Homological Perturbation Theory and Associativity

Pedro Real Universidad de Sevilla, Facultad de Informática y Estadística, Dpto. Matemática Aplicada I, Avenida Reina Mercedes, s/n 41012 - Sevilla (Spain) Fax: 34-5-4557878 e-mail: real@cica.es

January, 1997

Abstract:In this paper, we prove various results concerning DGA-algebras and DGA-coalgebras in the context of the Homological Perturbation Theory. More precisely, we establish here a perturbation machinery that is a generalization of the Algebra Perturbation Lemma of [11] and [15]. We show three important cases in which these results are essential for the construction of algebra or coalgebra structures in various chain complexes.

Math. Subject Classification (1991): Primary: 18G99, 55U15, 55U99. Secondary: 18G10

Contents

1 Introduction. 3

2 Notations and definitions 4

3 The Basic Perturbation Lemma 9

4 Perturbation machinery for algebra and coalgebra categories 11

5 Applications 22

5.1 Example 1: A Hopf algebra structure on the bar construction

contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.2 Example 2: A particular Eilenberg-Zilber contraction . . . . . . 24 5.3 Example 3: Cartan’s little constructions . . . . . . . . . . . . . 29

6 Related questions 42

7 Appendix: The Shih operator 43 7.1 Prerequisites. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 7.2 Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7.3 Proof of the theorem. . . . . . . . . . . . . . . . . . . . . . . . 44

This work has been partially supported by various Junta de Andalucía grants. The author wishes to express his gratitude to Laboratoire de Mathématiques de l’Institut Fourier de Grenoble (France) for support during the completion of this manuscript.

1 Introduction.

Homological Perturbation Theory ([32], [2], [8], [13], [18], [10], [14], [11], [15], …) is a set of efficient techniques for the transference of structures from one object to another up to homotopy. Its most important element is the Basic Perturbation Lemma which can be considered as an actual algorithm; its main input is a contraction between two DG-modules (an especial homotopy equivalence) and its output is a new contraction as well.

Homological Perturbation Theory has been largely considered and results about preservation of structures (DG-algebras, DG-coalgebras, Lie algebras, …) have already been proved. The technique for obtaining these “homological perturbation machines” is to determine in which conditions the Basic Perturbation Lemma allows to establish preservation results for the data structures. In this way, a (Co)algebra Perturbation Lemma is described in [11] and [15] in which the (co)algebra laws are preserved.

In this paper, assuming weaker hypotheses for the data of the Basic Perturbation Lemma, we prove analogous theorems, where the resulting contraction, of course, has weaker properties too. More precisely, we analyse conditions under which the Basic Perturbation Lemma preserves strict associativity. Apart from the intrinsic interest of this problem, in several important situations (see Section 6) our results allow us to determine and construct essential algebra or coalgebra structures on various chain complexes.

Here is a summary of the present paper. Notation and terminology are introduced in Section 2. In Section 3, the basic ingredients in Homological Perturbation Theory are reviewed. Section 4 is devoted to prove the main results of this paper. Three applications of these theorems are given in Section 5. Finally, various open problems are proposed in Section 6. We think that our main theorems could play a key role in those questions. Finally, an appendix, written by Frederic Morace, is added. It is devoted to prove that the homotopy operator of an important contraction, the Eilenberg-Zilber contraction, is defined by an explicit formula discovered experimentally by Julio Rubio. I wish to express my sincere thanks to Frederic for having written down this result.

I would also like to express my gratitude to Francis Sergeraert for suggesting a problem which led to the present paper and for his continuing advice and encouragement. Finally, I wish to thank Julio Rubio for stimulating discussions and many valuable suggestions. These suggestions resulted in notable improvements to the exposition.

2 Notations and definitions

Let Λ be a commutative ring with 1 ≠ 0, taken henceforth as ground ring and fixed throughout. Graded modules are graded by the non-negative integers. The degree of an (homogeneous) element x of a graded module will be denoted by |x|. A morphism f : M → N of graded modules has degree i if it satisfies f(M_{n}) ⊂ N_{n+i} for all n.

All tensor products are over Λ. Given a graded module M, we will denote M^{⊗0} = Λ and M^{⊗n} = M ⊗ ⋯ ⊗ M, n times. The identity map of a graded module M will be denoted by 1_{M}. Throughout this paper, we adopt Koszul’s convention, which defines the tensor product of two (homogeneous) graded morphisms as:

(f ⊗ g)(x ⊗ y) = (−1)^{|g||x|}f(x) ⊗ g(y).

If f : M → N is a morphism of graded modules and n is a positive integer, the morphism f⊗ …^{n times} ⊗f; M^{⊗n} → N^{⊗n} will be denoted by f^{⊗n}. An element x ∈ M is called f-vanished if x ∈ Ker f.

Let M and N be two graded modules. The morphism of graded modules t : M ⊗ N → N ⊗ M is defined by t(x ⊗ y) = (−1)^{|x|·|y|}y ⊗ x.

If f : M^{⊗i} → M is a morphism of graded modules and n is a positive integer, it is possible to define the following morphism

f^{[n]} = ∑_{j=0}^{n−i} f^{[n,j]} = ∑_{j=0}^{n−i} 1⊗j_{M} ⊗ f ⊗ 1⊗n−i−j_{M};  (1)

in the summands for j = 0 and j = n − i, the morphism 1⊗0_{M} is understood to be 1_{Λ}. We define the morphism f^{[ ]} : ⊕_{j≥i} M^{⊗j} → ⊕_{k≥1} M^{⊗k} as f^{[ ]}|_{M^{⊗n}} = f^{[n]}

If f : M → M is a morphism of graded modules and n is a positive integer, the composite f∘ ⋯^{n times} ∘f will be denoted by f^{∘n}.

Let (M, d_{M}) be a DG-module. The suspension of M, denoted by S(M), is defined by S(M)_{n+1} = M_{n} and its differential is −d_{M}. Let f : N → M be a morphism of DG-modules (morphism of graded modules compatible with the differential structures) of degree i. We define the morphism S(f) : S(N) → S(M) by S(f)(a) = (−1)^{i}f(a).

A DGA-module (M, d_{M}, ϵ_{M}, η_{M}) is a DG-module (M, d_{M}) equipped with two morphisms of DG-modules: an augmentation and a coaugmentation morphisms ϵ_{M} : M → Λ and η_{M} : Λ → M respectively, so that ϵ_{M}η_{M} is the identity map of Λ; morphisms are then required to preserve this structure. The DG-module \overline{M} = Ker ϵ_{M} is defined in a natural way. Given a morphism of DGA-modules f : M → N, it is defined f̄: \overline{M} → \overline{N} by f̄(x) = f(x).

We shall use without further explanation the concepts of DGA-algebra, DGA-coalgebra, derivation, coderivation, Hopf DGA-algebra, etc. The structure maps of a DGA-algebra (A, d_{A}, μ_{A}, ϵ_{A}, η_{A}) and a DGA-coalgebra (C, d_{C}, Δ_{C}, ϵ_{C}, η_{C}) will be

μ_{A} : A ⊗ A → A (product), and η_{A} : Λ → A (unit)

and

Δ_{C} : C → C ⊗ C (coproduct), and ϵ_{C} : C → Λ (counit),

respectively. Moreover, the augmentation ϵ_{A} and the coaugmentation η_{C} preserve the respective structures.

We will denote the unit of a DGA-algebra A by θ_{A} = η_{A}(1).

Henceforth, if no possibility of confusion exists for the structure maps, the DG-modules, DGA-modules, DGA-algebras and DGA-coalgebras will be denoted only by the cap letter which defines the underlying graded module.

Given a DGA-algebra A, the reduced bar construction on A (see [20]) is the DGA-coalgebra

(B̄(A), dB̄, ΔB̄, ϵB̄, ηB̄) = (T (S(\overline{A})), d_{t} + d_{s}, Δ_{T}, ϵ_{T}, η_{T}),

where (T (M), d_{t}, Δ_{T}, ϵ_{T}, η_{T}) is the tensor coalgebra of a DGA-module M.

An homogeneous element S \overline{a_{1}} ⊗ ⋯ ⊗ S \overline{a_{n}} ∈ B̄(A) will be written in the form [a_{1}| ⋯ |a_{n}]. We also adopt the convention 1 = [ ]. The morphisms d_{s} and d_{t} are called tensor differential and simplicial differential, respectively. Both differentials are coderivations with respect to the canonical coproduct. Given an element z = [a_{1}| … |a_{n}], the tensor degree is defined by |z|_{t} = ∑_{i=1}^{n} |a_{i}|, and the simplicial degree is defined by |z|_{s} = n. It is clear that

|z| = |z|_{t} + |z|_{s}.

Let u = [a_{1}| … |a_{m}] and v = [b_{1}| … |b_{n}] be two elements of B̄(A). We define the following (non-commutative) product:

u • v = [a_{1}| … |a_{m}|b_{1}| … |b_{n}].  (2)

which is extended to B̄(A) by linearity.

If A is a commutative DGA-algebra, it is well known (see [20]) that it is possible to define a commutative product ∗ on B̄(A) (called shuffle product).

In this way, \overline{B}(A) enjoys a Hopf DGA-algebra structure and both simplicial and tensor differentials are derivations with respect to the shuffle product.

Let {A_{i}}_{i∈I} a set of commutative DGA-algebras. A twisted tensor product ˜⊗_{i∈I}^{ρ}A_{i} is a commutative DGA-algebra satisfying the following conditions:

i) as graded algebra, ˜⊗_{i∈I}^{ρ}A_{i} coincides with the tensor product ⊗_{i∈I}A_{i},

ii) and its differential is the sum of the differential of the banal tensor product and a derivation ρ.

In this paper, we deal with a special type of homotopy equivalence: a contraction.

A contraction is a data set r : {N, M, f, g, ϕ} where f : N → M, g : M → N are morphisms of DGA-modules (called the projection and the inclusion of the contraction r, respectively) and ϕ : N → N is a morphism of graded modules of degree +1 (called homotopy operator), and these data are required to satisfy the rules,

\begin{matrix}(c1) & f ∘ g = 1_{M}; \\ (c2) & f ∘ ϕ = 0; \\ (c3) & ϕ ∘ g = 0; \\ (c4) & ϕ ∘ d_{N} + d_{N} ∘ ϕ + g ∘ f = 1_{N}; \\ (c5) & ϕ ∘ ϕ = 0. \end{matrix}  (3)

Given a contraction r : {N, M, f, g, ϕ}, we have Ker ϕ = Im g ⊕ Im ϕ. In fact, the DGA-module N is a direct sum of M and an acyclic DGA-module. N is called the the big DGA-module of r , and M is called the small DGAmodule of r.

In this definition we follow Eilenberg-MacLane ([6]) terminology; we also find in literature: “strong deformation retraction” or SDR (Lambe-Stasheff [18], Gugenheim-Lambe [10] and Gugenheim-Stasheff [9]), “Eilenberg-Zilber data” (Gugenheim-Munkholm [12]), or “trivial extension” (Munkholm [24]).

An Eilenberg-Zilber contraction is a contraction from (L × L′)_{N} to L_{N} ⊗ L′_{N}, where L and L′ are augmented simplicial modules [19]. The subscript N means normalization in the simplicial structure. A least one Eilenberg-Zilber contraction exists:

Theorem 2.1. [5] Let L and L′ be two augmented simplicial Λ-modules. The Alexander-Whitney operator AW : (L × L′)_{N} → L_{N} ⊗ L′_{N}, the Eilenberg-MacLane operator EML : L_{N} ⊗ L′_{N} → (L × L′)_{N} and the Shih operator (of degree +1) SHI : (L × L′)_{N} → (L × L′)_{N} of L and L′ are defined by the following formulas:

AW (x_{n}, y_{n}) = ∑_{i=0}^{n} ∂_{i+1} … ∂_{n}x_{n} ⊗ ∂_{0} … ∂_{i−1}y_{n},  (4)

EML(x_{p} ⊗ y_{q}) = ∑_{(α,β)∈{(p,q)−shuffles}}(−1)^{σ(α,β)} (s_{β_{q}} … s_{β_{1}}x_{p}, s_{α_{p}} … s_{α_{1}}y_{q}) (5)

SHI(x_{n}, y_{n}) =

= ∑(−1)^{n−p−q+σ(α,β)}(s_{β_{q}+n−p−q} … s_{β_{1}+n−p−q}s_{n−p−q−1}∂_{n−q+1} … ∂_{n}x_{n},

s_{α_{p+1}+n−p−q} … s_{α_{1}+n−p−q}∂_{n−p−q} … ∂_{n−q−1}y_{n}),

where the last sum is taken over all the indices 0 ≤ q ≤ n − 1, 0 ≤ p ≤ n − q − 1, (α, β) ∈ {(p + 1, q) − shuffles} and σ(α, β) = ∑ [α_{i} − (i − 1)].

Then, the data

EZ_{L,L′} : {(L × L′)_{N}, L_{N} ⊗ L′_{N}, AW_{L,L′}, EML_{L,L′}, SHI_{L,L′}}

defines a contraction.

The first definition of the SHI operator is given in an inductive way in [7]. The above explicit formula for the Shih operator is given by J. Rubio in [29]. Rubio discovered experimentally this formula by means of a lisp program constructed together with F. Sergeraert. An appendix in this paper, written by F. Morace, is devoted to prove theoretically that this explicit formula satisfies the inductive definition given in [7].

Let X and Y be two simplicial sets. Using the last contraction, we construct

EZ_{X,Y} : {C_{N}(X × Y), C_{N}(X) ⊗ C_{N}(Y), AW_{X,Y} , EML_{X,Y} , SHI_{X,Y} },

where C_{N}(Z) denotes the normalized chain complex of a simplicial set Z.

Several basic and known results ([18], [10]) about constructions of contractions will be used in this paper. We record that if r : {N, M, f, g, ϕ} is a contraction, the suspension version S(r) : {S(N), S(M), S(f), S(g), S(ϕ)} and the contraction r̄: \{\overline{N}, \overline{M}, f̄, ḡ, ϕ̄\} can be formed. If we have two contractions

r_{i} : {N_{i}, M_{i}, f_{i}, g_{i}, ϕ_{i}} i = 1, 2

1. and N_{2} = M_{1}, the following composition contraction can be constructed:

r_{2} ∘ r_{1} : {N_{1}, M_{2}, f_{2} ∘ f_{1}, g_{1} ∘ g_{2}, ϕ_{1} + g_{1} ∘ ϕ_{2} ∘ f_{1}}

2. the tensor product contraction can be constructed:

r_{1}⊗r_{2} : {N_{1}⊗N_{2}, M_{1}⊗M_{2}, f_{1}⊗f_{2}, g_{1}⊗g_{2}, ϕ_{1}⊗g_{2}∘f_{2}+1_{M_{1}} ⊗ϕ_{2}}. (6)

Notation 2.2. Of course, if we have a contraction r : {N, M, f, g, ϕ}, it is possible to form:

r^{⊗n} = \overbrace{r ⊗ … ⊗ r}^{n times} : \{N^{⊗n}, M^{⊗n} , f^{⊗n}, g^{⊗n}, ϕ^{[⊗n]}\}

where

ϕ^{[⊗n]} = ∑_{i=0}^{n−1} ϕ^{[⊗n,i]}  (7)

and ϕ^{[⊗n,i]} = 1⊗i_{N} ⊗ ϕ ⊗ (gf)^{⊗(n−i−1)}; in the case i = 0, the morphism 1⊗i_{N} is understood to be the identity 1_{Λ}, while, for i = n − 1, (gf)^{⊗n−i−1} similarly designates the identity 1_{Λ}.

Let us also note that

ϕ^{[⊗n,i]} ∘ ϕ^{[⊗n,j]} = 0  (8)

for 0 ≤ i, j ≤ n − 1.

Then, using this notation, the tensor module contraction of r will be T (r) : {T (N), T (M), T (f), T (g), T (ϕ)}, where T (f), T (g) and T (ϕ) are defined in each degree by:

T (f)_{n} = f^{⊗n}; T (g)_{n} = g^{⊗n} and T (ϕ)_{n} = ϕ^{[⊗n]}.

3 The Basic Perturbation Lemma

The Basic Perturbation Lemma (for its genesis, see [32], [2], [8]; with regard to later developments see [13], [18],[14], [10], [15], [11], [17]) is a systematic and efficient technique for transferring structures from one object to another up to homotopy. This “algebraic fixed point theorem” [1] is a powerful device for obtaining chain complexes that represent a given homotopy type.

First we recall the concept of perturbation datum. Let M be a graded module N and let f : N → N be a morphism of graded modules. The morphism f is pointwise nilpotent if for all x, being a non-null element of N, a positive integer n exists (in general, the number n depends on the element x) such that f^{∘n}(x) = 0. A perturbation of a DGA-module N is a morphism of graded modules δ : N → N of degree −1, such that (d_{N} + δ)^{∘2} = 0 and ϵ_{N} ∘ δ = 0. For instance, if A is a DGA-algebra, the simplicial differential d_{s} is a perturbation of T (S(\overline{A})). A perturbation datum of the contraction r : {N, M, f, g, ϕ} is a perturbation δ of the DGA-module N, which satisfies the composition ϕ ∘ δ is pointwise nilpotent.

Theorem 3.1. (Basic Perturbation Lemma) [32]

Let r : {N, M, f, g, ϕ} be a contraction and δ : N → N a perturbation datum of this contraction. Then a new contraction

r_{δ} : {(N, d_{N} + δ, ϵ_{N}, η_{N}), (M, d_{M} + d_{δ}, ϵ_{M}, η_{M}), f_{δ}, g_{δ}, ϕ_{δ}}

is defined by the following formulas:

d_{δ} = f ∘ δ ∘ Σ_{r}^{δ} ∘ g;  (9)
f_{δ} = f ∘ (1 − δ ∘ Σ_{r}^{δ} ∘ ϕ);  (10)
g_{δ} = Σ_{r}^{δ} ∘ g;  (11)
ϕ_{δ} = Σ_{r}^{δ} ∘ ϕ;  (12)

where

Σ_{r}^{δ} = ∑_{i≥0}(−1)^{i} (ϕ ∘ δ)^{∘i} = 1 − ϕ ∘ δ + ϕ ∘ δ ∘ ϕ ∘ δ − ⋯ + (−1)^{i}(ϕ ∘ δ)^{∘i} + ⋯

Let us note that Σ_{r}^{δ}(x) is a finite sum for each x ∈ N, because of the pointwise nilpotency of the composition ϕ ∘ δ. Moreover, it is obvious that the morphism d_{δ} is a perturbation of the DGA-module (M, d_{M}, ϵ_{M}, η_{M}). From now on, the Basic Perturbation Lemma will be called BPL.

For later references, we state two special cases.

Proposition 3.2. [10] Under the conditions of Theorem 3.1:

1. If the projection f verifies f ∘ ρ ∘ ϕ = 0, then f_{ρ} = f and d_{ρ} = f ∘ ρ ∘ g.

2. If the inclusion g verifies ϕ ∘ ρ ∘ g = 0, then g_{ρ} = g and d_{ρ} = f ∘ ρ ∘ g.

It is well known that BPL has a good behaviour when we use classical algebraic constructions:

Proposition 3.3. ([11], [15])

Let r : {N, M, f, g, ϕ} be a contraction and let δ, δ′ and δ + δ′ be perturbation data of r. Then, the following commutativity properties are satisfied:

\overline{r_{δ}} = r̄δ̄;  (13)
S(r_{δ}) = [S(r)]_{S(δ)};  (14)
(r ⊗ r)_{(δ⊗1+1⊗δ′)} = r_{δ} ⊗ r_{δ′};  (15)
T (r_{δ}) = [T (r)]_{δ^{[ ]}};  (16)
(r_{δ})_{δ′} = (r_{δ′})_{δ} = r_{(δ+δ′)}.  (17)

4 Perturbation machinery for algebra and coalgebra categories

In [11] and [15] it is shown that BPL accepts algebra or coalgebra data without a change in the formulas, provided that the initial contraction satisfies a strong compatibility condition with respect to the underlying structures. In this section, we assume weaker conditions over the morphisms of the initial contraction such that BPL produces a new contraction between two DGAalgebras. The main motivation for doing this is the fact that the (Co)Algebra Perturbation Lemma of [11] and [15] can not be applied to several important cases in Homological Algebra and Algebraic Topology (see Section 5).

We will set up in this section perturbation theorems for DGA-algebras and DGA-coalgebras and give the details for DGA-algebras.

Definition 4.1. Let A and A′ be two DGA-algebras and let r : {A, A′, f, g, ϕ} be a contraction. We say that r is an algebra contraction if f or g are morphisms of DGA-algebras

The set of the algebra contractions will be denoted by AC. This set is closed by composition and tensor product of contractions.

We define hereby the notion of algebra homotopy.

Definition 4.2. ([10])

Let A and A′ be two DGA-algebras and let r : {A, A′, f, g, ϕ} be a contraction. The homotopy ϕ is an algebra homotopy if

μ_{A} ∘ ϕ = ϕ^{[⊗2]} ∘ μ_{A}.

An easy way for determining that a homotopy operator of one contraction is not a (co)algebra homotopy is shown by the following lemma. Its proof is left to the reader.

Lemma 4.3. Let r : {A, A′, f, g, ϕ} be an algebra contraction. If the composite

ϕ ∘ μ_{A} ∘ ϕ^{[⊗2]}

is not the morphism identically zero, then ϕ is not an algebra homotopy.

Now, we define two notions that will allow us to distinguish algebra contractions.

Definition 4.4. Let r : {A, A′, f, g, ϕ} an algebra contraction. We say that ϕ is a quasi algebra homotopy if the following conditions hold:

\begin{matrix}ϕ ∘ μ_{A} ∘ (ϕ ⊗ ϕ) = 0, \\ ϕ ∘ μ_{A} ∘ (g ⊗ ϕ) = 0, \\ ϕ ∘ μ_{A} ∘ (ϕ ⊗ g) = 0. \end{matrix}  (18)

We say that f is a quasi algebra projection if the following conditions hold:

\begin{matrix}f ∘ μ_{A} ∘ (ϕ ⊗ ϕ) = 0, \\ f ∘ μ_{A} ∘ (g ⊗ ϕ) = 0, \\ f ∘ μ_{A} ∘ (ϕ ⊗ g) = 0. \end{matrix}  (19)

Let us note that the product of the DGA-algebra A′ is not involved in the previous definitions.

We now characterize various algebra contractions. The reason is, as we will see later, that they will have good properties under perturbation.

Definition 4.5. Let r : {A, A′, f, g, ϕ} be an algebra contraction. We say that r is

• a full algebra contraction if f and g are morphisms of DGA-algebras and ϕ is an algebra homotopy.

• an almost-full algebra contraction if f and g are morphisms of DGAalgebras and ϕ is a quasi algebra homotopy.

• a semi-full algebra contraction if f is a quasi algebra projection, g is a morphism of DGA-algebras and ϕ is a quasi algebra homotopy.

The sets of the full, almost-full and semi-full algebra contractions will be denoted respectively by F AC, AF AC and SF AC.

We have the following chain of strict contentions F AC ⊂ AF AC ⊂ SF AC ⊂ AC. Let us observe that each of these set is closed by composition and tensor product of contractions. We will make use of the following notation:

\begin{aligned}\overline{AC} & = & AC \ SF AC; \\ \overline{SF AC} & = & SF AC \ AF AC; \\ \overline{AF AC} & = & AF AC \ F AC. \end{aligned}

Examples of full algebra contraction are given in [11], using the “tensor trick”. On the other hand, it is very easy to find almost-full (non-full) algebra contractions in Algebraic Topology. For instance, Eilenberg and MacLane determined in [7] explicit algebra contractions of this type from B̄( [ ]) to the exterior algebra E(u, 1) on one generator u of degree 1, and from B̄( [ _{}]) to the twisted tensor product E(u, 1) ˜⊗^{δ_{h}}Γ(v, 2), where Γ(v, 2) is the polynomial power algebra on one “generator” v of degree 2 and δ_{h}(v) = h · u is a derivation. An explicit almost-full algebra contraction can also be established from B̄(P (u, 2n)) to E(v, 2n + 1), where P (u, 2n) is the polynomial algebra on one generator u of degree 2n. Finally, it is not difficult either to establish algebra contractions of \overline{SF AC}. An example is the explicit algebra contraction that can be constructed from Q_{(2)}(u, 4) ˜⊗^{δ}E(v, 5) to E(w, 9), where Q_{(2)}(u, 4) = P (u, 4)/(u^{2}) is the truncated polynomial algebra on one generator u of degree 4 and δ(v) = u. In the following section, we will also deal with several important almost-full and semi-full algebra contractions that will appear in a natural way in the context of the homology theory of commutative DGA-algebras.

Proposition 4.6. An algebra contraction r : {A, A′, f, g, ϕ} in which ϕ is an algebra homotopy, is a full algebra contraction.

Proof.

Suppose, for instance, that the projection f is a morphism of DGAalgebras. We will prove that g is also a morphism of DGA-algebras.

Making use of the fact that ϕ is an algebra homotopy:

ϕ ∘ μ_{A} = μ_{A} ∘ ϕ^{[⊗2]},  (20)

and composing (20) respectively with d_{A} from the left and with d[2]_{A} from the right, we obtain the following relations

\begin{aligned}d_{A} ∘ ϕ ∘ μ_{A} & = & μ_{A} ∘ d[2]_{A} ∘ ϕ^{[⊗2]} \\ ϕ ∘ d_{A} ∘ μ_{A} & = & μ_{A} ∘ ϕ^{[⊗2]} ∘ d[2]_{A} \end{aligned}

If we sum these two last equalities, we have:

(d_{A} ∘ ϕ + ϕ ∘ d_{A}) ∘ μ_{A} = μ_{A} ∘ (d[2]_{A} ∘ ϕ^{[⊗2]} + ϕ^{[⊗2]} ∘ d[2]_{A})  (21)

Plugging the rules (r4) of (3) for the contractions r and r ⊗ r into (21), we get

(1 − g ∘ f) ∘ μ_{A} = μ_{A} ∘ (1_{A^{⊗2}} − g^{⊗2} ∘ f^{⊗2})

or, simplifying

g ∘ f ∘ μ_{A} = μ_{A} ∘ g^{⊗2} ∘ f^{⊗2}.  (22)

Taking account that f is a morphism of DGA-algebras, (22) can be rewritten as

g ∘ μ_{B} ∘ f^{⊗2} = μ_{A} ∘ g^{⊗2} ∘ f^{⊗2}.

Thus, since the morphism f is onto, the preceeding equality is equivalent to the following one:

g ∘ μ_{B} = μ_{A} ∘ g^{⊗2}.

Hence, we have deduced that r is a full algebra contraction.

In an analogous way, we can prove that if g is a morphism of DGAalgebras, then f must also be a morphism of DGA-algebras.

Proposition 4.7. Let A be a DGA-algebra, M a DGA-module and r : {A, M, f, g, ϕ}. We have the following equalities:

f ∘μ_{A} −(f ∘μ_{A} ∘g^{⊗2})∘f^{⊗2}) = d_{M} ∘(f ∘μ_{A} ∘ϕ^{[⊗2]})+(f ∘μ_{A} ∘ϕ^{[⊗2]})∘d[2]_{A}, (23)

μ_{A} ∘ g^{⊗2} − g ∘ (f ∘ μ_{A} ∘ g^{⊗2}) = d_{A} ∘ (ϕ ∘ μ_{A} ∘ g^{⊗2}) + (ϕ ∘ μ_{A} ∘ g^{⊗2}) ∘ d_{A}, (24)

ϕ∘μ_{A}−μ_{A}∘ϕ^{[⊗2]} = ϕ∘μ_{A}∘ϕ^{[⊗2]}∘d[2]_{A}−d_{A}∘ϕ∘μ_{A}∘ϕ^{[⊗2]}−g∘f ∘μ_{A}∘ϕ^{[⊗2]}. (25)

Proof.

We will only verify here the equality (25). The equalities (23) and (24) can be obtained in a analogous way.

First, we use the property (c4) of the contraction r^{⊗2}:

1⊗2_{A} − g^{⊗2} ∘ f^{⊗2} = d[2]_{A} ∘ ϕ^{[⊗2]} + ϕ^{[⊗2]} ∘ d[2]_{A}.  (26)

We compose (26) with μ_{A} from the left and using the facts that g is multiplicative and μ_{A} commutes with the differential d_{A}, we obtain:

μ_{A} − g ∘ μ_{A′} ∘ f^{⊗2} = d_{A} ∘ μ_{A} ∘ ϕ^{[⊗2]} + μ_{A} ∘ ϕ^{[⊗2]} ∘ d[2]_{A}.  (27)

Composing again (27) with ϕ from the left and using the annihilation property (c3) of r, we have

ϕ ∘ μ_{A} = ϕ ∘ d_{A} ∘ μ_{A} ∘ ϕ^{[⊗2]} + ϕ ∘ μ_{A} ∘ ϕ^{[⊗2]} ∘ d[2]_{A}.  (28)

Now, we apply to the second term of (28) the property (c4) of r

ϕ ∘ μ_{A} = (1_{A} − g ∘ f − d_{A} ∘ ϕ) ∘ μ_{A} ∘ ϕ^{[⊗2]} + ϕ ∘ μ_{A} ∘ ϕ^{[⊗2]} ∘ d[2]_{A}. (29)

Finally, (29) can be rewritten in the following form

ϕ∘μ_{A}−μ_{A}∘ϕ^{[⊗2]} = ϕ∘μ_{A}∘ϕ^{[⊗2]}∘d[2]_{A}−d_{A}∘ϕ∘μ_{A}∘ϕ^{[⊗2]}−g∘f ∘μ_{A}∘ϕ^{[⊗2]}. (30)

————–

To differenciate almost-full algebra contractions from full algebra contractions, we can use the following result: Proposition 4.8. Let r : {A, A′, f, g, ϕ} be an almost-full algebra contraction. r belongs to F AC if and only if ϕ ∘ μ_{A} ∘ (1 ⊗ ϕ) = 0.

We also leave the proof of the following result to the reader. Proposition 4.9. Let r : {A, A′, f, g, ϕ} be a semi-full algebra contraction. r belongs to AF AC if and only if f ∘ μ_{A} ∘ (1 ⊗ ϕ) = 0.

The following proposition states that an equivalent definition of semifull algebra contraction may be an algebra contraction in which Ker ϕ is a sub-algebra of A. Proposition 4.10. Let r : {A, A′, f, g, ϕ} be a semi-full algebra contraction. Let u, v be two elements of Ker ϕ. Then

f ∘ μ_{A}(u ⊗ v) = μ_{A′} ∘ (f ⊗ f)(u ⊗ v),  (31)

and

ϕ ∘ μ_{A}(u ⊗ v) = μ_{A} ∘ ϕ^{[⊗2]}(u ⊗ v).  (32)

Proof.

We will only prove here the equality (32). (31) can be demonstrated in an analogous way.

We consider here the equality (25). Since ϕ is a quasi algebra homotopy and f is and quasi algebra projection, the second term of (25) turns out to be:

φ = ϕ ∘ μ_{A} ∘ (1_{A} ⊗ ϕ) ∘ d[2]_{A} − d_{A} ∘ ϕ ∘ μ_{A} ∘ (1_{A} ⊗ ϕ) − g ∘ f ∘ μ_{A} ∘ (1_{A} ⊗ ϕ) (33)

If u and v are elements of Ker ϕ = Im g ⊕ Im ϕ, the relation φ(u⊗v) = 0 is immediately proved.

Hence, for these particular arguments, we get (32).

——————

In order to determining the pairs of elements for which the projection and the homotopy operator of a semi-full algebra contraction become multiplicative, we may refine the last result.

Proposition 4.11. Let r : {A, A′, f, g, ϕ} be a semi-full algebra contraction. Let u and v be two elements of A. If u or v are in Im g, then

f ∘ μ_{A}(u ⊗ v) = μ_{A′} ∘ (f ⊗ f)(u ⊗ v),  (34)

and

ϕ ∘ μ_{A}(u ⊗ v) = μ_{A} ∘ ϕ^{[⊗2]}(u ⊗ v).  (35)

We also leave the proof of the following result to the reader. In spite of its simplicity, this proposition will be essential in the proof of ours main theorems.

Proposition 4.12. Let A be a DGA-algebra, M a DGA-module and r : {A, M, f, g, ϕ} a contraction. Let us suppose that

\begin{matrix}ϕ ∘ μ_{A} ∘ g^{⊗2} = 0. \\ (resp. f ∘ μ_{A} ∘ ϕ^{[⊗2]} = 0) \end{matrix}  (36)

Then the morphism υ_{M} = f ∘ μ_{A} ∘ g^{⊗2} defines a product on M (being its unit η_{M}) and r is an algebra contraction from (A, d_{A}, μ_{A}, ϵ_{A}, η_{A}) to (M, d_{M}, υ_{M}, ϵ_{M}, η_{M}) with multiplicative inclusion (resp. multiplicative projection).

It is time to introduce the notion of algebra perturbation datum.

Definition 4.13. Let A and A′ be two DGA-algebras and r : {A, A′, f, g, ϕ} be a contraction. An algebra perturbation datum δ of this contraction is a perturbation datum of r which is also a derivation.

At this point, it is natural to pose this question: if we have an algebra contraction r : {A, A′, f, g, ϕ} and an algebra perturbation datum δ of r, BPL can be applied; under which conditions, can we state that the composition

υ_{A′} = f_{δ} ∘ μ_{A} ∘ (g_{δ})^{⊗2}  (37)

on A′, that is defined by transferring the product on A to A′ by means of the perturbes of f and g, is an actual product on (A′)_{δ}?

In [11] and [15], the perturbation problem for algebras is solved assuming a strong condition for the homotopy of the contraction: to be a algebra homotopy. In this way, they establish a preservation result of (co)algebra laws. The following theorem for DGA-algebras (that we will call Full Algebra Perturbation Lemma or, more briefly, F -APL) is a translation, using our language, of this result.

Theorem 4.14 (F -APL) [11] Let r : {A, A′, f, g, ϕ} be a full algebra contraction and δ an algebra perturbation datum of r.

Then the contraction r_{δ} obtained by applying BPL (teor. 3.1) is a full algebra contraction from (A, d_{A} + δ, μ_{A}, ϵ_{A}, η_{A}) to (A′, d_{A′} + d_{δ}, μ_{A′}, ϵ_{A′}, η_{A′}). Moreover, the perturbation d_{δ} for the DGA-module A′ is a derivation.

The preceeding theorem says us that F AC is closed by perturbation. The goal of this section and the main aim of this paper is to establish perturbation machineries for other algebra contractions. The idea is to generalize the following proposition (that we will call Special Algebra Perturbation Lemma or S-APL) which is an enriched version of Proposition 3.2:

Proposition 4.15 (S-APL) [10]

Let A and A′ be two DGA-algebras, r : {A, A′, f, g, ϕ} a contraction and δ an algebra perturbation datum of r. If f (resp. g) is a morphism of DGAalgebras and

ϕ ∘ δ ∘ g = 0 (resp.: f ∘ δ ∘ ϕ = 0)  (38)

Then the morphism g_{δ} = g (resp. f_{δ} = f) of the contraction r_{δ}, obtained by applying BPL, is, consequentely, a morphism of DGA-algebras; the perturbation d_{δ} = f ∘ δ ∘ g obtained in A′ is a derivation.

In order to obtain perturbation results preserving the DGA-algebra category, we use the proposition 4.12, taking as datum the perturbed contraction r_{δ}. In this way, we state the following result (that we will call General(inclusion) Algebra Perturbation Lemma or, more briefly GI-APL) that guarantees the preservation of the strict associativity in a perturbed contraction:

Theorem 4.16 (GI-APL) Let A and A′ two DGA-algebras, r : {A, A′, f, g, ϕ} a contraction and δ an algebra perturbation datum of r.

Let us assume that g is a morphism of DGA-algebras and

ϕ ∘ μ_{A} ∘ (ϕ^{[⊗2]} ∘ δ^{[2]})^{∘n} ∘ g^{⊗2} = 0. ∀n ≥ 1  (39)

Then the inclusion g_{δ} of r_{δ}, obtained by applying BPL, is a morphism of DGA-algebras from (A′, d_{A′} + d_{δ}, υ_{A′}, ϵ_{A′}, η_{A′}) to (A, d_{A} + δ, μ_{A}, ϵ_{A}, η_{A}), where the product υ_{A′} is given by:

\begin{aligned}υ_{A′} & = & f ∘ μ_{A} ∘ (g_{δ})^{⊗2} \\ & = & f ∘ μ_{A} ∘ [∑_{m≥0}(−1)^{m}(ϕ^{[⊗2]} ∘ δ^{[2]})^{∘m} ∘ g^{⊗2}]. \end{aligned}  (40)

Moreover, if we assume that f is also a morphism of DGA-algebras, the product υ_{A′} coincides with the initial product μ_{A′} of A′.

Proof.

Considering the prop. 3.3 (15), we have that

(g_{δ})^{⊗2} = (g^{⊗2})_{δ^{[2]}} = ∑_{n≥0}(−1)^{n} (ϕ^{[⊗2]} ∘ δ^{[2]})^{∘n} ∘ g^{⊗2}.

Hence, the condition (39) and the fact that g is a morphism of DGA-algebras imply that ϕ ∘ μ_{A} ∘ (g_{δ})^{⊗2} = 0, and, consequently that ϕ_{δ} ∘ μ_{A} ∘ (g_{δ})^{⊗2} = 0. If we now apply Prop. 4.12 to the contraction r_{δ}, we endow A′ with an algebra structure

υ_{A′} = f_{δ} ∘ μ_{A} ∘ (g_{δ})^{⊗2}
= (f − f ∘ δ ∘ Σ_{r}^{δ} ∘ ϕ) ∘ μ_{A} ∘ (g^{⊗2})_{δ^{[2]}}
= f ∘ μ_{A} ∘ [∑_{m≥0}(−1)^{m}(ϕ^{[⊗2]} ∘ δ^{[2]})^{∘m}] ∘ g^{⊗2}

Furthermore, the contraction r_{δ} is an algebra contraction from (A, d_{A} + δ, μ_{A}, ϵ_{A}, η_{A}) to (A′, d_{A′} + d_{δ}, υ_{A′}, ϵ_{A′}, η_{A′}) with multiplicative inclusion.

If f is also a morphism of DGA-algebras, and making use of the properties (c1) and (c2) of the contraction r, we get

υ_{A′} = f ∘ μ_{A} ∘ [∑_{m≥0}(−1)^{m}(ϕ^{[⊗2]} ∘ δ^{[2]})^{∘m}] ∘ g^{⊗2}
= μ_{A} ∘ f ^{ot2} ∘ [∑_{m≥0}(−1)^{m}(ϕ^{[⊗2]} ∘ δ^{[2]})^{∘m}] ∘ g^{⊗2}
= μ_{A′} ∘ f^{⊗2} ∘ g^{⊗2}
= μ_{A′}

The previous result was already enunciated in [26].

We now give a similar result to the preceding one without proof.

Theorem 4.17 (GP -APL) Let A and A′ be two DGA-algebras, r : {A, A′, f, g, ϕ} an algebra contraction and δ an algebra perturbation datum of r.

Let us suppose that f is a morphism of DGA-algebras and

f ∘ (δ ∘ ϕ)^{⊗n} ∘ μ_{A} ∘ ϕ^{[⊗2]} = 0,  ∀ n ≥ 1.  (41)

Then the projection f_{δ} of the contraction r_{δ}, obtained by applying BPL, is a morphism of DGA-algebras from (A, d_{A} + δ, μ_{A}, ϵ_{A}, η_{A}) to (A′, d_{A′} + d_{δ}, υ_{A′}, ϵ_{A′}, η_{A′}), where the product υ_{A′} is given by the following formula:

υ_{A′} = f_{δ} ∘ μ_{A} ∘ g^{⊗2}.  (42)

Moreover, if we assume that g is also a morphism of DGA-algebras, the product υ_{A′} coincides with the initial product μ_{A′} of A′.

The last theorems can be considered as respective generalizations of S-APL Theorem. On the other hand, when we work with a full algebra contraction as initial datum, GI-APL and GP -APL theorems only tell us that the perturbed morphims f_{δ} and g_{δ} are morphisms of DGA-algebras, giving us less information than the F -APL Theorem.

Let us note that, in general, the perturbation datum δ is involved in an essential way in the last perturbation results. However, if we take contractions of AF AC or SF AC (in which its homotopy operator is a quasi-algebra homotopy), we can prove the following perturbation result:

Theorem 4.18 (SF -APL) Let r : {A, A′, f, g, ϕ} be a semi-full algebra contraction and δ an algebra perturbation datum of r. Then the perturbed contraction r_{δ} is a semi-full algebra contraction from (A, d_{A} + δ, μ_{A}, ϵ_{A}, η_{A}) to (A′, d_{A′} + d_{δ}, μ_{A′}, ϵ_{A′}, η_{A′}).

Proof.

Taking into account the formulas of the morphisms of the perturbed contraction r_{δ}, we directely see that f_{δ} and ϕ_{δ} are, respectively, a quasi algebra projection and a quasi algebra homotopy.

Concerning to g_{δ}, taking into account that ϕ is a quasi algebra homotopy and g is a morphism of DGA-algebras, we obtain

ϕ ∘ μ_{A} ∘ (g_{δ} ⊗ g_{δ}) = 0.  (43)

Now, by GI-APL we determine that g_{δ} is a morphism of DGA-algebras from (A′, d_{A′} + d_{δ}, υ_{A′}, ϵ_{A′}, η_{A′}) to (A, d_{A} + δ, μ_{A}, ϵ_{A}, η_{A}), where the product υ_{A′} is given by (40):

\begin{aligned}υ_{A′} & = & f ∘ μ_{A} ∘ (g_{δ} ⊗ g_{δ} \\ & = & f ∘ μ_{A} ∘ [(∑_{i≥0} (−1)^{i}(ϕ ∘ δ)^{∘i} ∘ g) ⊗ (∑_{i≥0} (−1)^{j}(ϕ ∘ δ)^{∘j} ∘ g)]. \end{aligned}

(44)

Using that f is a quasi algebra projection, g is a morphism of DGAalgebras and the property (c1) of the contraction r, the formula (44) turns out to be:

υ_{A′} = f ∘ μ_{A}(g ⊗ g) = f ∘ g ∘ μ_{A′} = μ_{A′}.

—————

Hence, to be a semi-full algebra reduction is a hereditary property under homological perturbation.

The development of the ideas above for DGA-coalgebras is analogue and we can uniformly replace the notions of bar construction, derivation, product, …, with their duals. Note, however, that some care must be taken in establishing the condition which must satisfy the inputs of the coalgebra perturbation results. In this way, GI-CPL, GP -CPL and SF -CPL Theorems can be enunciated.

5 Applications

In this section, we will show several examples in which F -A(C)PL can not be applied, but the hypotheses of SF -A(C)PL or GI-A(C)PL Theorems are verified.

5.1 Example 1: A Hopf algebra structure on the bar construction contraction

Let r : {A, A′, f, g, ϕ} be a contraction where A and A′ be two commutative DGA-algebras. We here study the multiplicative behaviour of the contraction B̄(r) with respect to the shuffle product ∗. Let us recall that the bar construction of a commutative DGA-algebra A is a Hopf DGA-algebra denoted by (B̄(A), dB̄, ∗, ΔB̄, ϵB̄, ηB̄).

We begin with the following result:

Theorem 5.1. [11]

Let A and M be a DGA-algebra and a DGA-module respectively, and r : {A, M, f, g, ϕ} a contraction. Hence, the following full coalgebra contraction can be established:

B̄(r) = [T (S(r̄)]_{d_{s}^{A}} : {B̄(A), B̃(M), B̄(f), B̄(g), B̄(ϕ)}  (45)

where ds^{A} is the simplicial differential of B̄(A) and the DG-module B̃(M) is the bar tilde construction of Stasheff [33].

In the next result, we state the compatibility of the bar construction contraction and the perturbation machinery.

Theorem 5.2. Let r : {A, M, f, g, ϕ} be a contraction where A and M are a DGA-algebra and a DGA-module, respectively. If δ is an algebra perturbation datum of r, then the following commutativity property is satisfied:

B̄(r_{δ}) = [B̄(r)]_{S(δ̄)^{[ ]}}

Proof.

First, by applying Prop. 5.1 to r_{δ}, we establish the perturbed contraction B̄(r_{δ}) = [T (S(\overline{r_{δ}})]_{d_{s}^{A}}.

On the other hand, using the product of A, we define the following morphism of DG-modules of degree -1:

S̃(\overline{μ_{A}}) : S(Ā) ⊗ S(Ā) ⟶ S(Ā)

defined by S̃(\overline{μ_{A}})(Sa′ ⊗ Sa′′) = (−1)^{|Sa′|}μ_{A}(Sa′ ⊗ Sa′′).

It is easy to verify that the simplicial differential ds^{A} of B̄(A) is the morphism S̃(\overline{μ_{A}})^{[ ]}. Since δ is a derivation, then

S(δ̄)^{[ ]} ∘ S̃(\overline{μ_{A}})^{[ ]} + S̃(\overline{μ_{A}})^{[ ]} ∘ S(δ̄)^{[ ]} = 0.

Finally, taking account of the good behaviour of BPL in (13), (14), (16) and (17), we have:

B̄(r_{δ}) = [T (S(\overline{r_{δ}}))]_{d_{s}}
= [T (S(r̄δ̄))]_{d_{s}}
= [T (S(r̄)_{S(δ̄)})]_{d_{s}}
= [T (S(r̄))_{S(δ̄)^{[ ]}}]_{d_{s}}
= [T (S(r̄))_{d_{s}}]_{S(δ̄)^{[ ]}}
= B̄(r)_{S(δ̄)^{[ ]}}

And the theorem follows.

If M = A′ is a DGA-algebra and r is an algebra contraction, then the contraction B̄(r) “connects” two bar constructions (see [11]). For example, if g is a morphism of DGA-algebras, then B̄(r) is a contraction from B̄(A) to B̄(A′) and its inclusion B̄(g) coincides with T (S(ḡ)). Since this morphism preserves shuffle products, B̄(r) is an algebra contraction with multiplicative inclusion. Without making use of any assumption, we show here that B̄(g) always preserves shuffle products.

Theorem 5.3. Let r : {A, M, f, g, ϕ} be a contraction where A is a commutative DGA-algebra and M is a DGA-module. Then

B̄(r) : {B̄(A), B̃(M), B̄(f), B̄(g), B̄(ϕ)}

is a semi-full algebra contraction.

Proof.

It is known (see [11]) that B̄(r) is a full coalgebra contraction. It suffices to prove that the inclusion of B̄(r) is multiplicative with respect to the shuffle product.

First, both differentials dt^{A} and ds^{A} are ∗-derivations. Secondly, T (S(r̄)) is an almost-full algebra contraction. Indeed, it is obvious that both T (S(f̄)) and T (S(ḡ)) are compatible with the shuffle products. It is also a simple exercise to prove that T (S(ϕ̄)) is a quasi algebra homotopy. However,

T (S(ϕ̄)) ∘ ∗ ∘ T (S(ϕ̄))^{[⊗2]}([Sa] ⊗ [Sb]) ≠ 0

for a, b ∈ Ā with ϕ(a) ≠ 0 ≠ ϕ(b). Hence, it follows from Proposition 4.8 that T (S(ϕ̄)) is not an algebra homotopy.

With these ingredients, the result directely follows from SF -APL.

——————

5.2 Example 2: A particular Eilenberg-Zilber contraction

A contraction R_{B̄⊗}, described by Eilenberg-MacLane in [7], from B̄(A ⊗ A′) to B̄(A) ⊗ B̄(A′), where A and A′ are two commutative DGA-algebras, is recorded here. We will show here that this contraction is an almost-full algebra contraction and belongs to the set of coalgebra contractions \overline{CC}.

Moreover, taking this contraction as initial datum, if we perturb the differential of A ⊗ A′ or we modify the product of A ⊗ A′ in such a way that we still have a new commutative DGA-algebra, we will determine the good (co)multiplicative behaviour of the perturbed contraction by making use of SF -APL and GI-CPL Theorems.

The study done in this section can be generalized in a natural way to the contraction from B̄(⊗_{i=1}^{n}A_{i}) to ⊗_{i=1}^{n}B̄(A_{i}), for n = 3, 4, . . ..

Theorem 5.4. ([7], pag. 59-60)

If A and A′ are two commutative DGA-algebras, then there is an algebra contraction R_{B̄⊗} with multiplicative projection and inclusion from B̄(A ⊗ A′) to B̄(A) ⊗ B̄(A′).

R_{B̄⊗} is also a coalgebra contraction with comultiplicative inclusion.

Eilenberg and MacLane in [6] assign to each commutative DGA-algebra A, a construction B(A) that may be regarded, in particular, as an augmented simplicial Λ-algebra (that we will call B^{smp}(A)) and, hence, they establish the normalized construction Bsmp_{N}(A). This last construction endowed with the degree, differential, product and coproduct induced by B(A) coincides with our definition of bar construction B̄(A).

Eilenberg and Maclane exploit here the Eilenberg-Zilber Theorem (Th. 2.1) and they establish a contraction EZ_{B^{smp}(A),B̄^{smp}(A′)} from (B̄^{smp}(A) × B̄^{smp}(A′))_{N} to B̄N^{smp}(A) ⊗ B̄N^{smp}(A′). The desired contraction

R_{B̄⊗} : {B̄(A ⊗ A′), B̄(A) ⊗ B̄(A′), f_{B̄⊗}, g_{B̄⊗}, ϕ_{B̄⊗}}.

is obtained composing the preceeding contraction with iso-contractions which amounts largerly to sign changes.

The explicit formulas of the projection and inclusion of this contraction are given by Eilenberg-MacLane in [7]. The formula of the homotopy operator is:

ϕ_{B̄⊗} = ς^{−1} SHI_{B̄^{smp}A,B̄^{smp}A′} ς.  (46)

The morphism SHI_{B̄^{smp}A,B̄^{smp}A′} is the homotopy operator of the contraction EZ_{B̄^{smp}A,B̄^{smp}A′} which is defined in Theorem 2.1, and the isomorphism of DG-modules ς : B̄(A ⊗ A′) → (B̄^{smp}(A) × B̄^{smp}(A′))_{N} is defined for homogeneous elements of B̄(A ⊗ A′) of the form

[a_{1} ⊗ a′_{1}| ⋯ |a_{m} ⊗ a′_{m}],

with a_{i} and a′_{i}, homogeneous elements of A and A′, respectively, by the formula

ς [a_{1} ⊗ a′_{1}| ⋯ |a_{m} ⊗ a′_{m}] = (−1)^{∑_{j>k} |a_{j} | |a′_{k}|} [a_{1}| ⋯ |a_{m}] ⊗ [a′_{1}| ⋯ |a′_{m}].

It is easy to see that ϕ_{B̄⊗} is not an algebra homotopy.

In addition, the morphism ϕ_{B̄⊗} is not a coalgebra homotopy. In fact, the projection f_{B̄⊗} is not a morphism of DGA-coalgebras. Since g_{B̄⊗} is a morphism of DGA-coalgebras and taking into account Proposition 4.6, we deduce that ϕ_{B̄⊗} is not a coalgebra homotopy.

We will employ the rest of this subsection to prove that R_{B̄⊗} is an almostfull algebra contraction.

First, we introduce the following definitions:

Definition 5.5. Let A and A′ be two DGA-algebras. An element a of Ā (resp. a′ of Ā′) can be regarded as an element of the tensor product A ⊗ A′ by the morphism of DGA-algebras:

\begin{matrix}a ↦ c = a ⊗ θ_{A′}, \\ (resp. a′ ↦ c′ = θ_{A} ⊗ a′). \end{matrix}

The elements of c ∈ A ⊗ A′ (resp. c′ ∈ A ⊗ A′) which are obtained in this way are called simple elements of A ⊗ A′.

Remark 5.6. We can consider the DGA-algebras A and A′ as sub-DG-algebras of A ⊗ A′. In the same way, the constructions B̄(A) and B̄(A′) can be viewed as sub-DG-algebras of B̄(A ⊗ A′) and of B̄(A) ⊗ B̄(A′). From now on, we will make use of these identifications when they may be considered appropriate.

Definition 5.7. Let A and A′ be two DGA-algebras. Let u = [c_{1}| ⋯ |c_{m}] be an element of B̄(A ⊗ A′). We say that u is simple if for all 1 ≤ i ≤ m, c_{i} is a simple element of A ⊗ A′. In this case, this element has s inversions if there is exactly s elements c_{r_{i}} ∈ \overline{A′} with i = 1, … s and r_{1} < r_{2} < … < r_{s}, such that c_{r_{s}+1} ∈ \overline{A}. The graded module generated by the simple elements of B̄(A ⊗ A′) will be denoted by SB̄(A ⊗ A′).

Of course, if u is a f_{B̄⊗}-vanished simple element of B̄(A⊗A′), this element has at least 1 inversion.

Definition 5.8. Let A and A′ be two DGA-algebras. Let u = [c_{1}| ⋯ |c_{m}] be an element of B̄(A ⊗ A′). We say that u is 1-simple if there is only one subscript k ∈ \{1, …, m\} such that c_{i} is simple, for i ≠ k, and c_{k} ∈ \overline{A} ⊗ \overline{A′}. In this case, this element has s + 1 inversions if there is exactly s elements c_{r_{i}} ∈ \overline{A′} with i = 1, … m and r_{1} < r_{2} < … < r_{s}, such that c_{r_{s}+1} ∈ \overline{A} or c_{r_{s}+1} = c_{k} ∈ \overline{A} ⊗ \overline{A′}. The graded module generated by the 1-simple elements of B̄(A ⊗ A′) will be denoted by S1B̄(A ⊗ A′).

Definition 5.9. Let A and A′ be two DGA-algebras. Let u = [c_{1}| ⋯ |c_{m}] be an element of B̄(A ⊗ A′). We say that u finishes with an inversion if there is a subscript k ∈ {1, …, m} such that [c_{k}|c_{k+1}| ⋯ |c_{m}] is an element of SB̄(A ⊗ A′) with 1 inversion. The graded module generated by these elements will be denoted by IB̄(A ⊗ A′).

The proof of the following proposition follows from the explicit description of the morphisms f_{B̄⊗}, g_{B̄⊗} and ϕ_{B̄⊗} of the contraction R_{B̄⊗}.

Proposition 5.10. Let A and A′ be two commutative DGA-algebras. Then,

a) S1B̄(A ⊗ A′) ⊂ Ker f_{B̄⊗}.

b) Im g_{B̄⊗} ⊂ SB̄(A ⊗ A′) ⊂ Ker ϕ_{B̄⊗}.

c) SB̄(A ⊗ A′) ∗ SB̄(A ⊗ A′) ⊂ SB̄(A ⊗ A′).

d) ΔB̄(SB̄(A ⊗ A′)) ⊂ SB̄(A ⊗ A′) ⊗ SB̄(A ⊗ A′).

e) The homotopy operator ϕ_{B̄⊗} carries 1-simple elements with s inversions to zero or to sums of f_{B̄⊗}-vanished simple elements with at least s inversions.

f) IB̄(A ⊗ A′) = Im ϕ_{B̄⊗} ⊂ Ker ϕ_{B̄⊗}.

Most of the proposition follows from the explicit description of the homotopy operator ϕ_{B̄⊗} (see (46)).

From the proposition above, we easily get Proposition 5.11. (1) The contraction R_{B̄⊗} is an almost-full algebra con-

traction.

(2) The contraction R_{B̄⊗} is a coalgebra contraction with comultiplicative inclusion. This contraction is in \overline{CC} because neither f_{B̄⊗} is a quasi coalgebra projection nor ϕ_{B̄⊗} is a quasi coalgebra homotopy.

The following theorem directely follows from SF -APC and Proposition 5.11 (1).

Theorem 5.12. Let A and A′ be two commutative DGA-algebras and δ an algebra perturbation datum of the contraction R_{B̄⊗} from B̄(A ⊗ A′) to B̄(A) ⊗ B̄(A). Then the perturbed contraction (R_{B̄⊗})_{δ} is a semi-full algebra contraction.

Now, we define two special types of perturbation data of the contraction R_{B̄⊗}.

Definition 5.13. Let A and A′ be two commutative DGA-algebras and δ a perturbation datum of the contraction R_{B̄⊗}. We say that δ is a simple perturbation datum if it is a derivation, a coderivation and carries simple elements into simple or 1-simple elements.

The perturbations for R_{B̄⊗} arising from perturbations of the differential of A ⊗ A′ or from “modifications” of the product of A ⊗ A′ so that the new modified tensor product is still a commutative DGA-algebra, are simple perturbation data. We will find specific perturbation data of this type in the next section.

The use of the coalgebra perturbation result GI-CPL is essential in the following theorem.

Theorem 5.14. Let A and A′ be two commutative DGA-algebras. Assume that δ is a simple perturbation datum of R_{B̄⊗}.

Then the inclusion of the perturbed contraction (R_{B̄⊗})_{δ} is a morphism of Hopf DGA-algebras from

(B̄(A) ⊗ B̄(A′), d[2]_{B̄} + d_{δ}, (∗_{A} ⊗ ∗_{A′}) ∘ (1 ⊗ t ⊗ 1), Δ̃_{⊗B̄}, ϵ⊗2_{B̄}, η⊗2_{B̄})

to

(B̄(A ⊗ A′), dB̄ + δ, ∗_{A⊗A′}, ΔB̄, ϵB̄, ηB̄),

where

Δ̃_{⊗B̄} = (f_{B̄⊗} ⊗ f_{B̄⊗}) ∘ ΔB̄ ∘ (g_{B̄⊗})_{δ}.

Moreover, the perturbation d_{δ} over B̄(A) ⊗ B̄(A′) is a derivation and a coderivation.

Proof.

Concerning to algebra structures, all have been proved in Theorem 5.12 which is valid for every algebra perturbation datum δ.

To complete the proof, it suffices to verify that the following condition holds:

(1 ⊗ ϕ_{B̄⊗} + ϕ_{B̄⊗} ⊗ g_{B̄⊗}f_{B̄⊗}) ∘ ΔB̄ ∘ (ϕ_{B̄⊗} ∘ δ)^{∘n} ∘ g_{B̄⊗} = 0, ∀ n ≥ 0.

(47)

First, by prop. 5.10 b) we have that Img_{B̄⊗} ⊂ SB̄(A ⊗ A′). From the fact that δ is a simple perturbation datum and prop. 5.10 e, the composite ϕ_{B̄⊗} ∘ δ)^{∘n} carries any element of SB̄(A ⊗ A′) into an element of this same graded module. Finally, (47) follows from prop. 5.10 d) and b).

Thus, we are able to apply GI-CPL and this proves the theorem.

5.3 Example 3: Cartan’s little constructions

Let us recall that in 1954, Henri Cartan [4] determined the integer homology algebra of Eilenberg-MacLane spaces. To do this, he introduced the notion of construction and several homological operations (suspension, transpotence and divided powers) constructing, thanks to them, tensor products of a certain number of “elementary complexes” (or “Cartan’s little constructions”), one for each appropriated homological operation (admissible words). The homology of this tensor product reduced by several identifications gave the desired integer homology of an Eilenberg-MacLane space. On the other hand, a deep study of the p-primary (p prime) algebra homology of Cartan’s little constructions was given by John C. Moore in [22]. A short account of the work of Moore on this subject is given in [34].

In the setting of the homological perturbation theory and taking Λ = _{(p)} ( localized at a prime p) as our ground ring, we will show here a method for computing the homology algebras of Cartan’s little constructions.

We must point out that our method is written essentially in the same way in Cartan and Moore papers, with the difference that we are interested in obtaining explicit contractions, while they only work with the inclusions of these. In our process, a case appears in which SF -APL will be the unique algebra perturbation theorem that could be applied. In the method developed by Cartan, the generators of the elementary algebras appearing in its iterative process admit an interpretation in terms of homological operations. Here, we maintain this notation in order to provide a comparison with the Cartan method.

We begin with the following definitions:

Definition 5.15. [4] Let A be a commutative DGA-algebra. We define the following morphisms of graded modules:

• The suspension,

σ : A → B̄(A) defined by
σ(a) = [a], where a ∈ A.

• The p-transpotence, where p is a prime number,

φ_{p} : A → B̄(A), defined by
φ_{p}(a) = [a|a^{p−1}] where a ∈A.

Let us notice that Cartan defines the additive functions suspension and transpotence with values in H_{∗}(B̄(A)).

We define now Cartan’s little constructions.

Definition 5.16. [4] Let p be a prime number. An elementary complex is a DGA-algebra of the form:

• Type I: an exterior algebra E(u, 2n − 1); • Type II: a divided power algebra Γ(u, 2n);

• Type III: a twisted tensor product E(u, 2n − 1) ˜⊗^{δ_{±p^{r}}} Γ(v, 2n);

• Type IV: a twisted tensor product Γ(u, 2n) ˜⊗^{δ_{±p^{r}}} E(v, 2n + 1);

where n ≥ 1 and the derivation δ_{±p^{r}} is given by the formula

δ_{±p^{r}}(v) = ±p^{r} u, r ≥ 1.

Definition 5.17. ([15]) Let p be a prime number and let Λ = _{(p)} be our ground ring. A DGA-module (M, d_{M}, ϵ_{M}, η_{M}) is called minimal if it is free, of finite type as a graded module and d_{M}(M) ⊂ p · M.

Definition 5.18. Let p be a prime number and let Λ = _{(p)} be our ground ring. We say that H is a minimal homological model of a commutative DGAalgebra A if there is an algebra contraction from the bar construction B̄(A) to the minimal DGA-algebra H.

Working on _{(p)}, our goal in this subsection is to obtain by perturbation semi-full algebra contractions from bar constructions of Cartan’s little constructions to minimal DGA-algebras.

A first result about the homology of elementary complexes is given by Eilenberg-MacLane:

Theorem 5.19. [7] Let Λ be a commutative ring with 1 ≠ 0 and let n be a natural number. There is an isomorphism of Hopf DGA-algebras between the bar construction of an exterior algebra B̄(E(u, 2n − 1)) and the divided power algebra Γ(σ(u), 2n).

In an obvious way, the last isomorphism can be considered as a full algebra contraction:

R_{B̄I} : \{\overline{B}(E(u, 2n − 1)), Γ(σ(u), 2n), f_{B̄I}, g_{B̄I}, 0\}  (48)

where

f_{B̄I}(\overbrace{[u| … |u]}^{k times}) = γ_{k}(σ(u));

and

g_{B̄I}(γ_{k}(σ(u)) = \overbrace{[u| … |u]}^{k times}

From now on, in order to compare later our method with the one developed by Cartan, we will denote the generators of the small DGA-algebras of the contractions that may appear, by making use of the suspension, transpotence and divided power morphisms . For example, the generator v of the divided power algebra in the last contraction has been denoted by σ(u). The notation σ(u) means that g_{B̄I}(v) = σ(u).

Now, it is about to find an explicit contraction for the bar construction of a divided power algebra (complex of type II). This case is quite complicated. This was the principal problem Eilenberg and MacLane encountered at the time of obtaining a good result for the homology of K(π, n).

Here, we can resolve this question if we work with coefficients in _{(p)} (p prime). We will only deal with odd primes p. For p = 2, similar results can be obtained. In the sequel of this subsection, we will take as ground ring (p).

First, we establish the following

Proposition 5.20. Let n and p be a natural number and a prime number, respectively. There is an isomorphism of DGA-algebras between the divided power algebra Γ(u, 2n) and

˜⊗_{i≥0}Q_{(p)}(u_{i}, 2np^{i}).

As _{(p)}-module, this last DGA-algebra is equal to the ordinary tensor product ⊗_{i≥0}Q_{(p)}(u_{i}, 2np^{i}). Its multiplicative law is

u_{i}^{k} u_{j}^{h} = \{ \begin{matrix}u_{i}^{k} ⊗ u_{j}^{h} & if i ≠ j \\ u_{i}^{k+h} & if i = j and k + h < p \\ −pu_{i}^{t}u_{i+1} & if i = j and k + h = p + t \end{matrix}  (49)

Proof.

We can consider the above isomorphism as a full algebra contraction:

R_{Γ} : {Γ(u, 2n), ˜⊗_{i≥0}Q_{(p)}(u_{i}, 2np^{i}), f_{Γ}, g_{Γ}, 0}  (50)

We take the function S_{p}(n) = \frac{p^{n}−1}{p−1}, ∀n = 0, 1, 2, . . .. The explicit morphisms of R_{Γ} are defined by:

f_{Γ} : Γ(u, 2n) → ˜⊗_{i≥0}Q_{(p)}(u_{i}, 2np^{i})

f_{Γ}(γ_{k}(u)) = \frac{(−p)^{∑_{i=1}^{r} k_{i}S_{p}(i)}}{k!} u_{0}^{k_{0}} … u_{r}^{k_{r}},

where k = k_{0} + k_{1}p + … + k_{r}p^{r} (0 ≤ k_{i} < p) is the p-adic development of k, and

g_{Γ} : ˜⊗_{i≥0}Q_{(p)}(u_{i}, 2np^{i}) → P (u, 2n)

g_{Γ}(u_{n}^{k}) = \frac{(−1)^{nk}k(kp^{n} − 1)!}{p^{kS_{p}(n)−n}} γ_{kp^{n}}(u), 0 ≤ k < p.

Notice that the numbers \frac{(−p)^{∑_{i=1}^{r} k_{i}S_{p}(i)}}{k!} and \frac{(−1)^{nk}k(kp^{n} − 1)!}{p^{kS_{p}(n)−n}} are in-

vertible elements in _{(p)}.

Remark 5.21. From now on, we will identify the generators u_{i} of the truncated algebras with the elements γ_{p^{i}}(u) of Γ(u, 2n); in fact, g_{Γ}(u_{i}) coincides with γ_{p^{i}}(u), excluding the coefficient.

This simplification of the DGA-algebra Γ(u, 2n) tells us that to know the homology of the truncated algebras is essential for obtaining by perturbation the algebra homology of a divided power algebra. In order to get an explicit contraction for the bar construction of a truncated algebra, we have lightly modified an argument used by Eilenberg-MacLane in [7].

Proposition 5.22. Let n and p be a natural number and an odd prime number, respectively. There is an almost-full algebra contraction:

\begin{matrix}R_{BQ} : \{B̄(Q_{(p)}(u, 2n)), E(σ(u), 2n + 1) ⊗ P (φ_{p}(u), 2np + 2), \\ f_{BQ}, g_{BQ}, ϕ_{BQ}\} \end{matrix} (51)

Proof.

We denote an element of B̄(Q_{(p)}(u, 2n)) of the form [u^{r_{1}}| … |u^{r_{m}}] by [r_{1}| … |r_{m}], where 0 ≤ r_{i} < p.

The explicit morphisms of R_{BQ} are the following:

\begin{matrix}f_{BQ}[r_{1}|t_{1}| … |r_{m}|t_{m}] = \{∏_{k=1}^{n} δ_{p,r_{k}+t_{k}}\}γ_{m}(φ_{p}(u)), \\ f_{BQ}[r_{1}|t_{1}| … |r_{m}|t_{m}|l] = δ_{1,l}\{∏_{k=1}^{n} δ_{p,r_{k}+t_{k}}\}σ(u) · γ_{m}(φ_{p}(u)), \end{matrix}

where the symbols δ_{i,j} are defined by:

δ_{i,j} = \{ \begin{matrix}0 & i ≠ j \\ 1 & i = j \end{matrix}

The morphism g_{BQ} : E(σ(u), 2n+1)⊗Γ(φ_{p}(u), 2np+2) → B̄(Q_{(p)}(u, 2n)) is defined over the generators as follows:

\begin{matrix}g_{BQ}(σ(u)) = [1], \\ g_{BQ}(γ_{k}(φ_{p}(u))) = [1|p − 1| {k times \atop …} |1|p − 1]. \end{matrix}

The homotopy operator ϕ_{BQ} is defined by:

ϕ_{BQ}1 = 0; ϕ_{BQ}[1] = 0;
ϕ_{BQ}[x] = −[1|x − 1] 1 < x < p;
ϕ_{BQ}[x|y] = −[1|x − 1|y];
ϕ_{BQ}[x|y|z] = −[1|x − 1|y|z] − δ_{p,x+y}[1|p − 1|ϕ(z)]

where z ∈ B̄(Q_{(p)}(u, 2n)).

Remark 5.23. For p = 2, the truncated algebra Q_{(2)}(u, 2n) coincides with E(u, 2n) as DGA-algebras.

We are ready to establish:

Theorem 5.24. Let n and p be a natural number and an odd prime number, respectively. There is a semi-full algebra contraction R_{B̄II} from \overline{B}(Γ(u, 2n)) to the DGA-algebra

E(σ(u), 2n + 1) ⊗ (⊗_{i≥1}[E(σγ_{p^{i}}(u), 2np^{i} + 1) ˜⊗^{δ_{p}}Γ(φ_{p}γ_{p^{i−1}}(u), 2np^{i} + 2)])

Proof.

For the sake of clarity, we will write the DGA-algebras without noting the degree of the generators.

Proposition 5.20 tells us that there is an isomorphism of DGA-algebras between Γ(u) and a tensor product ˜⊗_{i≥0}Q_{(p)}(γ_{p^{i}}(u)) (note that we are using the identifications of (5.21)). Then, this enables us to state that the respective bar constructions of these two algebras are isomorphics too. This isomorphism of Hopf DGA-algebras is denoted by B̄(R_{Γ}), where the construction B̄( ) over a contraction is defined in Theorem 5.1 and the contraction R_{Γ} is determined by (50).

Then, for the construction of R_{B̄II}, we need to establish a contraction for the Hopf DGA-algebra B̄( ˜⊗_{i≥0}Q_{(p)}(γ_{p^{i}}(u))). To this end, we first consider the ordinary tensor product - excluding “torsión” (49)- of the truncated algebras ⊗_{i≥0}Q_{(p)}(γ_{p^{i}}(u)), and we take:

• the almost-full algebra contraction:

R_{B̄⊗Q} : {B̄(⊗_{i≥0}Q_{(p)}(γ_{p^{i}}(u))), ⊗_{i≥0}B̄(Q_{(p)}(γ_{p^{i}}(u))), f_{B̄⊗Q}, g_{B̄⊗Q}, ϕ_{B̄⊗Q}},

which is constructed thanks to Theorem 5.4,

• and the almost-full algebra contraction:

\begin{matrix}R_{⊗B̄Q} : \{⊗_{i≥0}B̄(Q_{(p)}(γ_{p^{i}}(u))), ⊗_{i≥0} E(σγ_{p^{i}}(u)) ⊗ Γ(φ_{p}γ_{p^{i}}(u)), \\ f_{⊗B̄Q}, g_{⊗B̄Q}, ϕ_{⊗B̄Q}\}, \end{matrix}

which is constructed thanks to Propositions 5.22 and 6. In fact, R_{⊗B̄Q} is the contraction ⊗_{i≥0}Ri_{BQ}, where

Ri_{BQ} : {B̄(Q_{(p)}(γ_{p^{i}}(u)), E(σγ_{p^{i}}(u)) ⊗ Γ(φ_{p}γ_{p^{i}}(u)), f i_{BQ}, gi_{BQ}, ϕi_{BQ}}

is a contraction of type (51).

It is time to apply the machinery of homological perturbation to the composition contraction R_{BP} = R_{⊗B̄Q} ∘ R_{B̄⊗Q}, where the perturbation datum δ is the difference between the differentials of the bar constructions B̄(⊗_{i≥0}Q_{(p)}(γ_{p^{i}}(u))) and B̄( ˜⊗_{i≥0}Q_{(p)}(γ_{p^{i}}(u))). It is clear that δ is a derivation and represent the perturbation induced in the simplicial differential of B̄(⊗_{i≥0}Q_{(p)}(γ_{p^{i}}(u))) by the modification produced in the product of the algebra ⊗_{i≥0}Q_{(p)}(γ_{p^{i}}(u)). The projection, inclusion and homotopy operator of the contraction R_{BP} will be denoted by f_{BP}, g_{BP} and ϕ_{BP}, respectively.

From SF -APL, we directly deduce that (R_{B̄P})_{δ} is a semi-full algebra contraction. Moreover, S-APL can be applied in this case. Our aim here is to verify the following relation:

ϕ_{BP} ∘ δ ∘ g_{BP} = 0,

or, written in a more developed form:

(ϕ_{B̄⊗Q} + g_{B̄⊗Q} ∘ ϕ_{⊗B̄Q} ∘ f_{B̄⊗Q}) ∘ δ ∘ (g_{B̄⊗Q} ∘ g_{⊗B̄Q}) = 0.  (52)

First, the composite g_{B̄⊗Q} ∘ g_{⊗B̄Q} carries any generator w into an element z of the form [1] or [1|p − 1] of B̄(Q_{(p)}(γ_{p^{r_{i}}}(u)) ⊂ B̄(⊗_{i≥0}Q_{(p)}(γ_{p^{i}}(u))). Obviously, z is a simple element of B̄(⊗_{i≥0}Q_{(p)}(γ_{p^{i}}(u))).

We study now the image of z under δ. It is not hard to see that this image is zero if z = [1] and [γ_{p^{i+1}}(u)] if z = [1|p − 1]. In any case, it happens that the image of any generator under δ ∘ g_{B̄⊗Q} ∘ g_{⊗B̄Q} lies in S̄(⊗_{i≥0}Q_{(p)}^{i}) and, hence, in Ker ϕ_{B̄⊗Q} (see Prop. 5.10).

Thus, it is clear that f_{B̄⊗Q} ∘ δ ∘ g_{BP}(w) is zero or an element [1] ∈ Ker ϕr_{i}_{BQ} ⊂ Ker ϕ_{⊗B̄Q}. This completes the proof of (52).

Therefore, Theorem s-APL says us that it can be constructed the perturbed contraction R_{B̄II} = (R_{BP})_{δ} from B̄( ˜⊗_{i≥0}Q_{(p)}(γ_{p^{i}}(u))) to E(σ(u)) ⊗ (⊗_{i≥1} [E(σγ_{p^{i}}(u)) ˜⊗^{δ_{p}}Γ(φ_{p}γ_{p^{i−1}}(u))]), where its inclusion g_{B̄II} is precisely g_{BP} = g_{B̄⊗Q} ∘ g_{⊗B̄Q} and, consequently, it is a morphism of DGAalgebras.

We can easily check that the formula given by BPL for the twisted differential of the pairs

E(σγ_{p^{i}}(u), 2np^{i} + 1) ˜⊗^{δ_{p}}Γ(φ_{p}γ_{p^{i−1}}(u), 2np^{i} + 2)

is the following:

\begin{aligned}δ_{p}(γ_{k}φ_{p}γ_{p^{i−1}}(u)) & = & f_{BΓ} ∘ δ ∘ g_{BP}(γ_{k}φ_{p}γ_{p^{i−1}}(u)) \\ & = & p (σγ_{p^{i}}(u) ⊗ γ_{k−1}φ_{p}γ_{p^{i−1}}(u)), \end{aligned}

∀k ≥ 1, ∀i ≥ 1.

This yields the desired result.

Therefore, a complex of type II presents as minimal homological model a tensor product of complexes of type I and III. We are now concerned with an explicit algebra contraction for the bar construction of a complex of type III.

Theorem 5.25. Let n and p be a natural number and a odd prime number, respectively. There is a semi-full algebra contraction R_{B̄III} from B̄(E(u, 2n + 1) ˜⊗^{δ_{p^{r}}} Γ(v, 2(n + 1)) to the DGA-algebra

\begin{matrix}[Γ(σ(u), 2(n + 1)) ˜⊗^{δ_{−p^{r}}} E(σ(v), 2n + 3)]⊗ \\ (⊗_{i≥1}[E(σ̃γ_{p^{i}}(v), 2(n + 1)p^{i} + 1) ˜⊗^{δ_{p}}Γ(φ_{p}γ_{p^{i−1}}(v), 2(n + 1)p^{i} + 2)]). \end{matrix} (53)

The notation σ̃γ_{p^{i}}(v) means that the generators σγ_{p^{i}}(v) have been modified by a basis change.

Proof.

For brevity, we write the algebras omitting the degree of the generator. First, we consider:

• the almost-full algebra contraction:

R_{B⊗}^{EΓ} : {B̄(E(u) ⊗ Γ(v)), B̄(E(u)) ⊗ B̄(Γ(v)), f_{B̄⊗}, g_{B̄⊗}, ϕ_{B̄⊗}}

which is constructed using Theorem 5.4,

• and the semi-full algebra contraction R_{⊗B̄}^{E,Γ} = R_{B̄I} ⊗ R_{B̄II} from B̄(E(u)) ⊗ B̄(Γ(v)) to the DGA-algebra

Γ(σu) ⊗ E(σv) ⊗ [⊗_{i≥1}E(σγ_{p^{i}}v) ˜⊗^{δ_{p}}Γ(φ_{p}γ_{p^{i−1}}v)].  (54)

This contraction is obtained from the contractions (48) and (5.24). We will denote the projection, inclusion and homotopy operator of the contraction R_{⊗B̄}^{E,Γ} by f_{⊗B̄}, g_{⊗B̄} and ϕ_{⊗B̄}, respectively.

Hence, we are interested in the semi-full algebra contraction R_{BEΓ} = R_{⊗B̄}^{E,Γ} ∘ R_{B⊗}^{EΓ}. The projection, inclusion and homotopy operator of this last contraction will be denoted as f_{BEΓ}, g_{BEΓ} and ϕ_{BEΓ}, respectively.

Here, the perturbation datum ρ for R_{BEΓ} will be the modification generated by δ_{p^{r}} and induced in the tensor differential of the bar construction B̄(E(u) ˜⊗^{δ_{p^{r}}} Γ(v)). The morphism ρ is a derivation because it is the difference between the differentials of the constructions B̄(E(u) ˜⊗^{δ_{p^{r}}} Γ(v)) and B̄(E(u) ⊗ Γ(v)). Additionally, it is not hard to prove the pointwise nilpotency of the composite ϕ_{BEΓ} ∘ δ.

Hence, the SF -APL perturbation machine guarantees that (R_{BEΓ})_{ρ} = R_{BIII} is a semi-full algebra contraction from

B̄(E(u) ˜⊗^{δ_{p^{r}}} Γ(v))

to

(Γ(σ(u)) ⊗ E(σ(v)) ⊗ ⊗_{i≥1}[E(σγ_{p^{i}}(v)) ˜⊗^{δ_{p}}Γ(φ_{p}γ_{p^{i−1}}(v))], d′)

where d′ denotes the differential obtained by perturbation.

We now compute the differential d′ in the small DGA-algebra of the perturbed contraction R_{BIII}. First at all, we analyze the inclusion of R_{BIII}:

(g_{BEΓ})_{ρ} = ∑_{n≥0} (−1)^{n}(ϕ_{BEΓ} ∘ ρ)^{n} ∘ g_{BEΓ}  (55)

The morphism g_{BEΓ} is defined by:

g_{BEΓ} = g_{B̄⊗} ∘ g_{⊗B̄} = g_{B̄⊗} ∘ (g_{B̄I} ⊗ g_{B̄II}) = g_{B̄⊗} ∘ (g_{B̄I} ⊗ g_{BΓ}).

The previous morphism carries any generator of the small DGA-algebra (53) to an element of B̄(E(u)) or an element w of the form [1] or [1|p − 1] of B̄(Q_{(p)}(γ_{p^{i}}(v))) ⊂ B̄(Γ(v)) ⊂ B̄(E(u) ⊗ Γ(v)). Obviouly, this image lies in Ker ϕ_{B̄⊗}.

We can now concentrate our study to determining the image under ρ of an argument w of the form [1] or [1|p − k] (with k < p) of B̄(Q_{(p)}(γ_{p^{i}}(v))). Here we distinguish two cases. This image is, in general, a sum of two 1-simples elements of B̄(E(u) ⊗ Γ(v)). The only term with 1 inversion in this sum is

w̄ = [u ⊗ b] • z̄  (56)

where b ∈ Γ(v) and z̄ is [ ] if w = [1] or [p − k] if w = [1|p − k] ∈ B̄(Q_{(p)}(γ_{p^{i}}(v))). Let us note that in the case i = 0, w̄ is the simple element [u] • z̄.

Now, the homotopy operator of the contraction R_{BEΓ} has the form:

ϕ_{BEΓ} = ϕ_{B̄⊗} + g_{B̄⊗} ∘ ϕ_{⊗B̄} ∘ f_{B̄⊗} = ϕ_{B̄⊗} + g_{B̄⊗} ∘ (1 ⊗ ϕ_{B̄II}) ∘ f_{B̄⊗}.

By Proposition 5.10 g), the image of a 1-simple element of B̄(E(u)⊗Γ(v)) with r inversions (r ≥ 2) under ϕ_{B̄⊗} is zero or a sum with terms f_{B̄⊗}-vanished simple elements with at least r inversions. It is easy to check that if we take a f_{B̄⊗}-vanished simple element with r inversions (r ≥ 2), the image of this element under ρ is a sum in which its terms may be 1-simple elements with at least r inversions or f_{B̄⊗}-vanished simple elements. On the other hand, f_{B̄⊗} carries a 1-simple element into zero. Furthermore, from Prop. 5.10b), the morphism ϕ_{B⊗} carries a f_{B̄⊗}-vanished simple element into zero.

Now we pick w′ = [u ⊗ b] • [u| …^{r times} |u] • z′ with r ≥ 0, b ∈ \overline{Γ(v)} and z′ = [ ] or z′ = [h] ∈ B̄(Q_{(p)}(γ_{p^{i}}(v))).

In ϕ_{B̄⊗}(w′), the unique simple element with 1 inversion (excluding the sign):

[b] • [u| …^{r + 1 times} |u] • z′.  (57)

Now the morphism ρ maps an argument of the form (57) onto an element of B̄(E(u) ⊗ Γ(v)) which only has one term with at most 1 inversion:

[u ⊗ b′] • [u| …^{r + 1 times} |u] • z′,  (58)

where δ_{p^{r}}(b) = p^{r} (u ⊗ b′).

In fact, if b = γ_{1}(v), the element (58) is the simple element x = [u| …^{r + 1 times} |u] • z′.

Since ϕ_{BI} = 0, we determine that ϕ_{BEΓ} carries the element x into zero if z′ = [ ] or z′ = [1], and into the shuffle product y = [u| …^{r + 1 times} |u]∗[1|h−1] if z′ = [h] with h > 1. On the other hand, f_{BEΓ} maps x to γ_{r+1}σ(u) if z′ = [ ], to zero if z′ = [h], with h > 1 and to γ_{r+1}σ(u) ⊗ σγ_{p^{i}}(v) if z′ = [1].

Taking into account that ρ is a derivation and Proposition 4.11 (35), we have

(ϕ_{BEΓ} ∘ ρ)^{n}(y) = [u| …^{r + 1 times} |u] ∗ (ϕ_{BEΓ} ∘ ρ)^{n}([1|h − 1]).

Summarizing, if we examine d′, we observe that the torsion does not appear exactly by pairs, but is more complicated. To be more precise, we determine the image (up to invertible coefficient in _{(p)}) of different generators of (53) under d′ = f_{⊗B} ρ Σ^{ρ} g_{⊗B}:

d′(σ(v)) = − p^{r} σ(u);  (59)
d′(σγ_{p^{i}}(v)) = − p^{r p^{i}}γ_{p^{i}}σ(u)  (60)
d′(φ_{p}γ_{p^{i−1}}(v)) = − p^{rp^{i−1}(p−1)}γ_{p^{i−1}(p−1)}σ(u) ⊗ σγ_{p^{i−1}}(v).  (61)

That is, a scheme of the behaviour of the differential d′ can be the following,

d′(Γ(σ(u))) = 0;
d′(E(σ(v))) ⊂ Γ(σ(u));  (62)
d′(E(σγ_{p^{i}}(v))) ⊂ Γ(σ(u));  (63)
d′(Γ(φ_{p}γ_{p^{i−1}}(v))) ⊂ Γ(σ(u)) ⊗ E(σγ_{p^{i−1}}(v)).  (64)

We see that (62) corresponds to (59), (63) corresponds to (60), and (64) corresponds to (61).

Now the torsions (63) and (64) can be eliminated, making a basis change. This basis change can be seen as a full algebra contraction R_{norm} (with homotopy operator zero) from

([Γ(σ(u)) ⊗ E(σ(v))] ⊗ (⊗_{i≥1}[E(σγ_{p^{i}}(v)) ˜⊗^{δ_{p}}Γ(φ_{p}γ_{p^{i}}(v))]), d′),

to

(Γ(σ(u)) ˜⊗^{δ_{−p^{r}}} E(σ(v))) ⊗ (⊗_{i≥1}[E(σ̃γ_{p^{i}}(v)) ˜⊗^{δ_{p}}Γ(φ_{p}γ_{p^{i}}(v))]).

The notation σ̃γ_{p^{i}}(v) means that this generator has been modified in this basis change.

This process of “normalisation” was already determined by Moore [22] in his study of the p-adic homology of the Eilenberg-MacLane spaces.

The torsion (62) is neither eliminated nor modified by R_{norm} and coincides with the derivation δ_{−p^{r}}. In the same way, the torsion δ_{p} of the pairs E(σγ_{p^{i}}(v)) ˜⊗^{δ_{p}}Γ(φ_{p}γ_{p^{i−1}}(v)) remains unchanged too.

Therefore, the desired contraction is:

R_{BIII} = R_{norm} ∘ (R_{BEΓ})_{ρ}.

———–

Thus, an elementary complex of type III has a tensor product of elementary complexes of type III and IV as minimal homological model.

Proceeding in an analogous way to the previous proof, we obtain an explicit semi-full algebra contraction for an elementary complex of type IV. In this case, it is possible to apply S-APL Theorem. Its minimal homological model will be a tensor product of elementary complexes of type III. We leave the details to the reader.

Theorem 5.26. Let n and p be a natural number and an odd prime number, respectively. There is a semi-full algebra contraction R_{B̄IV} from B̄(Γ(u, 2n) ˜⊗^{δ_{p^{r}}} E(v, 2n + 1)) to

\begin{matrix}[E(σ(u), 2n + 1) ˜⊗^{δ_{−p^{r}}} Γ(σ(v), 2n + 2)]⊗ \\ (⊗_{i≥1}[E(σγ_{p^{i}}(u), 2np^{i} + 1) ˜⊗^{δ_{p}}Γ(φ_{p}γ_{p^{i−1}}(u), 2np^{i} + 2)]) \end{matrix}

Remark 5.27. For the prime p = 2, all the previous results are valid, with the only modification in their statements of changing the symbol φ_{p} for the composition γ_{2}σ and the inclusions of contractions are, in this case, morphisms of Hopf DGA-algebras.

The good behaviour of complexes of any type when we apply to them the bar construction, corresponds to symbols rules for its generators; that is, as we have denoted the generators in relation to the operations of suspension σ, transpotence ϕ_{p} and k-ieth divided power γ_{k}, these generators are in oneto-one correspondence with “admissible words” in the alphabet composed by the indicated three letters. These results agree with those described by Cartan (see [4]).

6 Related questions

We can see the last example of the previous section as a particular case of a more general phenomenon. On one hand, in [3], it is established that every commutative DGA-algebra (A, d_{A}) has a model ϕ : (Λ(V), d) → (A, d_{A}), where (Λ(V), d) is a free commutative DGA-algebra (FCA) generated by the free graded module V. On the other hand, being Λ = our ground ring, an almost-full algebra contraction from B̄(P (u)) to E(σ(u)) can be constructed. Using the algebra perturbation machinery of this paper, it can be developed a satisfactory theory for obtaining small homological models for commutative DGA-algebras.

Theorem 6.1. There is a semi-full algebra contraction from B̄(Λ(V)) to Γ(S(V)) (the divided power algebra generated by S(V)).

Taking into account the ideas of [17], we think that these good results can be extended without problem to Hochschild homology.

We have seen in the previous section that the Cartan-Moore method to obtain the homology algebras of K(π, n) can be translated to our framework. In this way, we have obtained explicit semi-full algebra contractions from bar constructions of Cartan’s littles constructions to tensor products of these same DGA-algebras. That says us that it is possible to control the differential of the minimal homological models of these particular algebras: this torsion can be given by pairs. The problem of generalizing these results to a large set of twisted tensor products (TTP) of exterior and divided power algebras woud be interesting. Given a twisted tensor product A of n DGA-algebras, the question is if it is possible to establish that a tensor product of TTP of i DGA-algebras, with i ≤ n, should be a minimal homological model of A.

On another hand, a situation in which we think that it is possible to apply GP -CPL is when we construct by perturbation the explicit contraction from the chain complex of the classifying space of a simplicial group G to the bar construction of the chain complex of G (see [27]). In this case, we conjecture that the projection of this perturbed contraction is a morphism of DGAcoalgebras.

7 Appendix: The Shih operator

7.1 Prerequisites.

The complex of simplicial chains (respectively of normalized simplicial chains) of a simplicial set X will be denoted C(X) (respectively C_{N}(X)).

The Eilenberg-Mac Lane operator is the unique (see [25]) natural transformation C_{N}(X) ⊗ C_{N}(Y) ⟶^{EML} C_{N}(X × Y) ; it has been defined by (5). The Alexander-Whitney natural transformation, C_{N}(X×Y) ⟶^{AW} C_{N}(X)⊗C_{N}(Y), has been defined in (4).

Recall that to any simplicial operator D = s_{i_{r}}...s_{i_{1}}∂_{j_{s}}...∂_{j_{1}} of degree r − s and initial dimension p, one can associate its derived simplicial operator D′ of degree r − s and initial dimension p + 1, defined by

D′ = s_{i_{r}+1}...s_{i_{1}+1}∂_{j_{s}+1}...∂_{j_{1}+1}

We call Shih natural transformation, the natural transformation C(X × Y) ⟶^{Φ} C(X × Y) satisfying the inductive formula

\{ \begin{matrix}Φ_{0} = 0 \\ Φ_{n} = −(Φ_{n−1})′ + (EML ∘ AW)′ ∘ s_{0}. \end{matrix}

It induces a natural transformation on normalized complexes.

Let us recall Rubio’s formula for C(X × Y) ⟶^{SHI} C(X × Y)

SHI(x_{n}, y_{n}) =  (65)

∑(−1)^{n−p−q+σ(α,β)}(s_{β_{q}+n−p−q}...s_{β_{1}+n−p−q}s_{n−p−q−1}∂_{n−q+1}...∂_{n}x_{n}, s_{α_{p+1}+n−p−q}...s_{α_{1}+n−p−q}∂_{n−p−q}...∂_{n−q−1}y_{n}).

where the sum is taken over all 0 ≤ q ≤ n − 1, 0 ≤ p ≤ n − q − 1 and (α, β) ∈ {(p + 1, q) − shuffles}.

Remark 7.1. Note that this formula makes sense as soon as n ≥ 1, we then

set SHI = 0 : C0^{N}(X × Y) ⟶ C1^{N}(X × Y).

This explicit formula has already been used in [29], [30] and [28].

To get more information about the previous definitions, the reader may have a look at [29], [19] and [21].

7.2 Theorem.

The Shih natural transformation is characterized by Rubio’s formula.

7.3 Proof of the theorem.

Thanks to remark 7.1, In dimension 0 both definitions correspond to Φ_{0} = 0. It remains to show that Rubio’s formula satisfies the inductive formula given by Shih :

SHI = −SHI′ + (EML ∘ AW)′ ∘ s_{0}.

We are going to make calculations in C(X × Y) and only at the end of this proof, we will show that the expected result is true in C_{N}(X × Y).

To begin with, let us apply the first term of the right member of this identity to any n-simplex (x_{n}, y_{n})

−SHI′(x_{n}, y_{n}) = − ∑(−1)^{(n−1)−p−q+σ(α,β)}

(s_{β_{q}+(n−1)−p−q+1}...s_{β_{1}+(n−1)−p−q+1}s_{(n−1)−p−q−1+1}∂_{(n−1)−q+1+1}...∂_{(n−1)+1}x_{n}, s_{α_{p+1}+(n−1)−p−q+1}...s_{α_{1}+(n−1)−p−q+1}∂_{(n−1)−p−q+1}...∂_{(n−1)−q−1+1}y_{n})

where the sum is taken over all 0 ≤ q ≤ n − 2, 0 ≤ p ≤ n − q − 2 and (α, β) ∈ {(p + 1, q) − shuffles}.

The previous expression may be simplified and can be rewritten as

−SHI′(x_{n}, y_{n}) =  (66)

∑(−1)^{n−p−q+σ(α,β)}(s_{β_{q}+n−p−q}...s_{β_{1}+n−p−q}s_{n−p−q−1}q∂_{n−q+1}...∂_{n}x_{n}, s_{α_{p+1}+n−p−q}...s_{α_{1}+n−p−q}∂_{n−p−q}...∂_{n−q−1}y_{n})

where the sum is taken over q, p and (α, β) describing the same sets. Let us now compute (EML ∘ AW)′ ∘ s_{0} on any n-simplex (x_{n}, y_{n}). First,

EML ∘ AW (x_{n}, y_{n}) =

∑_{i=0}^{n} ∑_{(α,β)}(−1)^{σ(α,β)}(s_{β_{n−i}}...s_{β_{1}}∂_{i+1}...∂_{n}x_{n}, s_{α_{i}}...s_{α_{1}}∂_{0}...∂_{i−1}y_{n})

where the second sum is taken over all the (i, n − i)-shuffles.

We can derive this operator and apply it on the 0-degenerate simplex (s_{0}x_{n}, s_{0}y_{n})

(EML ∘ AW)′(s_{0}x_{n}, s_{0}y_{n}) =

∑_{i=0}^{n} ∑_{(α,β)}(−1)^{σ(α,β)}(s_{β_{n−i}+1}...s_{β_{1}+1}∂_{i+2}...∂_{n+1}s_{0}x_{n}, s_{α_{i}+1}...s_{α_{1}+1}∂_{1}...∂_{i}s_{0}y_{n})

where the second sum is still taken over all the (i, n − i)-shuffles. Using the identities ∂_{i}s_{j} = sj∂_{i−1} whenever i > j + 1, since i + 2 > 1, we can write

(EML ∘ AW)′(s_{0}x_{n}, s_{0}y_{n}) =

∑_{i=0}^{n} ∑_{(α,β)}(−1)^{σ(α,β)}(s_{β_{n−i}+1}...s_{β_{1}+1}s_{0}∂_{i+1}...∂_{n}x_{n}, s_{α_{i}+1}...s_{α_{1}+1}∂_{1}s_{0}∂_{1}...∂_{i−1}y_{n})

Using the identity ∂_{1}s_{0} = 1, we get

(EML ∘ AW)′(s_{0}x_{n}, s_{0}y_{n}) =  (67)

∑_{i=0}^{n} ∑_{(α,β)}(−1)^{σ(α,β)}(s_{β_{n−i}+1}...s_{β_{1}+1}s_{0}∂_{i+1}...∂_{n}x_{n},

s_{α_{i}+1}...s_{α_{1}+1}∂_{1}...∂_{i−1}y_{n})

When, in the previous sum, i = 0, there is only one (0, n)-shuffle corresponding to the identical permutation of n terms (β_{k} = k − 1, for all 0 ≤ k ≤ n − 1). The single corresponding term in the previous sum (s_{n}...s_{1}s_{0}∂_{1}...∂_{n}x_{n}, s_{0}y_{n}) is clearly 0-degenerate. Indeed, using the identities s_{j+1}s_{0} = s_{0}s_{j}, whenever j ≥ 0, we can write

(s_{n}...s_{1}s_{0}∂_{1}...∂_{n}x_{n}, s_{0}y_{n}) = (s_{0}s_{n−1}...s_{0}∂_{1}...∂_{n}x_{n}, s_{0}y_{n})

= s_{0}(s_{n−1}...s_{0}∂_{1}...∂_{n}x_{n}, y_{n}).

Let us set z = (s_{n−1}...s_{0}∂_{1}...∂_{n}x_{n}, y_{n}).

Remark 7.2. the s_{0}z class is null in the normalized complex.

(67) may be written

(EML ∘ AW)′(s_{0}x_{n}, s_{0}y_{n}) =  (68)
s_{0}z + ∑(−1)^{σ(α,β)}(s_{β_{j}+1}...s_{β_{1}+1}s_{0}∂_{n−j+1}...∂_{n}x_{n},
s_{α_{i}+1}...s_{α_{1}+1}∂_{1}...∂_{n−j−1}y_{n})

where the sum is now taken over all 0 ≤ j ≤ n − 1, and all the (i, j)-shuffles (α, β) with i = n − j.

Note that when i = 1, there are no face operators in the second terms of the couples. In fact, we find again those couples in Rubio’s formula (see (65)) when q = n − 1 and p = 0. This suggests a change of indices : Let us now consider the new indices p and q, defined by p = i − 1 and q = j. (68) becomes

(EML ∘ AW)′(s_{0}x_{n}, s_{0}y_{n}) =  (69)
s_{0}z + ∑(−1)^{σ(α,β)}(s_{β_{q}+1}...s_{β_{1}+1}s_{0}∂_{n−q+1}...∂_{n}x_{n},
s_{α_{p+1}+1}...s_{α_{1}+1}∂_{1}...∂_{n−q−1}y_{n})

where the sum is taken over all 0 ≤ q ≤ n − 1, and all the (p + 1, q)-shuffles with p = n − q − 1.

If we add the right terms of (66) and (69) and subtract s_{0}z, we get exactly (65). Indeed, as in (69), p = n − q − 1 then n − p − q = 1 and (7) becomes

(EML ∘ AW)′(s_{0}x_{n}, s_{0}y_{n}) =  (70)
s_{0}z + ∑(−1)^{σ(α,β)}(s_{β_{q}+n−p−q}...s_{β_{1}+n−p−q}s_{0}∂_{n−q+1}...∂_{n}x_{n},
s_{α_{p+1}+n−p−q}...s_{α_{1}+n−p−q}∂_{n−p−q}...∂_{n−q−1}y_{n})

where this last sum is still taken over all 0 ≤ q ≤ n − 1, and all the (p + 1, q)- shuffles with p = n − q − 1. The expression of the summands in (70) is now the same (except the term s_{0}z but it does not matter since it is degenerate) as in (66) and (65), it suffices then to notice that the set of shuffles over which the sum is taken in (65) is the disjoint union of those over which the sums are taken in (66) and (70).

Thanks to remark 7.2, we get the expected result in the normalized complex.

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