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Partial cohomology of groups

We develop a cohomology theory of groups based on partial actions and explore its relation with the partial Schur multiplier as well as with cohomology of inverse semigroups.


Introduction
In [12] R. Exel initiated a new and successful method to study C * -algebras by showing that a separable C * -algebra possessing a semi-saturated stable action of the circle group is a crossed product by a partial action of the infinite cyclic group. This permitted him to investigate the internal structure, representations and K-theory of such algebras. Exel's approach is based on new concepts, namely, those of a partial action, the corresponding crossed product and a partial representation, as well as on the interaction between them. Relevant classes of C *algebras, endowed with the structure of non-trivial crossed product by a partial action, include the Bunce-Deddens and the Bunce-Deddens-Toeplitz algebras [13], the approximately finite dimensional algebras [14], the Toeplitz algebras of quasi-ordered groups, the Cuntz-Krieger algebras [18], [27] and, more recently, the C * -algebras O m,n related to dynamical systems in which n copies of a topological space is homeomorphic to m copies of the same space [1].
The first algebraic results on the above mentioned notions were established in [16], [5], [29], [22] and [4], which together with the development of the partial Galois theory in [8] stimulated an intensive algebraic activity on partial actions, corresponding crossed products and partial representations (see the surveys [3] and [20]). In particular, applications were obtained to graded algebras in [4] and [6], to Hecke algebras in [17] and to Leavitt path algebras in [21].
The general notion of a (continuous) twisted partial action of a locally compact group on a C * -algebra (a twisted partial C * -dynamical system) was given in [15]. The new construction permitted one to show that any second countable C * -algebraic bundle, which satisfies a certain regularity condition (automatically verified if the unit fiber algebra is stable), is a C * -crossed product of the unit fiber algebra by a continuous partial action of the base group. A purely ring theoretic treatment of twisted partial group actions, including an analogue of the above mentioned fact, was given in [6]. It involves a general twisting which satisfies the 2-cocycle identity in some restricted sense, and one naturally wonders what kind of cohomology theory would fit this. It is complicated to give such a theory which would embrace the full generality of [6], since the twisting takes its values in multiplier algebras of products of some ideals indexed by group elements. Nevertheless, imposing a reasonable restriction on the partial action, namely, that it is unital, makes it possible to define partial cohomology. Actually, this restriction is assumed in almost all algebraic papers around partial actions, because it provides an appropriate technical framework.
One feels more confident in developing such a cohomology theory if one shows that it matches some other concepts in a way how the usual group cohomology does. In particular, the Schur multiplier of a group G over the complex numbers C is isomorphic to H 2 (G, C * ) with trivial action of G on C * . Thus a theory of partial projective group representations could be a testing field for our cohomology. Such a theory was developed in [9], [10] and [11], in particular the structure of the partial Schur multiplier was explored. Further results on the latter topic were obtained in [25] and [26].
A twisted partial action of a group G on a commutative ring A falls into two parts: a partial action θ of G on A and its twisting. With an additional assumption that θ is unital (which means that the domains involved in θ are generated by central idempotents), one can derive the concept of a partial 2-cocycle (the twisting) whose values belong to groups of invertible elements of appropriate ideals of A. The concept of a partial 2-coboundary then follows from that of an equivalence of twisted partial actions introduced in [7]. Replacing A by a commutative multiplicative monoid, one comes to the definition of the second cohomology group H 2 (G, A). Thus instead of a usual G-module we deal with a partial G-module, which is a commutative monoid A with a unital partial action θ of G on A. The groups H n (G, A) with arbitrary n are defined in a similar way (see Section 1). Replacing A by an appropriate submonoid one may actually assume that A is inverse (see Remark 3.13). Next one asks how to obtain these groups using, say, projective resolutions. It turns out that the category of partial G-modules is not abelian since some sets of morphisms may be empty. Fortunately, our cohomology can be related to cohomology of inverse semigroups, as developed by H. Lausch in [23], via the inverse monoid S(G) introduced by R. Exel in [16] to serve partial actions and partial representations of G. From a partial action of G one comes to an action of S(G) and then to an "almost" Lausch's S(G)-module. The latter can be seen as a module in the sense of H. Lausch over an epimorphic image of S(G). Thus our category is made up of abelian "components" which are categories of Lausch's modules over epimorphic images of S(G) (see Remark 3.49). This way we are able to define free objects and free resolutions which lead to H n (G, A).
The article is organized as follows. After establishing the basic concept in Section 1, we show in Section 2 that the partial Schur multiplier pM (G) is a union of 2-cohomology groups with values in some, in general non-trivial, partial G-modules (Theorem 2.13). The passage from partial G-modules to S(G)-modules via unital actions of S(G) is done in Section 3 (Theorem 3.7). Our notion of a module A over an inverse semigroup S is more general than that introduced in [23] (Definition 3.8), and the relation with modules in the sense of [23] (strict S-modules) is given in Proposition 3.30. Here a crucial role is played by the semi-direct product A ⋊ S (Definition 3.22), which is the semigroup analogue of the crossed product introduced in [19]. If S = S(G) then A ⋊ S is isomorphic to the crossed product A * θ G (Remark 3.32) defined in [9]. In the same section we also define free S-modules and show their existence in Proposition 3.47 using Lausch's construction of (strict) free S-modules. Finally, in Section 4 we give a free resolution in order to obtain the cohomology groups H n (G, A) (Theorem 4.11).
There is a very natural point to be clarified, namely, the relation of low dimensional partial group cohomology to extensions. The constraints of the present article do not allow us to treat it here, and it is being considered in a separate paper under elaboration.

The notions
Let G be a group and A a semigroup. Recall from [9] that a partial action θ of G on A is a collection of semigroup isomorphisms θ x : We consider the case when A is a commutative monoid and each ideal A x is unital, i. e. A x is generated by an idempotent 1 x = 1 A x , which is central in A. In this case we shall say that θ is a unital partial action. Then A x ∩ A y = A x A y , so the properties (ii) and (iii) from the above definition can be replaced by Note also that (ii') implies a more general equality which easily follows by observing that Definition 1.1. A commutative monoid A with a unital partial action θ of G on A will be called a (unital) partial G-module.
Here the inverse elements are taken in the corresponding ideals. If n = 0 and a is an invertible element of A, we set (δ 0 a)(x) = θ x (1 x −1 a)a −1 .
for any f ∈ C n (G, A).
Proof. Let f ∈ C n (G, A). We check first that δ n f ∈ C n+1 (G, A). Indeed, for ..,x n+1 ) . Then being multiplied by 1 x −1 Definition 1.6. The map δ n is called a coboundary homomorphism. As in the classical case we define the abelian groups Z n (G, A) = ker δ n , B n (G, A) = im δ n−1 and H n (G, A) = ker δ n / im δ n−1 of partial n-cocycles, n-coboundaries and n-cohomologies of G with values in A, n ≥ 1 (H 0 (G, A) = Z 0 (G, A) = ker δ 0 ).
For example, (here and below U (A) denotes the group of invertible elements of A). Notice that H 0 (G, A) is the subgroup of θ-invariants of U (A) (see [8, p. 79]). Furthermore, For n = 2 we have with f ∈ C 2 (G, A), and Observe that if one takes a unital twisted partial action (see [6,Def. 2.1]) of G on a commutative ring A then it is readily seen that the twisting is a 2-cocycle with values in the partial G-module A, and the concept of equivalent unital twisted partial actions from [7, Def. 6.1] is exactly the notion of cohomologous 2-cocycles from Definition 1.6. Proposition 1.7. The map which sends a partial G-module A to the sequence is a functor from pMod(G) to the category of complexes of abelian groups.
The remaining properties of a functor are straightforward.
Composing the above functor with the homology functor we have the next.
Corollary 1.8. For any n ≥ 0 the map A → H n (G, A) determines a functor from pMod(G) to the category Ab of abelian groups.
2 H 2 (G, A) and the partial Schur multiplier In this section we assume that M is a K-cancellative monoid [9], where K is a field. Recall from [9, Theorem 6] that each partial projective representation Γ : G → M , Γ(1) = 1, with (partial) factor set σ : One extends σ to a fully defined map σ : G × G → K by setting σ(x, y) = 0 for each (x, y) ∈ dom σ. Indeed, by (21) from [9] σ(x, x −1 ) = σ(x −1 , x), and in the global case e x = 1 M , so A Γ x = A Γ = K · 1 M and θ Γ x = id for all x ∈ G. Any f ∈ C n (G, A Γ ) is identified with a map G n → K * , i. e. with a classical n-cochain, and (2) becomes the classical coboundary homomorphism under the trivial G-action on K * . Definition 2.2. By a K-linear partial G-module we shall mean (A, θ) ∈ pMod(G) such that A is K-cancellative and each θ x : Evidently, for any partial projective representation Γ of G in a K-cancellative monoid M, θ Γ gives a structure of a K-linear partial G-module on A Γ .
Clearly, for any partial projective representation Γ : G → M the K-linear partial G-module (A Γ , θ Γ ) is adjusted. We recall from [9] the next.
Definition 2.4. Let A be a K-linear partial G-module. A K-valued twisting related to A is a function σ : G × G → K satisfying the following conditions: (iii) A x A xy A xyz = 0 ⇒ σ(x, y)σ(xy, z) = σ(y, z)σ(x, yz) with x, y, z ∈ G.
By (iv) of Theorem 3 from [10] for any adjusted A ∈ pMod(G) and any twisting σ related to it there is a partial projective representation Γ : G → M with factor set σ such that A is isomorphic to A Γ . Thus, among the partial G-modules, the adjusted ones are precisely those, which come from partial projective representations of G.
Lemma 2.5. Let A be an adjusted (K-linear) partial G-module and A x 1 . . . A xn = 0. Then for any invertible element a of A x 1 . . . A xn there exists a unique α ∈ K * such that a = α1 x 1 . . . 1 xn .
Proof. The uniqueness of α obviously follows from the K-cancellative property of A. Suppose that a = α1 x 1 . . .
Since the left-hand side of this equality is stable under the multiplication by 1 y 1 . . . 1 ym , we conclude that Note that in fact a −1 = α −1 1 x 1 . . . 1 xn , so the group of invertible elements of A x 1 . . . A xn = 0 is isomorphic to K * . Furthermore, if A x 1 . . . A xn = 0 then A x 1 . . . A xn can be considered as the identity monoid, so it has the unique invertible element 1 x 1 . . . 1 xn = 0. Given The subgroup of Z 2 (G, A) formed by the normalized partial 2-cocycles will be denoted by N Z 2 (G, A).
For one can take f (x, y) = f (x, y)f (1, 1) −1 and g(x) = f (1, 1)1 x ∈ U (A x ) for all x, y ∈ G. Recall from [9] that the factor sets of all partial projective representations of G form a commutative inverse monoid pm(G) under pointwise multiplication. The quotient semigroup for some function η : G → K * , is called the partial Schur multiplier of G.
Proposition 2.7. Let (A, θ) be an adjusted K-linear partial G-module. Then N Z 2 (G, A) is isomorphic to the subgroup of pm(G) consisting of the factor sets of all partial projective repre- In particular, f (x, y) = σ f (x, y)1 x 1 xy . By Lemma 2.5 our definition of σ f is correct. Note that σ f is a K-valued twisting related to A. Indeed, (i) of Definition 2.4 immediately follows from (5), whereas (ii) is a consequence of (4). Next, suppose that A x A xy A xyz = 0, i. e. 1 x 1 xy 1 xyz = 0.
Using the 2-cocycle equality (δ 2 f )(x, y, z) = 1 x 1 xy 1 xyz , the fact that θ x : A x −1 → A x is a morphism of K-monoids and the K-cancellative property, we obtain so (iii) of Definition 2.4 is also true.
Having the adjusted K-linear partial G-module A and the twisting σ f related to it, by Theorem 8 from [9] one can construct a partial projective representation ∆ : G → M whose factor set is σ f . Using ∆ we obtain the K-linear partial G-module A ∆ . Since A is generated by α1 x as a semigroup (α ∈ K, x ∈ G), Corollary 11 from [9] (or (iv) of Theorem 3 from [10]) implies that A ∆ is isomorphic to A.
Conversely, let ∆ : G → M be a partial projective representation with factor set τ , such that A ∆ is isomorphic to A. Define g τ : G × G → A by Note that by Theorem 6 from [9] τ is a twisting related to A ∆ , so τ (x, y) = 0 ⇔ A ∆ x A ∆ xy = 0 which is equivalent to A x A xy = 0, because A ∆ is isomorphic to A. Therefore, g τ (x, y) is an invertible element of A x A xy and thus g τ ∈ C 2 (G, A). We show next that If (7) follows from the 2-cocycle identity for τ (see (iii) of Definition 2.4). If A x A xy A xyz = 0, then 1 x 1 xy 1 xyz = 0, so both sides of (7) are zero. Thus, g τ ∈ Z 2 (G, A). Moreover, g τ ∈ N Z 2 (G, A) because τ (1, 1) = 1 by (ii) of Definition 2.4.
In order to check that the map it is enough to notice that by (5) and (6) one has that g σ f = f. Clearly , by means of which N Z 2 (G, A) can be identified with a subgroup of pm(G), as desired.
Remark 2.8. Given two normalized 2-cocycles f and g it is readily seen that f is cohomologous to g if and only if σ f ∼ σ g .
This follows from Remarks 2.6 and 2.8.
Given an adjusted K-linear partial G-module A, denote by R(G, A) the subgroup of pM (G) formed by the equivalence classes of K-valued twistings related to A.
Indeed, this immediately follows from the fact that if τ is a twisting related to θ and τ ′ is a factor set equivalent to τ , then τ ′ is also a twisting related to θ. The latter is a direct consequence of Definition 2.4.

For by (i) of Definition 2.4 all twistings related to
Definition 2.12. Let A and A ′ be K-linear partial G-modules. We shall say that A and A ′ are twisting-equivalent if the sets of K-valued twistings related to A and A ′ coincide.
The relation between the partial Schur multiplier and partial cohomology is summarized in the next.
where A runs over the representatives of twisting-equivalence classes of adjusted Klinear partial G-modules such that Proof. Let σ be a factor set of a partial projective representation Γ : G → M, whose equivalence class belongs to pM D (G). Then σ is a K-valued twisting related to A Γ whose equivalence class , thanks to Corollary 2.9 and Remark 2.10.

Partial modules and modules over Exel's monoid
Recall that a semigroup S is called inverse if for any s ∈ S there is a unique s −1 ∈ S (called the inverse of s) such that ss −1 s = s and s −1 ss −1 = s −1 . A natural example of an inverse semigroup is the monoid I(X) of all partial bijections (i. e. bijections between subsets) of X. The product ϕψ in I(X) is the composition ϕ • ψ defined for all x ∈ X, such that ϕ(ψ(x)) makes sense. More precisely, ϕψ is the bijection Note that the inverse of ϕ is simply ϕ −1 : ran ϕ → dom ϕ.
Following Exel [16] by an action of an inverse semigroup S on X we mean a homomorphism S → I(X) (in semigroup theory such homomorphisms are called representations of S, see [2, 7.3]). If S is a monoid, then the term "action" will always mean "unital action", i. e. a homomorphism of monoids S → I(X). It was proved in [16] that there is a one-to-one correspondence between the partial actions of a group G on a set X and the (unital) actions of S(G) on X, where S(G) is the semigroup generated by [x] (x ∈ G) subject to the following relations: , all x i are different, nonidentity and not equal to y. This expression is unique up to a permutation of the idempotents ε x i (note that all ε x commute). It follows that each idempotent of S(G) has the form ε x 1 . . . ε xn for some (uniquely defined) set x 1 , . . . , x n ∈ G. It is also known that S(G) is isomorphic to the Birget-Rhodes expansion of G [22].
Notice that given a semigroup A, the composition of two partial isomorphisms of A (i. e. isomorphisms between arbitrary ideals of A) is not a partial isomorphism in general, since the domain of the composition is not necessarily an ideal. However, the set I ui (A) of all isomorphisms between unital ideals of A (i. e. ideals generated by central idempotents of A) forms an inverse monoid. By [16,Proposition 4.1] a partial action of G on a set X can be seen as a partial representation in I(X). This immediately implies that unital partial actions of G on a semigroup A can be identified with the partial representations of G in I ui (A). Then [16, Proposition 2.2] yields the next.
There is a one-to-one correspondence between the partial G-modules and the (unital) actions of S(G) on commutative monoids.
More precisely, let (A, θ) ∈ pMod(G). Then the corresponding action τ of S(G) on A is given as follows. For arbitrary s = ε x 1 . . . ε xn [y] ∈ S(G), τ s is the bijection This correspondence is in fact an isomorphism of categories as explained below.
, τ and τ ′ be the corresponding actions of S(G) on the semigroups A and A ′ . Denote by 1 s and 1 ′ s the identities of ran τ s and ran τ ′ s , respectively (s ∈ S(G)). Then a homomorphism of semigroups ϕ : A → A ′ is a morphism of partial G-modules if and only if for any s ∈ S(G) we have It is reasonable to give the next.

Definition 3.4. Let τ and τ ′ be actions of an inverse semigroup S on semigroups
such that for all s ∈ S the conditions (i) and (ii) from Remark 3.3 are true (as above ran τ s = 1 s A and ran τ ′ Thus, actions of an inverse semigroup S on commutative monoids form a category which will be denoted by A(S).
Indeed, Proposition 3.2 and Remark 3.3 give a functor pMod(G) → A(S(G)) which is bijective on objects and identity on morphisms.
We are going to give a characterization of actions of an inverse semigroup S on a semigroup A in terms of endomorphisms of A. For any semigroup A denote by E(A) the set of idempotents of A. The center of A will be denoted by C(A). Note that C(A) is a subsemigroup of A, while E(A) might not be a subsemigroup. But it is necessarily a subsemigroup when the idempotents commute. In particular, E(A) is a subsemigroup of an inverse A, and E(C(A)) is a subsemigroup of A for arbitrary semigroup A.
Proof. Equality (i) follows by observing that τ e is a bijection of dom τ e = ran τ e , which coincides with its square. For (ii) note that 1 s −1 1 t is the identity of Theorem 3.7. Let S be an inverse semigroup and A a semigroup. There is a one-to-one correspondence between the actions of S on A and the pairs (λ, α), where λ is a homomorphism S → End A, s → λ s , and α is a homomorphism E(S) → E(C(A)) such that (i) λ e (a) = α(e)a for all e ∈ E(S) and a ∈ A; (ii) λ s (α(e)) = α(ses −1 ) for all s ∈ S and e ∈ E(S).
Proof. Let τ : S → I ui (A), s → τ s , be an action of S on A. Obviously, the identity 1 s of ran τ s belongs to E(C(A)), so we may define α τ : According to (i) and (ii) of Lemma 3.6 we have 1 ef = τ e (1 e 1 f ) = 1 e 1 f , hence α τ is a homomorphism. Furthermore, for any s ∈ S and a ∈ A set The right-hand side of (9) makes sense, because 1 s −1 is the identity of ran τ s −1 = dom τ s and dom τ s is an ideal. Since 1 s −1 is a central idempotent and τ s is a homomorphism, we see that Here we again applied (ii) of Lemma 3.6 and the fact that ran τ st ⊆ ran τ s . Now we verify that the pair (λ τ , α τ ) satisfies (i) and (ii). The property (i) obviously follows from (i) of Lemma 3.6. For (ii) observe that using (i) and (ii) of Lemma 3.6. Conversely, given a pair (λ, α) with (i) and (ii), define 1 s ∈ E(C(A)) by Note that in view of (ii) Then It remains to show that τ → (λ τ , α τ ) and (λ, α) → τ (λ,α) are mutually inverse. Consider τ (λ τ ,α τ ) . For any s ∈ S the identity of dom τ which is the identity of dom τ ss −1 . But using (ii) of Lemma 3.6 we see that , which is α(ee −1 ) = α(e). Furthermore, using (i) and (ii) we get Definition 3.8. Given an inverse semigroup S, an S-module is a triple (A, λ, α) (often written shortly as A), where A is a commutative semigroup, λ : S → End A and α : E(S) → E(A) are homomorphisms satisfying (i) and (ii) of Theorem 3.7. If α is an isomorphism, then we say that A is strict.
Notice that in [23] the term S-module means our strict inverse S-module. Proposition 3.9. Let τ and τ ′ be actions of an inverse semigroup S on semigroups A and Proof. By (10) condition (i) is equivalent to ϕ(1 s ) = 1 ′ s , while (ii) is the same as ϕ(τ s (1 s −1 a)) = τ ′ s (1 ′ s −1 ϕ(a)) for all a ∈ A by (9). In view of (i) the latter can be replaced by ϕ(τ s (1 s −1 a)) = τ ′ s (ϕ(1 s −1 a)) for all a ∈ A, i. e. ϕ • τ s = τ ′ s • ϕ on 1 s −1 A = dom τ s . Given an inverse semigroup S, denote by Mod(S) the category of S-modules and their morphisms. Then Theorem 3.7 and Proposition 3.9 yield the next.
Proposition 3.11. For any inverse semigroup S the categories A(S) and Mod(S) are isomorphic.
In [23] H. Lausch considered (strict) S-module structures on semilattices of abelian groups A. The following remark shows that in the case of partial G-modules it is also reasonable to restrict our consideration to an inverse A.
Sinceθ is the restriction of θ, the coboundary homomorphisms of these two cochain complexes also coincide and thus, their cohomology groups are equal. Notice where x 1 , x 2 , . . . , x n run over G. Observe that if e, f ∈ A are idempotents with e = f then U (eA) ∩ U (f A) = ∅. Hence the above union can be made disjoint by removing repetitions, and then it will give a representation ofÃ as a semilattice of abelian groups. In particular, any idempotent ofÃ, being an element of some group component, is identity of this component, i. e. has a form 1 x 1 . . . 1 xn for x 1 , . . . , x n ∈ G.  Thus, passing from an inverse partial G-module to an S(G)-module, we see that the only reason why we might not obtain an S(G)-module in the sense of Lausch is that the map α θ : E(S(G)) → E(A) is not injective in general. For example, considering a classical (i. e. usual) G-module A (which is readily seen to be a strict inverse G-module) we have |E(A)| = 1, while |E(S(G))| = 2 |G|−1 . Nevertheless, an inverse partial G-module A may be seen as a strict inverse module over some epimorphic image of S(G). We shall see that this epimorphic image can be chosen to be E(A) * θ G as explained in what follows (see Corollary 3.33). In particular, in the above example this construction leads to the initial strict inverse G-module A.
Given an inverse semigroup S and an S-module (A, λ, α) denote by L(A, S) the set of aδ s , where s ∈ S, a ∈ α(ss −1 )A and δ s is a symbol. Proof. It is directly verified that L(A, S) can be seen as the λ-semidirect product defined in [24, p. 148], so that the above multiplication is well-defined and associative.  Inspired by [19], instead of L(A, S) we shall deal with a quotient of L(A, S) which takes into account the natural partial order on S. This idea has important advantages, one of them being the fact that in the C * -algebraic context the analogous construction fits the definition of the crossed product by an inverse semigroup given by N. Sieben in [28] (see [19,Theorem 5.6]).
It is well-known that any inverse semigroup S is partially ordered in a following natural way: s ≤ t if and only if there is e ∈ E(S) such that s = et (and in this case one can take e = ss −1 ). This is the same as to say that s = tf for some f ∈ E(S). Notice that for e, f ∈ E(S) we have e ≤ f ⇔ ef = e. Denote by σ the congruence generated by ≤. Since ≤ is compatible with multiplication in S ([2, Lemma 7.2]), σ is in fact the minimal equivalence relation containing the partial order ≤.
Lemma 3.21. For any inverse semigroup S and for all s, t ∈ S we have (s, t) ∈ σ if and only if there is u ∈ S such that u ≤ s, t.
Proof. We shall prove the "only if" part (the "if" part is obvious). As it was mentioned before the lemma, σ is the symmetric transitive closure of ≤. Thus, (s, t) ∈ σ means that there are s = s 1 , . . . , s n = t ∈ S such that for all i = 1, . . . , n−1 we have s i ≤ s i+1 or s i+1 ≤ s i . Therefore, s i = e i s i+1 or s i+1 = e i s i for some e i ∈ E(S) (i = 1, . . . , n − 1). In any case e i s i = e i s i+1 for all i = 1, . . .
The quotient semigroup L(A, S)/ρ will be denoted by A ⋊ S and called the semidirect product of A and S.
Example 3.26. Take the cyclic group Z 2 = a | a 2 = 1 and set S = Z 2 ∪ {0}. Then S is a commutative inverse monoid, whose group components are Z 2 and {0}. We can define a structure of a (strict) S-module on S as follows: λ s (t) = ss −1 t and α(e) = e for all s, t ∈ S and e ∈ E(S). Then (a, 1) ∈ σ, but (aδ a , aδ 1 ) ∈ ρ.
Recall that an inverse semigroup S is called E-unitary if es ∈ E(S) implies s ∈ E(S) for all s ∈ S and e ∈ E(S). Equivalently, σ coincides with the compatibility relation ∼ on S (see [24, pp. 24, 66]). For instance, S(G) is E-unitary for any group G, while the semigroup S from Example 3.26 is not E-unitary, because 0a = 0 ∈ E(S) and a / ∈ E(S). We need only explain the "if" part. Suppose (s, t) ∈ σ and a = b ∈ α(ss −1 tt −1 )A. Since S is E-unitary, s ∼ t and hence t −1 s ∈ E(S). Set v = (tt −1 )s = t(t −1 s) ≤ s, t. Note that vv −1 = ss −1 tt −1 and thus a = b ∈ α(vv −1 )A.
Definition 3.31. Given (A, θ) ∈ pMod(G), define the crossed product of A and G to be the set A * θ G of aδ x , where x ∈ G, a ∈ 1 x A and δ x is a symbol. It is a semigroup under multiplication The associativity of multiplication follows from the fact that all domains of θ are unital ideals of A (see [4,Corollary 3.2]). On one hand, this follows from Remark 3.23 and [19,Theorem 3.7] and, on the other hand, taking into account Remark 3.27, an easy direct verification shows that the map given by Indeed, there is π : S(G) → A ⋊ S(G) as in Proposition 3.30, namely which can be identified with 1 x 1 . . . 1 xn 1 y δ y ∈ E(A) * θ G by Remark 3.32. Obviously, each element of E(A) * θ G has this form.
Definition 3.36. Given an inverse semigroup S and an epimorphism π : S → S ′ we define the concept of a π-strict S-module as a pair (A, π) where A = (A, λ, α) is a strict module over S ′ = π(S).
Note that id-strict S-modules are precisely strict S-modules in the sense of Definition 3.8. Sometimes we shall omit π in (A, π) and call A a π-strict S-module. Moreover, if π is not specified, we say that A is an epi-strict S-module.
Epi-strict S-modules and their morphisms form a category (under the coordinatewise composition of morphisms) which will be denoted by ESMod(S). Definition 3.38. An epi-strict S-module isomorphic to (A,λ,α) given in Proposition 3.30 will be called standard.
Corollary 3.40. Under the conditions of Proposition 3.39 if (A, π ′ ) is standard, then there is an epimorphism of semigroups η : S ′ → S/σ such that η • π ′ is the natural map σ ♮ : S → S/σ. In particular, if A is the standard S(G)-module from Corollary 3.33, then η can be viewed as an epimorphism S ′ → G by means of η(1 x 1 . . . 1 xn 1 y δ y ) = y.
From now on we shall work only with inverse modules.
Definition 3.41. An epi-strict S-module (A, π) will be called inverse, if A is an inverse semigroup. The subcategory of inverse epi-strict S-modules is denoted by InvESMod(S).
Following Lausch [23] we are going to define a concept of a free object in InvESMod(S). Let us first agree on what will be a base of a free inverse epi-strict S-module.
In the classical case [23] a base of a free S-module is a so-called E(S)-set. It is a disjoint union of sets indexed by E(S). A morphism of E(S)-sets is a map which agrees with the partitions of these sets (i. e. it sends the e-component of one set to the subset of the e-component of another set for all e ∈ E(S)). Such sets appear in our situation and they will be called strict E(S)-sets. However, we need a more general concept of an E(S)-set.
Definition 3.42. Let S and S ′ be inverse semigroups and π : S → S ′ an epimorphism. A π-strict E(S)-set is a pair (T, π) where T is a strict E(S ′ )-set.
In particular, id-strict E(S)-sets can be identified with the "ordinary" (strict) E(S)-sets. As above, π will be often omitted and a π-strict E(S)-set, whose π is not specified, will be called an epi-strict E(S)-set. We shall also use the standard notation T e for the e-component of T (e ∈ E(S ′ )).
Remark 3.44. Each inverse π-strict S-module is a π-strict E(S)-set and each morphism of inverse epi-strict S-modules is a morphism of epi-strict E(S)-sets.
We recall Lausch's construction of a free (strict) S-module. Given an E(S)-set T , define F = F (T ) to be the disjoint union of components F e , where each F e is the free abelian group (written additively) over The sum of elements (s, t) ∈ F e and (s ′ , t ′ ) ∈ F e ′ of different components is the formal sum (e ′ s, t) + (es ′ , t ′ ) in F ee ′ . The structure of an S-module is defined by λ F s ′ (s, t) = (s ′ s, t) and α F (e) = 0 e , where 0 e is the zero of F e . The embedding ι : T → F is given by Remark 3.46. Notice that each (s, t), where t ∈ T f , can be written as λ  Proof. Since (T, π : S → S ′ ) is an E(S ′ )-set in the sense of Lausch, then by [23,Proposition 3.1] there is the free Lausch's S ′ -module (F (T ), λ F (T ) , α F (T ) ) which therefore can be considered as a π-strict S-module. We shall show that F (T ) is free over T in InvESMod(S).
Definition 3.48. The full subcategory of InvESMod(S) formed by the objects (A, π) with the same π : S → S ′ will be called the π-component of InvESMod(S).
Remark 3.49. Note that any morphism between elements of a fixed π-component of InvESMod(S), where π : S → S ′ , has form (ϕ, id S ′ ), so this component is isomorphic to the category of strict inverse S ′ -modules. In particular, it is an abelian category (see [23]).
Remark 3.50. The free inverse epi-strict module F (T ) over (T, π : S → S ′ ) belongs to the π-component of InvESMod(S). Moreover, it is projective in this component.
Indeed, by Corollary 3.2 from [23] F (T ) is projective as a strict inverse S ′ -module.
Corollary 3.51. There are non-standard epi-strict modules.
For an example consider an inverse semigroup S with σ = S 2 and the epimorphism π : S → 0 to the zero semigroup. Clearly S 2 = ker π ⊆ σ. Then for any π-strict set T the free inverse π-strict S-module F (T ) is not standard. According to Propositions 3.2, 3.30, Theorem 3.7 and Remark 3.17, each inverse partial G-module can be seen as an inverse epi-strict S(G)-module.
Definition 3.52. Let A be an inverse partial G-module and T an epi-strict E(S(G))-set. The module A is said to be free over T if the corresponding inverse epi-strict S(G)-module is free over T in InvESMod(S(G)).
We shall now examine the construction of a free inverse partial G-module in more detail. Write e x = Γ(x)Γ(x −1 ) = π(ε x ). Then the elements e x are pairwise commuting idempotents and one has that Γ(x)e y = e xy Γ(x) for all x, y ∈ G (see 2 [9, Section 7]). Let E be the submonoid of Γ(G) generated by the idempotents e x . We know from [9,Theorem 6] is the partial action of G on E which corresponds to Γ, with E x = e x E and θ x (e) = Γ(x)eΓ(x −1 ), e ∈ E x −1 . It follows that Γ(G) is a (not necessarily disjoint) union Moreover, since we have an epimorphism S(G) → Γ(G), we obtain that E = E(Γ(G)), the set of all idempotents of Γ(G). Given an arbitrary element s = eΓ(x) (e ∈ E x ) of Γ(G), write e = e x e y 1 . . . e y k (y 1 , . . . , y k ∈ G). Then ss −1 = e and Given a strict E-set T = e∈E T e , the free inverse partial G-module F (T ) over T can be specified as follows. Taking a fixed e ∈ E set The condition e ∈ E x means that e can be written as a product e = e x e y 1 . . . e y k for some y 1 , . . . , y k ∈ G. Observe that Γ e consists of all s ∈ Γ(G) with ss −1 = e. Next for an arbitrary s = eΓ(x) ∈ Γ e let E(s) be the set of all f ∈ E such that f ≥ s −1 s. In particular, E(s) contains all f ∈ E which can be written in the form f = (e x −1 ) ν j∈J e x −1 y j , where ν ∈ {0, 1} and J is a (possibly empty) subset of {1, . . . , k} (if ν = 0 and J = ∅ we assume f = 1 M ). Write T (s) = f ∈E(s) T f . Then the e-component F (T ) e of F (T ) is the free abelian group with free basis B e = {(s, t) | s ∈ Γ e , t ∈ T (s)}. Denote by 1 (e) the identity element of F (T ) e and write 1 x = 1 (ex) , x ∈ G. The elements from e∈E B e will be called the canonical generators of F (T ). The product of canonical generators (s, t) ∈ F (T ) e and (s ′ , t ′ ) ∈ F (T ) e ′ from different components is (e ′ s, t)(es ′ , t ′ ) as an element of F (T ) ee ′ .
The corresponding partial action θ F (T ) of G on F (T ) consists of the isomorphisms θ x ∈ G and θ F (T ) x (s, t) = (Γ(x)s, t) for any canonical generator (s, t) which is contained in D x −1 . If (s, t) ∈ D x −1 , then s = e x −1 eΓ(y) for some e ∈ E and y ∈ G such that e x −1 ee y = e x −1 e. Then

xy), t).
Writing e = e y e z 1 . . . e z k , where z 1 , . . . , z k ∈ G, we have θ x (e x −1 e) = e x e xy e xz 1 . . . e xz k . An arbitrary element a ∈ D x −1 belongs to some component F (T ) e and can be written as a combination of canonical generators of F (T ) e : a = (s 1 , t 1 ) n 1 . . . (s r , t r ) nr , n i ∈ Z, and one has x (s r , t r )) nr .

Partial cohomology of groups and cohomology of inverse semigroups
Definition 4.1. The cohomology groups of an inverse semigroup S with values in an inverse epi-strict S-module (A, π : S → S ′ ) can be defined in the following way: H n (S, A) = H n S ′ (A), where on the right-hand side A is meant to be a strict S ′ -module (see [23]).
Under the conditions of the above definition denote by Hom π (−, A) the restriction of Hom(−, A) to the π-component of InvESMod(S). Then it is an additive functor from an abelian category to Ab, and according to [23] the above cohomology is its derived functor, applied to the inverse π-strict S-module Z S ′ with (Z S ′ ) e = {n e | n ∈ Z}, n e + m f = (n + m) ef , λ Z S ′ s (n e ) = n ses −1 , α Z S ′ (e) = 0 e . Hence, H n (S, A) can be calculated by taking an appropriate projective resolution of Z S ′ in the π-component of InvESMod(S). It turns out that for S = S(G) and (A, π : S → S ′ ), which comes from an inverse (A, θ) ∈ pMod(G), we can construct a free resolution of Z S ′ , such that the corresponding cohomology groups with values in (A, π) are precisely H n (G, A). Definition 4.2. Let Γ : G → S be a partial homomorphism of a group G in an inverse monoid S. For any positive integer n denote by V n the strict E(S)-set, whose e-component is the (possibly empty) set of ordered n-tuples (x 1 , . . . , x n ) ∈ G n , such that e (x 1 ,...,xn) = e, where e (x 1 ,...,xn) is the idempotent For n = 0 define (V n ) e to be the one-element set {( )} if e = 1 S , and ∅ otherwise. The free S-module over V n will be denoted by R n .  Let (A, θ) be an inverse partial G-module and (A, λ, α) the corresponding (standard) inverse π-strict S(G)-module, π : S(G) → S. Set Γ = π • [ ] : G → S and consider R n = R n (Γ) as an element of the π-component of InvESMod(S(G)). Then the abelian group Hom π (R n , A) is isomorphic to C n (G, A).
Proof. Since R n is the free S-module over V n = V n (Γ), each morphism from R n to A is fully determined by its values on V n . So, Hom π (R n , A) can be identified with the set of morphisms of π-strict E(S(G))-sets from V n to A. A pair (ϕ, ψ) is such a morphism if and only if ψ = id S and ϕ(v) ∈ A e for all v ∈ (V n ) e . If n = 0, then ϕ is identified with the image of ( ) ∈ (V n ) 1 S , which should belong to A 1 S , i. e. be invertible with respect to Thus, Hom π (R 0 , A) ∼ = U (A) = C 0 (G, A). For n > 0 the function ϕ can be viewed as a map G n → A, such that ϕ(x 1 , . . . , x n ) ∈ A e , where e = e (x 1 ,...,xn) . By definition A e consists of a ∈ A, which are invertible with respect to So, A e = U (A (x 1 ,...,xn) ) and hence Hom π (R n , A) ∼ = C n (G, A). we have δ n f = f • ∂ n+1 , where f and δ n f are considered as homomorphisms of π-strict S(G)modules R n → A and R n+1 → A respectively.
Indeed, by construction the module (A, π) is standard. Since S(G) is E-unitary, by Proposition 3.39 any module from the π-component of the category InvESMod(S(G)) is standard, in particular all R n have this property. Thus without loss of generality we may assume that (R n , π) is obtained from some inverse partial G-module (R n , θ n ) for all n ≥ 0. Now taking A ′ = R n−2 and f = id R n−2 ∈ Hom π (R n−2 , A ′ ) by Lemma 4.5 and Proposition 1.5 we see that In view of Lemma 4.3 the cochain e n can be seen as the morphism from R n to R n−2 which maps each element of (V n ) e to the zero of (R n−2 ) e , e ∈ E(S). So, e n = 0 ∈ Hom π (R n , R n−2 ). Definition 4.7. Under the conditions of Lemma 4.3 let η : S → G be as in Corollary 3.40 and for all n ≥ 0 define the morphism σ n : R n → R n+1 of π-strict E(S(G))-sets as follows. On each e-component of R n it is the homomorphism of abelian groups (R n ) e → (R n+1 ) e given by σ n (s, (x 1 , . . . , x n )) = (ss −1 , (η(s), x 1 , . . . , x n )).
Furthermore, note that se = s implies sf = s for all f ≥ e, because sf = sef = se = s.
where 0 is the zero of the π-strict component of InvESMod(S(G)), is a free resolution of Z S whose cohomology groups with values in A are isomorphic to H n (G, A).
Proof. In view of Lemmas 4.3, 4.5 and Corollaries 4.6, 4.10 it only remains to prove the inclusion im ∂ 1 ⊆ ker ǫ (the exactness in Z S is obvious, because n e = ǫ(n(e, ( ))) for all n ∈ Z and e ∈ E(S)). This can be done by direct computation of ǫ • ∂ 1 . For any x ∈ G the image ǫ(∂ 1 (x)) is