Homotopy properties of differential lie modules over curved coalgebras and Koszul duality

ISSN 0001-4346, Mathematical Notes, 2013, Vol. 94, No. 3, pp. 335–350. © Pleiades Publishing, Ltd., 2013.Original Russian Text © S. V. Lapin, 2013, published in Matematicheskie Zametki, 2013, Vol. 94, No. 3, pp. 354–372.

Homotopy Propertiesof Differential Lie Modules over Curved Coalgebras

and Koszul Duality

S. V. Lapin*

Self-employed, Saransk, RussiaReceived December 12, 2011; in final form, April 8, 2012

Abstract—The notion of differential Lie module over a curved coalgebra is introduced. Thehomotopy invariance of the structure of a differential Lie module over a curved coalgebra is proved. Arelationship between the homotopy theory of differential Lie modules over curved coalgebras and thetheory of Koszul duality for quadratic-scalar algebras over commutative unital rings is determined.

DOI: 10.1134/S0001434613090058

Keywords: differential Lie module over a curved coalgebra, Koszul duality, quadratic-scalaralgebra, co-B-construction, differential module over a Clifford algebra, differential moduleover an exterior algebra, SDR-data for differential modules.

In [1], a theory of Koszul duality for inhomogeneous quadratic algebras and, in particular, quadratic-scalar algebras over fields, was constructed by using the B-construction of a curved differential algebra.This theory generalizes Koszul’s duality theory to quadratic algebras, provided that the latter are treatedas inhomogeneous quadratic algebras [2]. In [3], on the basis of the ideas underlying the constructionsand results of [1], a Koszul duality theory was developed for inhomogeneous quadratic operads overfields of characteristic zero. This theory generalizes that of Koszul duality to quadratic operads overfields of characteristic zero, provided that the latter are treated as inhomogeneous quadratic operads [4].The main advantage of the theory developed in [3] is that its methods make it possible to constructhomotopy invariant structures of algebras over operads from algebras over inhomogeneous quadraticKoszul operads in the case of a field of characteristic zero. However, since the domain of application ofthe methods of [3] is restricted to inhomogeneous quadratic Koszul operads over fields of characteristiczero, these methods do not generally apply to constructing homotopy invariant structures of algebrasover operads, such as the structure of a differential E∞-algebra, over arbitrary commutative rings.

On the other hand, in [5], a homotopy theory of differential Lie modules over coalgebras wasdeveloped, which gives a method for a homotopy invariant transfer of the structure of a differentialmodule over the co-B-construction of any coalgebra. In particular, the theory from [5] makes itpossible to construct homotopy invariant structures of differential modules over differential algebrasfrom the structure of a differential module over any quadratic algebra by using the Koszul duality functor.Moreover, the quadratic algebra under consideration is not required to be Koszul, and the commutativering over which this quadratic algebra is defined it not required to be a field of characteristic zero. Thus,the application of the homotopy theory of differential Lie modules over coalgebras developed in [5] toarbitrary quadratic algebras is the first step toward the development a theory of Koszul duality for anyquadratic operads over arbitrary commutative rings. This gives rise to the problem of constructinga homotopy theory of differential Lie modules over curved coalgebras which extends the homotopytheory of [5]. The great interest in this problem is mainly caused, on the one hand, by the problem ofconstructing a theory of Koszul duality for any quadratic-scalar operads over arbitrary commutativerings and, on the other hand, by the possibility of applying the homotopy theory of differential Liemodules over curved coalgebras to the construction of the homotopy counterpart Δop

∞ of a quadratic-scalar colored algebra Δop of simplicial faces and degeneracies, on which the majority of constructionsin algebraic topology are based (see, e.g., [6], [7]).

*E-mail: [email protected]

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In this paper, we introduce the notion of a differential Lie module over a curved coalgebra, whichgeneralizes the notion of a differential Lie module over a coalgebra introduced in [5]. We describethe basic homotopy and categorical properties of differential Lie modules over curved coalgebras andapplications of the homotopy technique of differential Lie modules over curved coalgebras to studyinghomotopy properties of differential modules over co-B-constructions of curved coalgebras Koszul dualto quadratic-scalar algebras. These quadratic-scalar algebras are not required to be Koszul, and thecommutative rings over which they are defined are not required to be fields of characteristic zero. Wealso exemplify the application of the homotopy constructions presented in this paper to constructinghomotopy invariant counterparts of the structures of a differential module over a Clifford algebra and adifferential module over an exterior algebra over any commutative unital ring. The homotopy theoryfor the homotopy invariant analog of the structure of a differential module over an exterior algebraconstructed in this paper generalizes the homotopy theory of D∞-differential modules [8], which wasapplied in [9] and [10] to study A∞-structures and E∞-structures in spectral sequences of fibrations.

We proceed to precise definitions and statements. All modules and maps of modules considered inthis paper are assumed, unless otherwise specified, to be K-modules and K-linear maps of modules,respectively, where K is any commutative unital ring.

1. CURVED COALGEBRAS KOSZUL DUAL TO QUADRATIC-SCALAR ALGEBRAS

First, we recall that a differential graded module, or, for short, simply a differential module (X, d)is a graded module X = {Xn}, n ∈ Z, endowed with a differential d : X• → X•−1, that is, a map ofgraded modules which has degree (−1) and satisfies the condition d2 = 0. A map f : (X, d) → (Y, d)is called a map of differential modules if this is a degree-0 map of graded modules satisfying thecondition df = fd. A homotopy h : X → Y between maps f, g : (X, d) → (Y, d) of differential modulesis a degree-1 map h : X• → Y•+1 of graded modules satisfying the condition dh + hd = f − g.

Let η : X � Y : ξ be maps of differential modules such that ηξ = 1Y , and let h : X → X be ahomotopy between the maps ξη and 1X of differential modules which satisfies the conditions ηh = 0,ξh = 0, and hh = 0. Any such triple (η : X � Y : ξ, h) is called SDR-data for differential modules.

Many examples of SDR-data for differential modules arise in considering the homology of differentialmodules over a field. Indeed, let H(X) = Ker d/ Im d be the homology module of any differentialmodule (X, d) over a field. Regarding the graded module H(X) as a differential module with zerodifferential and using a fixed direct sum decomposition Ker d = H(X) ⊕ Im d, we obtain SDR-data(η : X � H(X) : ξ, h) for differential modules. In what follows, we refer to these SDR-data ashomological SDR-data for the differential module (X, d).

The tensor product of differential modules (X, d) and (Y, d) is the differential module (X ⊗ Y, d)whose grading and differential are defined by

(X ⊗ Y )n =⊕

p+q=n

Xp ⊗ Yq, d(x ⊗ y) = d(x) ⊗ y + (−1)px ⊗ d(y), p, q ∈ Z.

For any differential modules (X, d) and (Y, d), a differential module (Hom(X;Y ), d) is always defined.The elements of the module Hom(X;Y )n, n ∈ Z, are any degree-n maps f : X• → Y•+n of gradedmodules, and the value of the differential

d : Hom(X;Y )n → Hom(X;Y )n−1 at f ∈ Hom(X;Y )n

is defined by

d(f) = df + (−1)n+1fd : X• → Y•+n−1.

A differential algebra (A, d, π) is a differential module (A, d), where A = {An}, n ∈ Z, endowedwith a multiplication π : A ⊗ A → A, that is, a map of differential modules satisfying the associativitycondition π(π ⊗ 1) = π(1 ⊗ π). A map f : (A′, d, π) → (A′′, d, π) of differential algebras is a mapf : (A′, d) → (A′′, d) of differential modules satisfying the condition π(f ⊗ f) = fπ. In the case d = 0,the differential algebra (A, d, π) is denoted by (A,π) and called a graded algebra.

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HOMOTOPY PROPERTIES OF DIFFERENTIAL LIE MODULES 337

An example of a differential algebra is the triple (Hom(X;X), d, π), where (Hom(X;X), d) is thedifferential module considered above and the associative multiplication

π : Hom(X;X) ⊗ Hom(X;X) → Hom(X;X)

in the differential algebra (Hom(X;X), d, π) is defined as the composition of maps. The unit of thedifferential algebra (Hom(X,X), d, π) is the identity map of the graded module X.

A unital graded algebra A is called a quadratic algebra if this algebra is isomorphic to the gradedquotient algebra R = T (M)/(Q), where T (M) is the tensor algebra of some graded module M = {Mn},n ∈ Z, and (Q) is the two-sided ideal of T (M) generated by a submodule Q of the graded moduleM ⊗ M .

It is easy to see that, for any quadratic algebra R = T (M)/(Q), the composition of the obvious em-bedding M → T (M) and the projection T (M) → T (M)/(Q) is an embedding M → R = T (M)/(Q)of graded modules. In what follows, we identify the image of this embedding with the module M andalways assume that M is a submodule of R.

A graded coalgebra (C,∇) is, by definition, a graded module C = {Cn}, n ∈ Z, n ≥ 0, endowedwith a comultiplication ∇ : C → C ⊗ C, that is, a degree-0 map of graded modules satisfying theassociativity condition

(∇⊗ 1)∇ = (1 ⊗∇)∇.

A graded coalgebra (C,∇) is said to be connected if C0 = K.The counit of a graded coalgebra (C,∇) is a degree-0 map ε : C → K of graded modules satisfying

the condition

(1 ⊗ ε)∇ = 1 = (ε ⊗ 1)∇,

where K is treated as a graded module concentrated in dimension 0 and the graded modules C ⊗ K andK ⊗ C are identified in a standard way with the graded module C.

A degree-0 map f : C ′ → C ′′ of graded modules is called a map of graded coalgebras and denotedby

f : (C ′,∇) → (C ′′,∇) if ∇f = (f ⊗ f)∇.

The coaugmentation of a graded coalgebra (C,∇) is a map of graded coalgebras

ν : (K,∇) → (C,∇), where ∇(1) = 1 ⊗ 1 for 1 ∈ K.

It is easy to see that any graded coalgebra (C,∇) with counit ε : C → K and coaugmentation admitsa decomposition into the direct sum C = K ⊕ C of graded modules, where C = ker(ε). It is also clearthat comultiplication in a graded counital coaugmented coalgebra (C,∇) determines a comultiplication∇ : C → C ⊗ C in the graded submodule C, which is defined by

∇(c) = ∇(c) − (1 ⊗ c + c ⊗ 1).

Thus, for any graded coaugmented counital coalgebra (C,∇), the graded noncounital coalgebra (C,∇)is always defined.

A graded unital algebra (A,π) is said to be quadratic-scalar [1] if this algebra is isomorphic toa graded quotient algebra R = T (M)/(Q), where T (M) is the tensor algebra of some graded moduleM = {Mn}, n ∈ Z, and (Q) is the two-sided ideal of T (M) generated by a submodule Q of the gradedmodule M⊗2 ⊕ K and satisfying the condition Q ∩ K = 0.

For any quadratic-scalar algebra R = T (M)/(Q), the corresponding quadratic algebra

qR = T (M)/(p(Q))

is defined, where p : T (M) → M⊗2 is the obvious projection and

p(Q) = Im(p : Q ⊂ T (M) → M⊗2).

Moreover, for each quadratic-scalar algebra R = T (M)/(Q), the rule ϑ(x) = k, where x ∈ p(Q), k ∈K, and x − k ∈ Q is the unique element for which p(x − k) = x, defines a map ϑ : p(Q) → K of graded


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modules. In other words, all those relations in the algebra (R,π) which contain scalar terms from Khave the form x + ϑ(x) = 0, where x ∈ p(Q)0.

It is easy to see that if a quadratic-scalar algebra R = T (M)/(Q) is quadratic, then qR = R andϑ = 0.

Definition 1.1. A curved graded coalgebra (C,∇, ϑ), or, for short, a graded ϑ-coalgebra, is a gradedcoalgebra (C,∇) endowed with a map ϑ : C2 → K, called the curvature of the coalgebra (C,∇), whichsatisfies the condition

ϑ(c′2)c′′n−2 = c′n−2ϑ(c′′2)

for each element cn ∈ Cn, n ≥ 2; here the elements c′2, c′′2 ∈ C2 and c′n−2, c

′′n−2 ∈ Cn−2 are determined

from cn by using the decomposition

∇(cn) = · · · + c′2 ⊗ c′′n−2 + · · · + c′n−2 ⊗ c′′2 + · · · ∈ (C ⊗ C)n.

In what follows, by the counit and the coaugmentation of a graded curved coalgebra (C,∇, ϑ) wemean, respectively, the counit and the coaugmentation of the graded coalgebra (C,∇). A gradedϑ-coalgebra (C,∇, ϑ) is said to be connected if the graded coalgebra (C,∇) is connected.

Note that the notion of a graded curved coalgebra introduced above is dual to the graded version ofthe notion of a curved differential algebra introduced in [1].

The suspension of a graded module X is a graded module SX with grading defined by the equality(SX)n+1 = Xn. Let us denote the elements of SX by [x], where x ∈ X; then the dimension of [x] isequal to n + 1 if the dimension of x is n. For any submodule Y of the graded module X ⊗ X, by S⊗2Ywe denote the corresponding submodule of the graded module SX ⊗ SX.

Definition 1.2. The curved coalgebra Koszul dual to a quadratic-scalar algebra (R = T (M)/(Q), π)is the graded ϑ-coalgebra (R!,∇, ϑ) with counit ε and coaugmentation ν defined by

R! =⊕

k≥0

(R!)(k), (R!)(0) = K, (R!)(1) = SM,

(R!)(k) =⋂

i+2+j=k

(SM)⊗i ⊗ S⊗2p(Q) ⊗ (SM)⊗j , k ≥ 2, i ≥ 0, j ≥ 0,

∇(1) = 1 ⊗ 1,

∇([x1 ⊗ · · · ⊗ xk]) = 1 ⊗ [x1, . . . , xk] + [x1, . . . , xk] ⊗ 1 +k−1∑

i=1

[x1, . . . , xi] ⊗ [xi+1, . . . , xk],

1 ∈ K, [x1, . . . , xk] ∈ (R!)(k),

ϑ : R!2 → K, ϑ([x1, x2]) = ϑ(x1 ⊗ x2), [x1, x2] ∈ (R!)(2)2 = (S⊗2p(Q))2,

ϑ([x]) = 0, [x] ∈ (R!)(1)2 = (SM)2,ε(1) = 1, ε([x1, . . . , xk]) = 0, k ≥ 1, ν(1) = 1;

here ϑ : p(Q) → K is the map determined by the quadratic-scalar algebra R = T (M)/(Q) in the sameway as above.

Let us check that the curvature ϑ : R!2 → K of the coalgebra (R!,∇) satisfies the condition

ϑ(c′2)c′′n−2 = c′n−2ϑ(c′′2)

in Definition 1.1. Obviously, this condition holds for elements of the form [x1, x2] ∈ (R!)(2). Considerany element [x1, x2, x3] ∈ (R!)(3). It follows from the definition of (R!)(3) that

x1 ⊗ x2 ∈ p(Q) and x2 ⊗ x3 ∈ p(Q);

therefore,

π(x1 ⊗ x2) + ϑ(x1 ⊗ x2) = 0 and π(x2 ⊗ x3) + ϑ(x2 ⊗ x3) = 0,



because these equalities are relations in the quadratic-scalar algebra (R,π). Multiplying the firstequality by x3 on the right and the second by x1 on the left, we obtain the following equalities in thealgebra (R,π):

π(π(x1 ⊗ x2) ⊗ x3) + π(ϑ(x1 ⊗ x2) ⊗ x3) = 0,π(x1 ⊗ π(x2 ⊗ x3)) + π(x1 ⊗ ϑ(x2 ⊗ x3)) = 0.

They imply

π(ϑ(x1 ⊗ x2) ⊗ x3) = π(x1 ⊗ ϑ(x2 ⊗ x3)),

or, equivalently, ϑ(x1 ⊗ x2)x3 = x1ϑ(x2 ⊗ x3). Thus, if the algebra R = T (M)/(Q) is quadratic-scalar,then the last equality in (R,π) implies the equality ϑ(x1 ⊗ x2)x3 = x1ϑ(x2 ⊗ x3) in M , or, equivalently,the equality ϑ([x1, x2])[x3] = [x1]ϑ([x2, x3]) in SM = (R!)(1). Hence the required condition holds forall elements [x1, x2, x3] ∈ (R!)(3). Now, take any element [x1, x2, x3, x4] ∈ (R!)(4) and consider theelements [x1, x2, x3] ∈ (R!)(3) and [x2, x3, x4] ∈ (R!)(3). Applying the same argument as above to theseelements, we obtain the equalities

ϑ([x1, x2])[x3] = [x1]ϑ([x2, x3]) and ϑ([x2, x3])[x4] = [x2]ϑ([x3, x4])

in SM = (R!)(1). Taking the tensor products of the first equality and [x4] ∈ SM and of [x1] ∈ SM andthe second equality, we obtain the following equalities in (SM)⊗2:

ϑ([x1, x2])[x3] ⊗ [x4] = [x1]ϑ([x2, x3]) ⊗ [x4],[x1] ⊗ ϑ([x2, x3])[x4] = [x1] ⊗ [x2]ϑ([x3, x4]);

they imply the equality

ϑ([x1, x2])[x3, x4] = [x1, x2]ϑ([x3, x4])

in (R!)(2). Thus, the required condition holds for all elements [x1, x2, x3, x4] ∈ (R!)(4). Proceeding byinduction, we consider all elements [x1, . . . , xn] ∈ (R!)(n), n ≥ 5, and arrive at the conclusion that therequired condition holds for all elements of the coalgebra (R!,∇).

Now, let us introduce the notion of the co-B-construction for a curved graded coalgebra with counitand coaugmentation, which is dual to the graded version of the notion of the B-construction for a curveddifferential algebra introduced in [1].

The desuspension of a graded module X is the graded module S−1X for which (S−1X)n−1 = Xn.The elements of S−1X are traditionally denoted by [x], where x ∈ X; an element [x] has dimension n− 1if the dimension of x is n.

Suppose given any graded coaugmented counital ϑ-coalgebra (C,∇, ϑ). Let (C,∇) be the gradednoncounital coalgebra determined by the graded coalgebra (C,∇), and let ϑ : C• → K•−2 be the map ofgraded modules of degree (−2) defined by

ϑ(c) =

{ϑ(c), c ∈ C2,

0, c ∈ Cn, n = 2;

the base ring K is treated as the graded module K• = {Kn}, n ∈ Z, where K0 = K and Kn = 0 forn = 0.

Definition 1.3. The co-B-construction for a curved graded coalgebra (C,∇, ϑ) is the differentialalgebra (F (C,ϑ), d, π) defined as follows:

F (C,ϑ) =⊕

k≥0

(S−1C )⊗k, (S−1C )⊗0 = K, d = d1 + d2,

d1([c1, . . . , ck]) =k∑

i=1

(−1)i+μi−1+p′i [c1, . . . , ci−1,∇(ci), ci+1, . . . , ck],


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∇(ci) =∑

(ci)

c′i ⊗ c′′i (Sweedler’s notation),

d2([c1, . . . , ck]) =k∑

i=1

(−1)i−1+μi−1 ϑ(ci)[c1, . . . , ci−1, ci+1, . . . , ck],

π([c1, . . . , cq] ⊗ [cq+1, . . . , ck]) = [c1, . . . , cq, cq+1, . . . , ck],

where [c1, . . . , ck] = [c1] ⊗ · · · ⊗ [ck], μi−1 = p1 + · · · + pi−1, pt is the dimension of the element ct ∈ C

for 1 ≤ t ≤ i − 1, p′i is the dimension of the element c′i ∈ C, and

[c1, . . . , ci−1,∇(ci), ci+1, . . . , ck] =∑

(ci)

[c1, . . . , c′i, c

′′i , . . . , ck].

It is easy to see that, if the curvature ϑ vanishes, then the definition of the co-B-construction fora curved graded coalgebra coincides with the usual definition of the co-B-construction for a gradedcoalgebra.

Given any graded ϑ-coalgebra (C,∇, ϑ) and any unital differential algebra (A, d, π), we regard themap ϑ : C• → K•−2 specified above as the map ϑ : C• → K•−2 ⊂ A•−2 defined by

ϑ(c) =

{ϑ(c) ∈ K ⊂ A0, c ∈ C2,

0 ∈ An−2, c ∈ Cn, n = 2.

Definition 1.4. A twisting curved cochain, or, briefly, a twisting ϑ-cochain, from a graded ϑ-coalgebra (C,∇, ϑ) to a unital differential algebra (A, d, π) is defined as a map ϕ : C• → A•−1 of gradedmodules which has degree (−1) and satisfies the ϑ-cochain twisting condition

dϕ + ϕ ∪ ϕ + ϑ = 0,

where ϕ ∪ ϕ : C• → A•−2 is the map defined by ϕ ∪ ϕ = π(ϕ ⊗ ϕ)∇.

It is easy to see that if the curvature ϑ vanishes, then the definition of a twisting curved cochaincoincides with the usual definition of a twisting cochain.

The simplest example of a twisting ϑ-cochain from a graded ϑ-coalgebra (C,∇, ϑ) to a differentialalgebra (F (C,ϑ), d, π) is the map

ϕF : C• → F (C,ϑ)•−1 given by ϕF (c) = [c] for c ∈ C.

It is easy to show that, for any twisting curved cochain ϕ : C• → A•−1 from a graded coaugmentedcounital ϑ-coalgebra (C,∇, ϑ) to a differential algebra (A, d, π), the map

F (ϕ) : (F (C,ϑ), d, π) → (A, d, π)

defined at the generators [c1, . . . , ck] of the graded module F (C,ϑ) by

F (ϕ)([c1, . . . , ck]) = π(k)(ϕ(c1) ⊗ · · · ⊗ ϕ(ck))

is a map of differential algebras; here π(1) is the identity map of A and π(k) = π(1⊗ π(k−1)), k ≥ 2, is theiterated multiplication in A.

Let (R,π) be any quadratic-scalar algebra, and let (R!,∇, ϑ) be the ϑ-coalgebra Koszul dual to(R,π). Note that the rule

ϕ!ϑ([x1, . . . , xk]) =

{x1 if k = 1,0 if k > 0

defines a twisting ϑ-cochain ϕ!ϑ : R!

• → R•−1. It follows that, as mentioned above, for each quadratic-scalar algebra (R,π), a map of differential algebras

F (ϕ!ϑ) : (F (R!, ϑ), d, π) → (R, d = 0, π)



is defined, which makes it possible to regard any differential R-module as a differential F (R!, ϑ)-module.To conclude this section, we mention that, given a Koszul [1] quadratic-scalar algebra (R,π), it can

be shown for the co-B-construction F (R!, ϑ) by using the spectral sequence of a bicomplex that themap

F (ϕ!ϑ) : (F (R!, ϑ), d, π) → (R, d = 0, π)

of differential algebras specified above is a quasi-isomorphism, i.e., it induces an isomorphism ofhomology modules. We do not prove this assertion here, because the fact that F (ϕ!

ϑ) is a quasi-isomorphism is nowhere used in what follows.

2. HOMOTOPY PROPERTIES OF DIFFERENTIAL LIE MODULESOVER CURVED COALGEBRAS

First, we recall that a left differential module (X, d, μ) over a differential algebra (A, d, π), or, briefly,a differential A-module, is a differential module (X, d) endowed with a left action μ : A ⊗ X → X,which is a map of differential modules and satisfies the condition μ(π ⊗ 1) = μ(1 ⊗ μ).

Given a differential module (X, d), consider the corresponding differential algebra (Hom(X;X), d, π).It is easy to see that endowing (X, d) with the structure of a differential module (X, d, μ) over a differ-ential algebra (A,π) is equivalent to specifying a map

μ : (A, d, π) → (Hom(X;X), d, π) of differential

algebras. Indeed, μ andμ uniquely determine each other by (

μ(a))(x) = μ(a ⊗ x).

Now, suppose that (X, d) and (Y, d) are any differential modules and (C,∇) is a graded coalgebra. Let(Hom(C ⊗X;Y ), d) be the differential module corresponding to the differential modules (C ⊗X, 1⊗ d)and (Y, d). For any differential modules (X, d), (Y, d), and (Z, d), we define a map

∪ : (Hom(C ⊗ Y ;Z) ⊗ Hom(C ⊗ X;Y ))• → Hom(C ⊗ X;Z)•

of graded modules by setting

f ∪ g = f(1 ⊗ g)(∇⊗ 1), where f ∈ Hom(C ⊗ Y ;Z), g ∈ Hom(C ⊗ X;Y ).

It is easy to show that ∪ is a map of differential modules with associativity property, i.e.,

d(f ∪ g) = d(f) ∪ g + (−1)deg(f)f ∪ d(g), (f ∪ g) ∪ l = f ∪ (g ∪ l).

To describe the homotopy properties of differential F (R!, ϑ)-modules, we introduce the notion ofdifferential Lie module over a curved graded coalgebra, which generalizes the corresponding notion ofdifferential Lie modules over graded coalgebras in homotopy theory [5].

Definition 2.1. A differential Lie module (X, d, ψ) over a curved graded counital coalgebra (C,∇, ϑ)with coaugmentation ν : K → C, or, briefly, a differential Lie (C,ϑ)-module, is a differential module(X, d) together with a map ψ : (C ⊗ X)• → X•−1 satisfying the conditions

d(ψ) + ψ ∪ ψ + ϑ = 0, ψ(ν ⊗ 1) = 0,

where ϑ is defined by ϑ = ϑ ⊗ 1: (C ⊗ X)• → (K ⊗ X)•−2 = X•−2.

Note that if ϑ = 0, then the definition of a differential Lie module over a graded ϑ-coalgebra (C,∇, ϑ)coincides with the definition of a differential Lie module over a graded coalgebra (C,∇) given in [5].

Definition 2.2. We define a morphism

f = (f ′, f ′′) : (X, d, ψ) → (Y, d, ψ)

of differential Lie (C,ϑ)-modules as a pair of maps f ′ : X• → Y• and f ′′ : (C ⊗ X)• → Y•, where f ′ is amap of differential modules and f ′′ satisfies the conditions

d(f ′′) − f ′′ ∪ ψ + ψ ∪ f ′′ − f ′ψ + ψ(1 ⊗ f ′) = 0, f ′′(ν ⊗ 1) = 0.


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Note that the condition

d(f ′′) − f ′′ ∪ ψ + ψ ∪ f ′′ − f ′ψ + ψ(1 ⊗ f ′) = 0

in the definition of a morphism of differential (C,ϑ)-modules implies

f ′′ ∪ ϑ − ϑ ∪ f ′′ = ϑ(1 ⊗ f ′) − f ′ϑ.

The composition gf : X → Z of morphisms

f = (f ′, f ′′) : (X, d, ψ) → (Y, d, ψ) and g = (g′, g′′) : (Y, d, ψ) → (Z, d, ψ)

of differential Lie (C,ϑ)-modules is defined by

gf = (g′, g′′)(f ′, f ′′) = (g′f ′, g′′ ∪ f ′′ + g′′(1 ⊗ f ′) + g′f ′′).

Clearly, the operation of composition of morphisms of differential Lie (C,ϑ)-modules is associative;moreover, for each differential Lie (C,ϑ)-module (X, d, ψ), the identity morphism

1X = (1′X , 1′′X) : (X, d, ψ) → (X, d, ψ)

is defined, where 1′X = idX is the identity map of the module X and 1′′X = 0. Thus, the class of alldifferential Lie (C,ϑ)-modules over any commutative unital ring K and their morphisms form a category,which we denote by L(C,ϑ)(K).

Consider the problem of characterizing an isomorphism f : (X, d, μ) → (Y, d, μ) in the categoryL(C,ϑ)(K) in terms of its component f ′ : (X, d) → (Y, d).

Proposition 2.1. Suppose that (C,∇, ϑ) is a connected graded ϑ-coalgebra,

f = (f ′, f ′′) : (X, d, μ) → (Y, d, μ)

is a morphism in the category L(C,ϑ)(K), and f ′ : (X, d) → (Y, d) is an isomorphism of differen-tial modules. Then the morphism f = (f ′, f ′′) is an isomorphism in the category L(C,ϑ)(K), andthe inverse isomorphism g = (g′, g′′) : (Y, d, μ) → (X, d, μ) in the category L(C,ϑ)(K) is defined by

g′ = (f ′)−1, g′′ =∑

n≥1

(−1)n((g′f ′′) ∪ · · · ∪ (g′f ′′)︸︷︷︸n

)(1 ⊗ g′).

Proof. First, let us show that the expression for g′′ indeed defines a map. Since the ϑ-coalgebra(C,∇, ϑ) is connected and ψ(ν ⊗ 1) = 0, it follows that the values of the summands on the right-handside of the expression for g′′ at any element are nonzero for only finitely many n. Now, let us verify that(g′, g′′)(f ′, f ′′) = (idX , 0). Obviously, we have g′f ′ = idX . For the map g′′ ∪ f ′′ + g′′(1 ⊗ f ′) + g′f ′′,the following relations hold (we use the notation (g′f ′′)∪k = (g′f ′′) ∪ · · · ∪ (g′f ′′) (k factors)):

g′′ ∪ f ′′ + g′′(1 ⊗ f ′) + g′f ′′

=∑

n≥1

(−1)n(g′f ′′)∪n(1 ⊗ g′)(1 ⊗ f ′′)(∇⊗ 1) +∑

n≥1

(−1)n(g′f ′′)∪n(1 ⊗ g′)(1 ⊗ f ′) + g′f ′′

=∑

n≥1

(−1)n(g′f ′′)∪(n+1) +∑

n≥1

(−1)n(g′f ′′)∪n + g′f ′′ = −g′f ′′ + g′f ′′ = 0.

The equality (f ′, f ′′)(g′, g′′) = (idY , 0) is verified in a similar way.

Definition 2.3. By a homotopy h = (h′, h′′) : (X, d, ψ) → (Y, d, ψ) between morphisms

f = (f ′, f ′′) : (X, d, ψ) → (Y, d, ψ) and g = (g′, g′′) : (X, d, ψ) → (Y, d, ψ)

of differential Lie (C,ϑ)-modules we mean a pair of maps h′ : X• → Y•+1 and h′′ : (C ⊗ X)• → Y•+1,where h′ is a homotopy between the maps f ′ and g′ of differential modules and h′′ satisfies the conditions

d(h′′) + h′′ ∪ ψ + ψ ∪ h′′ + h′ψ + ψ(1 ⊗ h′) = f ′′ − g′′, h′′(ν ⊗ 1) = 0.



Note that the condition

d(h′′) + h′′ ∪ ψ + ψ ∪ h′′ + h′ψ + ψ(1 ⊗ h′) = f ′′ − g′′

in the definition of a homotopy between morphisms of differential (C,ϑ)-modules implies

h′′ ∪ ϑ − ϑ ∪ h′′ + h′ϑ − ϑ(1 ⊗ h′) = ψ ∪ (f ′′ − g′′) − (f ′′ − g′′) ∪ ψ.

Let η : (X, d, ψ) � (Y, d, ψ) : ξ be any morphisms of differential Lie (C,ϑ)-modules such thatηξ = 1Y , and let h : (X, d, ψ) → (X, d, ψ) be a homotopy between the morphisms ξη and 1X ofdifferential Lie (C,ϑ)-modules which satisfies the conditions

ηh = (η′, η′′)(h′, h′′) = (η′h′, η′′ ∪ h′′ + η′′(1 ⊗ h′) + η′h′′) = (0, 0),

hξ = (h′, h′′)(ξ′, ξ′′) = (h′ξ′, h′′ ∪ ξ′′ + h′′(1 ⊗ ξ′) + h′ξ′′) = (0, 0),

hh = (h′, h′′)(h′, h′′) = (h′h′, h′′ ∪ h′′ + h′′(1 ⊗ h′) + h′h′′) = (0, 0).

We refer to any such triple (η : (X, d, ψ) � (Y, d, ψ) : ξ, h) as SDR-data for differential Lie(C,ϑ)-modules.

The following theorem asserts the homotopy invariance of the structure of a differential Lie moduleover a curved graded coalgebra under homotopy equivalences of the type of SDR-data for differentialmodules.

Theorem 2.1. Let (X, d, ψ) be a differential Lie module over a connected graded counitalcoaugmented ϑ-coalgebra (C,∇, ϑ), and let (η : (X, d) � (Y, d) : ξ, h) be any SDR-data for dif-ferential modules. Then the differential module (Y, d) admits the structure of a differential Lie(C,ϑ)-module (Y, d, ψ ), which is defined by

ψ = η

(∑

n≥0

ψ ∪ (hψ) ∪ · · · ∪ (hψ)︸︷︷︸n

)(1 ⊗ ξ). (2.1)

Moreover, the relations

ξ′ = ξ, ξ

′′ = h

(∑

n≥0

ψ ∪ (hψ) ∪ · · · ∪ (hψ)︸︷︷︸n

)(1 ⊗ ξ), (2.2)

η ′ = η, η ′′ = η

(∑

n≥0

ψ ∪ (hψ) ∪ · · · ∪ (hψ)︸︷︷︸n

)(1 ⊗ h), (2.3)

h′ = h, h

′′ = h

(∑

n≥0

ψ ∪ (hψ) ∪ · · · ∪ (hψ)︸︷︷︸n

)(1 ⊗ h) (2.4)

define the SDR-data (η : (X, d, ψ) � (Y, d, ψ ) : ξ, h ) for differential Lie (C,ϑ)-modules.

Proof. First, note that (2.1) indeed defines a map, because the connectedness of ϑ-coalgebra (C,∇, ϑ)and the equality ψ(ν ⊗ 1) = 0 imply the vanishing of all but finitely many summands on the right-handside of (2.1) at any element. Similar considerations apply to (2.2)–(2.4). Now, let us show that the map ψ

satisfies the relation d(ψ ) + ψ ∪ψ + ϑ = 0. Using the abbreviated notation (hψ)∪k = (hψ)∪ · · · ∪ (hψ)(k factors), we can write

d(ψ ) = η

(∑

n≥0

d(ψ) ∪ (hψ)∪n

)(1 ⊗ ξ) − η

(∑

n≥1

n∑

i=1

ψ ∪ (hψ)∪(i−1) ∪ (d(hψ)) ∪ (hψ)∪(n−i)

)(1 ⊗ ξ).

Since

d(ψ) = −ψ ∪ ψ − ϑ, d(h) = ξη − 1 and d(hψ) = d(h)ψ − hd(ψ),

it follows that

d(hψ) = ξηψ − ψ + h(ψ ∪ ψ) + hϑ.


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Using this equality and the relations

h(ψ ∪ ψ) = (hψ) ∪ ψ, ψ ∪ (ξηψ) = (ψ(1 ⊗ ξ)) ∪ (ηψ),(hψ) ∪ (ξηψ) = ((hψ)(1 ⊗ ξ)) ∪ (ηψ),

we obtain

d(ψ ) = −ψ ∪ ψ − η

(∑

n≥0

ϑ ∪ (hψ)∪n

)(1 ⊗ ξ)

− η

(∑

n≥1

n∑

i=1

ψ ∪ (hψ)∪(i−1) ∪ (hϑ) ∪ (hψ)∪(n−i)

)(1 ⊗ ξ).

The relation ϑ = ϑ ⊗ 1 implies hϑ ∪ hψ = (hϑ(1 ⊗ h)) ∪ ψ = 0, because

hϑ(1 ⊗ h) = (1K ⊗ h)(ϑ ⊗ 1)(1 ⊗ h) = hhϑ = 0.

Moreover, we have hϑ(1 ⊗ ξ) = hξϑ = 0 and

η(ϑ ∪ (hψ)) = (ηϑ(1 ⊗ h)) ∪ ψ = (ηhϑ) ∪ ψ = 0.

The last two equalities imply the required relation d(ψ ) + ψ ∪ψ + ϑ = 0, because ηϑ(1⊗ ξ) = ηξϑ = ϑ.It is easy to see that ψ satisfies the relation ψ(ν ⊗ 1) = 0. Indeed, from ψ(ν ⊗ 1) = 0, we obtain

(hψ)(1 ⊗ ξ)(ν ⊗ 1) = hψ(ν ⊗ 1)(1 ⊗ ξ) = 0,

which implies the required relation ψ(ν ⊗ 1) = 0. A similar argument proves that the morphisms ξ andη and the homotopy h satisfy the required relations and that the SDR-data

(η : (X, d, ψ) � (Y, d, ψ ) : ξ, h )

for differential Lie (C,ϑ)-modules are defined.

Now, let us show that the homotopy properties of any morphism f = (f ′, f ′′) : (X, d, ψ) → (Y, d, ψ)of differential Lie (C,ϑ)-modules are completely determined by the homotopy properties of the compo-nent f ′ : (X, d) → (Y, d).

Proposition 2.2. Let (C,∇, ϑ) be a connected graded coaugmented counital ϑ-coalgebra. Thena morphism f = (f ′, f ′′) : (X, d, ψ) → (Y, d, ψ) of differential Lie (C,ϑ)-modules determinesSDR-data (f : (X, d, ψ) � (Y, d, ψ) : g, s) for differential Lie (C,ϑ)-modules if and only if themap f ′ : (X, d) → (Y, d) of differential modules determines SDR-data (f ′ : (X, d) � (Y, d) : g′, s′)for differential modules.

Proof. Clearly, SDR-data (f : (X, d, ψ) � (Y, d, ψ) : g, s) for differential Lie (C,ϑ)-modules deter-mine the SDR-data (f ′ : (X, d) � (Y, d) : g′, s′) for differential modules. Now, suppose that

f = (f ′, f ′′) : (X, d, ψ) → (Y, d, ψ)

is a morphism of differential Lie (C,ϑ)-modules and (f ′ : (X, d) � (Y, d) : g′, s′) is SDR-data for differ-ential modules. Let (η : (X, d, ψ) � (Y, d, ψ ) : ξ, h ) be the SDR-data for differential Lie (C,ϑ)-modulesdefined by (2.1)–(2.4), where η ′ = f ′, ξ

′ = g′, and h′ = s′. Consider the morphism γ = fξ : (Y, d, ψ ) →

(Y, d, ψ) of differential Lie (C,ϑ)-modules. Since γ′ = f ′ξ′= f ′g′ = 1Y , it follows by Proposition 2.1

that the morphism γ is an isomorphism of differential Lie (C,ϑ)-modules. Let ζ : (Y, d, ψ) → (Y, d, ψ )denote the isomorphism of differential Lie (C,ϑ)-modules inverse to γ. Considering the morphism g =ξζ : (Y, d, ψ) → (X, d, ψ) of differential Lie (C,ϑ)-modules, setting s = (1X − gf)h, and performingdirect calculations, we see that (f : (X, d, ψ) � (Y, d, ψ) : g, s) is SDR-data for differential Lie (C,ϑ)-modules, as required.



The following proposition asserts the uniqueness of the structure of a differential Lie (C,ϑ)-module(Y, d, ψ ) defined in Theorem 2.1 for a differential module (Y, d).

Proposition 2.3. The structure of a differential Lie (C,ϑ)-module (Y, d, ψ ) on a differentialmodule (Y, d) defined in Theorem 2.1 is determined uniquely up to isomorphism of differentialLie (C,ϑ)-modules.

Proof. Let (η : (X, d) � (Y, d) : ξ, h) be any SDR-data for differential modules, and let

(η : (X, d, ψ) � (Y, d, ψ ) : ξ, h )

be the SDR-data for differential Lie (C,ϑ)-modules defined in Theorem 2.1. Suppose that

(η : (X, d, ψ) � (Y, d, ψ) : ξ, h)

is yet other SDR-data for differential Lie (C,ϑ)-modules satisfying the conditions η ′ = η, ξ ′ = ξ, andh ′ = h. It follows by Proposition 2.1 that the morphism η ξ : (Y, d, ψ) → (Y, d, ψ ) of differential Lie(C,ϑ)-modules is an isomorphism of differential Lie (C,ϑ)-modules, because η ′ξ ′ = ηξ = 1Y .

3. HOMOTOPY PROPERTIES OF DIFFERENTIAL F (R!, ϑ)-MODULESFOR QUADRATIC-SCALAR ALGEBRAS (R,π) AND DIFFERENTIAL MODULES

OVER CLIFFORD ALGEBRAS

First, consider the homotopy properties of differential F (R!, ϑ)-modules, where (R!,∇, ϑ) is the ϑ-coalgebra Koszul dual to a quadratic-scalar algebra (R,π).

It is easy to see that defining the structure of a differential F (R!, ϑ)-module (X, d, μ) on a dif-ferential module (X, d) is equivalent to defining the structure of a differential Lie (R!, ϑ)-module(X, d, ψ) on (X, d). Indeed, given a differential F (R!, ϑ)-module (X, d, μ), we have the differential Lie(R!, ϑ)-module (X, d, ψμ) with structure map

ψμ = μ(ϕF ⊗ 1): (R! ⊗ X)• → X•−1.

Conversely, if a differential Lie (R!, ϑ)-module (X, d, ψ) is given, then its structure map

ψ : (R! ⊗ X)• → X•−1

determines a twisting ϑ-cochain

ψ : R!

• → Hom(X;X)•−1. This twisting ϑ-cochain induces a mapμψ = F (

ψ) : F (R!, ϑ) → Hom(X;X) of differential algebras, which, as mentioned above, determines

the structural map μψ : F (R!, ϑ) ⊗ X → X of the differential F (R!, ϑ)-module (X, d, μψ).

Definition 3.1. By an F (R!, ϑ)-map f : (X, d, μ) → (Y, d, μ) of differential F (R!, ϑ)-modules for agiven any quadratic-scalar algebra (R,π) we mean a morphism f : (X, d, ψμ) → (Y, d, ψμ) of thecorresponding differential Lie (R!, ϑ)-modules.

Thus, an F (R!, ϑ)-map f : (X, d, μ) → (Y, d, μ) of differential F (R!, ϑ)-modules, where (R,π) is anyquadratic-scalar algebra, is a pair of maps f = (f ′, f ′′), where f ′ : X• → Y• is a map differential modulesand f ′′ : (R! ⊗ X)• → Y• is a map satisfying the conditions

d(f ′′) − f ′′ ∪ ψμ + ψμ ∪ f ′′ − f ′ψμ + ψμ(1 ⊗ f ′) = 0, f ′′(ν ⊗ 1) = 0.

Definition 3.2. By an F (R!, ϑ)-homotopy h : (X, d, μ) → (Y, d, μ) between given F (R!, ϑ)-mapsf, g : (X, d, μ) → (Y, d, μ) of differential F (R!, ϑ)-modules for any quadratic-scalar algebra (R,π), wemean a homotopy h : (X, d, ψμ) → (Y, d, ψμ) between the morphisms f, g : (X, d, ψμ) → (Y, d, ψμ) ofthe corresponding differential Lie (R!, ϑ)-modules.


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Thus, an F (R!, ϑ)-homotopy h : (X, d, μ) → (Y, d, μ) between given F (R!, ϑ)-maps

f = (f ′, f ′′), g = (g′, g′′) : (X, d, μ) → (Y, d, μ)

of differential F (R!, ϑ)-modules is a pair of maps h = (h′, h′′), where h′ : X• → Y•+1 is a homotopybetween the maps f ′, g′ : X• → Y• of differential modules and h′′ : (R! ⊗X)• → Y•+1 is a map satisfyingthe conditions

d(h′′) + h′′ ∪ ψμ + ψμ ∪ h′′ + h′ψμ + ψμ(1 ⊗ h′) = f ′′ − g′′, h′′(ν ⊗ 1) = 0.

Suppose that (R,π) is a quadratic-scalar algebra. Suppose that η : (X, d, μ) � (Y, d, μ) : ξ areF (R!, ϑ)-maps of differential F (R!, ϑ)-modules such that ηξ = 1Y and h : (X, d, μ) → (X, d, μ) is anF (R!, ϑ)-homotopy between the F (R!, ϑ)-maps ξη and 1X of differential F (R!, ϑ)-modules whichsatisfies the conditions ηh = 0, ξh = 0, and hh = 0. We refer to any such triple

(η : (X, d, μ) � (Y, d, μ) : ξ, h)

as F (R!, ϑ)-SDR-data for differential F (R!, ϑ)-modules.

The following theorem, which follows from Theorem 2.1, asserts the F (R!, ϑ)-homotopy invarianceof the structure of a differential F (R!, ϑ)-module under homotopy equivalences of the type of SDR-datafor differential modules.

Theorem 3.1. Let (R!,∇, ϑ) be the ϑ-coalgebra Koszul dual to a quadratic-scalar algebra (R,π),and let (X, d, μ) be a differential F (R!, ϑ)-module. Suppose that (η : (X, d) � (Y, d) : ξ, h) is anySDR-data for differential modules. Then the differential module (Y, d) admits the structure ofa differential F (R!, ϑ)-module (Y, d, μψ), which is defined by (2.1). Moreover, expressions (2.2)–

(2.4) define F (R!, ϑ)-SDR-data (η : (X, d, μ) � (Y, d, μψ) : ξ, h ) for differential F (R!, ϑ)-modules.

For any differential module over a field, homological SDR-data are defined; therefore, the followingassertion is a special case of Theorem 3.1.

Corollary 3.1. Suppose that the base ring K is a field, (R!,∇, ϑ) is the ϑ-coalgebra Koszul dualto a quadratic-scalar algebra (R,π), and (X, d, μ) is a differential F (R!, ϑ)-module over K. Thenrelations (2.1)–(2.4) determine the structure of a differential F (R!, ϑ)-module (H(X), d = 0, μψ)on the graded homology K-module H(X) of the differential K-module (X, d) and defineF (R!, ϑ)-SDR-data (η : (X, d, μ) � (H(X), d = 0, μψ) : ξ, h ) for differential F (R!, ϑ)-modules,which extend the homology SDR-data (η : (X, d) � (H(X), d = 0) : ξ, h) for the differentialmodule (X, d).

Yet another important special case of Theorem 3.1 is the following assertion, which is based onthe possibility of interpreting each differential module over any quadratic-scalar algebra (R,π) asan F (R!, ϑ)-module by using the map F (ϕ!

ϑ) : (F (R!, ϑ), d, π) → (R, d = 0, π) of differential algebrasspecified in Section 1.

Corollary 3.2. Suppose that (R,π) is a quadratic-scalar algebra, (X, d, μ) is a differentialR-module, and (η : (X, d) � (Y, d) : ξ, h) is SDR-data for differential modules. If the differentialR-module (X, d, μ) is treated as an F (R!, ϑ)-module (X, d, μ(F (ϕ!

ϑ) ⊗ 1)), then relations (2.1)–(2.4) determine the structure of a differential F (R!, ϑ)-module (Y, d, μψ) on (Y, d) and define

F (R!, ϑ)-SDR-data

(η : (X, d, μ(F (ϕ!ϑ) ⊗ 1)) � (Y, d, μψ) : ξ, h )

for differential F (R!, ϑ)-modules.



Now, consider the application of the homotopy technique of differential F (R!, ϑ)-modules to con-structing a homotopy invariant analog over any commutative ring for a differential module over a Cliffordalgebra.

By a generalized Clifford algebra (Cl(n, ai), π) over a commutative unital ring K we mean anassociative K-algebra with generators ε1, . . . , εn satisfying the relations

εiεj + εjεi =

{0, i = j, 1 ≤ i ≤ n, 1 ≤ j ≤ n,

ai, 1 ≤ i = j ≤ n,(3.1)

where εkεm = π(εk ⊗ εm) for 1 ≤ k ≤ n and 1 ≤ m ≤ n and a1, . . . , an are any fixed elements of thering K (possibly, all or some of the ai are zero).

It is easy to see that any Clifford algebra Cl(n, ai) is a quadratic-scalar algebra, because Cl(n, ai)is isomorphic to the quotient algebra T (M)/(Q), where M is the free K-module with basis ε1, . . . , εn

and Q is the free K-submodule with basis

εi ⊗ εj + εj ⊗ εi, i = j, 1 ≤ i ≤ n, 1 ≤ j ≤ n, 2εi ⊗ εi − ai, 1 ≤ i ≤ n,

of the module M⊗2 ⊕ K. Clearly, Q ∩ K = 0.

Definition 3.3. A differential (n, ai)-Clifford module is defined as a differential module (X, d)together with a family of maps εi : (X, d) → (X, d) of differential modules (1 ≤ i ≤ n) which satisfyrelations (3.1).

Clearly, defining the structure of a differential (n, ai)-Clifford module (X, d, εi) on a differentialmodule (X, d) is equivalent to defining the structure of a differential Cl(n, ai)-module on (X, d).

Consider the curved graded coalgebra (Cl(n, ai)!,∇, ϑ) Koszul dual to the quadratic-scalar algebraCl(n, ai). Since the submodule p(Q) of M⊗2 is a free module and has basis εi ⊗ εj + εj ⊗ εi, 1 ≤ i ≤ n,

1 ≤ j ≤ n, it follows that the module (Cl(n, ai)!)(k)k , k ≥ 1, is free as well, and its basis is formed by the

elements

[εi1 ] ∨ · · · ∨ [εik ] =∑

σ∈Σk

[εσ(i1), . . . , εσ(ik)], 1 ≤ i1 ≤ · · · ≤ ik ≤ n,

where Σk is the symmetric group on k elements. Moreover, obviously, we have

(Cl(n, ai)!)(k)m = 0 for k = m.

In what follows, we refer to a k-tuple (i1, . . . , ik) of positive integers in which 1 ≤ i1 ≤ · · · ≤ ik ≤ n asan ordered tuple. It is easy to see that comultiplication in the graded coalgebra (Cl(n, ai)!,∇) is defined

at the basis elements [εi1 ] ∨ · · · ∨ [εik ] of the module (Cl(n, ai)!)(k)k , k ≥ 1, by

∇([εi1 ] ∨ · · · ∨ [εik ]) = 1 ⊗ ([εi1 ] ∨ · · · ∨ [εik ]) + ([εi1 ] ∨ · · · ∨ [εik ]) ⊗ 1

+∑

σ∈Σk

∑

Iσ

([εσ(i1)] ∨ · · · ∨ [εσ(im)]) ⊗ ([εσ(im+1)] ∨ · · · ∨ [εσ(ik)]),

where Iσ is the set of all partitions of the n-tuple (σ(i1), . . . , σ(ik)) into two ordered tuples

(σ(i1), . . . , σ(im)) and (σ(im+1), . . . , σ(ik)), 1 ≤ m ≤ k − 1.

It is also clear that the curvature ϑ : Cl(n, ai)!2 → R of the graded coalgebra (Cl(n, ai)!,∇) is defined at

the basis elements of (Cl(n, ai)!)(2)2 by

ϑ([εi1 ] ∨ [εi2 ]) =

{0, 1 ≤ i1 < i2 ≤ n,

−ai, 1 ≤ i1 = i2 ≤ n.

Now, let (F (Cl(n, ai)!, ϑ), d, π) be the co-B-construction for the curved coalgebra (Cl(n, ai)!,∇, ϑ)Koszul dual to the quadratic-scalar algebra (Cl(n, ai), π). It follows from the above description of


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the structure of the curved graded coalgebra (Cl(n, ai)!,∇, ϑ) that the algebra (F (Cl(n, ai)!, ϑ), π) isgenerated by the elements

[[εi1 ] ∨ · · · ∨ [εik ]] ∈ F (Cl(n, ai)!, ϑ)k−1, 1 ≤ i1 ≤ · · · ≤ ik ≤ n,

at which the differential is defined by

d([[εi1 ] ∨ · · · ∨ [εik ]]) =∑

σ∈Σk

∑

Iσ

π([[εσ(i1)] ∨ · · · ∨ [εσ(im)]] ⊗ [[εσ(im+1)] ∨ · · · ∨ [εσ(ik)]]

),

k ≥ 1, k = 2,

d([[εi] ∨ [εj ]]) =

{π([[εi]] ⊗ [[εj ]]) + π([[εj ]] ⊗ [[εi]]), 1 ≤ i < j ≤ n,

π(2[[εi]] ⊗ [[εi]]) − ai, 1 ≤ i = j ≤ n,

where π is multiplication in the algebra F (Cl(p, q)!, ϑ) and Iσ is the set of all partitions of the k-tuple(σ(i1), . . . , σ(ik)) into two ordered tuples (σ(i1), . . . , σ(im)) and (σ(im+1), . . . , σ(ik)), 1 ≤ m ≤ k − 1.

Definition 3.4. By a differential Cl(n, ai)∞-module (X, d, ε(i1 ,...,ik)) we mean a differential module(X, d) together with a family of maps ε(i1,...,ik) : X• → X•+k−1, 1 ≤ i1 ≤ · · · ≤ ik ≤ n, k ≥ 1, satisfyingthe relations

d(ε(i)) = 0, 1 ≤ i ≤ n, d(ε(i,i)) = 2ε2i − ai, 1 ≤ i ≤ n,

d(ε(i,j)) = ε(i)ε(j) + ε(i)ε(j), 1 ≤ i < j ≤ n,

d(ε(i1,...,ik)) =∑

σ∈Σk

∑

Iσ

ε(σ(i1),...,σ(im))ε(σ(im+1),...,σ(ik)), k ≥ 3,(3.2)

where Iσ is the set of all partitions of the k-tuple (σ(i1), . . . , σ(ik)) into two ordered tuples

(σ(i1), . . . , σ(im)) and (σ(im+1), . . . , σ(ik)), 1 ≤ m ≤ k − 1.

It is obvious that we can regard any differential (n, ai)-Clifford module (X, d, εi) as a differentialCl(n, ai)∞-module (X, d, ε(i1 ,...,ik)); it suffices to set ε(i) = εi for 1 ≤ i ≤ n and ε(i1,...,ik) = 0 for k ≥ 2.On the other hand, relations (6) show that, for each differential Cl(n, ai)∞-module (X, d, ε(i1 ,...,ik)), thetriple (X, d, ε(i)) is a differential (n, ai)-Clifford module (up to homotopy). In particular, for each gradedCl(n, ai)∞-module (X, d = 0, ε(i1,...,ik)), the triple (X, d = 0, ε(i)) is a graded (n, ai)-Clifford module.

It is easy to see that endowing (X, d) with the structure of a differential Cl(n, ai)∞-module(X, d, ε(i1 ,...,ik)) is equivalent to endowing (X, d) with the structure of the differential F (Cl(n, ai)!, ϑ)-module (X, d, μ), where the structure map μ : F (Cl(n, ai)!, ϑ) ⊗ X → X is defined for

[[εi1 ] ∨ · · · ∨ [εik ]] ∈ F (Cl(n, ai)!, ϑ)

and x ∈ X by

μ([[εi1 ] ∨ · · · ∨ [εik ]] ⊗ x) = ε(i1,...,ik)(x).

Definition 3.5. A morphism f : (X, d, ε(i1 ,...,ik)) → (Y, d, ε(i1,...,ik)) of differential Cl(n, ai)∞-modulesis an F (Cl(n, ai)!, ϑ)-map f : (X, d, μ) → (Y, d, μ) of the corresponding differential F (Cl(n, ai)!, ϑ)-modules. The composition of morphisms of differential Cl(n, ai)∞-modules is defined as the compo-sition of the corresponding F (Cl(n, ai)!, ϑ)-maps.

Definition 3.6. A homotopy h : (X, d, ε(i1 ,...,ik)) → (Y, d, ε(i1 ,...,ik)) between morphisms

f, g : (X, d, ε(i1 ,...,ik)) → (Y, d, ε(i1 ,...,ik))

of differential Cl(n, ai)∞-modules is defined as an F (Cl(n, ai)!, ϑ)-homotopy h : (X, d, μ) → (Y, d, μ)between the corresponding F (Cl(n, ai)!, ϑ)-maps f, g : (X, d, μ) → (Y, d, μ). SDR-data for dif-ferential Cl(n, ai)∞-modules are the corresponding F (Cl(n, ai)!, ϑ)-SDR-data for differentialF (Cl(n, ai)!, ϑ)-modules.



The following theorem, which is implied by Theorem 3.1, asserts the homotopy invariance of thestructure of a differential Cl(n, ai)∞-module under homotopy equivalences of the type of SDR-data fordifferential modules.

Theorem 3.2. Let (X, d, ε(i1 ,...,ik)) be a differential Cl(n, ai)∞-module, and let

(η : (X, d) � (Y, d) : ξ, h)

be any SDR-data for differential modules. Then relation (2.1) defines the structure of adifferential Cl(n, ai)∞-module (Y, d, ε(i1,...,ik)) on the differential module (Y, d). Moreover, re-lations (2.2)–(2.4) define SDR-data (η : (X, d, ε(i1 ,...,ik)) � (Y, d, ε(i1,...,ik)) : ξ, h ) for differentialCl(n, ai)∞-modules.

Corollary 3.3. Let (X, d, ε(i1 ,...,ik)) be a differential Cl(n, ai)∞-module over an arbitrary field.Then relations (2.1)–(2.4) endow the graded homology module H(X) of the differential module(X, d) with the structure of a differential Cl(n, ai)∞-module (H(X), d = 0, ε(i1,...,ik)) and defineSDR-data

(η : (X, d, ε(i1 ,...,ik)) � (H(X), d = 0, ε(i1,...,ik)) : ξ, h )

for differential Cl(n, ai)∞-modules, which extend the homological SDR-data for the differentialmodule (X, d).

Corollary 3.4. Let (X, d, εi) be a differential (n, ai)-Clifford module, and let

(η : (X, d) � (Y, d) : ξ, h)

be SDR-data for differential modules. If (X, d, εi) is treated as a differential Cl(n, ai)∞-module (X, d, ε(i1 ,...,ik)), then relations (2.1)–(2.4) endow (Y, d) with the structure of a differentialCl(n, ai)∞-module (Y, d, ε(i1,...,ik)) and define SDR-data

(η : (X, d, ε(i1 ,...,ik)) � (Y, d, ε(i1,...,ik)) : ξ, h )

for differential Cl(n, ai)∞-modules, which extend the given SDR-data

(η : (X, d) � (Y, d) : ξ, h)

for differential modules.

Below, we give interesting interpretations of Corollary 3.4 in some special cases of (n, ai)-Cliffordmodules.

It is easy to see that, in the in case K = Z, differential (1,−2)-Clifford modules (X, d, ε) aredifferential Z-modules (X, d) over the algebra Z[i] of Gaussian integers; the action of i on (X, d) isdetermined by the map ε. Applying Corollary 3.4 to the structure of a differential Z-module over thealgebra Z[i], we obtain the homotopy invariant structure of a differential Z[i]∞-module over the ring Z.

Clearly, in the case K = R, differential (1,−2)-Clifford modules (X, d, ε) are differential R-modules(X, d) over the algebra C = R[i] of complex numbers. Applying Corollary 3.4 to the structure of adifferential R-module over the algebra C, we obtain the homotopy invariant structure of a differentialC∞-module over the field R.

Differential (n, {0, . . . , 0})-Clifford modules (X, d, εi) are differential modules over the exterioralgebra Λ(ε1, . . . , εn). Obviously, a differential module over Λ(ε1, . . . , εn) is a differential module (X, d)endowed with a family of additional degree-0 differentials di = εi : X• → X• for 1 ≤ i ≤ n (d2

i = 0)satisfying the differential compatibility conditions didj + djdi = 0, for 0 ≤ i ≤ n and 0 ≤ j ≤ n, whered0 = d. Applying Corollary 3.4 to the structure of a differential module over the exterior algebraΛ(ε1, . . . , εn), we obtain the homotopy invariant structure of a differential Λ(ε1, . . . , εn)∞-module.Note that the homotopy theory of differential Λ(ε1)∞-modules is equivalent to that of D∞-differentialmodules [8], which was applied in [9] and [10] to study E∞-structures and A∞-structures in spectralsequences of fibrations.


350 LAPIN

REFERENCES1. L. E. Positsel’skii, “Nonhomogeneous quadratic duality and curvature,” Funktsional. Anal. i Prilozhen. 27

(3), 57–66 (1993) [Functional Anal. Appl. 27 (3), 197–204 (1993)].2. S. B. Priddy, “Koszul resolutions,” Trans. Amer. Math. Soc. 152, 39–60 (1970).3. J. Hirsh and J. Milles, Curved Koszul Duality Theory, arXiv:1008.5368v1.4. V. Ginzburg and M. Kapranov, “Koszul duality for operads,” Duke Math. J. 76 (1), 203–272 (1994).5. V. A. Smirnov, “Lie algebras over operads and their applications in homotopy theory,” Izv. Ross. Akad. Nauk

Ser. Mat. 62 (3), 121–154 (1998) [Izv. Math. 62 (3), 549–580 (1998)].6. J. P. May, Van Nostrand Mathematical Studies, Vol. 11: Simplicial Objects in Algebraic Topology (Van

Nostrand, Princeton, NJ, 1967).7. V. A. Smirnov, Simplicial and Operad Methods in Algebraic Topology (Factorial, Moscow, 2002) [in

Russian].8. S. V. Lapin, “Differential perturbations and D∞-differential modules,” Mat. Sb. 192 (11), 55–76 (2001) [Sb.

Math. 192 (11), 1639–1659 (2001)].9. S. V. Lapin, “D∞-Differential E∞-algebras and spectral sequences of fibrations,” Mat. Sb. 198 (10), 3–30

(2007) [Sb. Math. 198 (10), 1379–1406 (2007)].10. S. V. Lapin, “Multiplicative A∞-structure in terms of spectral sequences of fibrations,” Fund. Prikl. Mat. 14

(6), 141–175 (2008) [J. Math. Sci. 164 (1), 95–118 (2010)].


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