(1)⊗ C (11) (iii) Equivariance: For a discretionary component σ ∈ Σj , the accompanying graph com- mutes: O(j)⊗ C σ⊗id −→ O(j)⊗ C θ ↓ ↓ θ Cj −→ σ Cj (12) The morphism in O-coalgebras are the map commuting strictly with the above structure. The class O-coalgebras will be referred to by CoAlgO. We characterize W as the class semi-isomorphisms and Fib as the arrangement of sur- jective morphisms. Consider (C ◦ f) as the set of morphisms i to such an extent that it is present in each commutative square of strong bolts in Alg. Definition 11. [7] A graded coalgebra over K is graded K-module C with a comultiplication of degree 0, to such an extent that the accompanying diagram commutes: C 4 −→ C ⊗ C 4 ↓ ↓ 4 C ⊗ C 1⊗4 −→ C ⊗ C ⊗ C this called co-associativit A coderivation on co-algebra is the map G : C −→ C satisfying co-Leibnizs rule, that is, the accompanying diagram commutes: A. Noreldeen, S. Abo Quota / Eur. J. Pure Appl. Math, 14 (2) (2021), 480-492 486 C G −→ C 4 ↓ ↓ 4 C ⊗ C G⊗+1⊗G −→ C ⊗ C Definition 12. [4] A DG-coalgebra is graded coalgebra with co-derivation P : C −→ C of degree (-1) such that, P 2 = 0. Example 3. The fundamental cause of an evaluated graded co-algebra is co-tensor co- algebra of graded K-module: T (V ) = ∞∑ n=0 V ⊗n (13) The co-multiplication is formed as; 4(v1, · · · , vn) = n∑ i=0 ⊗(vi+1, · · · , vn) (14) since (v1, · · · , vn) stands for v1 ⊗ · · · ⊗ vn. So, for every graded co-algebra C and the linear map C −→ V , there is one of a kind expansion to co-algebra map C −→ T (V ) with the end goal that the diagram; C −→ T (V ) ↓ ↓ V = V is commutes. Definition 13. [12] A co-algebra C is a co-complete if the union of the compositions of the canonical projection as the kernel’s maps C −→ C⊗n −→ (CK )⊗n, n ≥ 2 with the iterated co-multiplication. Definition 14. [11] For the nth complex Hom•k(C,A) is n degree space of homogeneous K-linear maps f : C −→ A and differential maps f to (d ◦ f − (−1)nf ◦ d). This complex turns into a differential graded algebra for the convolution characterized by; f ∗ g = µ ◦ (f ⊗ g) ◦ 4 The maps τ : C −→ A, which is homogeneous k-linear of degree 1, is twisting cochain if it is homogeneous and fulfills d(τ) + τ ∗ τ = 0, ε ◦ τ ◦ ε = 0 (15) A. Noreldeen, S. Abo Quota / Eur. J. Pure Appl. Math, 14 (2) (2021), 480-492 487 Proposition 4. [14] Define Tw(C,A) a class of the twisting cochains. Then for A ∈ Alg, the functor C ◦ g −→ Sets, C 7−→ Tw(C,A) is representable. In the following section, we study the relation and morphisms in E∞-algebra and we introduce the definition of Massy sequence and Massy product. 4. Basic statement on E-infinity algebras In the current part,we consider the fundamental relations in the E∞-modules. The augmented E∞-algebra is equipped with morphism ε : A −→ k. Definition 15. [13] The complex H•k(C,A) becomes an augmented E∞-algebra for the convolution operation; bn(g1, · · · , gn) = bAn ◦ (g1 ⊗ · · · ⊗ g)n) ◦ 4(n) (16) Where, 4(n) is iterate of taking values in ⊗n for a coalgebra ∈ ◦ g and an augmented ∞-algebra A. Let ∞(, A) be the arrangement of all arrangement of the ”Maurer-Cartan equation”,∑ n≥1 bn(τ, · · · , τ) = 0 Proposition 5. [2] B∞A is T c(SA) enriched with the one of a kind co-derivation whose composition with a basic projection BA ↓ SA has the segments bn : (SA)⊗n −→ SA, n ≥ 1 (17) Example 4. Let A = TV , where V = k is concentrated with degree 1. Endow A with the novel differential whose confinement to V ⊂ TV is V = k∼−→k ⊗ k = V ⊗2 ⊂ TV (18) Then A is the semi isomorphic to its sub-algebra k, which is fibrant-cofibrant. If A was fibrant-cofibrant, then the inclusion k −→ TV should admit a left inverse up to homotopy in the feeling of C. Since there are non-zero maps h : A −→ k with degree -1 such that ε◦h = 0, then A cannot be fibrant or cofibrant, since A is the cobra construction on a non-complete dg-coalgebra. Definition 16. A C-comodule is a chain complex D together with chains maps, λ : O(j)⊗ D −→ D ⊗ Cj−1 for O is an operad and C is an O-coalgebra, fulfilling the conditions: (i) Associativity: ∑k s=1 js = j, and A. Noreldeen, S. Abo Quota / Eur. J. Pure Appl. Math, 14 (2) (2021), 480-492 488 O(k)⊗O(j1)⊗ · · ·O(jK)⊗D γ⊗id −→ O(j)⊗ C ↓ θ id⊗ λ ↓ D ⊗ Cj−1 ↑ λ⊗ θk−1 O(j1)⊗ · · ·O(jK)⊗D ⊗ Ck−1 −→ shuffle O(j1)⊗D ⊗ · · ·O(jK)⊗ C is commutes. (ii) Unity: the accompanying outline commutes: R⊗D ∼= −→ D γ ⊗ id ↓ ↗ θ O(1)⊗D (iii) Equivariance: Let σ ∈ Σj − 1 ⊂ Σj , then the accompanying outline is a commute: O(j)⊗D σ⊗id −→ O(j)⊗D θ ↓ ↓ θ D ⊗ Cj−1 −→ id⊗σ D ⊗ Cj−1 A morphism in C-comodules is a homeomorphism of abelian groups commuting with the above structure, see [10]. Definition 17. For all classes of all right dg- A-modules and category dg −C-comodules M . Then M is co-complete, M −→M ⊗ C̄⊗n, n ≥ 2 . Then the match ModA !⊗τ A ↑↓ !⊗τ C ComcC is a couple of adjoint functors. Theorem 3. . (i) The category ComcC concedes to a special structure of model category whose weak equivalences are the morphisms f such that f ⊗τ A be a semi isomorphism and whose cofibrations are injective morphisms. (ii) The functors !⊗τ C and !⊗τ A induce semi-inverse equivalences D(A)∼−→D(C) since D(C) the localization of ComcC is regarding classes of weak equivalences. By comparing the adjunction morphism, B∞A = C −→ BΩC = B∞(ΩC) to canonical E- infinity morphism A −→ ΩB∞A, which is a semi-isomorphism which is universal among the E-infinity morphisms from A to a dg-algebra, since A be an augmented E∞-algebra and B∞A = C. Definition 18. If we let U(A) = ΩB∞A, at that point there is canonical cyclic twisting cochain τ : B∞(A) −→ U(A). From theorem 3 we have an equivalence A. Noreldeen, S. Abo Quota / Eur. J. Pure Appl. Math, 14 (2) (2021), 480-492 489 D(U(A))∼−→D(B∞(A)) Definition 19. [6] Assume that Mod∞A is the grouping of E∞-modules over AbMn , n ≥ 2 which are vanish when one of the contentions is 1. The morphisms gn, n ≥ 2 are strictly unital (vanish if the arguments is 1). This category is isomorphic to the order of all E-infinity modules and all E-infinity morphisms (to more see [11]). Suppose that M is in (Mod∞A). The datum of the arbitrary E-infinity module structure over Ā and the datum of its strictly unital E-infinity module structure over A are equivalent each to other. it is also equivalent to a co-module differential in the induced co-module (M⊗B∞A). (B∞M) is the induced co-module supplied with the differential corresponding to a given E-infinity module structure on M . The functor, Mod∞A −→ ComcB∞(A),M −→ B∞M exists. Proposition 6. [1] The functor M −→ B∞M induces the following equivalence: (i) The equivalence onto subcategory fibrant (cofibrant) objects (ComcB∞(A))cf of ComcB∞(A). (ii) (Mod∞A)/homotopy∼−→D(C). Definition 20. [2] The derived classification for a non-augmented E-infinity algebra D∞A, is the kernel of the functor D∞(A+) −→ D∞(k), since A+ = A ⊕ k be the increased E-infinity algebra obtained by adjoining k and the augmentation A+ −→ k yields a functor Mod∞A + −→Mod∞k Proposition 7. The cohomology H∗(M) is unital H∗(A)-module i.e. M includes a place with the kernel iff M is homologically unital since A is homologically unital. The category D∞(A) is compactly created a triangulated category and has the free A-module of the rank one as a generator of the compact. Definition 21. [9] Let B be a differential polynormal algebra with, B = B1 ⊃ B2 ⊃ · · · ⊃ Bn ⊃ · · · (19) if a natural map f : B −→ B̂ is an isomorphism, then B is perfect algebra. Since B̂ is polynormal algebra and given by; B̂ = limB/Bn. Definition 22. Let the E∞-algebra A. Outline the Massy sequence (a2, · · · , an) of the elements ai ∈ SA⊗i such that, (π(2)⊗ · · · ⊗ 1 + 1⊗ · · · ⊗ π(2))(an) = 0 (π(3)⊗ · · · ⊗ 1 + 1⊗ · · · ⊗ π(3))(an) + (π(2)⊗ · · · ⊗ 1 + 1⊗ · · · ⊗ π(2))(an−1) = 0 A. Noreldeen, S. Abo Quota / Eur. J. Pure Appl. Math, 14 (2) (2021), 480-492 490 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (π(n− 1)⊗ 1 + 1⊗ π(n− 1))(an) + (π(n− 2)⊗ 1 + 1⊗ (π(n− 2))(an−1) + · · ·+ (π(2)⊗ 1 + 1⊗ π(2))(a3) = 0 and the Massy product is given by; µ(a2, · · · , an) = π(2)(a2) + · · ·+ π(2)(an) all parts in A are decomposable if they’re images of Massy product. And therefore, the module of the indecomposable elements JA is that factor A concerning the decomposable elements. Definition 23. Let B be the graded E∞-algebra and F̂B is that the B-construction. From the short exact sequence; 0 −→ F̂ 1Bi −→F̂B p −→ −→ 0 we’ve got the long exact sequence; · · · −→ Hn(F̂B)pn−→B vn −→Hn(F̂ 1B) −→ · · · (20) with the projections; i : F̂ 1B −→ F̂B, p : F̂B −→ B. And for all x ∈ B, if x ∈ kervn : B −→ Hn(F̂ 1B). Then x is a primitive element. Now we can present the results that we studied to clarify important relationships of morphisms in the homology and cohomology theory of E∞-algebra. 5. Main Result Through our study of E∞-algebra and providing some definitions of perfect algebra and primitive and indecomposable elements, we will study and prove the relationships between them in the (co)homology theory through the following theories: Theorem 4. Let A and NA are the perfect algebra and N -construction, respectively. For the homology of NA, the space of the primitive elements PHn(NA) is isomorphic to indecomposable elements space; PHn(NA) ∼= JA. Proof. Since PHn(NA) is primitive space then; PHn(NA) = Im{Hn(F̂Hn(NA)) 7−→ Hn(NA)} since Hn(F̂NA) ∼= A, then PHn(NA) ∼= ImA −→ Hn(NA) ∼= JA. Theorem 5. Consider JHn(FU) be the space of indecomposable components in Hn(FU), wherever U is perfect algebra and PU be the primitive space. Then JHn(FU) ∼= PU . Proof. For the indecomposable elements of E∞-algebra we tend to get; JHn(FU) = Im{Hn(FU) −→ Hn(BHn(FU))} since Hn(BHn(FU)) = U , we get JHn(FU) ∼= Im{Hn(FU) −→ U} ∼= PA A. Noreldeen, S. Abo Quota / Eur. J. Pure Appl. Math, 14 (2) (2021), 480-492 491 Theorem 6. For the ideal algebra U , and therefore the nth-homology Un = Hn(BA) is the graded space with the approximation property. Then we get of Un = Hn(BA) in E∞-algebra as, ∑ n≥0 πn( ¯q ⊗ · · · ⊗ q)π̄(n+ 2)(x) = 0, x ∈ K̄ (21) Now we present some examples as an application to what we got as results from previous theories. Example 5. Consider perfect algebra U = =(m) 1 . For the cohomology Hn(U), we discover the generator a1 ∈ H1(U) corresponding e1 ∈ =m1 and satisfy that, πn−2(a1 ⊗ · · · ⊗ a2) = 0, 2 ≤ n ≤ m and we get the even-dimensional of cohomologies have the generators an2 = a2 · · · a2 ∈ H2n(U) and isomorphic to C. However, the odd-dimensional cohomology has the genera- tors an2 · pa1 ∈ H2n+1(A) and isomorphic to C. Example 6. For the ideal algebra U = =(m) 1 and also the short exact sequence 0 −→ =(m) 1 −→ =1 −→ =m1 −→ 0, we discover that the cohomology H1(U) has generators am+1, · · · , a2m+1 that like em+1, · · · , e2m+1 ∈ U and satisfy that; am+1am+1 = 0, am+1am+2 + am+2am+1 = 0, · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · am+1a2m+1 + · · ·+ a2m+1am+1 = 0, π1(am+1 ⊗ am+1 ⊗ am+1) + am+2a2m+1 + · · ·+ a2m+1am+2 = 0 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 6. 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