Twists, crossed products and inverse semigroup cohomology
aa r X i v : . [ m a t h . R A ] J a n TWISTS, CROSSED PRODUCTS AND INVERSESEMIGROUP COHOMOLOGY
BENJAMIN STEINBERG
Abstract.
Twisted ´etale groupoid algebras have been studied recentlyin the algebraic setting by several authors in connection with an ab-stract theory of Cartan pairs of rings. In this paper, we show thatextensions of ample groupoids correspond in a precise manner to exten-sions of Boolean inverse semigroups. In particular, discrete twists overample groupoids correspond to certain abelian extensions of Booleaninverse semigroups and we show that they are classified by Lausch’s sec-ond cohomology group of an inverse semigroup. The cohomology groupstructure corresponds to the Baer sum operation on twists.We also define a novel notion of inverse semigroup crossed prod-uct, generalizing skew inverse semigroup rings, and prove that twistedSteinberg algebras of Hausdorff ample groupoids are instances of inversesemigroup crossed products. The cocycle defining the crossed productis the same cocycle that classifies the twist in Lausch cohomology. Introduction
Groupoid C ∗ -algebras have played an important role in the subject sinceRenault’s seminal monograph [32]. An important development was a seriesof works, principally by Renault [32, 33] and Kumjian [25], in which it wasshown that Cartan pairs of C ∗ -algebras correspond to twisted groupoid C ∗ -algebras for a certain class of ´etale groupoids. This was used to good effectby Matui and Matsumoto in the study of symbolic dynamics via Cuntz-Krieger algebras [29].An ´etale groupoid whose unit space is Hausdorff and has a basis of com-pact open sets is said to be ample [31]. The author introduced in [36]algebras over any base commutative ring associated to ample groupoids,nowadays called “Steinberg algebras”, which are ring theoretic analoguesof groupoid C ∗ -algebras. They include group algebras, inverse semigroupalgebras and Leavitt path algebras [1, 35]. In recent years they have beenthe subject of investigation by a number of authors. See for example [3, 7,10, 13–20, 30, 35–42]. Date : January 23, 2021.2010
Mathematics Subject Classification.
Key words and phrases. ample groupoids, twists, inverse semigroup cohomology, inversesemigroup crossed products.The author was supported by a PSC CUNY grant.
There were, in particular, a number of papers investigating when an amplegroupoid can be recovered from the pair of its Steinberg algebra and the“diagonal” subalgebra of functions on the unit space, cf. [2,11,13,39]. This issimilar in spirit to the work of Kumjian and Renault and may have motivatedthe authors of [5] to introduce twisted Steinberg algebras of ample groupoids.A theory of algebraic Cartan pairs was subsequently developed in [4] thatclosely parallels the Renault-Kumjian theory from the C ∗ -algebra setting.This paper arose from an attempt by the author to understand [4] fromthe point of view of inverse semigroup theory. It has been known for a longtime that there is a close connection between ample groupoids and inversesemigroups [22, 23, 28, 31, 32]. The strategy of the author to reconstruct anample groupoid from its Steinberg algebra and diagonal subalgebra in [39]was as follows. From a Steinberg algebra of an ample groupoid G and itsdiagonal subalgebra, one obtains an exact sequence of inverse semigroups K −→ S −→ S/K where S is the normalizer of the diagonal subalgebraand K is the semigroup of diagonal normalizers (which is a normal inversesubsemigroup of S ). It turns out that if the original groupoid is effective, forexample, then S/K is isomorphic to the inverse semigroup of compact openbisections of G . Since G can be reconstructed from its inverse semigroupof compact open bisections via Exel’s ultrafilter groupoid construction [22,23], this shows that G is determined by its Steinberg algebra and diagonalsubalgebra.Therefore, it is natural to attempt to show that discrete twists over amplegroupoids correspond to certain exact sequences of inverse semigroups. Thenone could reprove some of the results of [4] via purely inverse semigrouptheoretic means by showing that the exact sequence corresponding to thetwist is equivalent to the exact sequence coming from the Cartan pair. Thispaper sets the foundations for such an approach.The papers of Bice [8, 9] seem to be related to ours in that it wants tounderstand related algebras using inverse semigroup theory, but it does notapply the categorical and cohomological approach we do, nor the crossedproduct construction.We begin the paper by giving a covariant equivalence of categories be-tween ample groupoids and appropriately restricted functors and Booleaninverse semigroups and appropriately restricted homomorphisms. Moreover,we show these functors send exact sequences of ample groupoids to exactsequences of inverse semigroups, and vice versa. We then describe the ex-act sequences of inverse semigroups corresponding to twists over an amplegroupoid by a discrete abelian group A (written multiplicatively). We showthat A -twists over G correspond to extensions of the inverse semigroup Γ c ( G )of compact open bisections of G by the commutative inverse semigroup e A ofcompactly supported locally constant functions f : G (0) −→ A ∪ { } . Suchextensions are then classified by Lausch’s cohomology theory of inverse semi-groups [26]. In particular, we show that abelian group of equivalence classesof A -twists under Baer sum is isomorphic to Lausch’s second cohomology WISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY 3 group H (Γ c ( G ) , e A ). This is similar in spirit to how Lausch cohomologyarose in the work on Cartan pairs of von Neumann algebras in [21]. Wealso show that the question of whether the twist admits a continuous globalsection can be interpreted inverse semigroup theoretically using ideas of [21].In the final section of the paper, we introduce what we believe to be anew notion of inverse semigroup crossed product arising from an action ofan inverse semigroup on a ring and a 2-cocycle with respect to the action(a related, but different C ∗ -algebraic notion is in [12]). It generalizes theskew inverse semigroup ring construction for a large class of actions (thosein [24]), as well as group crossed products (with respect to an action). Afterdeveloping the basic properties of the crossed product (including a universalproperty), we show that twisted Steinberg algebras are inverse semigroupcrossed products with respect to the same Lausch 2-cocycle that classifiesthe twist under our bijection between twists and cohomology classes.2. Groupoids and inverse semigroups
Here we recall some basic notions about inverse semigroups and groupoids.2.1.
Inverse semigroups. An inverse semigroup is a semigroup S suchthat, for each s ∈ S , there is a unique element s ∗ ∈ S with ss ∗ s = s and s ∗ ss ∗ = s ∗ . In an inverse semigroup, the idempotents commute and henceform a subsemigroup E ( S ). Moreover, e ∗ = e for any idempotent e . Notethat ss ∗ , s ∗ s ∈ E ( S ) for any s ∈ S , as is ses ∗ for any e ∈ E ( S ). We alsoobserve that ( st ) ∗ = t ∗ s ∗ . An element s of a semigroup S is (von Neumann) regular if s = ss ′ s with s ′ ∈ S ; a semigroup S is (von Neumann) regular if all its elements are regular. It is well known that inverse semigroups areprecisely the regular semigroups with commuting idempotents.There is a natural partial order on S given by s ≤ t if s = te for someidempotent e ∈ E ( S ) or, equivalently, s = f t for some f ∈ E ( S ). One can,in fact, take e = s ∗ s and f = ss ∗ . The natural partial order is compati-ble with multiplication and preserved (not reversed) by the involution. If e, f ∈ E ( S ), then ef is the meet of e, f in the natural partial order andso E ( S ) is a meet semilattice. Any homomorphism ϕ : S −→ T of inversesemigroups automatically preserves the involution and order. We say that ϕ is idempotent separating if ϕ | E ( S ) is injective.A normal inverse subsemigroup of an inverse semigroup S is an inversesubsemigroup K with E ( K ) = E ( S ) and sKs ∗ ⊆ K for all s ∈ S . If ϕ : S −→ T is a homomorphism of inverse semigroups, then ker ϕ = ϕ − ( E ( T ))is a normal inverse subsemigroup and idempotent separating homomor-phisms are “determined” by their kernels. See [27] for basics on inversesemigroup theory.Many inverse semigroups have a zero element. If S is an inverse semigroupwith zero, then we say that s, t ∈ S are orthogonal if st ∗ = 0 = t ∗ s . A Boolean inverse semigroup is an inverse semigroup S with zero such that E ( S ) is a Boolean algebra, and which admits joins of orthogonal pairs of BENJAMIN STEINBERG elements. When we say that E ( S ) is a Boolean algebra, we mean that itadmits finite joins, and these joins distribute over meets, and it has relativecomplements (if f ≤ e , then there exists e \ f with f ( e \ f ) = 0 and e = f ∨ ( e \ f )). Note that if s, t are orthogonal, then ( s ∨ t )( s ∨ t ) ∗ = ss ∗ ∨ tt ∗ and( s ∨ t ) ∗ ( s ∨ t ) = s ∗ s ∨ t ∗ t . A homomorphism ϕ : S −→ T of Boolean inversesemigroups is called additive if it preserves joins of orthogonal idempotents,in which case it preserves all finite joins existing in S . There are a number ofother axiomatizations of Boolean inverse semigroups and we refer the readerto [43] for more details.If X is a topological space, then the set I X of all homeomorphisms betweenopen subsets of X is an inverse semigroup under composition of partialfunctions. An action of an inverse semigroup S on a space X by partialhomeomorphisms is just a homomorphism α : S −→ I X . The action is non-degenerate if X is the union of the domains of the elements of α ( S ).2.2. Groupoids. A groupoid G is a small category in which each arrow isinvertible. We take the approach here, popular in analysis, of viewing G asa set with a partially defined multiplication and a totally defined inversion.The objects of G are identified with the identity arrows (also called units)and the unit space is denoted G (0) . We use d : G −→ G (0) and r : G −→ G (0) for the domain and range maps, respectively.A topological groupoid is a groupoid endowed with a topology makingthe multiplication and inversion maps continuous. As d ( g ) = g − g and r ( g ) = gg − , the domain and range maps are also continuous where we givehere G (0) the subspace topology. When working with topological groupoids,we want our functors to be continuous.A topological groupoid G is ´etale if the domain map is a local homeomor-phism. This is equivalent to the range map being a local homeomorphismand also to the multiplication map being a local homeomorphism. The unitspace G (0) of an ´etale groupoid is open. See [34] for details. A (local) bi-section of an ´etale groupoid is an open subset U ⊆ G such that d | U and r | U are injective; some authors do not required bisections to be open andso we often say “open bisection” for emphasis. The origin of the term “bi-section” is that they correspond to subsets of G that are simultaneously theimage of a local section of d and of r . If U, V are bisections, then so are
U V = { gg ′ | g ∈ U, g ′ ∈ V } and U ∗ = { g − | g ∈ U } . The bisections forman inverse semigroup with respect to this product operation with U ∗ as theinverse of U . The idempotent bisections are the open subsets of G (0) andthe natural partial order is just containment.An ´etale groupoid G is called ample (following Paterson [31]) if G (0) isHausdorff and has a basis of compact open sets. In this case, the compactopen bisections form a basis for the topology of G . The collection Γ c ( G ) ofcompact open bisections of G is a Boolean inverse semigroup. Referencesfor ample groupoids include [22, 31, 36]. Ample groupoids arise in nature WISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY 5 as groupoids of germs of actions of inverse semigroups on Hausdorff spaceswith bases of compact open sets.3.
Extensions of groupoids and inverse semigroups
An equivalence of categories.
In this subsection we show that thebijection between isomorphism classes of ample groupoids and Boolean in-verse semigroups arising from the work of Exel [22, 23] and Lawson andLenz [28] can be turned into a covariant equivalence of categories if we re-strict our morphisms suitably. We will use this equivalence to show thatdiscrete twists over an ample groupoid correspond to certain idempotent-separating extensions of inverse semigroups that are classified by Lauschcohomology [26].Let us call a continuous functor ϕ : G −→ H of topological groupoids iso-unital if ϕ | G (0) : G (0) −→ H (0) is a homeomorphism. Note that if G and H are ´etale and ϕ is open, then this is equivalent to ϕ | G (0) : G (0) −→ H (0) beingbijective. Notice that if ϕ is iso-unital, then d ( ϕ ( g )) = d ( ϕ ( g ′ )) if and onlyif d ( g ) = d ( g ′ ), and dually r ( ϕ ( g )) = r ( ϕ ( g ′ )) if and only if r ( g ) = r ( g ′ ). Proposition 3.1.
Let ϕ : G −→ H be an iso-unital functor between ´etalegroupoids. Then the restriction of ϕ to any bisection is injective. Conse-quently, ϕ is open if and only if ϕ is a local homeomorphism.Proof. Let U ⊆ G be a bisection. If g , g ∈ U and ϕ ( g ) = ϕ ( g ), then since ϕ is iso-unital, we must have d ( g ) = d ( g ) and so g = g as U is a bisection.Thus ϕ | U is injective. Clearly, if ϕ is a local homeomorphism, it is open.Conversely, if ϕ is open and g ∈ G , then since G is ´etale, we can find anopen bisection U with g ∈ U . Then ϕ | U : U −→ ϕ ( U ) is injective and openand hence a homeomorphism. Therefore, ϕ is a local homeomorphism. (cid:3) The corresponding notion for inverse semigroups is the following. A ho-momorphism ϕ : S −→ T of inverse semigroups is idempotent bijective if ϕ | E ( S ) : E ( S ) −→ E ( T ) is a bijection. Note that if S and T are Booleaninverse semigroups, then an idempotent bijective homomorphism is clearlyadditive.The following lemma combines two well-known facts, but we include aproof for completeness. Lemma 3.2.
Let ϕ : S −→ T be an idempotent bijective homomorphism ofinverse semigroups and ψ : G −→ H an iso-unital functor between topolog-ical groupoids. (1) ϕ is injective if and only if ker ϕ = ϕ − ( E ( T )) = E ( S ) . (2) ψ is injective if and only if ψ − ( H (0) ) = G (0) .Proof. If ϕ is injective, then ϕ ( s ) = e ∈ E ( T ) implies ϕ ( s ∗ s ) = e ∗ e = e andso s = s ∗ s ∈ E ( S ). Conversely, if ϕ − ( E ( T )) = E ( S ) and ϕ ( s ) = ϕ ( s ),then since ϕ is idempotent bijective, s ∗ s = s ∗ s . Also ϕ ( s s ∗ ) = ϕ ( s s ∗ ) ∈ BENJAMIN STEINBERG E ( T ) and so s s ∗ ∈ E ( S ). Therefore, s = s s ∗ s = s s ∗ s ≤ s and dually s ≤ s and hence s = s . Thus ϕ is injective.Similarly, if ψ is injective, then ψ ( g ) = x ∈ H (0) implies ψ ( g ) = ψ ( d ( g ))and so g = d ( g ) ∈ G (0) . On the other hand, if ψ − ( H (0) ) = G (0) and ψ ( g ) = ψ ( g ), then d ( g ) = d ( g ) and r ( g ) = r ( g ), as ψ is iso-unital, and ψ ( g g − ) = ψ ( g g − ) ∈ H (0) and so g g − ∈ G (0) , whence g = g . (cid:3) To every ample groupoid G , we can associate the Boolean inverse semi-group Γ c ( G ). The inverse construction is more difficult to describe. If S is aBoolean inverse semigroup, then Spec( E ( S )) denotes the Stone space of theBoolean algebra E ( S ). It elements are the non-zero Boolean algebra homo-morphisms (characters) χ : E −→ { , } . A basis of compact open subsetsfor Spec( E ( S )) consists of the sets of the form D ( e ) = { χ ∈ Spec( E ( S )) | χ ( e ) = 1 } and in fact e D ( e ) is an isomorphism of Boolean algebras between E ( S )and the Boolean algebra of compact open subsets of Spec( E ( S )). Thereis a natural action of S on Spec( E ( S )) by partial homeomorphisms. Toeach s ∈ S , there is a homeomorphism β s : D ( s ∗ s ) −→ D ( ss ∗ ) given by β s ( χ )( e ) = χ ( s ∗ es ) and the assignment s β s is a non-degenerate actionof S by partial homeomorphisms. We often write sχ instead of β s ( χ ). Put G ( S ) = S ⋉ Spec( E ( S )), the groupoid of germs. Recall that G ( S ) = ( S × Spec( B )) / ∼ where ( s, χ ) ∼ ( t, λ ) if and only if χ = λ and there exists u ≤ s, t with χ ( u ∗ u ) = 1. We write [ s, χ ] for the equivalence class of ( s, χ ). The groupoidmultiplication is given by defining [ s, χ ][ t, λ ] if and only if χ = tλ , in whichcase the product is [ st, λ ]. A basis for the topology is given by the compactopen bisections D ( s ) = { [ s, χ ] | χ ∈ D ( s ∗ s ) } and this topology makes G ( S ) into an ample groupoid. The unit space G ( S ) (0) can be identified homeomorphically with Spec( E ( S )) via χ [ e, χ ]where e ∈ E ( S ) with χ ( e ) = 1. Under this identification, d ([ s, χ ]) = χ and r ([ s, χ ]) = sχ . Inversion is given by [ s, χ ] − = [ s ∗ , sχ ]. See [22, 31, 36]for more on groupoids of germs. (In [28], they work with a groupoid ofultrafilters on the poset S , but it is well known and easy to see that thisgroupoid is isomorphic to G ( S ).)The following theorem combines work of Exel [22,23], Lawson and Lenz [28]. Theorem 3.3 (Exel-Lawson-Lenz) . Let H be an ample groupoid and S aBoolean inverse semigroup. (1) There is an isomorphism η H : H −→ G (Γ c ( H )) given by η H ( h ) =[ U, χ d ( h ) ] where U is any compact open bisection containing h and χ x is the characteristic function for the set of compact open subsetsof H (0) containing x ∈ H (0) . (2) There is an isomorphism ε S : S −→ Γ c ( G ( S )) given by ε S ( s ) = D ( s ) . WISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY 7
We now show that the constructions H Γ c ( H ) and S
7→ G ( S ) can bepromoted to inverse equivalences between the category of ample groupoidswith open iso-unital functors and the category of Boolean inverse semigroupswith idempotent bijective homomorphisms. Theorem 3.4.
The category of ample groupoids with open iso-unital func-tors is equivalent to the category of Boolean inverse semigroups with idem-potent bijective homomorphisms. More precisely, Γ c is a functor from thecategory of ample groupoids with open iso-unital functors to the categoryof Boolean inverse semigroups with idempotent bijective homomorphisms,where if ϕ : G −→ H is an open iso-unital functor, then Γ c ( ϕ ) : Γ c ( G ) −→ Γ c ( H ) is given by Γ c ( ϕ )( U ) = ϕ ( U ) . Moreover, the groupoid of germs con-struction S
7→ G ( S ) provides a quasi-inverse functor where if ϕ : S −→ T isidempotent bijective, then G ( ϕ )([ s, χ ]) = [ ϕ ( s ) , χ ◦ ( ϕ | E ( S ) ) − ] .Proof. We begin by showing that Γ c is a functor. Let ϕ : G −→ H be anopen iso-unital functor. Let U ∈ Γ c ( G ). Then ϕ ( U ) is compact open; we justneed to show that it is a bisection. Let g , g ∈ U with d ( ϕ ( g )) = d ( ϕ ( g )).Then d ( g ) = d ( g ), as ϕ is iso-unital, and so g = g because U is abisection. Thus d | ϕ ( U ) is injective and the same argument applies to r | ϕ ( U ) .We conclude that ϕ ( U ) ∈ Γ c ( G ). Next we check that ϕ ( U V ) = ϕ ( U ) ϕ ( V )for U, V ∈ Γ c ( G ). If g ∈ U V , then g = g g with g ∈ U and g ∈ V , andso ϕ ( g ) = ϕ ( g ) ϕ ( g ) ∈ ϕ ( U ) ϕ ( V ). Thus ϕ ( U V ) ⊆ ϕ ( U ) ϕ ( V ). Conversely,suppose that h ∈ ϕ ( U ) ϕ ( V ). So h = ϕ ( g ) ϕ ( g ) with g ∈ U and g ∈ V . Since ϕ is iso-unital, this implies d ( g ) = r ( g ) and so h = ϕ ( g g )with g g ∈ U V . Thus ϕ ( U V ) = ϕ ( U ) ϕ ( V ) as required. Observe thatsince ϕ | G (0) : G (0) −→ H (0) is a homeomorphism and E (Γ c ( G )) is the setof compact open subsets of G (0) and E (Γ c ( H )) is the set of compact opensubsets of H (0) , it follows that Γ c ( ϕ ) is idempotent bijective. It is clearfrom the definition that Γ c is functorial.Next we show that if ϕ : S −→ T is an idempotent bijective homomor-phism of Boolean inverse semigroups, then G ( ϕ ) is a well-defined, openiso-unital functor. To ease notation, we write ϕ ∗ for G ( ϕ ). To checkthat ϕ ∗ is well defined, suppose that u ≤ s, s ′ with χ ( u ∗ u ) = 1. Then ϕ ( u ) ≤ ϕ ( s ) , ϕ ( s ′ ) and χ ◦ ( ϕ E ( S ) ) − ( ϕ ( u ) ∗ ϕ ( u )) = χ ( u ∗ u ) = 1. Thus[ ϕ ( s ) , χ ◦ ( ϕ E ( S ) ) − ] = [ ϕ ( s ′ ) , χ ◦ ( ϕ E ( S ) ) − ]. It’s immediate from ϕ be-ing a homomorphism that ϕ ∗ is a functor. It is clearly bijective on unitspaces as χ χ ◦ ( ϕ E ( S ) ) − is a bijection of Stone spaces. It remainsto show that ϕ ∗ is open and continuous. For continuity, we claim that ϕ − ∗ ( D ( t )) = S ϕ ( s ) ≤ t D ( s ). Indeed, if ϕ ( s ) ≤ t and χ ( s ∗ s ) = 1, then ϕ ∗ ([ s, χ ]) = [ ϕ ( s ) , χ ◦ ( ϕ E ( S ) ) − ] = [ t, χ ◦ ( ϕ | E ( S ) ) − ] ∈ D ( t ). Conversely, if ϕ ∗ ([ s, χ ]) = [ ϕ ( s ) , χ ◦ ( ϕ E ( S ) ) − ] ∈ D ( t ), then there exists u ≤ ϕ ( s ) , t with χ ◦ ( ϕ E ( S ) ) − ( u ∗ u ) = 1. Put e = ( ϕ E ( S ) ) − ( u ∗ u ) ≤ ( ϕ E ( S ) ) − ( ϕ ( s ) ∗ ϕ ( s )) = s ∗ s and let s = se . Then ϕ ( s ) = ϕ ( s ) ϕ ( e ) = ϕ ( s ) u ∗ u = u ≤ t and χ ( s ∗ s ) = χ ( e ) = χ ◦ ( ϕ E ( S ) ) − ( u ∗ u ) = 1. Thus [ s, χ ] = [ s , χ ] ∈ D ( s ) BENJAMIN STEINBERG with ϕ ( s ) ≤ t . To show that ϕ ∗ is open, we claim that ϕ ∗ ( D ( s )) = D ( ϕ ( s )). From what we have just shown, ϕ ∗ ( D ( s )) ⊆ D ( ϕ ( s )). Supposethat [ ϕ ( s ) , λ ] ∈ D ( ϕ ( s )). Let χ = λ ◦ ϕ | E ( S ) . Then χ ( s ∗ s ) = λ ( ϕ ( s ∗ s )) = 1and so [ s, χ ] ∈ D ( s ). Moreover, ϕ ∗ ([ s, χ ]) = [ ϕ ( s ) , χ ◦ ( ϕ | E ( S ) ) − ] = [ ϕ ( s ) , λ ],as required. It is clear that if ϕ, ψ are composable idempotent bijective ho-momorphisms, then G ( ϕψ ) = G ( ϕ ) ◦ G ( ψ ).It now remains to show that η and ε from Theorem 3.3 are natural iso-morphisms. Let ϕ : G −→ H be an open iso-unital functor. Let g ∈ G and U be a compact open bisection containing g . Note that ϕ ( U ) is a compactopen bisection containing ϕ ( g ) and G (Γ c ( ϕ ))( η G ( g )) = G (Γ c ( ϕ ))[ U, χ d ( g ) ] =[ ϕ ( U ) , χ d ( ϕ ( g )) ] = η H ( ϕ ( g )). Next we check that ε is a natural transforma-tion. If ϕ : S −→ T is an idempotent bijective homomorphism of inversesemigroups, then G ( ϕ )( ε S ( s )) = G ( ϕ )( D ( s )) = D ( ϕ ( s )) = ε T ( ϕ ( s )) by thecomputation of the previous paragraph. This completes the proof. (cid:3) We next want to show that our functors preserve injective and surjectivemaps. Presumably, there is a categorical description of these morphisms inour categories but we give a direct proof.
Proposition 3.5.
The functors Γ c and G preserve injective and surjectivemorphisms.Proof. Let ϕ : G −→ H be an open iso-unital functor. Suppose first that ϕ is injective. Then, for U ∈ Γ c ( G ), we have that ϕ ( U ) ∈ E (Γ c ( H )) implies ϕ ( U ) ⊆ H (0) and hence U ⊆ G (0) by Lemma 3.2. Thus U ∈ E (Γ c ( G )) andso we conclude Γ c ( ϕ ) is injective by another application of Lemma 3.2.Next suppose that ϕ is a surjective and let V ∈ Γ c ( H ). Note that V is compact Hausdorff, being homeomorphic to the compact Hausdorffspace d ( V ). We claim that, for each h ∈ V , there is a compact openneighborhood V h ⊆ V of h such that there is a continuous section s : V h −→ G of ϕ . Choose g ∈ G with ϕ ( g ) = h . Then since ϕ − ( V ) is open andthe compact open bisections form a basis for the topology of G , as G isample, we can find a compact open bisection U g with g ∈ U g and ϕ ( U g ) ⊆ V . Then V h = ϕ ( U g ) is compact open containing h (since ϕ is open).Moreover, since ϕ is iso-unital and U g is a bisection, ϕ | U g is injective byProposition 3.1. Then s = ( ϕ | U g ) − is our desired section, as ϕ is open.Since V is compact, we can cover V by finitely many compact open subsets V , . . . , V n such that there is a continuous section s i : V i −→ G of ϕ . Since V is compact Hausdorff, its compact open subsets form a Boolean algebra.The finitely many compact open subsets V , . . . , V n of V generate a finiteBoolean algebra with maximum element V = S ni =1 V i . Let W , . . . , W k be the atoms of this Boolean algebra. Then the W i are pairwise disjoint.Also since W i = V ∩ W i = ( V ∩ W i ) ∪ · · · ∪ ( V n ∩ W i ), we see using W i is an atom that W i ⊆ V j for some j . Hence there is a continuous section t i = s j | W i : W i −→ G of ϕ . Since the maximum of a finite Boolean algebra isthe join of its atoms, we deduce that V = W ∪ · · · ∪ W k and this is a disjoint WISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY 9 union. Hence we can define a continuous section s : V −→ G by putting s | W i = t i for i = 1 , . . . , k . Note that since ϕ is a local homeomorphism byProposition 3.1, every continuous section of ϕ defined on an open subset of H is an open mapping. Put U = s ( V ) and note that U is compact openand ϕ ( U ) = V . Moreover, ϕ | U is injective as ϕ ◦ s = 1 V . We claim that U ∈ Γ c ( G ). If g, g ′ ∈ U with d ( g ) = d ( g ′ ), then d ( ϕ ( g )) = d ( ϕ ( g ′ )) and so ϕ ( g ) = ϕ ( g ′ ) since V is a bisection. Therefore, g = g ′ as ϕ | U is injective.Similarly, r | U is injective and so U ∈ Γ c ( G ). Thus Γ c ( ϕ ) is surjective.Suppose now that ϕ : S −→ T is an idempotent bijective homomorphismof Boolean inverse semigroups. First assume that ϕ is injective and sup-pose that [ ϕ ( s ) , χ ◦ ( ϕ | E ( S ) ) − ] = G ( ϕ )([ s, χ ]) ∈ G ( T ) (0) . Then there is anidempotent f ∈ E ( T ) with f ≤ ϕ ( s ) and χ ◦ ( ϕ | E ( S ) ) − ( f ) = 1. Putting e = ( ϕ | E ( S ) ) − ( f ), we have that χ ( e ) = 1 and ϕ ( se ) = ϕ ( s ) f = f . Since ϕ − ( E ( T )) = E ( S ) by Lemma 3.2, we have that se ∈ E ( S ) and se ≤ s .Moreover, χ ( es ∗ se ) = 1 and so [ s, χ ] = [ se, χ ] ∈ G (0) . Therefore, G ( ϕ ) is in-jective by Lemma 3.2. Next assume that ϕ is surjective and let [ t, χ ] ∈ G ( T ).Then t = ϕ ( s ) for some s ∈ S and χ ◦ ϕ | E ( S ) ( s ∗ s ) = χ ( t ∗ t ) = 1. Thus[ s, χ ◦ ϕ | E ( S ) ] ∈ G ( S ) and we have G ( ϕ )([ s, χ ◦ ϕ | E ( S ) ]) = [ ϕ ( s ) , χ ] = [ t, χ ].Therefore, G ( ϕ ) is surjective. (cid:3) Remark . The proof of Proposition 3.5 shows that if ϕ : G −→ H is asurjective open iso-unital functor between ample groupoids, then a contin-uous local section of ϕ can be defined on any compact open bisection of H .Note that since Γ c and G are equivalences of categories and isomorphismsin these categories are bijective, it follows that ϕ : G −→ H is injective(respectively, surjective) if and only if Γ c ( ϕ ) : Γ c ( G ) −→ Γ c ( H ) is injective(respectively, surjective) and ϕ : S −→ T is injective (respectively, surjec-tive) if and only if G ( ϕ ) is injective (respectively, surjective).The question of whether a surjective open iso-unital functor admits acontinuous section preserving unit spaces (but not necessarily a functor)is important when determining whether a twist comes from a groupoid 2-cocycle [5]. We show that in the ample setting, this is equivalent to the ques-tion considered in [21] of when a surjective idempotent bijective inverse semi-group homomorphism admits an order-preserving and idempotent-preservingsection.The following is a minor variation of [21, Lemma 4.2]. Lemma 3.7.
Let ϕ : S −→ T be an idempotent bijective surjective homo-morphism of inverse semigroups. Then the following are equivalent. (1) There is an order-preserving map j : T −→ S with j ( E ( T )) ⊆ E ( S ) (i.e., j is idempotent preserving) and ϕ ◦ j = 1 T . (2) There is a map j : T −→ S with ϕ ◦ j = 1 T and j ( te ) = j ( t ) j ( e ) forall t ∈ T and e ∈ E ( T ) . (3) There is a map j : T −→ S with ϕ ◦ j = 1 T and j ( et ) = j ( e ) j ( t ) forall t ∈ T and e ∈ E ( T ) .Proof. We prove the equivalence of the first and second items as the equiva-lence of the first and third items is dual. Suppose first that j as in (2) exists.Then if e ∈ E ( T ), we have that j ( e ) = j ( ee ) = j ( e ) j ( e ) and so j preservesidempotents. Also, if t ≤ t , we may write t = t e with e ∈ E ( T ). There-fore, j ( t ) = j ( t e ) = j ( t ) j ( e ) ≤ j ( t ), as j preserves idempotents, and so j is order preserving. Thus (2) implies (1).Suppose that j is as in (1). Let t ∈ T and e ∈ E ( T ). First we claimthat j ( t ) ∗ j ( t ) j ( e ) = j ( te ) ∗ j ( te ). Indeed, applying ϕ to both sides yields t ∗ te . Consequently, j ( t ) ∗ j ( t ) j ( e ) = j ( te ) ∗ j ( te ) as j ( e ) ∈ E ( S ) and ϕ isidempotent bijective. Since te ≤ t , the fact that j is order preserving impliesthat j ( te ) ≤ j ( t ) and so j ( te ) = j ( t ) j ( te ) ∗ j ( te ) = j ( t ) j ( t ) ∗ j ( t ) j ( e ) = j ( t ) j ( e )as required. (cid:3) Proposition 3.8.
Let ϕ : G −→ H be a surjective open iso-unital functorbetween ample groupoids. There there is a continuous mapping f : H −→ G with ϕ ◦ f = 1 H and f ( H (0) ) ⊆ G (0) if and only if there is an order-preserving and idempotent-preserving mapping j : Γ c ( H ) −→ Γ c ( G ) with Γ c ( ϕ ) ◦ j = 1 Γ c ( H ) .Proof. Suppose first that f exists. Then since ϕ is a local homeomorphismby Proposition 3.1, the mapping f is open. Let U ∈ Γ c ( H ) and put j ( U ) = f ( U ). Then j ( U ) is compact open. It is also a bisection since if f ( h ) , f ( h ′ ) ∈ f ( U ) = j ( U ) with d ( f ( h )) = d ( f ( h ′ )) and h, h ′ ∈ U , then d ( h ) = d ( h ′ )because ϕ is a functor and ϕ ◦ f = 1 H . Thus h = h ′ because U is abisection. Similarly, r | j ( U ) is injective and so j ( U ) ∈ Γ c ( G ). Obviously, j isorder preserving and preserves idempotents since f ( H (0) ) ⊆ G (0) .For the converse, it suffices by Theorem 3.4 to show that if ϕ : S −→ T is a surjective idempotent bijective homomorphism of Boolean inverse semi-groups admitting an order-preserving section j : T −→ S with j ( E ( T )) ⊆ E ( S ), then G ( ϕ ) : G ( S ) −→ G ( T ) admits a continuous section f preservingunits. Put f ([ t, χ ]) = [ j ( t ) , χ ◦ ϕ ]. This is well defined since if [ t, χ ] = [ t ′ , χ ],then we can find u ≤ t, t ′ with χ ( u ∗ u ) = 1 and so j ( u ) ≤ j ( t ) , j ( t ′ ) as j isorder preserving and χ ( ϕ ( j ( u ) ∗ j ( u ))) = χ ( u ∗ u ) = 1. Thus [ j ( t ) , χ ◦ ϕ ] =[ j ( t ′ ) , χ ◦ ϕ ]. Trivially, G ( ϕ )( f ([ t, χ ])) = G ( ϕ )([ j ( t ) , χ ◦ ϕ ]) = [ ϕ ( j ( t )) , χ ] =[ t, χ ]. Also if [ e, χ ] ∈ G ( T ) (0) with e ∈ E ( T ), then f ([ e, χ ]) = [ j ( e ) , χ ◦ ϕ ] ∈G ( S ) (0) since j ( e ) ∈ E ( S ).For continuity, we claim that f − ( D ( s )) = S j ( t ) ≤ s D ( t ). Indeed, if j ( t ) ≤ s , then f ([ t, χ ]) = [ j ( t ) , χ ◦ ϕ ] = [ s, χ ◦ ϕ ] ∈ D ( s ). Conversely, if [ j ( t ) , χ ◦ ϕ ] = f ([ t, χ ]) ∈ D ( s ), then there exists u ≤ j ( t ) , s with χ ( ϕ ( u ∗ u )) = 1.Put e = u ∗ u and note that j ( t ) e = u = se . Then ϕ ( u ) ≤ ϕ ( j ( t )) = t and χ ( ϕ ( u ) ∗ ϕ ( u ))) = 1 and so [ t, χ ] = [ ϕ ( u ) , χ ] ∈ D ( ϕ ( u )). Since ϕ is WISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY 11 idempotent bijective and j is idempotent-preserving, we must have that j ( ϕ ( e )) = e . Thus tϕ ( e ) = ϕ ( j ( t ) u ∗ u ) = ϕ ( u ), and so j ( ϕ ( u )) = j ( tϕ ( e )) = j ( t ) j ( ϕ ( e )) = j ( t ) e = se ≤ s by Lemma 3.7. This completes the proof. (cid:3) It is shown in [21, Proposition 4.6] that if ϕ : S −→ T is an idempo-tent bijective surjective homomorphism of Boolean inverse monoids whoseidempotents form a complete Boolean algebra, then an order-preservingand idempotent-preserving section always exists. In the language of am-ple groupoids, these conditions mean that the unit spaces of G ( S ) and G ( T )are Stonean (compact and extremally disconnected).In [5] it was shown that twists over second countable Hausdorff amplegroupoids admit a unit-preserving global section; in [4, Lemma 2.4] and theremark thereafter, it was observed that the same proof works for paracom-pact Hausdorff ample groupoids. The following proposition generalizes theargument to open iso-unital functors. Proposition 3.9.
Let ϕ : G −→ H be a surjective open iso-unital functorbetween ample groupoids. If H is Hausdorff and H \ H (0) is paracompact,then there is a continuous section s : H −→ G with s ( H (0) ) ⊆ G (0) .Proof. Since H is Hausdorff, we have that H (0) is clopen in H . Therefore, H ′ = H \ H (0) is clopen. First we show that every σ -compact clopensubset X of H ′ admits a continuous section s X : X −→ G . Since X isa countable union of compact sets and each of these compact sets can becovered by finitely many compact open bisections of H contained in X ,as H is ample and X is clopen, we have that X = S ∞ n =1 U n with the U n compact open bisections of H . Put V = U and V n = U n \ S n − i =1 V i .Then since H is Hausdorff, each V n is a compact open bisection and byconstruction S ∞ n =1 V n = S ∞ n =1 U n = X . Moreover, the V n are pairwisedisjoint. By Remark 3.6, there is a continuous section s n : V n −→ G of ϕ and we can then define s X by s X | V n = s n for n ≥ σ -compactsubspaces to write H ′ = ` α ∈ A X α with X α clopen and σ -compact. We nowdefine s : H −→ G by s | H (0) = ( ϕ | G (0) ) − and s | X α = s X α . This completesthe proof. (cid:3) Extensions of inverse semigroups.
By an extension of inverse semi-groups we mean a sequence K ι −→ T ϕ −−→ S (3.1)of idempotent bijective homomorphisms with ι ( K ) = ker ϕ = ϕ − ( E ( S )).We call the extension abelian if K is commutative.Abelian extensions are classified by Lausch’s second cohomology group [26,Section 7]. We recall the setup. Let S be an inverse semigroup. An S -module consists of a commutative inverse semigroup K , an idempotent bijective ho-momorphism p : K −→ E ( S ) and a (total) left action of S on K (by en-domorphisms) such that p ( sk ) = sp ( k ) s ∗ and p ( k ) k = k for all k ∈ K and s ∈ S . The category of S -modules is an abelian category with enough pro-jectives and injectives and Lausch developed a corresponding cohomologytheory based on the derived functors of the functor taking an S -module p : K −→ E ( S ) to the S -equivariant sections q : E ( S ) −→ K of p (with re-spect to the conjugation action on E ( S )) [26]. We shall only need the secondcohomology group, which classifies extensions. Note that Lausch uses rightmodules, so we have dualized his results here.Given an extension (3.1) of S by a commutative inverse semigroup K wecan define an S -module structure on K by putting p = ϕι : K −→ E ( S ),choosing a set theoretic section j : S −→ T and setting sk = ι − ( j ( s ) ι ( k ) j ( s ) ∗ )for s ∈ S and k ∈ K . Note that ι ( K ) = ker ϕ is a normal inverse sub-semigroup of T and so this makes sense. Lausch shows that the modulestructure is independent of the choice of j . Also each element of t ∈ T canbe uniquely written as t = ι ( k ) j ( s ) with s ∈ S and k ∈ K , namely, s = ϕ ( t )and k = ι − ( tj ( ϕ ( t )) ∗ ).Given an S -module K with p : K −→ E ( S ), a 2 -cocycle is a mapping c : S × S −→ K satisfying the following properties:(1) p ( c ( s, t )) = stt ∗ s ∗ ;(2) ( sc ( t, u )) c ( s, tu ) = c ( s, t ) c ( st, u ) for s, t, u ∈ S .We say that c is normalized if c ( e, e ) ∈ E ( K ) for all e ∈ E ( S ). The trivial2-cocycle is defined by c ( s, t ) = ( p | E ( K ) ) − ( stt ∗ s ∗ ). The 2-cocycles form anabelian group under pointwise multiplication with the trivial cocycle as theidentity and with inversion done pointwise: c − ( s, t ) = c ( s, t ) ∗ . The groupof 2-cocycles will be denoted Z ( S, K ). The set of all mappings F : S −→ K with p ( F ( s )) = ss ∗ is an abelian group C ( S, K ) under pointwise operationswith identity F ( s ) = ( p | E ( K ) ) − ( ss ∗ ). There is a coboundary homomor-phism δ : C ( S, K ) −→ Z ( S, K ) given by ( δF )( s, t ) = F ( s )( sF ( t )) F ( st ) ∗ and the image B ( S, K ) is the group of 2 -coboundaries . We put H ( S, K ) = Z ( S, K ) /B ( S, K ) and call it the second Lausch cohomology group of S with coefficients in K . Proposition 3.10.
Every -cocycle is cohomologous to a normalized one.Proof. Let c : S × S −→ K be a 2-cocycle. Define F : S −→ K by F ( s ) = c ( ss ∗ , ss ∗ ) ∗ and put c ′ = cδF . Then, for e ∈ E ( S ), we have c ′ ( e, e ) = c ( e, e ) F ( e )( eF ( e )) F ( e ) ∗ = c ( e, e ) c ( e, e ) ∗ c ( e, e ) ∗ c ( e, e ) ∈ E ( K ). Thus c ′ isnormalized. (cid:3) In Lausch’s cohomology theory he calls this H ( S I , K ) where S I and K are obtainedby adjoining identities (see [26, Section 7]), but we avoid this extra notation for simplicity. WISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY 13
Given an abelian extension as per (3.1), we may choose a set theo-retic section j : S −→ T such that j | E ( S ) = ( ϕ | E ( T ) ) − . We can thendefine c : S × S −→ K by c ( s, s ′ ) = ι − ( j ( s ) j ( s ′ ) j ( ss ′ ) ∗ ) ∈ K ; equiva-lently, j ( s ) j ( s ′ ) = ι ( c ( s, s ′ )) j ( ss ′ ) for all s, s ′ ∈ S . One checks, cf. [26, Sec-tion 7], that c is a 2-cocycle. Moreover, it is normalized since c ( e, e ) = ι − ( j ( e ) j ( e ) j ( ee ) ∗ ) = ι − ( j ( e )) ∈ E ( K ), as j ( e ) ∈ E ( T ). Changing thesection j results in a cohomologous 2-cocycle. Lausch does not require j to preserve idempotents, which results in a not necessarily normalized 2-cocycle, but we find normalized 2-cocycles more convenient to work with.So to each extension of K by S , we obtain an S -module structure on K anda normalized 2-cocycle whose cohomology class is well defined.Two extensions K ι −→ T ϕ −−→ S, K ι ′ −−→ T ′ ϕ −−→ S are equivalent if there is an isomorphism ψ : T −→ T ′ such that the diagram K T SK T ′ S ι K ϕψ S ι ′ ϕ ′ commutes.Lausch proves that two extensions of K by S are equivalent if and only ifthe module structures on K are the same and the corresponding cohomologyclasses of 2-cocycles are the same. Moreover, for each 2-cocycle, he showsthat there is an extension (3.1) of K by S realizing the cohomology classof the 2-cocycle. If c : S × S −→ K is a normalized 2-cocycle, we can put T = { ( k, s ) ∈ K × S | p ( k ) = ss ∗ } with multiplication given by ( k, s )( k ′ , s ′ ) =( k ( sk ′ ) c ( s, s ′ ) , ss ′ ). Here, ι ( k ) = ( k, p ( k )) and ϕ ( k, s ) = s . The inverseis given by ( k, s ) ∗ = (( s ∗ k ∗ ) c ( s, s ∗ ) ∗ , s ∗ ). The class of the trivial cocyclecorresponds to the split extension of K by S and T , which, in this case,is what is termed the full restricted semidirect product K ⊲⊳ S of K by S in [27, Chapter 5.3]. More details can be found in [26, Section 7].We record here some properties of normalized 2-cocycles for later use. Proposition 3.11.
Let K be an S -module with idempotent bijective homo-morphism p : K −→ E ( S ) and c : S × S −→ K a normalized -cocycle. (1) c ( s, s ∗ s ) = c ( ss ∗ , s ) ∈ E ( K ) . (2) c ( s, s ∗ ) = sc ( s ∗ , s ) for all s ∈ S . (3) c ( e, ef ) = c ( ef, e ) ∈ E ( K ) for all e, f ∈ E ( S ) . (4) c ( e, f ) ∈ E ( K ) for all e, f ∈ E ( S ) . (5) c ( s, e ) = c ( s, s ∗ se ) for all e ∈ E ( S ) , s ∈ S . (6) c ( e, s ) = c ( ess ∗ , s ) for all e ∈ E ( S ) , s ∈ S . (7) c ( e, es ) , c ( se, e ) ∈ E ( K ) for all e ∈ E ( S ) and s ∈ S . (8) c ( e, s ) = c ( s, s ∗ es ) for all e ∈ E ( S ) and s ∈ S . (9) c ( s, e ) = c ( ses ∗ , s ) for all e ∈ E ( S ) and s ∈ S . (10) c ( u, s ) c ( ut, s ∗ u ∗ us ) ∗ = ( uc ( t, s ∗ s ) ∗ ) c ( u, t ) if s ≤ t ∈ S and u ∈ S . (11) c ( s, u ) c ( tu, u ∗ s ∗ su ) ∗ = c ( t, s ∗ s ) ∗ c ( t, u ) if s ≤ t ∈ S and u ∈ S .Proof. For e ∈ E ( S ), let k e denote the unique idempotent of K with p ( k e ) = e . By the 2-cocycle condition, the definition of a module and since c isnormalized, c ( s, s ∗ s ) = ( sc ( s ∗ s, s ∗ s )) c ( s, s ∗ s ) = c ( s, s ∗ s ) c ( s, s ∗ s ). Also, the2-cocycle condition and the definition of a module yields c ( ss ∗ , s ) = ( ss ∗ c ( ss ∗ , s )) c ( ss ∗ , s ) = c ( ss ∗ , ss ∗ ) c ( ss ∗ , s ) = c ( ss ∗ , s ) , as c is normalized. Thus c ( s, s ∗ s ) , c ( ss ∗ , s ) are idempotents with image ss ∗ under p and hence are equal as p is idempotent bijective. This yields (1).For (2), we have ( sc ( s ∗ , s )) c ( s, s ∗ s ) = c ( s, s ∗ ) c ( ss ∗ , s ). By the first item,we conclude that c ( s, s ∗ ) = sc ( s ∗ , s ).For (3), we compute c ( e, ef ) = ( ec ( e, ef )) c ( e, ef ) = c ( e, e ) c ( e, ef ) = c ( e, ef ) since c is normalized and c ( e, ef ) = ef c ( e, ef ) = ec ( e, ef ) = c ( e, e ) c ( e, ef ) . Similarly, we have c ( ef, e ) = ( ef c ( e, e )) c ( ef, e ) = c ( ef, e ) c ( ef, e )as ef c ( e, e ) = k ef because c is normalized. Since p ( c ( e, ef )) = ef = p ( c ( ef, e )) and p is idempotent bijective, we deduce that c ( e, ef ) = c ( ef, e ).To prove (4), observe that ( ec ( f, ef )) c ( e, ef ) = c ( e, f ) c ( ef, ef ) = c ( e, f )since c is normalized, and so c ( e, f ) ∈ E ( K ) by (3).For (5), we have ( sc ( s ∗ s, e )) c ( s, s ∗ se ) = c ( s, s ∗ s ) c ( s, e ). By (4), sc ( s ∗ s, e ) = k ses ∗ and, by (1), c ( s, s ∗ s ) = k ss ∗ ≥ k ses ∗ , and so the left hand side is c ( s, s ∗ se ) and the right hand side is c ( s, e ).For (6), we have ( ec ( ss ∗ , s )) c ( e, s ) = c ( e, ss ∗ ) c ( ess ∗ , s ). As c ( ss ∗ , s ) = k ss ∗ , c ( e, ss ∗ ) = k ess ∗ by (1) and (4), we deduce that c ( e, s ) = c ( ess ∗ , s ).For (7), we have c ( e, es ) = c ( ess ∗ , es ) ∈ E ( K ) and c ( se, e ) = c ( se, es ∗ s ) ∈ E ( K ) by (6) and (1).For (8), we note by the 2-cocycle condition( ec ( s, s ∗ es )) c ( e, es ) = c ( e, s ) c ( es, s ∗ es ) . But by (1), c ( es, s ∗ es ) = k ess ∗ and by (7) c ( e, es ) = k ess ∗ . Also since p ( c ( s, s ∗ es )) = ess ∗ ≤ e , we have that ec ( s, s ∗ es ) = ess ∗ c ( s, s ∗ es ) = c ( s, s ∗ es ).We conclude that c ( s, s ∗ es ) = c ( e, s ) as required.For (9), we note that ( ses ∗ c ( s, e )) c ( ses ∗ , se ) = c ( ses ∗ , s ) c ( se, e ) by the2-cocycle condition. But c ( ses ∗ , se ) = k ses ∗ = c ( se, e ) by (1) and (7), andso we deduce, since p ( c ( s, e )) = ses ∗ , that c ( s, e ) = c ( ses ∗ , s ).To prove (10), we compute( uc ( t, s ∗ s )) c ( u, s ) = ( uc ( t, s ∗ s )) c ( u, ts ∗ s ) = c ( u, t ) c ( ut, s ∗ s )= c ( u, t ) c ( ut, t ∗ u ∗ uts ∗ s )where the last equality uses (5). But t ∗ u ∗ uts ∗ s = s ∗ st ∗ u ∗ uts ∗ s = s ∗ u ∗ us .Thus we have ( uc ( t, s ∗ s )) c ( u, s ) = c ( u, t ) c ( ut, s ∗ u ∗ us ). Using p ( uc ( t, s ∗ s )) = WISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY 15 uss ∗ u ∗ = p ( c ( ut, s ∗ u ∗ us )), we deduce that c ( u, s ) c ( ut, s ∗ u ∗ us ) ∗ = ( uc ( t, s ∗ s ) ∗ ) c ( u, t ) , as required.We now turn to (11). From the 2-cocycle condition we obtain( tc ( u, u ∗ s ∗ su )) c ( t, s ∗ su ) = c ( t, u ) c ( tu, u ∗ s ∗ su ) , ( tc ( s ∗ s, u )) c ( t, s ∗ su ) = c ( t, s ∗ s ) c ( ts ∗ s, u )= c ( t, s ∗ s ) c ( s, u )Note that c ( s ∗ s, u ) = c ( u, u ∗ s ∗ su ) by (8). Therefore, c ( t, s ∗ s ) c ( s, u ) = c ( t, u ) c ( tu, u ∗ s ∗ su ). Then, using that we have p ( c ( t, s ∗ s )) = ss ∗ ≥ suu ∗ s ∗ = p ( c ( tu, u ∗ s ∗ su ))= p ( c ( s, u )) = p ( c ( t, s ∗ s ) ∗ c ( t, u )) , we deduce that c ( s, u ) c ( tu, u ∗ s ∗ su ) ∗ = c ( t, s ∗ s ) ∗ c ( t, u ), as required. (cid:3) Twists and extensions of groupoids.
By an extension of amplegroupoids we mean an exact sequence K ι −→ H ϕ −−→ G (3.2)of open iso-unital functors with ι injective, ϕ surjective and ϕ − ( G (0) ) = ι ( K ). A second extension K ι ′ −−→ H ′ ϕ ′ −−→ G is equivalent to (3.2) if there is an isomorphism ψ : H −→ H ′ making thediagram K H GK H ′ G ι K ϕψ G ι ′ ϕ ′ commute.We now wish to show that our functors Γ c and G are exact in the sensethat they preserve extensions. Theorem 3.12.
The functors Γ c and G preserve extensions. More precisely: (1) If K ι −→ H ϕ −−→ G is an extension of ample groupoids, then Γ c ( K ) Γ c ( ι ) −−−−→ Γ c ( H ) Γ c ( ϕ ) −−−−→ Γ c ( G ) is an extension of Boolean inverse semigroups. (2) If K ι −→ T ϕ −−→ S is an extension of Boolean inverse semigroups,then G ( K ) G ( ι ) −−−→ G ( T ) G ( ϕ ) −−−−→ G ( S ) is an extension of ample groupoids. Proof.
By Theorem 3.4 and Proposition 3.5, all that remains to check is thatΓ c ( ι )(Γ c ( K )) = Γ c ( ϕ ) − ( E (Γ c ( G ))) and G ( ι )( G ( K )) = G ( ϕ ) − ( G ( S ) (0) ).For the first item, note that since G (0) is open, we have that ϕ − ( G (0) ) is anopen subgroupoid of H and hence ample. Moreover, Γ c ( ϕ ) − ( E (Γ c ( G ))) =Γ c ( ϕ − ( G (0) )) . Since ι : K −→ ϕ − ( G (0) ) is surjective and open iso-unital,we deduce from Proposition 3.5 thatΓ( ι )(Γ c ( K )) = Γ c ( ϕ − ( G (0) )) = Γ c ( ϕ ) − ( E (Γ c ( G ))) , as required.For the second item, liet [ k, χ ] ∈ G ( K ). Then G ( ϕ ) G ( ι )([ k, χ ]) = [ ϕι ( k ) , χ ◦ ( ϕι | E ( K ) ) − ] ∈ G ( S ) (0) since ϕι ( k ) ∈ E ( S ). Conversely, if [ ϕ ( t ) , χ ◦ ( ϕ | E ( T ) ) − ] = G ( ϕ )([ t, χ ]) ∈G ( S ) (0) , then there is an idempotent f ∈ E ( S ) with f ≤ ϕ ( t ) and χ ◦ ( ϕ | E ( S ) ) − ( f ) = 1 . Putting e = ( ϕ | E ( S ) ) − ( f ), we have that χ ( e ) = 1 and ϕ ( te ) = ϕ ( t ) f = f .Since χ ( et ∗ te ) = 1, we deduce that [ t, χ ] = [ te, χ ]. As te ∈ ϕ − ( E ( S )),we have that te = ι ( k ) with k ∈ K . Then [ k, χ ◦ ι | E ( K ) ] ∈ G ( K ) (as χι ( k ∗ k ) = χ ( et ∗ te ) = 1) and G ( K )( ι )([ k, χ ◦ ι | E ( K ) ]) = [ te, χ ] = [ t, χ ]. Thus G ( ι )( G ( K )) = G ( ϕ ) − ( G ( S ) (0) ), as required. This completes the proof. (cid:3) It follows from Theorem 3.12 that classifying extensions of ample groupoidsis equivalent to classifying extensions of Boolean inverse semigroups. Onehas to have some care though in applying Lausch cohomology in order tomake sure that the extension coming from a 2-cocycle is Boolean.We briefly digress to examine how the Hausdorff property behaves un-der extensions of ample groupoids. It follows from the results of [38], seealso [28], that an ample groupoid G is Hausdorff if and only if each U ∈ Γ c ( G )has a maximum idempotent below it in the natural partial order. In the case G is Hausdorff, U ∩ G (0) is the maximum idempotent below U . Proposition 3.13.
Consider an extension of ample groupoids as in (3.2) .Then G is Hausdorff if and only if each element of Γ c ( H ) has a maximumelement of Γ c ( ι )(Γ c ( K )) below it. Moreover, if G and K are both Hausdorff,then so is H .Proof. Without loss of generality, we may assume that ι is an inclusion.Suppose that G is Hausdorff and U ∈ Γ c ( H ). Then ϕ ( U ) ∈ Γ c ( G ) has aunique maximum idempotent V ⊆ ϕ ( U ). Since V ⊆ G (0) , we have that ϕ − ( V ) ⊆ K . Also, since V is clopen (as G is Hausdorff), we have that ϕ − ( V ) ∩ U ⊆ K is clopen and hence compact (since U is compact). Thus W = ϕ − ( V ) ∩ U ∈ Γ c ( K ) and W ⊆ U . Moreover, if W ′ ∈ Γ c ( K ) with W ′ ⊆ U , then ϕ ( W ′ ) ⊆ ϕ ( U ) ∩ G (0) = V and so W ′ ⊆ W . Thus W is themaximum element of Γ c ( K ) below U . For the converse, by Proposition 3.5,every element of Γ c ( G ) is of the form ϕ ( U ) with U ∈ Γ c ( H ). Let W be the WISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY 17 maximum element of Γ c ( K ) below U . Then ϕ ( W ) ⊆ G (0) is an idempotentand ϕ ( W ) ⊆ ϕ ( U ). Suppose that V ⊆ G (0) is compact open with V ⊆ ϕ ( U ).Let x ∈ V and let h ∈ U with ϕ ( h ) = x . Since U ∩ ϕ − ( V ) is an open subsetof K containing h , we can find a compact open bisection W ′ ∈ Γ c ( K )with h ∈ W ′ ⊆ U ∩ ϕ − ( V ) ⊆ K . Then W ′ ⊆ W by choice of W and so ϕ ( W ′ ) ⊆ ϕ ( W ). As x = ϕ ( h ) ∈ ϕ ( W ′ ), we deduce that x ∈ ϕ ( W ) and hence V ⊆ ϕ ( W ). This shows that ϕ ( W ) is the maximum idempotent below ϕ ( U )and hence G is Hausdorff.Suppose now that G and K are Hausdorff. We continue to assume that ι is an inclusion. If U ∈ Γ c ( H ), then there is a unique maximum element V ofΓ c ( K ) with V ⊆ U by the above paragraph. Since K is Hausdorff, there isa unique maximum idempotent W ⊆ V in Γ c ( K ). Note that E (Γ c ( K )) = E (Γ c ( H )) as K (0) = H (0) . Thus any idempotent U ′ ⊆ U belongs toΓ c ( K ), and hence is below V , and therefore below W . Thus W is themaximum idempotent below U . It follows that H is Hausdorff. (cid:3) Let G be an ample groupoid and A a discrete abelian group with identity1 and with binary operation written multiplicatively (which we can think ofas a one-object ample groupoid). In practice, A = R × where R is a com-mutative ring with unity. A (discrete) A -twist over G is an exact sequenceof ample groupoids A × G (0) ι −→ Σ ϕ −−→ G (3.3)(with ϕι ( a, x ) = x ) which is central in the sense that ι ( a, r ( ϕ ( s ))) s = sι ( a, d ( ϕ ( s ))) (3.4)for all a ∈ A and s ∈ Σ. In this case, we often write as for ι ( a, r ( ϕ ( s ))) s = sι ( a, d ( ϕ ( s ))) and we note that ( as )( a ′ s ′ ) = ( aa ′ ) ss ′ for a, a ′ ∈ A and s, s ′ ∈ Σ with d ( s ) = r ( s ′ ). Also note that ( a, s ) as is a free action of A on Σby homeomorphisms (but not by groupoid automorphisms). Note that since A × G (0) is Hausdorff, if G is Hausdorff, then so is Σ by Proposition 3.13.This was already observed in [4, Corollary 2.3] using an argument specificto twists.Note that a different definition of twists is given in [5], where Σ is notrequired to be ample and ι is not required to be open, but where ϕ isrequired to be a locally trivial fiber bundle with fiber A . However, it isshown in [4, Proposition 2.2] that our definition is equivalent to the onein [5]. The first step in reformulating A -twists in terms of inverse semigroupsis to identify Γ c ( A × G (0) ) in semigroup theoretic terms. Note that A ∪ { } isan inverse semigroup, where 0 a = 0 = a for all a ∈ A . Note a ∗ = a − if a ∈ A and 0 ∗ = 0. Proposition 3.14.
Let e A = C c ( G (0) , A ∪ { } ) be the set of locally constantmappings f : G (0) −→ A ∪ { } with compact support (meaning supp( f ) = f − ( A ) is compact). Then e A is a commutative inverse semigroup under pointwise multiplication and there is an isomorphism γ : e A −→ Γ c ( A × G (0) ) given by f U f = { ( f ( x ) , x ) | x ∈ supp( f ) } .Proof. Clearly e A is a commutative inverse semigroup with f ∗ ( x ) = f ( x ) ∗ giving the inversion. Note thatsupp( f g ) = supp( f ) ∩ supp( g ) = supp( f ) supp( g )since A is a group. The map γ is a homomorphism because U f U g = { ( f ( x ) , x )( g ( x ) , x ) | x ∈ supp( f ) ∩ supp( g ) } = { ( f ( x ) g ( x ) , x ) | x ∈ supp( f g ) } = U fg . It is obvious that γ is injective since supp( f ) = d ( U f ) and f = π ◦ ( d | U f ) − on its support, where π is the projection to A . Conversely, given U ∈ Γ c ( A × G (0) ), we can define f by f ( x ) = ( π ◦ ( d | U ) − ( x ) , if x ∈ d ( U )0 , elseand U = U f . Thus γ is an isomorphism. (cid:3) We can make e A into a Γ c ( G )-module as follows. Define p : e A −→ E (Γ c ( G ))by p ( f ) = supp( f ). The action is given by( U f )( x ) = ( f ( d (( r | U ) − ( x ))) , if x ∈ r ( U )0 , elsefor U ∈ Γ c ( G ) and x ∈ G (0) . Proposition 3.15.
The commutative inverse semigroup e A is a Γ c ( G ) -modulewith respect to p and the action ( U, f ) U f .Proof.
The idempotents of e A are the characteristic functions 1 U with U ⊆ G (0) compact open and p (1 U ) = U . Since supp( f g ) = supp( f ) ∩ supp( g ) =supp( f ) supp( g ), we have that p is an idempotent bijective semigroup ho-momorphism. If W ⊆ G (0) is compact open and U ∈ Γ c ( G ), then U W U ∗ consists of those x ∈ G (0) such that there is an arrow g : y −→ x in U with y ∈ W . The support of U f consists of those x ∈ G (0) such that there isan arrow g : y −→ x with f ( y ) = 0. Hence supp( U f ) = U supp( f ) U ∗ . Alsonotice that if U ⊆ G (0) is compact open, then U f = 1 U f . It follows that p ( f ) f = 1 supp( f ) f = f . It remains to check that ( U, f ) U f is a semigroupaction by endomorphisms.Clearly, U ( f g ) = ( U f )( U g ) from the definition. If
U, V ∈ Γ c ( G ), then x ∈ supp( U ( V f )) if and only if there is g : y −→ x with g ∈ U and h ∈ V with h : z −→ y with f ( z ) = 0, in which case ( U ( V f ))( x ) = f ( z ). But ( U V ) f ( x )is non-zero if and only if there is k : z −→ x in U V with f ( z ) = 0, in whichcase ( U V ) f ( x ) = f ( z ). But then k = gh for unique g ∈ U and h ∈ V with,say, g : y −→ x and h : z −→ y . We deduce that ( U V ) f ( x ) = U ( V f ( x )).This concludes the proof that e A is a Γ c ( G )-module. (cid:3) WISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY 19
Given an extension (3.3), the question of whether it is central (i.e., satisfies(3.4)) can be determined from the corresponding Γ c ( G )-module structure onΓ c ( A × G (0) ) in the inverse semigroup extensionΓ c ( A × G (0) ) Γ c ( ι ) −−−→ Γ c (Σ) Γ c ( ϕ ) −−−→ Γ c ( G ) . (3.5) Proposition 3.16.
An extension (3.3) is central if and only if the semigroupisomorphism γ : e A −→ Γ c ( A × G (0) ) of Proposition 3.14 is an isomorphism of Γ c ( G ) -modules with respect to the module structure on Γ c ( A × G (0) ) comingfrom the extension (3.5) .Proof. Note that ϕι ( γ ( f )) = ϕ ( ι ( U f )) = supp( f ) by construction. So wejust need to check that Γ c ( G )-equivariance is equivalent to centrality. Fixa section j : Γ c ( G ) −→ Γ c (Σ) of Γ c ( ϕ ) with j | E (Γ c ( G )) = (Γ c ( ϕ ) | E (Γ c (Σ)) ) − .Then the action of V ∈ Γ c ( G ) on γ ( f ) = U f is given by V U f = ι − ( j ( V ) ι ( U f ) j ( V ) ∗ ) . This set consists of all ( a, x ) such that there is y ∈ supp( f ) and s ∈ j ( V )with ϕ ( s ) : y −→ x and ( a, x ) = ι − ( sι ( f ( y ) , y ) s − ). On the other hand,( V f )( x ) = 0 if and only if there is g ∈ V and y ∈ supp( f ), with g : y −→ x and then ( V f )( x ) = f ( y ). Since ϕ | j ( V ) : j ( V ) −→ V is a homeomorphism byProposition 3.1, this is equivalent to there being s ∈ j ( V ) and y ∈ supp( f )with ϕ ( s ) : y −→ x and then ( V f )( x ) = f ( y ). Thus V U f = U V f = γ ( V f )if and only if ι ( f ( d ( ϕ ( s ))) , r ( ϕ ( s ))) = sι ( f ( d ( ϕ ( s ))) , d ( ϕ ( s ))) s − for all s ∈ j ( V ) with d ( ϕ ( s )) ∈ supp( f ). This equivalent to ι ( f ( d ( ϕ ( s ))) , r ( ϕ ( s ))) s = sι ( f ( d ( ϕ ( s ))) , d ( ϕ ( s ))) (3.6)for all s ∈ j ( V ) with d ( ϕ ( s )) ∈ supp( f ), which clearly is satisfied if theextension is central. Hence if the extension is central, γ is an isomorphismof modules.Conversely, if γ is a module isomorphism and s ∈ Σ, a ∈ A , then wecan choose a bisection V containing ϕ ( s ) and we may assume that j ( V )was chosen to contain s . We can define f = a d ( V ) . Then, by (3.6), theequivariance of γ implies that ι ( a, r ( ϕ ( s ))) s = ι ( f ( d ( ϕ ( s ))) , r ( ϕ ( s ))) s = sι ( f ( d ( ϕ ( s ))) , d ( ϕ ( s ))) = sι ( a, d ( ϕ ( s ))). We conclude that the extension iscentral. (cid:3) We may now deduce from Theorem 3.4, Theorem 3.12 and Proposi-tion 3.16 the following.
Corollary 3.17.
There is a bijection between equivalence classes of A -twistsover Γ and extensions e A −→ T −→ Γ c ( G ) with T a Boolean inverse semi-group and with the Γ c ( G ) -module structure on e A as per Proposition 3.15. To complete our classification of A -twists by cohomology classes, we needthat every extension of e A by Γ c ( G ) is a Boolean inverse semigroup. In fact,it turns out every extension of a commutative Boolean inverse semigroup bya Boolean inverse semigroup is Boolean. Proposition 3.18.
Let S be a Boolean inverse semigroup and K a commu-tative Boolean inverse semigroup. If K ι −→ T ϕ −−→ S is any extension of S by K , then T is a Boolean inverse semigroup.Proof. Choose a section j : S −→ T with j | E ( S ) = ( ϕ | E ( T ) ) − . We canthen turn K into an S -module (with p = ϕι : K −→ E ( S )); denote by c : S × S −→ K the corresponding normalized 2-cocycle. Since E ( T ) ∼ = E ( S )is a Boolean algebra, we just need to show that if t, u ∈ T are orthogonal,then they have a join. In fact, it is enough to show they have a commonupper bound y for then, by [27, Proposition 1.4.18], one has y ( t ∗ t ∨ u ∗ u ) = yt ∗ t ∨ yu ∗ u = t ∨ u . Put e t = ϕ ( t ) and e u = ϕ ( u ). Then the fact that t, u areorthogonal is equivalent to t ∗ t, u ∗ u and tt ∗ , uu ∗ being orthogonal. This inturn is equivalent to e t, e u being orthogonal since ϕ is idempotent bijective.Thus x = e t ∨ e u exists. We can write t = ι ( k t ) j ( e t ) and u = ι ( k u ) j ( e u ) for unique k t , k u ∈ K with p ( k t ) = e t e t ∗ and p ( k u ) = e u e u ∗ . Then p ( k t c ( x, e t ∗ e t ) ∗ ) = e t e t ∗ and p ( k u c ( x, e u ∗ e u ) ∗ ) = e u e u ∗ and hence, since p is idempotent bijective, we have k = k t c ( x, e t ∗ e t ) ∗ ∨ k u c ( x, e u ∗ e u ) ∗ is defined and p ( k ) = e t e t ∗ ∨ e u e u ∗ = xx ∗ (since p is additive, being idempotent bijective). Let y = ι ( k ) j ( x ). We claim that t, u ≤ y . The argument is symmetric, so we just handle the case of t .Note that ι ( K ) centralizes E ( T ) as ι ( k ) e = ι ( kf ) = ι ( f k ) = eι ( k ) where f ∈ E ( K ) is unique with ι ( f ) = e (using that ι is idempotent bijective). Wethen have yt ∗ t = ι ( k ) j ( x ) j ( e t ∗ e t ) = ι ( k ) ι ( c ( x, e t ∗ e t )) j ( x e t ∗ e t ) = ι ( kc ( x, e t ∗ e t )) j ( e t ) . Therefore, it suffices to show that kc ( x, e t ∗ e t ) = k t . Note that p ( c ( x, e t ∗ e t )) = e t ∗ e t and so k u c ( x, e u ∗ e u ) ∗ c ( x, e t ∗ e t ) = 0. Therefore, kc ( x, e t ∗ e t ) = k t c ( x, e t ∗ e t ) ∗ c ( x, e t ∗ e t ) = k t , since multiplication distributes over joins in a Boolean inverse semigroup,as required. This completes the proof. (cid:3) Putting together Theorem 3.4, Theorem 3.12, Corollary 3.17, Proposi-tion 3.18, and [26, Section 7], we have the following classification of A -twists. Theorem 3.19.
Let G be an ample groupoid and A a discrete abelian group.Then there is a bijection between equivalence classes of discrete A -twists and H (Γ c ( G ) , C c ( G (0) , A ∪ { } )) . Note that the class of the trivial 2-cocycle corresponds to the trivial ex-tension A × G (0) −→ A × G −→ G . Indeed, Γ c ( A × G ) can be identifiedwith the full restricted semidirect product e A ⊲⊳ Γ c ( G ) via the mapping thatsends ( f, U ) ∈ e A ⊲⊳ Γ c ( G ) with supp( f ) = r ( U ) to the compact open bi-section { ( f ( r ( g )) , g ) | g ∈ U } of A × G , as is easily checked. The abeliangroup structure on H (Γ c ( G ) , C c ( G (0) , A ∪ { } )) corresponds to the Baersum operation on A -twists over G . We state the result, but omit many ofthe routine verifications as it will not be needed in the sequel.If A × G (0) Σ G , A × G (0) Σ ′ G ι ϕ ι ′ ϕ ′ WISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY 21 are A -twists, then since ϕ and ϕ ′ are local homeomorphisms by Propo-sition 3.1, the projections from the pullback Σ × ϕ,ϕ ′ Σ ′ along ϕ, ϕ ′ to Σand Σ ′ are local homeomorphisms and hence the pullback is an amplegroupoid. The Baer sum Σ ⊕ Σ ′ is the groupoid Σ × ϕ,ϕ ′ Σ ′ /A where A acts on Σ × ϕ,ϕ ′ Σ ′ by a ( s, s ′ ) = ( a − s, as ′ ) and we use the quotient topology.This a properly discontinuous action of a discrete group and so the quotientmap (Σ × ϕ,ϕ ′ Σ ′ ) −→ Σ ⊕ Σ ′ is a covering map, hence a local homeomor-phism. This equivalence relation is compatible with multiplication and sogives a groupoid structure which makes Σ ⊕ Σ ′ an ample groupoid. We leavethe straightforward verifications to the interested reader. The inclusion κ of A × G (0) into Σ ⊕ Σ ′ sends ( a, x ) to [ x, ax ] and the quotient map ρ fromΣ ⊕ Σ ′ to G sends [ s, s ′ ] to ϕ ( s ) = ϕ ′ ( s ′ ). Theorem 3.20.
The set of equivalence classes of A -twists over G underBaer sum is an abelian group isomorphic to H (Γ c ( G ) , C c ( G (0) , A ∪ { } )) .Proof. Using Theorem 3.19 we just need to show that the cohomology classcorresponding to the equivalence class of Σ ⊕ Σ ′ is the product of the co-homology classes corresponding to Σ and Σ ′ . Let j : Γ c ( G ) −→ Γ c (Σ)and j ′ : Γ c ( G ) −→ Γ c (Σ ′ ) be set theoretic sections with j and j ′ preserv-ing idempotents. Then the cohomology classes corresponding to Σ andΣ ′ are given by normalized 2-cycles c, c ′ , respectively, with j ( U ) j ( V ) = ι ( γ ( c ( U, V ))) j ( U V ) and j ′ ( U ) j ′ ( V ) = ι ′ ( γ ( c ′ ( U, V ))) j ′ ( U V ) where γ : e A −→ Γ c ( A × G (0) ) is the isomorphism of Proposition 3.16. Define e j : Γ c ( G ) −→ Γ c (Σ ⊕ Σ ′ ) by e j ( U ) = { [ g, g ′ ] | g ∈ j ( U ) , g ′ ∈ j ′ ( U ) } . This is a compact open set since it is the image of j ( U ) × ϕ,ϕ ′ j ′ ( U ), which iscompact open in Σ × ϕ,ϕ ′ Σ ′ . It is a bisection because if d ([ g, g ′ ]) = d ([ h, h ′ ])with g, h ∈ j ( U ) and g ′ , h ′ ∈ j ′ ( U ), then d ( g ) = d ( h ) and d ( g ′ ) = d ( h ′ )and so g = g ′ and h = h ′ . Similarly, r | e j ( U ) is injective. By construction ρ ( e j ( U )) = U . Also, if U ⊆ G (0) , then e j ( U ) ⊆ (Σ ⊕ Σ ′ ) (0) . Let us computethe normalized 2-cocycle e c corresponding to e j .Let U, V ∈ Γ c ( G ). Then e j ( U ) e j ( V )( e j ( U V )) ∗ consists of all composableproducts [ g, g ′ ][ h, h ′ ][ k − , ( k ′ ) − ] with g ∈ j ( U ), h ∈ j ( V ), k ∈ j ( U V ), g ′ ∈ j ′ ( U ), h ′ ∈ j ′ ( V ) and k ′ ∈ j ′ ( U V ). Then since
U, V are bisections, itfollows from composability that ϕ ( g ) ϕ ( h ) = ϕ ( k ) and ϕ ′ ( g ′ ) ϕ ′ ( h ′ ) = ϕ ′ ( k ′ ).Therefore, gh = ak and g ′ h ′ = a ′ k ′ for some a, a ′ ∈ A . But using that j ( U ) j ( V ) = ι ( γ ( c ( U, V ))) j ( U V ) and j ′ ( U ) j ′ ( V ) = ι ′ ( γ ( c ′ ( U, V ))) j ′ ( U V ),we deduce that a = c ( U, V )( r ( gh )) and a ′ = c ′ ( U, V )( r ( gh )). Thus[ g, g ′ ][ h, h ′ ][ k − , ( k ′ ) − ] = [ c ( U, V )( r ( gh )) r ( gh ) , c ′ ( U, V )( r ( gh ) r ( gh )]= [ r ( gh ) , ( c ( U, V ) c ′ ( U, V ))( r ( gh )) r ( gh )]= κ (( c ( U, V ) c ′ ( U, V ))( r ( gh )) , r ( gh )) . It follows that e c ( U, V ) = c ( U, V ) c ′ ( U, V ), as required. (cid:3)
We remark that the easiest way to build an A -twist over G is to beginwith a normalized locally constant 2-cocycle c : G (2) −→ A (where G (2) isthe space of composable pairs of elements of G ). Being a 2-cocycle meansthat c ( h, k ) c ( g, hk ) = c ( g, h ) c ( gh, k ) and being normalized means c ( x, x ) =1 for all units x ∈ G (0) . From a normalized 2-cocycle, you can build atwist Σ = A × G with the product topology, d ( a, g ) = (1 , d ( g )), r ( a, g ) =(1 , r ( g )) and ( a, g )( b, h ) = ( abc ( g, h ) , gh ). The inverse is given by ( a, g ) − =( a − c ( g, g − ) − , g − ). The corresponding exact sequence is A × G (0) −→ Σ −→ G where the first map is the inclusion and the second is the projection. Oneeasily adapts [5, Proposition 4.8] to general A to show that an A -twist asin (3.3) is equivalent to one coming from a locally constant 2-cocycle on G if and only if there is a continuous section s : G −→ Σ with s ( G (0) ) ⊆ Σ (0) .By Proposition 3.8 this occurs if and only if Γ c ( ϕ ) : Γ c (Σ) −→ Γ c ( G ) admitsan order-preserving and idempotent-preserving section. Proposition 3.9 re-covers the folklore result that any twist over a second countable Hausdorffample groupoid comes from a 2-cocycle [5], and extends it to the paracom-pact case. 4. Inverse semigroup crossed products
It was observed in [7] that Steinberg algebras of Hausdorff ample groupoidsare skew inverse semigroup rings (see [20] for the non-Hausdorff case). Wewill now define a notion of inverses semigroup crossed product that capturestwisted Steinberg algebras. Even without a twist, our definition will lookdifferent than the definition of a skew inverse semigroup ring found in theliterature [6, 7], but it coincides with that definition for the case of so-calledspectral actions [24], which is what arises in the ample groupoid setting.4.1.
Actions.
Let R be a ring (associative, but not necessarily unital orcommutative). We say that a ring endomorphism ψ : R −→ R is proper if ψ ( R ) = Re with e a central idempotent of R (necessarily unique). Note thatif e, f ∈ E ( Z ( R )) are central idempotents, then Re ∩ Rf = Ref = ReRf . Proposition 4.1.
The proper ring endomorphisms of R form a semigroup End c ( R ) under composition. The idempotent proper endomorphisms arethose of the form ϕ e ( r ) = re with e ∈ E ( Z ( R )) and hence E (End c ( R )) isa commutative subsemigroup isomorphic to the Boolean algebra E ( Z ( R )) ofcentral idempotents of R (under meet). Moreover, ϕ ( Z ( R )) ⊆ Z ( R ) for anyproper endomorphism ϕ .Proof. For ϕ ∈ End c ( R ), we put ϕ ( R ) = Re ϕ with e ϕ ∈ E ( Z ( R )). Firstwe claim that if ϕ ∈ End c ( R ) and z ∈ Z ( R ), then ϕ ( z ) ∈ Z ( R ). In-deed, if r ∈ R , then re ϕ ∈ Re ϕ = ϕ ( R ) and so re ϕ = ϕ ( r ′ ) with r ′ ∈ R .Thus ϕ ( z ) r = ( ϕ ( z ) e ϕ ) r = ϕ ( z )( re ϕ ) = ϕ ( z ) ϕ ( r ′ ) = ϕ ( zr ′ ) = ϕ ( r ′ z ) = ϕ ( r ′ ) ϕ ( z ) = re ϕ ϕ ( z ) = rϕ ( z ). So if ϕ, ψ ∈ End c ( R ), then ϕ ( ψ ( R )) = WISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY 23 ϕ ( Re ψ ) = ϕ ( R ) ϕ ( e ψ ) = Re ϕ ϕ ( e ψ ) and e ϕ ϕ ( e ψ ) is a central idempotent as ϕ ( Z ( R )) ⊆ Z ( R ).Clearly, ϕ e ( r ) = re with e a central idempotent is a proper endomorphismwith image Re . Conversely, if ϕ ∈ End c ( R ) is an idempotent, then ϕ fixes ϕ ( R ) = Re ϕ . So if r ∈ R , then ϕ ( r ) = ϕ ( r ) e ϕ = ϕ ( r ) ϕ ( e ϕ ) = ϕ ( re ϕ ) = re ϕ ,as required. This completes the proof. (cid:3) Since End c ( R ) has commuting idempotents, the von Neumann regular el-ements of End c ( R ) form an inverse semigroup. Thus it is natural to consideractions of inverse semigroups on rings by proper endomorphisms. Definition 4.2.
We define an action of an inverse semigroup S on a ring R to be a homomorphism α : S −→ End c ( R ), written s α s . Often wewrite sr for α s ( r ). If e ∈ E ( S ), let 1 e be the central idempotent with α e ( R ) = R e . We say that action is non-degenerate if α is idempotentseparating and R = P e ∈ E ( S ) R e .A set E of idempotents of a ring R is called a set of local units if, forall finite subsets F of R , there is an idempotent e ∈ E with F ⊆ eRe . If S admits a non-degenerate action on R , then it is easy to check that theBoolean algebra generated by the 1 e with e ∈ E ( S ) is a set of local units for R . If S is a Boolean inverse semigroup and if α is additive, then the set of1 e with e ∈ E ( S ) already is a set of local units. Proposition 4.3.
Let S have a non-degenerate action α on R . (1) α s ( R ) = R ss ∗ . (2) α s ( R e ) = R ses ∗ .Proof. We have that α s ( R ) = α ss ∗ ( α s ( R )) ⊆ R ss ∗ . On the other hand,since α ss ∗ ( r ) = r ss ∗ by Proposition 4.1, we have r ss ∗ = α ss ∗ ( r ) = α s ( α s ∗ ( r )) ∈ α s ( R ) . This establishes the first item. The second follows from the first because α s ( R e ) = α s α e ( R ) = α se ( R ) = R ses ∗ . (cid:3) We typically are interested in the case that S has a zero and α preserves0 (so α ( r ) = 0 for all r ∈ R ). We call such an action zero-preserving .One should note that α s restricts to an isomorphism R s ∗ s −→ R ss ∗ , asis easily checked.To define a notion of crossed product, we need to next consider twists. Solet S have a non-degenerate action on a ring R . Put e R = S e ∈ E ( S ) ( Z ( R )1 e ) × . Proposition 4.4.
The set e R is a commutative inverse semigroup undermultiplication with E ( e R ) = { e | e ∈ E ( S ) } . Moreover, if s ∈ S , then α s ( e R ) ⊆ e R and hence S acts on e R by endomorphisms.Proof. Given r, r ′ ∈ ( Z ( R )1 e ) × and u, u ′ ∈ ( Z ( R )1 f ) × with rr ′ = 1 e and uu ′ = 1 f , we have that ru, u ′ r ′ ∈ Z ( R )1 ef and ruu ′ r ′ = r f r ′ = rr ′ f = e f = 1 ef . Thus ru, u ′ r ′ ∈ e R and ( ru )( u ′ r ′ )( ru ) = ru . Therefore, e R ⊆ Z ( R ) is a commutative von Neumann regular semigroup and hence inverse.Clearly, E ( e R ) = { e | e ∈ E ( S ) } .If s ∈ S and r, r ′ ∈ ( Z ( R )1 e ) × with rr ′ = 1 e , then α s ( r ) α s ( r ′ ) = α s (1 e ) =1 ses ∗ , and so α s ( r ) ∈ ( Z ( R )1 ses ∗ ) × (using Propositions 4.1 and 4.3). (cid:3) We now have that if the action is non-degenerate, then e R is an S -modulewhere we put p ( r ) = e if r ∈ ( Z ( R )1 e ) × by Proposition 4.3(2).4.2. Crossed products.
We continue to work in the context of an inversesemigroup S with a non-degenerate action α : S −→ End c ( R ) on R . Fix anormalized 2-cocycle c : S × S −→ e R . To define the crossed product R ⋊ α,c S ,we proceed in two steps. Proposition 4.5.
Let S be an inverse semigroup and let α be a non-degenerate action of S on a ring R . Let c : S × S −→ e R be a normalized -cocycle. (1) The abelian group L s ∈ S Rδ s (here δ s is an indexing symbol) is aring with product defined by rδ s · r ′ δ t = r ( sr ′ ) c ( s, t ) δ st . (2) The additive subgroup I generated by rδ s − rc ( t, s ∗ s ) ∗ δ t with s ≤ t isa two-sided ideal. If S has a zero and the action is zero-preserving,then I contains Rδ .Proof. For the first item, it suffices to check associativity. We have(( r δ s )( r δ t )) r δ u = r ( sr ) c ( s, t )( str ) c ( st, u ) δ stu ,r δ s ( r δ t r δ u ) = r s ( r ( tr ) c ( t, u )) c ( s, tu ) δ stu = r ( sr )( str )( sc ( t, u )) c ( s, tu ) δ stu . Since c takes values in the center of R , associativity follows from c being a2-cocycle.The second item is more technical. Consider s ≤ t ∈ S and r ∈ R . If u ∈ S and a ∈ R , then aδ u ( rδ s − rc ( t, s ∗ s ) ∗ δ t ) = a ( ur ) c ( u, s ) δ su − a ( ur )( uc ( t, s ∗ s ) ∗ ) c ( u, t ) δ ut . We have su ≤ ut and so it suffices to show that c ( u, s ) c ( ut, s ∗ u ∗ us ) ∗ =( uc ( t, s ∗ s ) ∗ ) c ( u, t ), which is the content of Proposition 3.11(10). On theother hand,( rδ s − rc ( t, s ∗ s ) ∗ δ t ) aδ u = r ( sa ) c ( s, u ) δ su − rc ( t, s ∗ s ) ∗ ( ta ) c ( t, u ) δ tu . Note that c ( t, s ∗ s ) ∈ R ss ∗ and so c ( t, s ∗ s ) ∗ ( ta ) = c ( t, s ∗ s ) ∗ ss ∗ ( ta ) = c ( t, s ∗ s ) ∗ ( ta )1 ss ∗ = c ( t, s ∗ s ) ∗ ( ss ∗ ( ta )) = c ( t, s ∗ s ) ∗ ( sa ). Thus it suffices toshow that r ( sa ) c ( s, u ) δ su − r ( sa ) c ( t, s ∗ s ) ∗ c ( t, u ) δ tu ∈ I . Since su ≤ tu , itsuffices to show that c ( s, u ) c ( tu, u ∗ s ∗ su ) = c ( t, s ∗ s ) ∗ c ( t, u ), which followsfrom Proposition 3.11(11). This completes the proof that I is an ideal.If α is zero preserving, then since 0 ≤
0, we have that rδ − rc (0 , ∗ δ ∈ I .But c (0 ,
0) = 0 as 1 = 0 and so ( Z ( R )1 ) × = 0. Thus rδ ∈ I . (cid:3) WISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY 25
We can now define the crossed product.
Definition 4.6.
Let R be a ring, S an inverse semigroup and α : S −→ End c ( R ) a non-degenerate action. Let c : S × S −→ e R be a normalized 2-cocycle. Then the crossed product R ⋊ α,c S is the ring L s ∈ S Rδ s /I wherewe retain the notation of Proposition 4.5.It is relatively straightforward to verify that two cohomologous normalized2-cocycles yield isomorphic crossed products. Proposition 4.7.
Let α : S −→ End c ( R ) be a non-degenerate action of S on R . Let c, c ′ : S × S −→ e R be normalized two-cocycles. If c and c ′ arecohomologous, then R ⋊ α,c S ∼ = R ⋊ α,c ′ S .Proof. Let F : S −→ e R be a mapping with F ( s ) ∈ ( Z ( R )1 ss ∗ ) × , for all s ∈ S , and with c = c ′ · δF . So c ( s, t ) = c ′ ( s, t ) F ( s )( sF ( t )) F ( st ) ∗ . Since c, c ′ are normalized, c ( e, e ) = 1 e = c ′ ( e, e ) for e ∈ E ( S ). Thus we have that F ( e ) = c ′ ( e, e ) F ( e )( eF ( e )) F ( ee ) ∗ = c ( e, e ) = 1 e and so F is idempotent preserving.We put A = L s ∈ S Rδ s with product rδ s · r ′ δ t = r ( sr ′ ) c ( s, t ) δ st and let I be the additive subgroup generated by all rδ s − rc ( t, s ∗ s ) ∗ δ t with s ≤ t . Sim-ilarly, we let A ′ = L s ∈ S Rδ s with multiplication rδ s · r ′ δ t = r ( sr ′ ) c ′ ( s, t ) δ st and I ′ be the additive subgroup generated by all rδ s − rc ′ ( t, s ∗ s ) ∗ δ t with s ≤ t . We first construct a homomorphism Φ : A −→ A ′ with Φ( I ) ⊆ I ′ .Define Φ( rδ s ) = rF ( s ) δ s . Note that Φ( rδ s · r ′ δ t ) = r ( sr ′ ) c ( s, t ) F ( st ) δ st ,whereas Φ( rδ s )Φ( r ′ δ t ) = rF ( s ) δ s · r ′ F ( t ) δ t = r ( sr ′ ) F ( s )( sF ( t )) c ′ ( s, t ) δ st .But F ( st ) ∗ F ( st ) = 1 stt ∗ s ∗ and c ′ ( s, t ) ∈ R stt ∗ s ∗ , whence c ′ ( s, t ) = F ( st ) ∗ F ( st ) c ′ ( s, t ) . Therefore, we have thatΦ( rδ s )Φ( r ′ δ s ) = r ( sr ′ ) F ( s )( sF ( t )) F ( st ) ∗ c ′ ( s, t ) F ( st ) δ st = r ( sr ′ ) c ( s, t ) F ( st ) δ st = Φ( rδ s · r ′ δ t ) . We conclude that Φ is a homomorphism.Suppose now that s ≤ t and note that Φ( rδ s − rc ( t, s ∗ s ) ∗ δ t ) = rF ( s ) δ s − rc ( t, s ∗ s ) ∗ F ( t ) δ t . But c ( t, s ∗ s ) ∗ = c ′ ( t, s ∗ s ) ∗ F ( t ) ∗ ( tF ( s ∗ s )) ∗ F ( ts ∗ s ) = c ′ ( t, s ∗ s ) ∗ F ( t ) ∗ ss ∗ F ( s )as F ( s ∗ s ) = 1 s ∗ s and ts ∗ s = s . Thus, using 1 tt ∗ ≥ ss ∗ , we have rc ( t, s ∗ s ) ∗ F ( t ) = rF ( s ) c ′ ( t, s ∗ s ) ∗ F ( t ) ∗ F ( t )1 ss ∗ = rF ( s ) c ′ ( t, s ∗ s ) ∗ . Therefore, rF ( s ) δ s − rc ( t, s ∗ s ) ∗ F ( t ) δ t = rF ( s ) δ s − rF ( s ) c ′ ( t, s ∗ s ) ∗ δ t ∈ I ′ ,and so we have a well-defined homomorphism ϕ : R ⋊ α,c S −→ R ⋊ α,c ′ S given by ϕ ( rδ s + I ) = Φ( rδ s ) + I ′ = rF ( s ) δ s + I ′ . Similarly, we have awell-defined homomorphism ψ : R ⋊ α,c ′ S −→ R ⋊ α,c S given by ψ ( rδ s + I ′ ) = rF ( s ) ∗ δ s + I . We show that these two homomorphisms are inverseto each other. By symmetry, it suffices to consider ψ ◦ ϕ . We have that ψϕ ( rδ s + I ) = ψ ( rF ( s ) δ s + I ′ ) = rF ( s ) F ( s ) ∗ δ s + I = r ss ∗ δ s + I . Butsince s ≤ s and c ( s, s ∗ s ) = 1 ss ∗ by Proposition 3.11(1), it follows that rδ s − r ss ∗ δ s = rδ s − rc ( s, s ∗ s ) ∗ δ s ∈ I . Thus ψϕ ( rδ s + I ) = rδ s + I . Thiscompletes the proof. (cid:3) When the 2-cocycle c is trivial, then R ⋊ α,c S is called a skew inversesemigroup ring . Although it looks superficially different, this notion of askew inverse semigroup ring coincides with the standard one for what arecalled spectral actions in [24]. Note that when c is trivial, I is generated byall differences rδ s − r ss ∗ δ t with s ≤ t .The following two results yield a partial generalization of a result for skewinverse semigroup rings [6, Proposition 3.1]. Proposition 4.8.
Let α : S −→ End c ( R ) be a non-degenerate action and c : S × S −→ e R a normalized -cocycle. Then the additive subgroup of thering R ⋊ α,c S generated by the elements rδ e + I with r ∈ R and e ∈ E ( S ) is a subring isomorphic to a quotient of R . If there is an additive grouphomomorphism τ : R ⋊ α,c S −→ R with τ ( rδ e + I ) = r for all r ∈ R e and e ∈ E ( S ) , then this subring is isomorphic to R .Proof. Let A be the additive subgroup generated by the rδ e + I with r ∈ R and e ∈ E ( S ). Since rδ e · r ′ δ f + I = r ( er ′ ) c ( e, f ) δ ef + I , it is clear that R isa subring. Suppose that r ∈ R . Since the action is non-degenerate, we maywrite r = P ni =1 r i with r i ∈ R e i and e i ∈ E ( S ), for i = 1 , . . . , n . Define ρ ( r ) = P ni =1 r i δ e i + I . We claim that ρ is a well-defined homomorphism from R to A . To show that ρ is well defined, it suffices to show that if 0 = P ni =1 r i with r i ∈ R e i , then P ni =1 r i δ e i ∈ I . We proceed by induction on n , withthe case n = 1 being trivial, as then r = 0 and so r δ e = 0 ∈ I . Supposenow that the result is true for n − P ni =1 r i with r i ∈ R e i with n ≥
2. Then r n = − P n − i =1 r i and so r n = r n e n = − P n − i =1 r i e n . Note that r i − r i e n ∈ R e i and 0 = P n − i =1 ( r i − r i e n ). By induction, we deduce that P n − i =1 ( r i − r i e n ) δ e i ∈ I , i.e., n − X i =1 r i δ e i + I = n − X i =1 r i e n δ e i + I. (4.1)Since e i e n ≤ e i , we have that r i δ e i e n − r i e n δ e i ∈ I , as r i c ( e i , e i e n ) ∗ = r i e i e n = r i e n by Proposition 3.11(4). On the other hand, since e i e n ≤ e n ,we have that r i δ e i e n − r i e n δ e n ∈ I as r i c ( e n , e i e n ) ∗ = r i e i e n = r i e n .Therefore, r i e n δ e i + I = r i e n δ e n + I . Thus, we deduce from (4.1) that n − X i =1 r i δ e i + I = n − X i =1 r i e n δ e n + I = − r n δ e n + I. It follows that P ni =1 r i δ e i ∈ I , as required.To verify that ρ is a homomorphism, let r = P ni =1 r i and r ′ = P mj =1 r ′ j with r i ∈ R e i and r j ∈ R f j with e i , f j ∈ E ( S ) for 1 ≤ i ≤ n , 1 ≤ j ≤ WISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY 27 m . Then rr ′ = P ni =1 P mj =1 r i r ′ j and r i r ′ j ∈ R e i f j . Therefore, ρ ( rr ′ ) = P ni =1 P mj =1 r i r ′ j δ e i f j + I . On the other hand, ρ ( r ) ρ ( r ′ ) = n X i =1 r i δ e i m X j =1 r j δ f j + I = n X i =1 m X j =1 r i ( e i r j ) c ( e i , f j ) δ e i f j + I. But e i r j = r j e i and c ( e i , f j ) = 1 e i f j by Proposition 3.11(4) and so r i ( e i r j ) c ( e i , f j ) = r i r j e i e i f j = r i r j . Therefore, ρ is a homomorphism.To check that ρ is surjective, notice that since e ≤ e , we have that rδ e + I = rc ( e, e ) ∗ δ e + I = r e δ e + I = ρ ( r e ). Suppose that τ , as in the final statement,exists. If r ∈ R with r = P ni =1 r i with r i ∈ R e i , for i = 1 , . . . , n , then τ ( ρ ( r )) = τ ( P ni =1 r i δ e i + I ) = P ni =1 r i = r and so ρ is injective. (cid:3) We suspect that ρ is injective in general, but we could only prove theexistence of the additive homomorphism τ in certain cases. Proposition 4.9.
Let α : S −→ End c ( R ) be a non-degenerate action and c : S × S −→ e R a normalized -cocycle. Then an additive group homomor-phism τ : R ⋊ α,c S −→ R with τ ( rδ e + I ) = r for all r ∈ R e exists in thefollowing cases. (1) If c ( t, s ∗ s ) = 1 ss ∗ whenever s ≤ t (i.e., if c is trivial, that is, R ⋊ α,c S is a skew inverse semigroup ring). (2) If, for each s ∈ S , either the set E ( s ) = { e ∈ E ( S ) | e ≤ s } is emptyor has a unique maximum element e ( s ) .Proof. Suppose the first item holds. Define a homomorphism of additivegroups τ ′ : L s ∈ S Rδ s −→ R by rδ s r ss ∗ . Then a generator for I is ofthe form rδ s − r ss ∗ δ t with s ≤ t , as c ( t, s ∗ s ) ∗ = 1 ss ∗ , and τ ′ ( rδ s − r ss ∗ δ t ) = r ss ∗ − r ss ∗ tt ∗ = 0 since s ≤ t . Therefore, τ ′ induces an additive grouphomomorphism τ : R ⋊ α,c S −→ R . Moreover, τ ( rδ e + I ) = r e = r for r ∈ R e with e ∈ E ( S ).Now, suppose that each set E ( s ) is either empty or has a maximum e ( s ).Define a homomorphism of additive groups τ ′ : L s ∈ S Rδ s −→ R by τ ′ ( rδ s ) = ( rc ( s, e ( s )) , if E ( s ) = ∅ , else.We need to check that I ⊆ ker τ ′ . Suppose that s ≤ t . Obviously E ( s ) ⊆ E ( t ). On the other hand, if e ∈ E ( t ), then es ∗ s ≤ ts ∗ s = s and so es ∗ s ∈ E ( s ). Thus E ( s ) = ∅ if and only if E ( t ) = ∅ . In particular, if s ≤ t and E ( s ) = ∅ = E ( t ), then τ ′ ( sδ s − sc ( t, s ∗ s ) ∗ δ t ) = 0. So let us assumethat E ( s ) and E ( t ) are non-empty. Then e ( s ) ≤ s ≤ t implies e ( s ) ≤ e ( t )and e ( s ) ≤ s ∗ s . Thus e ( s ) ≤ e ( t ) s ∗ s . But e ( t ) s ∗ s ≤ ts ∗ s = s , and so e ( t ) s ∗ s ≤ e ( s ). Therefore, e ( t ) s ∗ s = e ( s ). A similar argument shows that ss ∗ e ( t ) = e ( s ). We now compute ( tc ( s ∗ s, e ( s ))) c ( t, s ∗ se ( s )) = c ( t, s ∗ s ) c ( ts ∗ s, e ( s )), whichyields c ( t, e ( s )) = c ( t, s ∗ s ) c ( s, e ( s )) since c is normalized (using Proposi-tion 3.11(4)). But( tc ( e ( t ) , s ∗ s )) c ( t, e ( t ) s ∗ s ) = c ( t, e ( t )) c ( te ( t ) , s ∗ s ) = c ( t, e ( t )) c ( e ( t ) , s ∗ s ) . As e ( t ) s ∗ s = e ( s ), we deduce, using Proposition 3.11(4), that c ( t, e ( s )) = c ( t, e ( t ))1 e ( s ) = e ( s ) c ( t, e ( t )). Therefore, e ( s ) c ( t, e ( t )) = c ( t, s ∗ s ) c ( s, e ( s )).Also note that p ( c ( t, e ( t ))1 ss ∗ ) = ss ∗ e ( t ) = e ( s ) and so c ( t, e ( t ))1 ss ∗ = e ( s )( c ( t, e ( t ))1 ss ∗ ) = e ( s )( ss ∗ c ( t, e ( t ))) = e ( s ) c ( t, e ( t )) = c ( t, s ∗ s ) c ( s, e ( s )).Thus we have τ ′ ( rδ s − rc ( t, s ∗ s ) ∗ δ t ) = rc ( e, s ) − rc ( t, s ∗ s ) ∗ c ( t, e ( t ))= rc ( e, s ) − rc ( t, s ∗ s ) ∗ ss ∗ c ( t, e ( t ))= rc ( e, s ) − rc ( t, s ∗ s ) ∗ c ( t, s ∗ s ) c ( s, e ( s ))= 0as ss ∗ ≥ e ( s ). Therefore, τ ′ induces a well defined map τ : R ⋊ α,c S −→ R with τ ( rδ e + I ) = τ ′ ( rδ e ) = rc ( e, e ) = r e = r if r ∈ R e , as c isnormalized. (cid:3) In addition to the case that c is trivial, we note that if (3.1) is an extensionof inverse semigroups which admits an order-preserving and idempotent-section j : S −→ T , then the corresponding normalized 2-cocycle c satisfies c ( t, s ∗ s ) = 1 ss ∗ when s ≤ t since j ( t ) j ( s ∗ s ) = j ( ts ∗ s ) = j ( s ) by Lemma 3.7.Recall that an inverse semigroup S is E -unitary if s ≥ e with e ∈ E ( S )implies s ∈ E ( S ) and S is 0- E -unitary if S has a zero and s ≥ e = 0with e ∈ E ( S ) implies s ∈ E ( S ); see [27] for details. Examples of inversesemigroups satisfying the second condition of Proposition 4.9 include E -unitary and 0- E -unitary inverse semigroups and the inverse semigroup ofcompact open bisections of a Hausdorff ample groupoid G .We now establish a universal property for the crossed product, general-izing that of skew inverse semigroup rings [24]. Let S be an inverse semi-group with a non-degenerate action α : S −→ End c ( R ) on a ring R and let c : S × S −→ e R be a normalized 2-cocycle. Then a covariant representation of ( α, c ) in a ring A consists of a ring homomorphism ρ : R −→ A and a map ψ : S −→ A such that:(C1) ψ ( s ) ρ ( r ) = ρ ( sr ) ψ ( s );(C2) ρ (1 e ) = ψ ( e );(C3) ψ ( s ) ψ ( t ) = ρ ( c ( s, t )) ψ ( st ); and(C4) ψ ( ss ∗ ) ψ ( s ) = ψ ( s )for all s, t ∈ S , r ∈ R and e ∈ E ( S ). Note that (C4) is equivalent to(C4’) ψ ( s ) ψ ( s ∗ s ) = ψ ( s )in the presence of (C1), (C2). Indeed, ψ ( s ) ψ ( s ∗ s ) = ψ ( s ) ρ (1 s ∗ s ) = ρ (1 ss ∗ ) ψ ( s ) = ψ ( ss ∗ ) ψ ( s ) = ψ ( s ) WISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY 29 by (C1), (C2) and (C4). So (C4) implies (C4’) and the reverse implicationis dual. In the case that c is trivial, the conjunction of (C3) and (C4) isequivalent to ψ being a semigroup homomorphism, and so the notion of co-variant representation for skew inverse semigroup rings given here coincideswith that considered in [24]. Proposition 4.10.
Let S be an inverse semigroup with a non-degenerateaction α : S −→ End c ( R ) on a ring R and c : S × S −→ e R a normalized -cocycle. Let ρ : R −→ R ⋊ α,c S be the homomorphism from the proof ofProposition 4.8 and let ψ : S −→ R ⋊ α,c S be given by ψ ( s ) = 1 ss ∗ δ s + I .Then ( ρ, ψ ) is a covariant representation and it is the universal covariantrepresentation, that is, if ρ ′ : R −→ A and ψ ′ : S −→ A give a covariantrepresentation, then there is a unique homomorphism π : R ⋊ α,c S −→ A such that ρ ′ = πρ and ψ ′ = πψ . Namely, one has π ( rδ s + I ) = ρ ′ ( r ) ψ ′ ( s ) .Proof. We first check that ρ, ψ yield a covariant representation. If r ∈ R with r = P ni =1 r i with r i ∈ R e i , then ρ ( r ) = P ni =1 r i δ e i + I and so ψ ( s ) ρ ( r ) = n X i =1 ss ∗ δ s · r i δ e i + I = n X i =1 ( sr i ) c ( s, e i ) δ se i + I. (4.2)One the other hand, sr = P ni =1 sr i and sr i ∈ R se i s ∗ . Therefore, ρ ( sr ) ψ ( s ) = n X i =1 sr i δ se i s ∗ · ss ∗ δ s + I = n X i =1 ( sr i ) c ( se i s ∗ , s ) δ se i + I. (4.3)It follows that the right hand sides of (4.2) and (4.3) are equal by Proposi-tion 3.11(9), yielding (C1). For (C2), note that ρ (1 e ) = 1 e δ e + I = ψ ( e ) bydefinition. For (C3), we compute ψ ( s ) ψ ( t ) = 1 ss ∗ δ s · tt ∗ δ t + I = 1 ss ∗ ( s tt ∗ ) c ( s, t ) δ st + I = c ( s, t )1 stt ∗ s ∗ δ st + I = c ( s, t ) δ stt ∗ s ∗ · stt ∗ s ∗ δ st + I = ρ ( c ( s, t )) ψ ( st )since c ( s, t ) ∈ R stt ∗ s ∗ . Here we have used Proposition 4.3 to obtain s (1 tt ∗ ) =1 stt ∗ s ∗ and we have used Proposition 3.11(1) to get c ( stt ∗ s ∗ , st ) = 1 stt ∗ s ∗ .This verifies (C3). Finally, (C4) follows because ψ ( ss ∗ ) ψ ( s ) = 1 ss ∗ δ ss ∗ · ss ∗ δ s + I = 1 ss ∗ δ s + I by Proposition 3.11(1).Suppose now that ρ ′ : R −→ A and ψ ′ : S −→ A give rise to a covariantrepresentation. First we check that rδ s ρ ′ ( r ) ψ ′ ( s ) yields a well-definedhomomorphism π ′ : L s ∈ S Rδ s −→ A . Indeed, π ′ ( r δ s ) π ′ ( r δ s ) = ρ ′ ( r ) ψ ′ ( s ) ρ ′ ( r ) ψ ′ ( s ) = ρ ′ ( r ) ρ ′ ( s r ) ψ ′ ( s ) ψ ′ ( s )= ρ ′ ( r ( s r ) c ( s , s )) ψ ′ ( s s ) = π ′ ( r δ s · r δ s )by (C1) and (C3). We now check that I ⊆ ker π ′ . Let s ≤ t and r ∈ R . First notethat ts ∗ st ∗ = ss ∗ and so c ( t, s ∗ s ) = c ( ss ∗ , t ) = c ( ss ∗ , t )1 ss ∗ by Proposi-tion 3.11(9). Therefore, π ′ ( rc ( t, s ∗ s ) ∗ δ t ) = ρ ′ ( rc ( ss ∗ , t ) ∗ ss ∗ ) ψ ′ ( t ) = ρ ′ ( rc ( ss ∗ , t ) ∗ ) ρ ′ (1 ss ∗ ) ψ ′ ( t )= ρ ′ ( rc ( ss ∗ , t ) ∗ ) ψ ′ ( ss ∗ ) ψ ′ ( t ) = ρ ′ ( rc ( ss ∗ , t ) ∗ c ( ss ∗ , t )) ψ ′ ( s )= ρ ′ ( r ss ∗ ) ψ ′ ( s ) = ρ ′ ( r ) ρ ′ (1 ss ∗ ) ψ ′ ( s )= ρ ′ ( r ) ψ ′ ( ss ∗ ) ψ ′ ( s ) = ρ ′ ( r ) ψ ′ ( s )= π ′ ( rδ s )by (C2), (C3) and (C4). Thus I ⊆ ker π ′ and so π : R ⋊ α,c S −→ A givenby π ( rδ s + I ) = ρ ′ ( r ) ψ ′ ( s ) is well defined. If r ∈ R with r = P ni =1 r i where r i ∈ R e i , then π ( ρ ( r )) = P ni =1 π ′ ( r i δ e i ) = P ni =1 ρ ′ ( r i ) ψ ′ ( e i ) = P ni =1 ρ ′ ( r i ) ρ ′ (1 e i ) = ρ ′ ( r ) using (C2). Also π ( ψ ( s )) = π ′ (1 ss ∗ δ s ) = ρ ′ (1 ss ∗ ) ψ ′ ( s ) = ψ ′ ( ss ∗ ) ψ ′ ( s ) = ψ ′ ( s )using (C2) and (C4).For uniqueness, if γ : R × α,c S −→ A is a homomorphism with γρ = ρ ′ and γψ = ψ ′ , then since rδ s + I = r ss ∗ δ s + I as s ≤ s and c ( s, s ∗ s ) = 1 ss ∗ = c ( ss ∗ , s ) by Proposition 3.11(1), we have that γ ( rδ s + I ) = γ ( r ss ∗ δ s + I ) = γ ( r ss ∗ δ ss ∗ · ss ∗ δ s + I )= γ ( ρ ( r ss ∗ )) γ ( ψ ( s )) = ρ ′ ( r ss ∗ ) ψ ′ ( s )= ρ ′ ( r ) ψ ′ ( ss ∗ ) ψ ′ ( s ) = ρ ′ ( r ) ψ ′ ( s )by (C2) and (C4), as required. This completes the proof. (cid:3) We note that our notion of crossed product seems related to, but differ-ent from, the C ∗ -algebraic twisted action crossed product in [12]. In theirnotion, the action of an element of the inverse semigroup on a ring is onlypartially defined. Also, they loosen the requirement on how the action of aproduct of semigroup elements behaves. We could also allow greater gen-erality by letting α : S −→ End c ( R ) just be a map, not a homomorphism,and allowing the 2-cocycle c to take values in the non-commutative inversesemigroup S e ∈ E ( S ) ( R e ) × , but the conditions on α and c that yield associa-tivity are slightly more complicated and we shall not need this more generalconstruction here in any event as we shall primarily be interested in the casethat R is commutative, in which case the general construction will reduceto ours.4.3. Twisted Steinberg algebras.
Our goal is to show that twisted Stein-berg algebras of Hausdorff ample groupoids are inverse semigroup crossedproducts, generalizing the result of [7] for the case of untwisted Steinbergalgebras and skew inverse semigroup rings. Let G be a Hausdorff amplegroupoid and let R × × G (0) ι −→ Σ ϕ −−→ G WISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY 31 be a discrete R × -twist. The associated twisted Steinberg algebra A R ( G ; Σ)consists of all locally constant functions f : Σ −→ R which are R × -anti-equivariant, in the sense that f ( rs ) = r − f ( s ) for s ∈ Σ and r ∈ R × , andhave compact support modulo R × , that is, ϕ (supp( f )) is compact. Additionand the R -module structure are pointwise. The product is defined as follows.Choose a set theoretic section p : G −→ Σ (we do not assume continuity of p ). Then the convolution is given by f ∗ f ′ ( s ) = X r ( g )= r ( ϕ ( s ))) f ( p ( g )) f ′ ( p ( g ) − s ) . This does not depend on the choice of p . Note that we are following theconventions of [4] and not [5], which uses equivariant functions.Let A = C c ( G (0) , R ) be the ring of compactly supported locally constantmappings G (0) −→ R with pointwise operations. There is a natural zero-preserving action α : Γ c ( G ) −→ End c ( A ) given by α U ( f )( x ) = ( f ( d (( r | U ) − ( x ))) , if x ∈ r ( U )0 , else.Notice that the image of α U is A r ( U ) and that the action is non-degenerateand additive. Moreover, e A = C c ( G (0) , R × ∪ { } ) with the action we previ-ously considered in Proposition 3.15. (That α is an action is essentially thesame proof as in Proposition 3.15.) Hence, we can consider a normalized 2-cocycle c associated to the twist as per Theorem 3.19 and form the crossedproduct A ⋊ α,c Γ c ( G ) and the crossed product ring is independent of thechoice of 2-cocycle up to isomorphism. More precisely, we fix a set theoreticsection j : Γ c ( G ) −→ Γ c (Σ) with j | E (Γ c ( G )) = (Γ c ( ϕ ) | E (Γ c (Σ)) ) − . One thenhas j ( U ) j ( V ) = ι ( { ( c ( U, V )( x ) , x ) | x ∈ r ( U V ) } ) j ( U V )= { c ( U, V )( r ( ϕ ( s ))) s | s ∈ j ( U V ) } . Our goal it to show that the crossed product A ⋊ α,c Γ c ( G ) is isomorphicto A R ( G ; Σ). Note that L U ∈ Γ c ( G ) Aδ U and A ⋊ α,c Γ c ( G ) are R -algebraswith r ( aδ U ) = ( ra ) δ U and the corresponding induced R -algebra structureon the crossed product, as is easily checked. Note that Proposition 4.9 andProposition 4.8 apply in this setting to embed A into the crossed productsince G is Hausdorff.We proceed by first establishing a number of lemmas. We retain thenotation I for the two-sided ideal in Proposition 4.5(2). Lemma 4.11.
Let a ∈ A and U ∈ Γ c ( G ) . Let V = supp( a ) ∩ r ( U ) . Then aδ U + I = ac ( U, d ( V U )) δ V U + I and supp( ac ( U, d ( V U )) = r ( V U ) .Proof. We have that U ≤ U and so aδ U + I = ac ( U, d ( U )) ∗ δ U + I = a r ( U ) δ U + I by Proposition 3.11(1), as c is normalized. Then a r ( U ) = a V = a r ( V U ) by definition of V . Since V U ≤ U we have that ac ( U, d ( V U )) δ V U + I = ac ( U, d ( V U )) c ( U, d ( V U )) ∗ δ U + I = a r ( V U ) δ U + I = aδ U + I, by the above. Since supp( c ( U, d ( V U ))) = r ( V U ) ⊆ supp( a ), we have thatsupp( ac ( U, d ( V U )) = r ( V U ). (cid:3) Thus A ⋊ α,c Γ c ( G ) is spanned by elements of the form aδ U + I withsupp( a ) = r ( U ). Lemma 4.12.
Suppose that U ∈ Γ c ( G ) is a union of pairwise disjointcompact open bisections U , . . . , U k and supp( a ) = r ( U ) . Then aδ U + I = P ki =1 a i δ U i + I with supp( a i ) = r ( U i ) for i = 1 , . . . , k .Proof. Let a i = ac ( U, d ( U i )). Then supp( a i ) = supp( a ) ∩ supp( c ( U, d ( U i ))) =supp( a ) ∩ r ( U i ) = r ( U i ) as U i = U · d ( U i ). Next observe that since U i ≤ U , wehave that a i δ U i + I = ac ( U, d ( U i )) δ U i + I = ac ( U, d ( U i )) c ( U, d ( U i )) ∗ δ U + I = a r ( U i ) δ U + I and so P ki =1 a i δ U i + I = P ki =1 a r ( U i ) δ U + I = a r ( U ) δ U + I = aδ U + I as the r ( U i ) are pairwise disjoint (since U is a bisection) with union r ( U ). (cid:3) Our final lemma finds us a sort of normal form for A ⋊ α,c Γ c ( G ). Lemma 4.13.
Every element of A ⋊ α,c Γ c ( G ) is equal to one of the form P ki =1 a i δ U i + I where U , . . . , U k are pairwise disjoint non-empty compactopen bisections and supp( a i ) = r ( U i ) , for i = 1 , . . . , k with possibly k = 0 .Proof. By Lemma 4.11, each element of A ⋊ α,c Γ c ( G ) can be written in theform P mj =1 b j δ V j + I with the V j ∈ Γ c ( G ) and supp( b j ) = r ( V j ). Since Aδ ∅ ⊆ I by Proposition 4.5, we may assume that all the V j are non-empty,unless our element is 0, in which case there is nothing to prove. The compactopen subsets of G form a Boolean algebra because G is Hausdorff. TheBoolean algebra generated by V , . . . , V m is finite and hence is generated byits atoms W , . . . , W k . Moreover, since V = V ∪ · · · ∪ V m is the maximumof this Boolean algebra, we have W i = V ∩ W i = ( V ∩ W i ) ∪ · · · ∪ ( V m ∩ W i )and hence, since W i is an atom, we must have W i = V j ∩ W i for some j ,that is, W i ⊆ V j . Thus the W i are compact open bisections. Each V j is thedisjoint union of the W i contained in it and so Lemma 4.12 lets us write b j δ V j + I = P W i ⊆ V j b ji δ W i + I with supp( b ji ) = r ( W i ). Thus m X j =1 b j δ V j + I = m X j =1 X W i ⊆ V i b ji δ W i + I = k X i =1 d i δ W i + I with supp( d i ) ⊆ r ( W i ). Using Lemma 4.11, we can replace d i δ W i by a i δ U i with U i ⊆ W i and supp( a i ) = r ( U i ). As Aδ ∅ ⊆ I by Proposition 4.5, wemay remove also the empty U i . This completes the proof. (cid:3) We are now ready to prove the main theorem of this section.
WISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY 33
Theorem 4.14.
Let R be a commutative ring, G a Hausdorff ample groupoidand R × × G (0) ι −→ Σ ϕ −−→ G a twist. Let j : Γ c ( G ) −→ Γ c (Σ) be anidempotent-preserving set theoretic section and let c : Γ c ( G ) × Γ c ( G ) −→ C c ( G (0) , R × ∪ { } ) the corresponding normalized -cocycle. Then the R -algebras C c ( G (0) , R ) ⋊ α,c Γ c ( G ) and A R ( G ; Σ) are isomorphic.Proof. We retain the above notation. Define a mapping from L U ∈ Γ c ( G ) Aδ U (where A = C c ( G (0) , R )) to A R ( G ; Σ) as follows. Let π : R × × G (0) −→ R × be the projection. For a ∈ A and U ∈ Γ c ( G ), put f a,U ( s ) = ( a ( r ( ϕ ( s )))( π ( ι − ( s (( ϕ | j ( U ) ) − ( ϕ ( s ))) − ))) − , if ϕ ( s ) ∈ U , else.(This makes sense by Proposition 3.1.) So f a,U ( s ) is 0, unless ϕ ( s ) ∈ U , inwhich case it is a ( r ( ϕ ( s ))) t where t ∈ R × with ts ∈ j ( U ). Since U is clopen,it is obvious that f a,U is continuous (i.e., locally constant). Notice that ϕ (supp( f a,U )) = supp( a ) U is compact. For anti-equivariance, we compute f a,U ( ts ) = ( a ( r ( ϕ ( s )))( π ( ι − ( ts (( ϕ | j ( U ) ) − ( ϕ ( s ))) − ))) − , if ϕ ( s ) ∈ U , elseas ϕ ( ts ) = ϕ ( s ). Thus f a,U ( ts ) = t − f a,U ( s ), and so f a,U ∈ A R ( G ; Σ).We claim that ψ ( aδ U ) = f a,U is a homomorphism L U ∈ Γ c ( G ) Aδ U onto A R ( G ; Σ). It is clearly R -linear. We first compute f a,U ∗ f b,V ( s ). We choosea section p : G −→ Σ such that p | U = ( ϕ | j ( U ) ) − . Then in order for f a,U ∗ f b,V ( s ) = 0 we need ϕ ( s ) ∈ U V . Suppose this is the case and put ϕ ( s ) = gh with g ∈ U and h ∈ V . Let e g = ( ϕ | j ( U ) ) − ( g ) and e h = ( ϕ | j ( V ) ) − ( h ).Put e s = ( ϕ | j ( UV ) ) − ( s ). Then we can write s = t e s with t ∈ R × and e g e h = c ( U, V )( r ( ϕ ( s ))) e s . We compute that f a,U ∗ f b,V ( s ) = a ( r ( g )) b ( r ( h )) π ( ι − ( e g − s e h − )) − . But e g − s e h − = t e g − e s e h − = tc ( U, V )( r ( ϕ ( s ))) − r ( h ). Thus f a,U ∗ f b,V ( s ) = a ( r ( g )) b ( r ( h )) t − c ( U, V )( r ( ϕ ( s ))) . On the other hand, aδ U bδ V = a ( U b ) c ( U, V ) δ UV . Now f a ( Ub ) c ( U,V ) ,UV ( s ) is 0unless ϕ ( s ) ∈ U V . Then, retaining the previous notation, we have f a ( Ub ) c ( U,V ) ,UV ( s ) = a ( r ( ϕ ( s )))( U b )( r ( ϕ ( s ))) c ( U, V )( r ( ϕ ( s ))) π ( ι − ( s e s − )) − = a ( r ( g )) b ( r ( h )) c ( U, V )( r ( ϕ ( s ))) t − This completes the proof that ψ is a homomorphism.If V ⊆ Σ is a compact open bisection (whence ϕ | V is injective by Propo-sitions 3.1), let e V ∈ A R ( G ; Σ) be given by e V ( s ) = ( π ( ι − (( ϕ | V ) − ( ϕ ( s )) s − )) , if ϕ ( s ) ∈ ϕ ( V )0 , else, that is, e V is supported on R × · V and sends s to t if ts ∈ V with t ∈ R × . Itis shown in [4, Proposition 2.8] that A R ( G ; Σ) is spanned as an R -module bythe ˜1 V . So to show that ψ is onto, we just need that e V is in the image of ψ for any V ∈ Γ c (Σ). Let U = ϕ ( V ) ∈ Γ c ( G ) and note that ϕ | j ( U ) : j ( U ) −→ U is a homeomorphism by Proposition 3.1. Then we can define a ∈ A by a ( x ) = ( π ( ι − (( ϕ | V ) − (( r | U ) − ( x ))( ϕ | j ( U ) ) − (( r | U ) − ( x )) − )) , x ∈ r ( U )0 , else.Note that a is supported on r ( U ) and so belongs to A . We claim that f a,U = e V . Both functions vanish on all s ∈ Σ with ϕ ( s ) / ∈ U . If ϕ ( s ) ∈ U ,let t , t ∈ R × with t s ∈ j ( U ) and t s ∈ V . Then e V ( s ) = t . On the otherhand, f a,U ( s ) = a ( r ( ϕ ( s ))) t = t t − t = t , as required. We conclude that ψ is onto.We next show that I = ker ψ . Suppose that a ∈ A and U ≤ V ∈ Γ c ( G ).We need to show that f a,U = f ac ( V, d ( U )) ∗ ,V . Note that supp( c ( V, d ( U )) ∗ ) = r ( V · d ( U )) = r ( U ). Thus both functions vanish on any s ∈ Σ with ϕ ( s ) / ∈ U .Assume that ϕ ( s ) ∈ U . Let e s ∈ j ( U ) with ϕ ( e s ) = ϕ ( s ) and s ∈ j ( V ) with ϕ ( s ) = ϕ ( s ). Since j ( d ( U )) = d ( U ), we then have that j ( V ) d ( U ) = { ι ( c ( V, d ( U ))( r ( g )) , r ( g )) | g ∈ U } j ( U ) , and so s = c ( V, d ( U ))( r ( ϕ ( s ))) e s . If s = u e s with u ∈ R × . Then s = uc ( V, d ( U ))( r ( ϕ ( s ))) − s . Therefore, f a,U ( s ) = a ( r ( ϕ ( s ))) u − and f a ( c ( V, d ( U )) ∗ ,V ( s ) = a ( r ( ϕ ( s ))( c ( V, d ( U ))( r ( ϕ ( s )))) − c ( V, d ( U ))( r ( ϕ ( s )) u − = a ( r ( ϕ ( s )) u − , as required. Thus I ⊆ ker ψ .Let z ∈ L U ∈ Γ c ( G ) Aδ U belong to ker ψ . Then z + I = P ki =1 a i δ U i + I with supp( a i ) = r ( U i ) and the U i pairwise disjoint and non-empty byLemma 4.13 with possibly k = 0. Since I ⊆ ker ψ , we conclude that 0 = ψ ( z ) = P ki =1 f a i ,U i . Since the U i are pairwise disjoint, we deduce that each f a i ,U i = 0. Since supp( a i ) = r ( U i ), the only way f a i ,U i vanishes is if U i = ∅ .We deduce that k = 0 and hence z ∈ I . This completes the proof. (cid:3) References [1] G. Abrams, P. Ara, and M. Siles Molina.
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