Homology of étale groupoids, a graded approach
aa r X i v : . [ m a t h . K T ] J a n HOMOLOGY OF ´ETALE GROUPOIDSA GRADED APPROACH
ROOZBEH HAZRAT AND HUANHUAN LI
Abstract.
We introduce a graded homology theory for graded ´etale groupoids. For Z -graded groupoids, we establish anexact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids,we prove that the graded zeroth homology group with constant coefficients Z is isomorphic to the graded Grothendieckgroup of the associated Leavitt path algebra. To do this, we consider the diagonal algebra of the Leavitt path algebra ofthe covering graph of the original graph and construct the group isomorphism directly. Considering the trivial grading,our result extends Matui’s on zeroth homology of finite graphs with no sinks (shifts of finite type) to all arbitrary graphs.We use our results to show that graded zeroth-homology group is a complete invariant for eventual conjugacy of shift offinite types and could be the unifying invariant for the analytic and the algebraic graph algebras. Introduction
The homology theory for ´etale groupoids was introduced by Crainic and Moerdijk [14] who showed that the homologygroups are invariant under Morita equivalences of ´etale groupoids and established spectral sequences which used for thecomputation of these groups. Matui [23, 24, 25] considered this homology theory in relation with the dynamical propertiesof groupoids and their full groups. In [23] Matui proved, using Lindon-Hochschild-Serre spectral sequence establishedby Crainic and Moerdijk, that for an ´etale groupoid G arising from shifts of finite type, the homology groups H ( G )and H ( G ) coincide with K -groups K ( C ∗ ( G )) and K ( C ∗ ( G )), respectively. Here C ∗ ( G ) is the groupoid C ∗ -algebraassociated to G which was first systematically studied by Renault in his seminal work [27]. The current emerging pictureis that the homology of groupoids should govern the behaviour of their associated algebras and dynamics.A groupoid G is called graded if one can naturally partition it by an index group Γ, namely, G equipped with cocycle c : G →
Γ. The majority of the important ´etale groupoids arising from combinatorial or dynamical data are naturallygraded. Taking into account the partitions and their rearrangements should give salient information about the groupoidand the structures associated to them.The theory of graded groupoids relates to the theory of graded rings via the vehicle of Steinberg algebras. Thesealgebras are the algebraic version of groupoid C ∗ -algebras which have been recently introduced in [10, 28]. For an amplegroupoid G , the Steinberg algebra A R ( G ), with arbitrary coefficient ring R with unit, is an algebra which encompassesvery interesting algebras such as inverse semigroup rings, Leavitt path algebras and their higher rank versions namelyKumjian-Pask algebras [5]. If the groupoid G is Γ-graded then its associated Steinberg algebra A R ( G ) is naturallyΓ-graded ring. The graded structure of these algebras, in return, allows us to consider the graded invariants associatedto these algebras such as graded K -theory, K gr n , n ≥ K -theory for graded rings, we initiate and study graded homologytheory for a Γ-graded ´etale groupoid G , introducing the groups H gr n ( G ), n ≥
0, which are naturally equipped with thestructure of Γ-modules. We prove that the graded homology of a strongly graded ample groupoid is isomorphic to thehomology of its ε -th component with ε the identity of the grade group. For Z -graded ample groupoid G we establish anexact sequence H gr0 ( G ) −→ H gr0 ( G ) −→ H ( G ) −→ , which is similar to the van den Bergh exact sequence for graded K -theory of Z -graded Noetherian regular rings ([19, § Z -grading, we prove that for the graph groupoid G E , arisingfrom an arbitrary graph E , we have an order preserving Z [ x, x − ]-module isomorphism H gr0 ( G E ) ∼ = K gr0 ( L R ( E )) , where L R ( E ) is the Leavitt path algebra associated to E with the coefficient field R . Considering the grade group tobe trivial, since the K -group of Leavitt and C ∗ -algebras associated to a graph coincide we obtain that H ( G E ) ∼ = K ( A ( G E )) ∼ = K ( L ( E )) ∼ = K ( C ∗ ( E )) ∼ = K ( C ∗ ( G E )) . Date : January 23, 2019.2010
Mathematics Subject Classification.
Key words and phrases. ´etale groupoid, homology theory, graded homology theory, Leavitt path algebra, diagonal of Leavitt path algebra,graded Grothendieck group.
Since shifts of finite type are characterised by finite graphs with no sinks [22] (see also [25, § X , denote by X + the one-sided shift associated to X , and by G X + the groupoid associated to X + (see [7] for a summary of concepts on symbolic dynamics). Then we establish that H gr0 ( G + X ) is a complete invariantcharacterising eventual conjugacy of X (Theorem 6.7).Throughout the note the emphasis is on the module structure of the graded homology groups which carries substantialinformation. We believe that graded homology theory allows us to unify the invariants proposed for the classificationof analytic and algebraic graph algebras, namely Leavitt path algebras and C ∗ -graph algebras. Combing previous workon this direction, it is plausible to formulate the following conjecture (see [18] and Theorem 6.6). Conjecture 1.1.
Let E and F be finite graphs and R a field. Then the following are equivalent. (1) There is a gauge preserving isomorphism φ : C ∗ ( E ) → C ∗ ( F ) ; (2) There is a graded ring isomorphism φ : L R ( E ) → L R ( F ) ; (3) There is an order preserving Z [ x, x − ] -module isomorphism H gr0 ( G E ) → H gr0 ( G F ) such that φ ([1 G E ]) = [1 G F ]) . This note is primarily concerned with the graded ample groupoids, parallel to graded ring theory. Three essentialconcepts in the study of graded ring theory are strongly graded rings, graded matrix rings and smash products [19, 26].We recall them in Section 2. We develop and study these concepts in the setting of graded ample groupoids in Section 3.In Section 4, we will then consider the Steinberg algebras of these groupoids and reconcile these constructions with theparallel concepts in graded ring theory. The comparison between these two theories lead us to the definition of gradedhomology for graded groupoids in Section 5. In fact these correspondences allow us to compute the graded homologygroups for certain groupoids, such as graph groupoids in Section 6.2.
Graded rings
We briefly review three main concepts in graded ring theory, namely, grading on matrices, graded Morita theory andthe smash product. We also recall the case of strongly graded rings. In Section 3 we do the parallel constructions inthe setting of topological groupoids. We will then observe that the concept of Steinberg algebras will tie together theseconcepts. They will be used to define graded homology theory in Section 5 and calculate them for graph groupoids inSection 6.2.1.
Graded rings.
Let Γ be a group with identity ε . A ring A (possibly without unit) is called a Γ -graded ring if A = L γ ∈ Γ A γ such that each A γ is an additive subgroup of A and A γ A δ ⊆ A γδ for all γ, δ ∈ Γ. The group A γ is calledthe γ - homogeneous component of A. When it is clear from context that a ring A is graded by group Γ , we simply saythat A is a graded ring . If A is an algebra over a ring R , then A is called a graded algebra if A is a graded ring and A γ is a R -submodule for any γ ∈ Γ. A Γ-graded ring A = L γ ∈ Γ A γ is called strongly graded if A γ A δ = A γδ for all γ, δ in Γ.The elements of S γ ∈ Γ A γ in a graded ring A are called homogeneous elements of A. The nonzero elements of A γ arecalled homogeneous of degree γ and we write deg( a ) = γ for a ∈ A γ \{ } . The set Γ A = { γ ∈ Γ | A γ = 0 } is called the support of A . We say that a Γ-graded ring A is trivially graded if the support of A is the trivial group { ε } —that is, A ε = A , so A γ = 0 for γ ∈ Γ \{ ε } . Any ring admits a trivial grading by any group. If A is a Γ-graded ring and s ∈ A ,then we write s α , α ∈ Γ for the unique elements s α ∈ A α such that s = P α ∈ Γ s α . Note that { α ∈ Γ : s α = 0 } is finitefor every s ∈ A .We say a Γ-graded ring A has graded local units if for any finite set of homogeneous elements { x , · · · , x n } ⊆ A ,there exists a homogeneous idempotent e ∈ A such that { x , · · · , x n } ⊆ eAe . Equivalently, A has graded local units, if A ε has local units with ε the identity of Γ and A ε A γ = A γ A ε = A γ for every γ ∈ Γ.For a Γ-graded ring A with graded local units, we denote by Gr - A the category of unital graded right A -moduleswith morphisms preserving the grading. For a graded right A -module M , we define the α - shifted graded right A -module M ( α ) as M ( α ) = M γ ∈ Γ M ( α ) γ , where M ( α ) γ = M αγ . That is, as an ungraded module, M ( α ) is a copy of M , but the grading is shifted by α . For α ∈ Γ, the shift functor T α : Gr - A −→ Gr - A, M M ( α )is an isomorphism with the property T α T β = T αβ for α, β ∈ Γ. OMOLOGY OF ´ETALE GROUPOIDS 3
Graded matrix rings.
Let A = L σ ∈ Γ A σ be a Γ-graded ring with Γ a group, n is a positive integer, and M n ( A )the ring of n × n -matrices with entries in A . Fix some σ = ( σ , · · · , σ n ) ∈ Γ n . To any γ ∈ Γ, we associate the followingadditive subgroup of M n ( A ) M n ( A ) γ ( σ ) = A σ γσ − A σ γσ − · · · A σ γσ − n A σ γσ − A σ γσ − · · · A σ γσ − n ... ... ... A σ n γσ − A σ n γσ − · · · A σ n γσ − n The family of additive subgroups (cid:8) M n ( A ) γ ( σ ) | γ ∈ Γ (cid:9) defines a Γ-grading of the ring M n ( A ) (see [26, Proposition2.10.4]). We will denote this graded ring by M n ( A )( σ ).We denote by M Γ ( A ) the ring of | Γ | × | Γ | -matrices with finitely many nonzero entries. Here, | Γ | is the cardinality ofΓ. To any γ ∈ Γ, we associate the following additive subgroup of M Γ ( A ) M Γ ( A ) γ = { ( x αβ ) α,β ∈ Γ | x αβ ∈ A αγβ − } . The family of additive subgroups { M Γ ( A ) γ | γ ∈ Γ } defines a Γ-grading for the ring M Γ ( A ), denoted by M Γ ( A )(Γ).2.3. Graded Morita theory.
We say a functor F : Gr - A −→ Gr - B is a graded functor provided F commutes with theshift functors, namely F ( M ( α )) = F ( M )( α ), for every α ∈ Γ and M ∈ Gr - A . A graded functor F : Gr - A −→ Gr - B issaid to be a graded equivalence provided there is a graded functor G : Gr - B −→ Gr - A which is the inverse to F . Whensuch a graded equivalence exists, we say that A and B are graded Morita equivalent rings .We also need to recall the definition of graded Morita context from [17, Definition 2.5]. Let A and B be Γ-gradedrings with graded local units. The six tuple ( A, B, M, N, ψ : M ⊗ B N −→ A, ϕ : N ⊗ A M −→ B ) is a graded Morita context provided:(1) A M B and B N A are graded bimodules in the following sense: A α M β B γ ⊆ M αβγ and B α N β A γ ⊆ N αβγ , for all α, β, γ ∈ Γ.(2) ψ : M ⊗ B N −→ A and ϕ : N ⊗ A M −→ B are graded homomorphisms in the sense that ψ ( M α ⊗ N β ) ⊆ A αβ and ϕ ( N α ⊗ M β ) ⊆ B αβ for all α, β ∈ Γ and that satisfy the following two conditions: ψ ( m ⊗ n ) m ′ = mϕ ( n ⊗ n ′ ) and ϕ ( n ⊗ m ) n ′ = nψ ( m ⊗ n ′ ) , for all m, m ′ ∈ M and n, n ′ ∈ N .By [17, Theorem 2.6] the graded rings A and B are graded Morita equivalent if and only if there is a Morita contextwith ψ and φ surjective. This will be used in the setting of Steinberg algebras (Theorem 4.6).2.4. The smash product of a graded ring.
For a Γ-graded ring A , the smash product ring A A and a finite group Γ in their seminal paper [13] (see also [26, Chapter7]). They established that the category of graded A -modules are equivalent to the category of A A to non-graded properties of A .Recall that for a Γ-graded ring A (possibly without unit), the smash product ring A P γ ∈ Γ r ( γ ) p γ , where r ( γ ) ∈ A and p γ are symbols. Addition is defined component-wise and multiplicationis defined by linear extension of the rule ( rp α )( sp β ) = rs αβ − p β , (2.1)where r, s ∈ A and α, β ∈ Γ.In [4, Proposition 2.5], the Cohen-Montgomery result on the equivalences of categories of graded modules of unitalrings graded by finite groups extended as follows: Let A be a Γ-graded ring with graded local units. Then there is anisomorphism of categories Gr - A −→ Mod - A . (2.2)The Equation (2.2) allows us to describe the graded functors for the graded ring A as a corresponding non-gradedversion on the category of A A as a usual K -groupof A K gr0 ( A ) ∼ = K ( A Graded Groupoids
In this section we develop the concepts of graded matrix groupoids and strongly graded groupoids. We then observethat semi-direct product of groupoids will replace the concept of smash product in the classical ring theory.
ROOZBEH HAZRAT AND HUANHUAN LI
Graded groupoids.
A groupoid is a small category in which every morphism is invertible. It can also be viewed asa generalisation of a group which has partial binary operation. Let G be a groupoid. If g ∈ G , s ( g ) = g − g is the source of g and r ( g ) = gg − is its range . The pair ( g , g ) is composable if and only if r ( g ) = s ( g ). The set G (0) := s ( G ) = r ( G )is called the unit space of G . Elements of G (0) are units in the sense that gs ( g ) = g and r ( g ) g = g for all g ∈ G . The isotropy group at a unit u of G is the group Iso( u ) = { g ∈ G | s ( g ) = r ( g ) = u } . Let Iso( G ) = F u ∈G (0) Iso( u ). For U, V ∈ G , we define
U V = (cid:8) g g | g ∈ U, g ∈ V and r ( g ) = s ( g ) (cid:9) . A topological groupoid is a groupoid endowed with a topology under which the inverse map is continuous, and suchthat composition is continuous with respect to the relative topology on G (2) := { ( g , g ) ∈ G ×G : s ( g ) = r ( g ) } inheritedfrom G × G . An ´etale groupoid is a topological groupoid G such that the domain map s is a local homeomorphism.In this case, the range map r is also a local homeomorphism. In this article, by an ´etale groupoid we mean a locallycompact Haudsorff groupoid such that the source map s is a local homeomorphism.An open bisection of G is an open subset U ⊆ G such that s | U and r | U are homeomorphisms onto an open subsetof G (0) . If G is a ´etale groupoid, then there is a base for the topology on G consisting of open bisections with compactclosure. As demonstrated in [10, 28], if G (0) is totally disconnected and G is ´etale, then there is a basis for the topologyon G consisting of compact open bisections. We say that an ´etale groupoid G is ample if there is a basis consisting ofcompact open bisections for its topology.Let Γ be a discrete group and G a topological groupoid. A Γ-grading of G is a continuous function c : G →
Γ suchthat c ( g ) c ( g ) = c ( g g ) for all ( g , g ) ∈ G (2) ; such a function c is called a cocycle on G . For γ ∈ Γ, setting G γ = c − ( γ )which are clopen subsets of G , we have G = F γ ∈ Γ G γ such that G β G γ ⊆ G βγ . In this case, we call G a Γ-graded groupoid.In this paper, we shall also refer to c as the degree map on G . For γ ∈ Γ, we say that X ⊆ G is γ -graded if X ⊆ G γ . Wealways have G (0) ⊆ G ε , so G (0) is ε -graded where ε is the identity of Γ. We write B co γ ( G ) for the collection of all γ -gradedcompact open bisections of G and B co ∗ ( G ) = [ γ ∈ Γ B co γ ( G ) . (3.1)A groupoid is trivially graded by considering the map c : G → { ε } . Throughout this paper, we consider the groupoidsto be Γ-graded. As soon as our grade group Γ is trivial, then we obtain the results in the non-graded setting. Manyimportant examples of ´etale groupoids have a natural graded structure. Example 3.1 ( Transformation groupoids) . Let φ : Γ y X be an action of a countable discrete group Γ on a Cantorset X by homeomorphisms, equivalently there is a group homomorphism Γ −→ Home( X ), where Home( X ) consists ofhomeomorphisms from X to X with the multiplication given by the composition of maps. Let G φ = Γ × X and definethe groupoid structure: ( γ, γ ′ x ′ ) · ( γ ′ , x ′ ) = ( γγ ′ , x ′ ), and ( γ, x ) − = ( γ − , γx ). Then G φ is an ´etale groupoid, calledthe transformation groupoid arising from φ : Γ y X . The unit space G (0) φ is canonically identified with X via the map( ε, x ) x . The natural cocyle G φ → Γ , ( γ, x ) γ makes G φ a Γ-graded groupoid. Example 3.2.
Let φ : Γ y G be an action of a discrete group Γ on an ´etale groupoid G , i.e., there is a grouphomomorphism α : Γ −→ Aut( G ) with Aut( G ) the group of homeomorphisms between G which respect the compositionof G . The semi-direct product G ⋊ φ Γ is
G ×
Γ with the following groupoid structure: ( g, γ ) and ( g ′ , γ ′ ) are composableif and only if g and φ γ ( g ′ ) are composable, ( g, γ )( g ′ , γ ′ ) = ( gφ γ ( g ′ ) , γγ ′ ) , and ( g, γ ) − = ( φ γ − ( g − ) , γ − ) . The unit space of G ⋊ φ Γ is G (0) × { ε } . The source map s : G ⋊ φ Γ −→ G (0) × { ε } is given by s ( g, γ ) = ( s ( φ γ − ( g )) , ε )and the range map r : G ⋊ φ Γ −→ G (0) × { ε } is given by r ( g, γ ) = ( r ( g ) , ε ) for g ∈ G and γ ∈ Γ. The semi-direct product G ⋊ φ Γ has a natural Γ-grading given by the cocycle c : G ⋊ φ Γ −→ Γ, c ( g, γ ) = γ .If G is an ample groupoid and φ : Γ y G is an action of a discrete group Γ on G , then G ⋊ φ Γ is again ample underthe product topology on
G ×
Γ. We will determine the Steinberg algebras of these construction in § Example 3.3 ( Groupoid of a dynamical system) . Let X be a locally compact Hausdorff space and σ : X → X a localhomeomorphism. Further suppose that d : X → Γ is a continuous map. We then have a Γ-graded groupoid defined asfollows: G ( X, σ ) := n ( x, s, y ) | σ n ( x ) = σ m ( y ) and s = n − Y i =0 d ( σ i ( x )) m − Y j =0 d ( σ j ( y )) − , n, m ∈ N o . Here the cocyle is defined by G ( X, σ ) → Γ , ( x, s, y ) s .Let E be a finite graph with no sinks and sources, and E ∞ the set of all infinite paths in E . Let σ : E ∞ → E ∞ ; e e e · · · 7→ e e . . . and the constant map d : E ∞ → Z , x
1. Then we have G ( E ∞ , σ ) = G E , where G E is thegraph groupoid associated to the graph E (see Section 6). OMOLOGY OF ´ETALE GROUPOIDS 5
Graded matrix groupoids.
Recall from § A is Γ-graded, then for a collection { γ , . . . , γ n } ofelements of Γ, we have a Γ-graded matrix ring M n ( A )( γ , . . . , γ n ).In this section we develop the parallel concept in the setting of groupoids, namely graded matrix groupoids with agiven shift. We will see that the Steinberg algebra of a graded matrix groupoid coincides with the graded matrix ringof the Steinberg algebra with the same shift (see Proposition 4.4).Let G be an ´etale groupoid. For f : G (0) −→ Z a map with f ≥
0, where Z is the discrete abelian group of integers,we let G f = { x ij | x ∈ G , ≤ i ≤ f ( r ( x )) , ≤ j ≤ f ( s ( x )) } , and equip G f with the induced topology from the product topology on G × Z × Z . We endow G f with the groupoidstructure as follows: G (0) f = { x ii | x ∈ G (0) , ≤ i ≤ f ( x ) } , ( x ij ) − = ( x − ) ji , two elements are x ij and y kl are composable if and only if s ( x ) = r ( y ), j = k and the product isgiven by x ij y jl = ( xy ) il . We call G f the matrix groupoid of G with respect to f .Suppose that G is a Γ-graded groupoid with the grading map c : G −→
Γ and f : G (0) −→ Z satisfies f ≥
0. Fix acontinuous map ψ : G (0) f −→ Γ. Define a map c ψ : G f −→ Γ , (3.2) x ij ψ ( r ( x ) ii ) c ( x ) ψ ( s ( x ) jj ) − . We claim that c ψ is a continuous cocycle for G f and thus G f is Γ-graded. Indeed, we observe that c ψ is the compositionmap G f µ −→ G (0) f × Γ × G (0) f ν −→ Γ , where µ ( x ij ) = ( r ( x ) ii , c ( x ) , s ( x ) jj ) for x ij ∈ G f and ν ( y kk , γ, z ll ) = ψ ( y kk ) γψ ( z ll ) − for y kk , z ll ∈ G (0) f and γ ∈ Γ. Since ψ , the source and range maps for the groupoid G f are continuous, µ and ν are continuous. Thus c ψ is continuous. Sincethe grading map c ψ of the Γ-graded groupoid G f is related to the continuous map ψ , we denote the Γ-graded groupoid G f by G f ( ψ ).For a locally compact Hausdorff space X and a toplogical group G , we denote by C c ( X, G ) the set of G -valuedcontinuous functions with compact support.For an ´etale groupoid G , we denote s − ( X ) ∩ r − ( X ) by G| X for a subset X ⊆ G (0) . A subset F ⊆ G (0) is said to be G -full , if r − ( x ) ∩ s − ( F ) is not empty for any x ∈ G (0) .The following lemma is the graded version of [23, Lemma 4.3]. Lemma 3.4.
Let G be an ´etale Γ -graded groupoid, graded by the cocyle c : G −→ Γ , whose unit space is compact and totallydisconnected and let Y ⊆ G (0) be a G -full clopen subset. There exist f ∈ C c ( Y, Z ) and a continuous map ψ : ( G| Y ) (0) f −→ Γ such that π : ( G| Y ) f −→ G satisfying π ( x ) = x for all x ∈ G| Y is a graded isomorphism.Proof. Let X = G (0) . There are compact open bisections V , V , · · · , V n such that s ( V j ) ⊆ Y for each j = 1 , · · · , n , r ( V ) , r ( V ) , · · · , r ( V n ) are mutually disjoint and their union is equal to X \ Y (see [23, Lemma 4.3]). For each subset λ ⊆ { , , · · · , n } we fix a bijection α λ : { k | ≤ k ≤ | λ |} −→ λ . For y ∈ Y put λ ( y ) = { k ∈ { , , · · · , n } | y ∈ s ( V k ) } . We define f ∈ C c ( Y, Z ) by f ( y ) = | λ ( y ) | . Since each s ( V k ) is clopen, f is continuous. A map θ : ( G| Y ) (0) f −→ G is definedby θ ( y ii ) = ( y, if i = 0;( s | V l ) − ( y ) otherwise , where l = α λ ( y ) ( i ) (see [23, Lemma 4.3]).We claim that θ : ( G| Y ) (0) f −→ G defined above is continuous. Recall that the collection of all the bisections of the´etale groupoid G forms a basis for its topology (see [15, Proposition 3.5] and [28, Proposition 3.4]). It suffices to showthat for any open bisection T of G , θ − ( T ) is open. Observe that θ − ( T ) = θ − ( T ∩ ∪ nj =1 V j ) ∪ ( T ∩ Y ) , where ( T ∩ Y ) = { x | x ∈ T ∩ Y } . Indeed, clearly θ − ( T ∩ ∪ nj =1 V j ) ∪ ( T ∩ Y ) ⊆ θ − ( T ) . On the other hand, take any y ii ∈ θ − ( T ). If i = 0, then θ ( y ) = y ∈ T ∩ Y , implying y ∈ ( T ∩ Y ) . If i = 0,then θ ( y ii ) = ( s | V l ) − ( y ) ∈ V l with l = α λ ( y ) ( i ). Since θ ( y ii ) belongs to T , we have θ ( y ii ) ∈ T ∩ ∪ nj =1 V j , implying y ii ∈ θ − ( T ∩ ∪ nj =1 V j ). Thus θ − ( T ) ⊆ θ − ( T ∩ ∪ nj =1 V j ) ∪ ( T ∩ Y ) . ROOZBEH HAZRAT AND HUANHUAN LI
Since ( T ∩ Y ) is an open subset of ( G| Y ) (0) f , it suffices to show that θ − ( T ∩ ∪ nj =1 V j ) is an open subset of ( G| Y ) (0) f . Infact, we have θ − ( T ∩ ∪ nj =1 V j ) = (cid:0) ( T ∩ ∪ nj =1 V j ) ∩ Y (cid:1) ⊔ ⊔ nj =1 s ( T ∩ V j ) with s ( T ∩ V j ) = { z | z ∈ s ( T ∩ V j ) } , since for each element x ∈ T ∩ ∪ nj =1 V j , we have f ( s ( x )) = 1.Let ψ be the composition map ( G| Y ) (0) f θ −→ G c −→ Γ. By (3.2), G f is Γ-graded. By the proof of [23, Lemma 4.3], thereis an isomorphism π : ( G| Y ) f −→ G such that π ( x ij ) = θ ( r ( x ) ii ) · g · θ ( s ( x ) jj ) − . It is evident that the isomorphism π : ( G| Y ) f −→ G preserves the grading. The proof is completed. (cid:3) Let Γ and Z be discrete groups, G a Γ-graded groupoid with c : G −→
Γ the grading map and f ∈ C c ( G (0) , Z ).Suppose that σ : Z −→ Z is a bijection such that for any x ∈ G (0) , σ ( D x ) = D x where D x = { , , · · · , f ( x ) } . Supposethat ψ : G (0) f −→ Γ is a continuous map. Let ψ σ : G (0) f −→ Γ be the continuous map given by ψ σ ( x ii ) = ψ ( x σ ( i ) σ ( i ) ) forany x ii ∈ G (0) f .In the following theorem, by Z (Γ) we mean the centre of the group Γ. Proposition 3.5.
Keep the above notation. We have (1) G f ( ψ ) ∼ = G f ( ψ σ ) as Γ -graded groupoids; (2) Suppose that f : G (0) −→ Z is a constant function such that f ( x ) = n with n a positive integer for any x ∈ G (0) .Let τ be a permutation of { , , · · · , n } . Let ψ : G (0) f −→ Γ be a continuous map and ψ τ : G (0) f −→ Γ be given by ψ τ ( x ii ) = ψ ( x τ ( i ) τ ( i ) ) for any x ii ∈ G (0) f . Then G f ( ψ ) ∼ = G f ( ψ τ ) as Γ -graded groupoids; (3) Let γ ∈ Z (Γ) and ψ : G (0) f −→ Γ a continuous map. Let ψ γ : G (0) f −→ Γ be the composition of ψ and the multiplicationmap Γ −→ Γ , β γβ with β ∈ Γ . Then G f ( ψ ) ∼ = G f ( ψ γ ) as Γ -graded groupoids.Proof. (1) Observe that G f ( ψ ) −→ G f ( ψ σ ) , x ij x σ − ( i ) σ − ( j ) is the desired Γ-graded groupoid isomorphism. For(2), we observe that in this case D x = { , , · · · , n } = for all x ∈ G (0) . Let σ : Z −→ Z be a bijection such that σ ( { , , · · · , n } ) = { , , · · · , n } . Applying (1), the result follows directly. For (3), G f ( ψ ) −→ G f ( ψ γ ) , x ij x ij is aΓ-graded groupoid isomorphism. (cid:3) Strongly graded groupoids.
Let Γ be a group with identity ε and let G be a Γ-graded groupoid, graded by thecocycle c : G −→
Γ. Parellel to the classical graded ring theory, we say G is a strongly Γ -graded groupoid if G α G β = G αβ for all α, β ∈ Γ. In [8] such groupoids were systematically studied. It was shown that G is strongly graded if and only if s ( G α ) = G (0) if and only if r ( G α ) = G (0) , for all α ∈ Γ (see [8, Lemma 2.3]). When G is strongly graded, the ε -componentof G , namely, G ε , contains substantial information about the groupoid G . In [8, Theorem 2.5] it was proved that anample Γ-graded groupoid G is strongly graded if and only if the Steinberg algebra A R ( G ) is strongly Γ-graded.3.4. Skew-product of groupoids.
In this subsection we first demonstrate that for the Γ-graded groupoid G , gradedby the cocycle c : G −→
Γ, the skew-product
G × c Γ takes the role of the smash product in the graded ring theory asdescribed in § Gr - R G ∼ = Mod - R G × c Γ , where Gr - R G is the category of graded G -sheaves of R -modules with graded morphisms and Mod - R G is the categoryof G -sheaves of R -modules (see § Gr - A −→ Mod - A , leads us to define the graded homology theory of a Γ-graded groupiod G as the homology theory of skew-product G × c Γ(see Definition 5.3).
Definition 3.6.
Let G be a locally compact Hausdorff groupoid, Γ a discrete group and c : G →
Γ a cocycle. The skew-product of G by Γ, denoted by G × c Γ, is the groupoid
G ×
Γ such that ( x, α ) and ( y, β ) are composable if x and y are composable and α = c ( y ) β . The composition is then given by (cid:0) x, c ( y ) β (cid:1)(cid:0) y, β (cid:1) = ( xy, β )with the inverse ( x, α ) − = ( x − , c ( x ) α ) . The source map s : G × c Γ −→ G (0) × Γ and the range map r : G × c Γ −→ G (0) × Γ of the skew-product
G × c Γ are givenby s ( x, α ) = ( s ( x ) , α ) and r ( x, α ) = ( r ( x ) , c ( x ) α ) for ( x, α ) ∈ G × c Γ.The skew-product
G × c Γ is a Γ-graded topological groupoid under the product topology on
G ×
Γ and with degreemap ˜ c ( x, γ ) := c ( x ). OMOLOGY OF ´ETALE GROUPOIDS 7
Remark 3.7.
Note that our convention for the composition of the skew-product here is slightly different from that in [27,Definition 1.6] and [23, § § G , we observe that a graded compact open bisection U ∈ B co α ( G × c Γ) with α ∈ Γ can be written as U = l G i =1 U i × { γ i } , (3.3)with γ i ∈ Γ distinct and U i an α -graded compact open bisection of G . Indeed, if U ⊆ ˜ c − ( α ), then for each γ ∈ Γ, theset U ∩ G × { γ } is a compact open bisection. Since these are mutually disjoint and U is compact, there are finitely many(distinct) γ , . . . , γ l ∈ Γ such that U = l G i =1 U ∩ ( G × { γ i } ) . Each U ∩ G × { γ i } has the form U i × { γ i } , where U i ⊆ G is compact open. The U i have mutually disjoint sources becausethe source map on G × c Γ is just s × id, and U is a bisection. So each U i ∈ B co α ( G ), and U = F li =1 U i × { γ i } . We observethat s ( U ) = l G i =1 s ( U i ) × { γ i } and r ( U ) = l G i =1 r ( U i ) × { αγ i } . Recall that for a locally compact, Hausdorff groupoid G , we say that G is effective if the interior Iso( G ) ◦ of Iso( G ) is G (0) . We say that G is principal if Iso( G ) = G (0) ([9]). Lemma 3.8.
Let G be a locally compact Hausdorff Γ -graded groupoid, with the cocycle c : G → Γ . We have Iso(
G × c Γ) =Iso( G ε ) × Γ . Thus the groupoid G ε is effective if and only if G × c Γ is effective. Furthermore, G ε is principal if and onlyif G × c Γ is principal.Proof. Let ( x, α ) ∈ Iso(
G × c Γ). Then ( s ( x ) , α ) = s ( x, α ) = r ( x, α ) = ( r ( x ) , c ( x ) α ). It immediately follows that x ∈ Iso( G ε ) and so Iso( G × c Γ) ⊆ Iso( G ε ) × Γ. The reverse inclusion is immediate. The rest of Lemma follows easily. (cid:3)
Groupoid spaces and groupoid equivalences.
In this subsection we recall the concepts of imprimitive groupoids,linking groupoids and equivalence spaces between two groupoids. In § G be a locally compact groupoid with Hausdorff unit space and X a locally compact space. We say G acts onthe left of X if there is a map r X from X onto G (0) and a map ( γ, x ) γ · x from G ∗ X := { ( γ, x ) ∈ G × X | s ( γ ) = r X ( x ) } to X such that(1) if ( η, x ) ∈ G ∗ X and ( γ, η ) is composable in G , then ( γη, x ) , ( γ, η · x ) ∈ G ∗ X and γ · ( η · x ) = ( γη ) · x ;(2) r X ( x ) · x = x for all x ∈ X .We call X a continuous left G -space if r X is an open map and both r X and ( γ, x ) γ · x are continuous. The action of G on X is free if γ · x = x implies γ = r X ( x ). It is proper if the map from G ∗ X → X × X given by ( γ, x ) ( γ · x, x )is a proper map in the sense that inverse images of compact sets are compact. Right actions can be defined similarly,writing s X for the map from X to G (0) , and X ∗ G := { ( x, γ ) ∈ X × G | s X ( x ) = r ( γ ) } . Suppose X is a left G -space. Define the orbit equivalence relation on X determined by G to be x ∼ y if and only ifthere exists γ ∈ G such that γ · x = y . The quotient space with respect to this relation is denoted G \ X , the elementsof G \ X are denoted [ x ], and the canonical quotient map is denoted by q . When X is a continuous G -space we willgive G \ X the quotient topology with respect to q . When X is a right G -space the orbit equivalence relation is definedsimilarly and we will use exactly the same notation. In some cases X will be both a left G -space and a right H -space,where G and H are groupoids, and in this situation we will denote the orbit space with respect to the G -action by G \ X and the orbit space with respect to the H -action by X/ H . Definition 3.9.
Let G and H be locally compact groupoids. A locally compact space Z is called a ( G , H ) -equivalence if(1) Z is a free and proper left G -space;(2) Z is a free and proper right H -space;(3) the actions of G and H on Z commute; ROOZBEH HAZRAT AND HUANHUAN LI (4) r Z induces a homeomorphism of Z/ H onto G (0) ;(5) s Z induces a homeomorphism of G \ Z onto H (0) .Let H be a locally compact Hausdorff groupoid, and let Z be a free and proper right H -space. Then H acts diagonallyon Z ∗ s Z = { ( x, y ) ∈ Z × Z | s ( x ) = s ( y ) } , (3.4)and we define Z H = ( Z ∗ s Z ) / H . Two equivalence classes [ z, w ] H , [ x, y ] H ∈ Z H are composable when [ w ] = [ x ] in Z/ H ,and we define [ z, w ] H [ w, y ] H = [ z, y ] H and [ z, w ] − H = [ w, z ] H . It can be checked that these operations make Z H intoa locally compact Hausdorff groupoid (see [29, Proposition 1.92]). Furthermore, the range and source maps are givenby r ([ z, w ] H ) = [ z, z ] H and s ([ z, w ] H ) = [ w, w ] H , which allows us to identify the unit space of Z H with Z/ H . Thegroupoid Z H is called the imprimitivity groupoid associated to the H -space Z . The imprimitivity groupoid associatedto a left action is defined analogously. The groupoid Z H admits a natural left action on Z , which makes Z into a( Z H , H )-equivalence (see [29, Proposition 1.93]). Thus any free and proper groupoid space gives rise to a groupoidequivalence in a canonical way. Indeed, this construction is prototypical: if Z is a ( G , H )-equivalence, then Z H and G are naturally isomorphic via the map [ z, w ] H G [ z, w ] , (3.5)where G [ z, w ] ∈ G is the unique element satisfying G [ z, w ] · w = z (see [29, Proposition 1.94]).Given a ( G , H )-equivalence Z , define the opposite space Z op = { z : z ∈ Z } to be a homeomorphic copy of Z , but let G and H act on the right and left of Z op as follows: r ( z ) = s ( z ) , s ( z ) = r ( z ) , η · z = z · η − , z · y = y − · z for η ∈ H, y ∈ G . It is straightforward to check that this makes Z op into an ( H , G )-equivalence. We then define the linking groupoid to be the topological disjoint union L = G ⊔ Z ⊔ Z op ⊔ H with r, s : L −→ L (0) := G (0) ⊔ H (0) inherited from the range and source maps on each of G , H , Z and Z op . The set ofcomposable pairs is L (2) = { ( x, y ) ∈ L × L : s ( x ) = r ( y ) } and multiplication ( k, l ) kl in L is given by • multiplication in G and H when ( k, l ) is a composable pair in G or H ; • kl = k · l when ( k, l ) ∈ ( Z ∗ H ) ⊔ ( G ∗ Z ) ⊔ ( H ∗ Z op ) ⊔ ( Z op ∗ G ); • kl = G [ k, h ] if k ∈ Z and l = h ∈ Z op , and kl = [ h, l ] H if j ∈ Z and k = h ∈ Z op .The inverse map is the usual inverse map in each of G and H and is given by z z on Z and z z on Z op . It is shownin [30, Lemma 2.1] that these operations make L into a locally compact Hausdorff groupoid.3.6. Graded groupoid equivalences.
Let Γ be a group and let G be a Γ-graded groupoid, graded by the cocycle c : G −→
Γ. Recall from [8, § G -space X is called graded G -space if there is a continuous map k : X −→ Γ suchthat k ( γ · x ) = c ( γ ) k ( x ), whenever ( γ, x ) ∈ G ∗ X .Let G and H be Γ-graded groupoids. A ( G , H )-equivalence Z is called a Γ -graded ( G , H ) -equivalence if Z is a Γ-graded G -space and also a Γ-graded H -space with respect to a continuous map k : Z −→ Γ.Let H be a Γ-graded groupoid graded by the cocycle c : H −→
Γ and X a Γ-graded right H -space with k : X −→ Γ thegrading map. Then the imprimitivity groupoid X H is a Γ-graded groupoid graded by the cocycle k : X H −→ Γ given by k ([ x, y ]) = k ( x ) k ( y ) − for [ x, y ] ∈ X H . The map k : X H −→ Γ is well defined. Observe that for [ x ′ , y ′ ] = [ x, y ] in X H there exists γ ∈ H suchthat x ′ = x · γ and y ′ = y · γ , and thus k ([ x ′ , y ′ ]) = k ( x ′ ) k ( y ′ ) − = k ( x · γ ) k ( y · γ ) = k ([ x, y ]). We observe that k is acontinuous map. Indeed, we have the commutative diagram X × X φ / / q (cid:15) (cid:15) Γ X ∗ s X ψ ; ; ①①①①①①①①① where φ ( x, y ) = k ( x ) k ( y ) − for ( x, y ) ∈ X × X , X ∗ s X is defined in (3.4), q : X × X −→ X ∗ s X is the quotient map, and ψ ( x, y ) = k ( x ) k ( y ) − . Observe that φ is continuous and that φ − ( γ ) = q − ( ψ − ( γ )) for any γ ∈ Γ. Thus ψ − ( γ ) is open,since φ − ( γ ) is open and q is the quotient map. Hence ψ is continuous. Similarly, we have the following commutativediagram X ∗ s X ψ / / q (cid:15) (cid:15) Γ X H k < < ①①①①①①①①① OMOLOGY OF ´ETALE GROUPOIDS 9 and thus k : X H −→ Γ is continuous. In the case that Z is a Γ-graded ( G , H )-equivalence, the groupoid isomorphismgiven in (3.5) preserves the grading and thus G and Z H are isomorphic as Γ-graded groupoids.If Z is a Γ-graded ( G , H )-equivalence, then the linking groupoid L of Z is Γ-graded with the cocycle map k : L −→ Γinherited from the grading maps of G , H and Z , and k ( z ) = k ( z ) − for z ∈ Z .3.7. The category of graded G -sheaves. Let G be a Γ-graded groupoid, graded by the cocycle c : G →
Γ and E aright G -space. For x ∈ G , the fibre s − E ( x ) of E is denoted by E x and is called the stalk of E at x .A G -space E is called a G -sheaf of sets if s E : E → G (0) is a local homeomorphism. For a commutative ring R withunit, we say the sheaf E is a graded G -sheaf of R -modules if each stalk E x is a (left) R -module such that(1) the zero section sending x ∈ G (0) to the zero of E x is continuous;(2) addition E × Γ (0) E → E is continuous;(3) scalar multiplication R × E → E is continuous, where R has the discrete topology;(4) for each g ∈ G , the map ψ g : E r ( g ) → E s ( g ) given by ψ g ( e ) = eg is R -linear.(5) for any x ∈ G (0) , E x = L γ ∈ Γ ( E x ) γ , where ( E x ) γ are R -submodules of E x ;(6) E γ := S x ∈G (0) ( E x ) γ is open in E for every γ ∈ Γ; and(7) E γ G δ ⊆ E γδ for every γ, δ ∈ Γ.We call ( E x ) γ the γ - homogeneous component of E x , and denote the homogeneous elements of E by E h := [ x ∈G (0) ,γ ∈ Γ ( E x ) γ . Note that the map k : E h → Γ, s γ , where s ∈ ( E x ) γ , is continuous, and (7) can be interpreted as k ( eg ) = k ( e ) c ( g )for every homogeneous e ∈ E h and any g ∈ G such that s E ( e ) = r ( g ). Unlike graded G -spaces, it does not make sense todefine a degree k on all of E . A morphism φ : E → F of G -sheaves of R -modules is a graded morphism if φ ( E γ ) ⊆ F γ forany γ ∈ Γ. The category of graded G -sheaves of R -modules with graded morphisms will be denoted Gr - R G . If from theoutset we set the grade group Γ the trivial group, then our construction gives the category of G -sheaves of R -modules,denoted by Mod - R G .3.8. Graded Kakutani equivalences and stable groupoids.
Matui [23] defines Kakutani equivalence for amplegroupoids with compact unit spaces. In [6], the authors extend this notion to ample groupoids with non-compact unitspaces. In this subsection, we consider graded groupoids and graded Kakutani equivalences. This allows us to showthat for a Γ-graded ample groupoid G , Gr - R G ∼ = Mod - R G × c Γ , (see (5.1) and compare this with (2.2)). This justifies the use of skew-product G × c Γ for the definition of the gradedhomology of the Γ-graded groupoid G (see Definition 5.3).Recall that two ample groupoids G and H are Kakutani equivalent if there are a G -full clopen X ⊆ G (0) and an H -fullclopen Y ⊆ H (0) such that G| X ∼ = H| Y . Definition 3.10.
Let Γ be a discrete group. Suppose that G and G are Γ-graded groupoids. The groupoids G and G are called Γ -graded Kakutani equivalent if there are full clopen subsets Y i ⊆ G i , for i = 1 ,
2, such that G | Y is gradedisomorphic to G | Y .Let G be a Γ-graded groupoid graded by the cocycle c : G →
Γ. The stable groupoid G Γ is defined as G Γ = (cid:8) g α,β | α, β ∈ Γ , g ∈ G (cid:9) , (3.6)with composition given by g α,β h β,γ = ( gh ) α,γ , if g and h are composable and inverse given by ( g α,β ) − = g − β,α . Here G (0)Γ = { x α,α | α ∈ Γ , x ∈ G (0) } , s ( g α,β ) = s ( g ) β,β and r ( g α,β ) = r ( g ) α,α . Observe that if G is an ample groupoid, so is G Γ .One defines a cocycle (called c again) c : G Γ −→ Γ ,g α,β α − c ( g ) β. This makes G Γ a Γ-graded groupoid. Furthermore, the ε -component of G Γ , ( G Γ ) ε = { g α,β | c ( g ) = αβ − } , is Γ-gradedgroupoid by the cocyle c : ( G Γ ) ε −→ Γ ,g α,β c ( g ) . Recall that for a Γ-graded ´etale groupoid G , the skew-product G × c Γ given in Definition 3.6 is Γ-graded withcomponents (
G × c Γ) γ = G γ × Γ, γ ∈ Γ (see [4, § Lemma 3.11.
Let G be a Γ -graded groupoid. We have the following. (1) If Γ is a trivial group then G Γ = G . (2) The groupoid G Γ is strongly Γ -graded. (3) There is a Γ -graded isomorphism G × c Γ ∼ = ( G Γ ) ε . (4) The groupoids G and G Γ are graded Kakutani equivalent.Proof. (1) This follows from the construction (3.6).(2) A Γ-groupoid G is strongly graded if and only if s ( G γ ) = G (0) for every γ ∈ Γ (see [8, Lemma 2.3]). Let x α,α ∈ G (0)Γ ,where α ∈ Γ and x ∈ G (0) . Then x αγ,α ∈ ( G Γ ) γ − , with s ( x αγ,α ) = x α,α . Thus G Γ is strongly graded.(3) Consider the map φ : G × c Γ −→ ( G Γ ) ε , (3.7)( g, γ ) g c ( g ) γ,γ . It is easy to show that this map is a Γ-graded groupoid isomorphism.(4) Note that if Y is a full clopen subset of G (0) if and only if Y ε := { x ε,ε | x ∈ Y } is a full clopen subset in G Γ . Themap ψ : G| Y −→ G Γ | Y ε , (3.8) g g ε,ε , is a Γ-graded groupoid isomorphism. Now considering Y = G (0) the claim follows. (cid:3) In [13, Theorem 2.12] it was shown that for a Γ-graded ring A , where Γ is a finite group, A is strongly gradedif and only if A ε and A Theorem 3.12.
Let G be a Γ -graded ´etale groupoid. The following statements are equivalent: (1) The groupoid G is strongly graded; (2) G ε and G × c Γ are Kakatuni equivalent with respect to some full clopen subset Y ⊆ G (0) ε in G ε and Y ε = { ( y, ε ) | y ∈ Y } a full clopen subset of G (0) × Γ in G × c Γ .Proof. (1) ⇒ (2): Suppose that G is strongly graded. We first claim that Y ε = { ( y, ε ) | y ∈ Y } is a full clopen subset of G (0) × Γ in
G × c Γ, when Y ⊆ G (0) ε is clopen full in G ε . Take any ( y, γ ) ∈ G (0) × Γ. For y ∈ G (0) , there exists g ∈ G γ suchthat r ( g ) = y , since G is strongly graded. For s ( g ) ∈ G (0) , there exists g ′ ∈ G ε such that s ( g ′ ) ∈ Y and r ( g ′ ) = s ( g ),since Y ⊆ G (0) is full in G ε . Then we have ( gg ′ , ε ) ∈ G × c Γ such that r ( gg ′ , ε ) = ( y, γ ) and s ( gg ′ , ε ) = ( s ( g ′ ) , ε ) ∈ Y ε .Since G × c Γ has product topology, Y ε is clopen in G (0) × Γ when Y is clopen in G (0) . Thus we complete the proof ofthe claim. Observe that ( G| Y ) ε = G ε | Y and ( G Γ | Y ε ) ε = ( G Γ ) ε | Y ε . By (3.8) and (3.7) , we have ( G ε ) | Y ∼ = ( G × c Γ) | Y ε asgroupoids. The unit space G (0) is a full clopen subset for G ε . Hence, (ii) holds.(2) ⇒ (1): Suppose that (ii) holds. Then there exist Y ⊆ G (0) ε a full clopen in G ε and Y ε = { ( y, ε ) | y ∈ Y } a fullclopen subset of G (0) × Γ in
G × c Γ. We need to show that r ( G γ ) = G (0) for any γ ∈ Γ. Take any γ ∈ Γ and any u ∈ G (0) .There exists ( g, η ) ∈ G × c Γ such that s ( g, η ) ∈ Y ε and r ( g, η ) = ( u, γ ). This implies η = ε and g ∈ G γ . Thus we have r ( G γ ) = G (0) for any γ ∈ Γ. (cid:3) Steinberg algebras
In this section we briefly recall the construction of the Steinberg algebra associated to an ample groupoid. We willthen tie together the concepts between graded groupoid theory and graded ring theory via Steinberg theory.Let G be an ample topological groupoid. Suppose that R is a commutative ring with unit. Consider A R ( G ) := C c ( G , R ), the space of compactly supported continuous functions from G to R with R given the discrete topology. Then A R ( G ) is an R -algebra with addition defined point-wise and multiplication by( f ∗ g )( γ ) = X αβ = γ f ( α ) g ( β ) . It is useful to note that 1 U ∗ V = 1 UV for compact open bisections U and V (see [28, Proposition 4.5(3)]). With this structure, A R ( G ) is an algebra called the Steinberg algebra associated to G . The algebra A R ( G ) can also be realised as the span of characteristic functions of the OMOLOGY OF ´ETALE GROUPOIDS 11 form 1 U , where U is a compact open bisection. By [11, Lemma 2.2] and [10, Lemma 3.5], every element f ∈ A R ( G ) canbe expressed as f = X U ∈ F a U U , (4.1)where F is a finite subset of mutually disjoint compact open bisections.Recall from [12, Lemma 3.1] that if G = F γ ∈ Γ G γ is a Γ-graded ample groupoid, then the Steinberg algebra A R ( G )is a Γ-graded algebra with homogeneous components A R ( G ) γ = { f ∈ A R ( G ) | supp( f ) ⊆ G γ } . The family of all idempotent elements of A R ( G (0) ) is a set of local units for A R ( G ) (see [9, Lemma 2.6]). Here, A R ( G (0) ) ⊆ A R ( G ) is a subalgebra. Note that any ample groupoid admits the trivial cocycle from G to the trivial group { ε } , which gives rise to a trivial grading on A R ( G ).Let A be an R -algebra. A representation of B co ∗ ( G ) in A is a family (cid:8) t U | U ∈ B co ∗ ( G ) (cid:9) ⊆ A satisfying(R1) t ∅ = 0;(R2) t U t V = t UV for all U, V ∈ B co ∗ ( G ), and(R3) t U + t V = t U ∪ V , whenever U, V ∈ B co γ ( G ) for some γ ∈ Γ, U ∩ V = ∅ and U ∪ V ∈ B co γ ( G ).We have the following statement that A R ( G ) is a universal algebra (see [10, Theorem 3.10] and [11, Proposition 2.3]),which will be used throughout the paper. Lemma 4.1.
Let G be a Γ -graded ample groupoid. Then (cid:8) U | U ∈ B co ∗ ( G ) (cid:9) ⊆ A R ( G ) is a representation of B co ∗ ( G ) which spans A R ( G ) . Moreover, A R ( G ) is universal for representations of B co ∗ ( G ) in the sense that for every representation (cid:8) t U | U ∈ B co ∗ ( G ) (cid:9) of B co ∗ ( G ) in an R -algebra A , there is a unique R -algebra homomorphism π : A R ( G ) −→ A such that π (1 U ) = t U , for all U ∈ B co ∗ ( G ) . The Steinberg algebra of a semi-direct product groupoid.
Let G be a Γ-graded ample groupoid, graded bythe cocycle c : G −→ Γ, R a unital commutative ring and G × c Γ the skew product groupoid (see Definition 3.6). By [4,Theorem 3.4] there is an isomorphism of Γ-graded algebras A R ( G × c Γ) −→ A R ( G ) , (4.2)1 U l X i =1 U i p γ i , where U = F li =1 U i × { γ i } ∈ B co ∗ ( G × c Γ) (see (3.3)). This will be used in the proof of Theorem 5.7 to calculate thegraded homology of graph groupoids.In this subsection we establish the dual of this statement. We show that the Steinberg algebra of the semi-directproduct groupoid of an ample groupoid (Example 3.2) is graded isomorphic to the partial skew group ring of its Steinbergalgebra (Proposition 4.2).Recall from [20, § φ = ( φ γ , X γ , X ) γ ∈ Γ of a discrete group Γ on a locally compact Hausdorfftopological space X . In the case X is an algebra or a ring then the subsets X γ should also be ideals and the maps φ γ should be isomorphisms of algebras. There is a Γ-graded groupoid G X = [ γ ∈ Γ γ × X γ , associated to a partial action ( φ γ , X γ , X ) γ ∈ Γ , whose composition and inverse maps are given by ( γ, x )( γ ′ , y ) = ( γγ ′ , x )if y = φ γ − ( x ) and ( γ, x ) − = ( γ − , φ γ − ( x )).The action φ : Γ y G is a partial action of Γ on G such that φ γ is a bijection from G to G . Thus the action φ inducesan action of Γ on A R ( G ), still denoted by φ , such that φ γ ( f ) = f ◦ φ γ − (4.3)for γ ∈ Γ and f ∈ A R ( G ) (see [20, § φ : Γ y G with G = G (0) , observe that the semi-direct product coincides with the groupoid G X = S γ ∈ Γ γ × X with X = G (0) .Let φ = ( φ γ , A γ , A ) γ ∈ Γ be a partial action of the discrete group Γ on an algebra A . The partial skew group ring A ⋊ φ Γ consists of all formal forms P γ ∈ Γ a γ δ γ (with finitely many a γ nonzero), where a γ ∈ A γ and δ γ are symbols, withaddition defined in the obvious way and multiplication being the linear extension of( a γ δ γ )( a γ ′ δ γ ′ ) = φ γ (cid:0) φ γ − ( a γ ) a γ ′ (cid:1) δ γγ ′ . Observe that A ⋊ φ Γ is always a Γ-graded ring with ( A ⋊ φ Γ) γ = A γ δ γ for γ ∈ Γ.Recall from [20, Proposition 3.7] that the Steinberg algebra of G X is the partial skew group ring C R ( X ) ⋊ φ Γ. Weshow here that the Steinberg algebra of the semi-direct product G ⋊ φ Γ is the partial skew group ring A R ( G ) ⋊ φ Γ. Proposition 4.2.
Let G be an ample groupoid and φ : Γ y G an action of a discrete group Γ on G and R a commutativering with unit. Then there is a Γ -graded R -algebra isomorphism A R ( G ⋊ φ Γ) ∼ = A R ( G ) ⋊ φ Γ . Proof.
We first define a representation { t U | U ∈ B co ∗ ( G ⋊ φ Γ) } in the algebra A R ( G ) ⋊ φ Γ. For any graded compactopen bisection U ∈ B co α ( G ⋊ φ Γ) of G ⋊ φ Γ, we may write U = U ∩ G × { α } = U α × α , where U α ⊆ G is a compact openbisection of G . We define t U = 1 U α δ α . We show that these elements t U satisfy (R1)–(R3). Certainly if U = ∅ , then t U = 0, giving (R1). For (R2), take V ∈ B co β ( G ⋊ φ Γ), and write V = V β × { β } . Then t U t V = 1 U α δ α · V β δ β = φ α ( φ α − (1 U α )1 V β ) δ αβ = 1 U α ∗ φ α (1 V β ) δ αβ = 1 U α ∗ φ α ( V β ) δ αβ = 1 U α φ α ( V β ) δ αβ . (4.4)Here, the second last equality holds since φ α (1 V β ) = 1 φ α ( V β ) by (4.3). On the other hand, by the composition of thesemi-direct product G ⋊ φ Γ, we have
U V = U α × { α } · V β × { β } = U α φ α ( V β ) × { αβ } . Observe that U α φ α ( V β ) is compact since U α and φ α ( V β ) are compact. It follows that t UV = 1 U a φ α ( V β ) δ αβ . Comparingwith (4.4), we have t U t V = t UV . For (R3), suppose that U and V are disjoint elements of B co γ ( G ⋊ φ Γ) for some γ ∈ Γsuch that U ∪ V is a bisection of G ⋊ φ Γ. Write them as U = U γ × { γ } and V = V γ × { γ } such that U γ and V γ aredisjoint. We have t U ∪ V = 1 U γ ∪ V γ δ γ = (1 U γ + 1 V γ ) δ γ = t U + t V .By the universality of Steinberg algebras, we have an R -homomorphism,Φ : A R ( G ⋊ φ Γ) −→ A R ( G ) ⋊ φ Γsuch that Φ(1 B ×{ α } ) = 1 B δ α for each compact open bisection B of G and α ∈ Γ. From the definition of Φ, it is evidentthat Φ preserves the grading. Hence, Φ is a homomorphism of Γ-graded algebras.It remains to show that Φ is an isomorphism. For injectivity, take f ∈ A R ( G ⋊ φ Γ) γ with γ ∈ Γ such that Φ( f ) = 0. Wemay write f = P ni =1 r i U i with U i ∈ B co γ ( G ⋊ φ Γ) such that U i ’s are mutually disjoint. Each U i = W i × { γ } with W i ⊆ G is a compact open bisection of G and W i ’s are mutually disjoint. Then we have Φ( f ) = P ni =1 r i W i δ γ = 0, and thus P ni =1 r i W i = 0, implying that each r i is zero. Hence f = 0. For the surjectivity of Φ, take f ′ δ γ ∈ A R ( G ) ⋊ φ Γ with γ ∈ Γsuch that f ′ = P mj =1 t j W j with t j ∈ R and W j disjoint compact open bisections of G . We have Φ( P mj =1 t j W j ×{ γ } ) = f ′ . (cid:3) Steinberg algebras of stable groupoids and graded matrix groupoids.
Next we relate the notion of gradedmatrix rings ( § § § Proposition 4.3.
Let Γ be a discrete group, G a Γ -graded ample groupoid and R a commutative ring with unit. Wehave A R ( G Γ ) ∼ = M Γ ( A R ( G ))(Γ) as Γ -graded R -algebras.Proof. We first define a representation { t U | U ∈ B co ∗ ( G Γ ) } in the algebra M Γ ( A R ( G ))(Γ). If U is a γ -graded compactopen bisection of G Γ , then for α, β ∈ Γ, the set U ∩ G αβ is a compact open bisection, where G αβ = { g αβ | g ∈ G} . Sincethese are mutually disjoint and U is compact, there are finitely many (distinct) pairs ( α, β ) such that U = G ( α,β ) ∈ Γ × Γ U ∩ G αβ . Each U ∩ G αβ has the form U αβ = { g αβ | g ∈ U ′ αβ } for U ′ αβ ⊆ G a compact open subset. Observe that U ′ αβ is a bisection,since U = F α,β ∈ Γ U αβ is a bisection. Take any x αβ ∈ U αβ . We have c ( x αβ ) = α − c ( x ) β = γ , implying c ( x ) = αγβ − for all x ∈ U ′ αβ . So each U ′ αβ ∈ B co αγβ − ( G ), and U = F ( α,β ) ∈ Γ × Γ U αβ . Using this decomposition, we define t U = X ( α,β ) ∈ Γ × Γ A αβ , (4.5)where A αβ is the matrix with 1 U ′ αβ in the ( α, β )-th position and all the other entries zero. Observe that each A αβ is in M Γ ( A R ( G )) γ . OMOLOGY OF ´ETALE GROUPOIDS 13
Obviously, t ∅ = 0 and thus (R1) holds. For (R2), take U, V ∈ B co ∗ ( G Γ ). We assume that U = F ( α,β ) U αβ for finitelymany distinct pairs ( α, β ) ∈ Γ × Γ and V = F ( µ,ν ) V µν finitely many distinct pairs ( µ, ν ) ∈ Γ × Γ. On one hand, we have t U t V = X ( α,β ) X ( µ,ν ) A αβ A µν = X ( α,β ) X ( µ,ν ) A αβ A µν = X ( α,β ) X { ( µ,ν ) ,β = µ } A αβ A µν . (4.6)On the other hand, we have U V = G ( α,β ) U αβ G ( µ,ν ) V µν = G ( α,β ) G ( µ,ν ) U αβ V µν = G ( α,β ) G { ( µ,ν ) ,β = µ } U αβ V µν , (4.7)where the last equality follows from the multiplication of the groupoid G Γ . If all the pairs ( α, ν ) appearing in (4.7)are pairwise distinct, then by comparing (4.6) and (4.7) we obtain t UV = t U t V , since 1 U ′ αβ V ′ µν = 1 U ′ αβ ∗ V ′ µν in A R ( G ).Assume that there are pairs ( α , β ) = ( α , β ) and ( µ , ν ) = ( µ , ν ) such that ( α , ν ) = ( α , ν ). In this case, wehave β = µ , β = µ , β = β and µ = µ . Since the ranges of U α β and U α β are disjoint, so are U α β V µ ν and U α β V µ ν . Observe that the matrix A α β A µ ν + A α β A µ ν has 1 U α β V µ ν + 1 U α jβ V µ ν in the ( α , ν )-thposition and all its other entries are zero. After combining pairs where ( α i , ν i ) = ( α i ′ , ν i ′ ) in (4.7), we have t UV = t U t V .For (R3), suppose that U and V are disjoint elements in B co γ ( G Γ ) with γ ∈ Γ such that U ∪ V is a bisection. Write U = F ( α,β ) U αβ and V = F ( µ,ν ) V µν as above. It follows that U ∪ V = F ( α,β ) F ( µ,ν ) U αβ ∪ V µν . Since U ∪ V is a bisection,we have U ′ αβ ∩ V ′ µν = ∅ if ( α, β ) = ( µ, ν ). In this case, A αβ + A µν is the matrix with 1 U ′ αβ ∪ V ′ µν = 1 U ′ αβ + 1 V ′ µν in the( α, β )th-position. This shows that after combining pairs that ( α, β ) = ( µ, ν ), we have t U ∪ V = t U + t V .By the universality of Steinberg algebra, there exists an R -algebra homomorphism φ : A R ( G Γ ) −→ M Γ ( A R ( G ))(Γ)such that φ (1 U ) = P ( α,β ) ∈ Γ × Γ A αβ given in (4.5) for each U ∈ B co ∗ ( G Γ ). The homomorphism φ preserves the grading.It remains to show that φ is an isomorphism. For the surjectivity of φ , for any α, β ∈ Γ, take a matrix E αβ ∈ M Γ ( A R ( G ))(Γ) with f ′ ∈ A R ( G ) in the ( α, β )-th position and all the other entries zero. It suffices to show that E αβ ∈ Im φ . We can write f ′ = P w ∈ F r w B w with F a finite set, r w ∈ R and B w ∈ B co ∗ ( G ). Let X w = { g α,β | g ∈ B w } for each w ∈ F . Then X w is a graded compact open bisection of G Γ . Observe that φ ( P w ∈ F r w X w ) = P w ∈ F φ ( r w X w ) = E αβ ∈ Im φ .To check the injectivity of φ , suppose that f ′′ ∈ A R ( G Γ ) such that φ ( f ′′ ) = 0. Recall from (4.1) that we can write f ′′ = P i r i U i with r i ∈ R and U i ∈ B co ∗ ( G Γ ) mutually disjoint graded compact open bisections. For any α, β ∈ Γ, let Y i = U i ∩ { g αβ | g ∈ G} for each i . There exists Y ′ i ⊆ G such that Y i = { g αβ | g ∈ Y ′ i } . Observe that Y ′ i ∩ Y ′ i ′ = ∅ for i = i ′ ; otherwise, U i ∩ U i ′ = ∅ . We write φ ( f ′′ ) = E ∈ M Γ ( A R ( G ))(Γ). It follows that the entry of E in the ( α, β )-thposition is P i r i Y ′ i ∈ A R ( G ). Thus P i r i Y ′ i = 0 in A R ( G ). Since all the Y ′ i ’s are disjoint, we obtain r i = 0 for all i ,implying f ′′ = 0 in A R ( G Γ ). Hence, φ is injective. (cid:3) Proposition 4.4.
Let Γ be a discrete group, G a Γ -graded ample groupoid and R a commutative ring with unit. Supposethat f : G (0) −→ Z with Z a discrete group is a constant function such that f ( x ) = n with n a positive integer for any x ∈ G (0) . Take ( γ , γ , · · · , γ n +1 ) ∈ Γ n +1 . Let ψ : G (0) f −→ Γ be the continuous map given by ψ ( x ii ) = γ i +1 for x ∈ G (0) and ≤ i ≤ n . Then A R ( G f ( ψ )) ∼ = M n +1 ( A R ( G ))( γ , γ , · · · , γ n +1 ) , as Γ -graded R -algebras.Proof. We observe that G f ( ψ ) −→ G Γ , x ij x γ − i +1 γ − j +1 is a Γ-graded isomorphism as groupoids. The proof now followsfrom Proposition 4.3. (cid:3) If Γ is a trivial group, we have the non-graded version of the above result.
Corollary 4.5.
Let G be an ample groupoid and R a commutative ring with unit. Suppose that f : G (0) −→ Z with Z a discrete group is a constant function such that f ( x ) = n with n a positive integer for any x ∈ G (0) . Then A R ( G f ) ∼ = M n +1 ( A R ( G )) . Steinberg algebras and graded Morita context.
Let G be a Γ-graded ample groupoid and R a commutativering with unit. Recall that the characteristic functions of compact open subsets of G (0) are graded local units for theSteinberg algebra A R ( G ).We show that the graded ( G , H )-equivalence induces a graded Morita equivalence on the Steinberg algebra level. Let G and H be Γ-graded ample groupoids. Suppose that Z is a Γ-graded ( G , H )-equivalence with linking groupoid L . Let i denote the inclusion maps from A R ( G ) and A R ( H ) into A R ( L ). Define M := (cid:8) f ∈ A R ( L ) | supp( f ) ⊆ Z (cid:9) and N := (cid:8) f ∈ A R ( L ) | supp( f ) ⊆ Z op (cid:9) . Let A R ( G ) and A R ( H ) act on the left and right of M and on the right and left of N by a · f = i ( a ) ∗ f and f · a = f ∗ i ( a ).Then there are Γ-graded bimodule homomorphisms ψ : M ⊗ i ( A R ( H )) N −→ A R ( G ) and ϕ : N ⊗ i ( A R ( G )) M −→ A R ( H )determined by iψ ( f ⊗ g ) = f ∗ g and iϕ ( g ⊗ f ) = g ∗ f .We are in a position to relate the graded equivalence of groupoids to equivalence of their associated Steinbergalgebras. Theorem 4.6.
Let G and H be Γ -graded ample groupoids. Suppose that Z is a Γ -graded ( G , H ) -equivalence with Γ -graded linking groupoid L . The tuple ( A R ( G ) , A R ( H ) , M, N, ψ, ϕ ) is a surjective graded Morita context, and so A R ( G ) and A R ( H ) are graded Morita equivalent.Proof. We refer to [12, Theorem 5.1] for the proof that ( A R ( G ) , A R ( H ) , M, N, ψ, ϕ ) is a surjective Morita context. Itis evident that the tuple ( A R ( G ) , A R ( H ) , M, N, ψ, ϕ ) is graded surjective Morita context. By [17, Theorem 2.6], A R ( G )and A R ( H ) are graded Morita equivalent as Γ-graded rings. (cid:3) Graded homology of ´etale groupoids
In this section, we introduce graded homology groups for a graded ´etale groupoid G and prove that when the groupoidis strongly graded and ample, its graded homology groups are isomorphic to the homology groups for the ε -th component G ε (Theorem 5.4). We then establish a short exact sequence H gr0 ( G ) −→ H gr0 ( G ) −→ H ( G ) −→ , for a graded ample groupoid G (Theorem 5.7). This is in line with the van den Bergh exact sequence for graded K -theoryof a Z -graded regular Noetherian ring R ([19, § −→ K gr n ( R ) −→ K gr n ( R ) −→ K n ( R ) −→ K gr n − ( R ) −→ · · · , and suggests that we might have a long exact sequence −→ H gr n ( G ) −→ H gr n ( G ) −→ H n ( G ) −→ H gr n − ( G ) −→ · · · . In Section 6 we show that for the graph groupoid G E associated to an arbitrary graph E , we have H gr0 ( G E ) ∼ = K gr0 ( L R ( E )), where L R ( E ) is the Leavitt path algebra with the coefficient field R . Theorem 5.1.
Let R be a commutative ring with unit and let G and G be Γ -graded ample groupoids. If G and G are Γ -graded Kakutani equivalent, then the categories Gr - R G and Gr - R G are graded equivalent.Proof. There are full clopen subsets Y i ⊆ G (0) i , for i = 1 ,
2, such that G | Y is graded isomorphic to G | Y . By [8,Proposition 3.3] the categories Gr - R G and Gr - A R ( G ) are equivalent. Since 1 Y i are homogeneous full idempotent of A R ( G i ), and 1 Y i A R ( G i )1 Y i ∼ = A R ( G i | Y i ), where i = 1 ,
2, we have Gr - R G ∼ = Gr - A R ( G ) ∼ = Gr -1 Y A R ( G )1 Y ∼ = Gr - A R ( G | Y ) ∼ = Gr - A R ( G | Y ) ∼ = Gr -1 Y A R ( G )1 Y ∼ = Gr - A R ( G ) ∼ = Gr - R G . (cid:3) One can provide an alternative proof for Theorem 5.1 as follows. There are full clopen subsets Y i ⊆ G (0) i , for i = 1 , G | Y is graded isomorphic to G | Y . By [12, Lemma 6.1] s − ( Y ) is Γ-graded ( G , G | Y )-equivalence. Similarly r − ( Y ) is Γ-graded ( G , G | Y )-equivalence. Graded groupoid equivalence is an equivalence relation. It follows that G and G are graded groupoid equivalent. By Theorem 4.6 and [17, Theorem 2.6] we have the category equivalece Gr - A R ( G ) ∼ = Gr - A R ( G ). Recall from [8, Proposition 3.3] that the categories Gr - R G and Gr - A R ( G ) are equivalent.Thus Gr - R G ∼ = Gr - A R ( G ) ∼ = Gr - A R ( G ) ∼ = Gr - R G .Since G and G Γ are graded Kakutani and the latter groupoid is strongly graded (Lemma 3.11), combining Theorem 5.1and [8, Theorem 3.10], we obtain Gr - R G ∼ = Gr - R G Γ ∼ = Mod - R ( G Γ ) ε ∼ = Mod - R G × c Γ . (5.1) OMOLOGY OF ´ETALE GROUPOIDS 15
The equivalence (5.1) justifies the use of
G × c Γ in order to define the graded version of homology of the Γ-gradedgroupoid G .5.1. Graded homology groups for ´etale groupoids.
Let X be a locally compact Hausdorff space and R a topologicalabelian group. Let Γ be a discrete group acting from right continuously on X , i.e., there is a group homomorphismΓ −→ Home( X ) op , where Home( X ) consists of homeomorphisms from X to X with the multiplication given by thecomposition of maps and Home( X ) op is the opposite group of Home( X ). Denote by C c ( X, R ) the set of R -valuedcontinuous functions with compact support. With point-wise addition, C c ( X, R ) is an abelian group. For γ ∈ Γ and f ∈ C c ( X, R ), defining γf ( x ) := f ( xγ ), makes C c ( X, R ) a left Γ-module.Let π : X → Y be a local homeomorphism between locally compact Hausdorff spaces which respect the right Γ-action,namely π ( xγ ) = π ( x ) γ . For f ∈ C c ( X, R ), define the map π ∗ ( f ) : Y → R by π ∗ ( f )( y ) = X π ( x )= y f ( x ) . One can check that π ∗ satisfies γπ ∗ ( f ) = π ∗ ( γf ) for f ∈ C c ( X, R ). Thus π ∗ is a Γ-homomorphism from C c ( X, R ) to C c ( Y, R ) which makes C c ( − , R ) a functor from the category of right Γ-locally compact Hausdorff spaces with Γ-localhomeomorphisms to the category of left Γ-modules.For a groupoid G , we denote by Aut( G ) the group of homeomorphisms between G which respect the composition of G . We say that a discrete group Γ continuously acts on G from right side if α : Γ −→ Aut( G ) op is a group homomorphismwith Aut( G ) op the opposite group of Aut( G ).Let G be an ´etale groupoid and Γ a discrete group acting continuously on G from right side. For n ∈ N , we write G ( n ) for the space of composable strings of n elements in G , that is, G ( n ) = { ( g , g , . . . , g n ) ∈ G n | s ( g i ) = r ( g i +1 ) for all i = 1 , , . . . , n − } . For i = 0 , , . . . , n , with n ≥ d i : G ( n ) → G ( n − be a map defined by d i ( g , g , . . . , g n ) = ( g , g , . . . , g n ) i = 0( g , . . . , g i g i +1 , . . . , g n ) 1 ≤ i ≤ n − g , g , . . . , g n − ) i = n. When n = 1, we let d , d : G (1) → G (0) be the source map and the range map, respectively. Clearly the maps d i arelocal homeomorphisms which respects the right Γ-actions.Define the Γ-homomorphisms ∂ n : C c ( G ( n ) , R ) → C c ( G ( n − , R ) by ∂ = s ∗ − r ∗ and ∂ n = n X i =0 ( − i d i ∗ . (5.2)One can check that the sequence0 ∂ ←− C c ( G (0) , R ) ∂ ←− C c ( G (1) , R ) ∂ ←− C c ( G (2) , R ) ∂ ←− · · · (5.3)is a chain complex of left Γ-modules.The following definition comes from [14, 23] with an added structure of a group acting on the given groupoid fromright. Definition 5.2. ( Homology groups of a groupoid G ) Let G be an ´etale groupoid and Γ a discrete group acting continuouslyfrom right on G . Define the homology groups of G with coefficients R , H n ( G , R ), n ≥
0, to be the homology groups ofthe Moore complex (5.3), i.e., H n ( G , R ) = ker ∂ n / Im ∂ n +1 , which are left Γ-modules. When R = Z , we simply write H n ( G ) = H n ( G , Z ). In addition, we define H ( G ) + = { [ f ] ∈ H ( G ) | f ( x ) ≥ x ∈ G (0) } , where [ f ] denotes the equivalence class of f ∈ C c ( G (0) , Z ).We are in a position to define a graded homology theory for a graded groupoid. Definition 5.3. ( Graded homology groups of a groupoid G ) Let G be a Γ-graded ´etale groupoid. Then the skew product G × c Γ is an ´etale groupoid with a right Γ-action ( g, α ) γ = ( g, αγ ). We define the graded homology groups of G as H gr n ( G , R ) := H n ( G × c Γ , R ) , n ≥ . When R = Z , we simply write H gr n ( G ) = H gr n ( G , Z ). In addition, we define H gr0 ( G ) + = H ( G × c Γ) + . Note that H gr n ( G , R ) is a left Γ-module. For a strongly Γ-graded ample groupoid G , we prove that the graded homology H gr n ( G ) of G is isomorphic to thehomology H n ( G ε ) for the groupoid G ε , where ε is the identity for the group Γ.Recall that a space is σ -compact if it has a countable cover by compact sets. Theorem 5.4.
Let G be a strongly Γ -graded ample groupoid such that G (0) is σ -compact. Then H gr n ( G ) ∼ = H n ( G ε ) foreach n ≥ , where ε is the identity of the group Γ .Proof. By Theorem 3.12, G ε and G × c Γ is Kakutani equivalent with respect to some full clopen subset of G (0) in G ε and Y ε = { ( y, ε ) | y ∈ Y } a full clopen subset of G (0) × Γ in
G × c Γ. Since G is ample, G ε and G × c Γ are ample groupoids.By [23, Theorem 3.6(2)] and [23, Proposition 3.5(2)], we have H n ( G ε ) ∼ = H n ( G ε | Y ) and H n ( G × c Γ) ∼ = H n (( G × c Γ) | Y ε )for each n ≥
0. Since G ε | Y ∼ = ( G × c Γ) | Y ε as groupoids, it follows that H n ( G × c Γ) ∼ = H n (( G × c Γ) | Y ε ) ∼ = H n ( G ε | Y ) ∼ = H n ( G ε )for each n ≥ (cid:3) Next we determine the image of the map ∂ in the Moore complex (5.3). Here we consider the coefficients of thehomology in the ring R , with discrete topology. Recall also from (3.1) that the B co ∗ ( G ) is the collection of all gradedcompact open bisections of G . Lemma 5.5.
Let G be a Γ -graded ample groupoid. We have Im ∂ = R - span (cid:8) s ( U ) − r ( U ) | U ∈ B co ∗ ( G ) (cid:9) as an abelian subgroup of C c ( G (0) , R ) .Proof. Since G is an ample groupoid, by Lemma 4.1 it suffices to show that ∂ (1 U ) = 1 s ( U ) − r ( U ) , for each U ∈ B co ∗ ( G ).For each x ∈ G (0) , we have ∂ (1 U )( x ) = X { g ∈G | s ( g )= x } U ( g ) − X { g ∈G | r ( g )= x } U ( g )= X { g ∈G | s ( g )= x } U ( g ) − X { g ∈G | s ( g )= x } U ( g − )= 1 U − U ( x ) − UU − ( x )= (1 s ( U ) − r ( U ) )( x ) . (cid:3) In the next section we will consider the homology groups of graph groupoids which have a natural Z -graded structure.For the remaining of this section, we will work with Z -graded groupoids. For a Z -graded ample groupoid, we establish anexact sequence relating the graded zeroth homology group to the non-graded version (Theorem 5.7). In order to achievethis, we will use the calculus of smash products on the level of Steinberg algebras first studied in [4] (see also § Lemma 5.6.
Let G be an abelian group. Then the following sequence of abelian groups is exact L Z G Λ −−−−→ L Z G Σ −−−−→ G −−−−→ , where Λ( { x i } i ∈ Z ) = { x i − } i ∈ Z − { x i } i ∈ Z and Σ( { x i } i ∈ Z ) = P i x i for any { x i } i ∈ Z in L Z G .Proof. Obviously, Σ is surjective and Σ ◦ Λ = 0. It suffices to show that Ker Σ ⊆ ImΛ. Take { x i } i ∈ Z ∈ Ker Σ. Denoteby j the minimal integer i such that x i is nonzero. Denote by k the maximal integer i such that x i is nonzero. We have P ki = j x i = 0, since { x i } i ∈ Z ∈ Ker Σ. Let { y i } i ∈ Z be given by y i = ( − P il = j x l , if j ≤ i ;0 , otherwise.Observe that if i > k , we have y i = − P il = j x l = − P kl = j x l − P il = k +1 x l = 0. Thus we have { y i } i ∈ Z ∈ L Z G . By thedefinition of Λ, we have Λ( { y i } i ∈ Z ) = { x i } i ∈ Z , implying Ker Σ ⊆ ImΛ. (cid:3)
We are in a position to prove the main theorem of this section.
Theorem 5.7.
Let G be a Z -graded ample groupoid. Then we have an exact sequence, H gr0 ( G ) e Λ −−−−→ H gr0 ( G ) e Σ −−−−→ H ( G ) −−−−→ . OMOLOGY OF ´ETALE GROUPOIDS 17
Proof.
Specialising the calculus of smash products to the case of Z -graded groupoids, by (4.2) there is an isomorphismof Z -graded algebras A R ( G × c Z ) ∼ = A R ( G ) Z which on the level of unit spaces gives rise to C c (cid:0) ( G × c Z ) (0) , Z (cid:1) ∼ = C c ( G (0) , Z ) Z . Observe that the multiplication in C c ( G (0) , Z ) Z is given by ( f p n )( gp m ) = δ n,m ( f g ) p n , for f, g ∈ C c ( G (0) , Z ) and n, m ∈ Z (see (2.1)). It follows that σ : C c (cid:0) ( G × c Z ) (0) , Z (cid:1) ∼ −→ C c ( G (0) , Z ) Z ∼ −→ M Z C c ( G (0) , Z )is an isomorphism of rings, sending 1 U to { f k } k ∈ Z such that f n i = 1 U i and f k = 0 for all other k ’s with U = F li =1 U i ×{ n i } satisfying n i = n j for i = j . Thus for any f ∈ C c (( G × c Z ) (0) , Z ) there is a unique element { f i } i ∈ Z in L Z C c ( G (0) , Z )corresponding to f . Note that there are only finitely many i ∈ Z such that f i are nonzero.Setting G = C c ( G (0) , Z ), by Lemma 5.6, we the following exact sequence, L Z C c ( G (0) , Z ) Λ −−−−→ L Z C c ( G (0) , Z ) Σ −−−−→ C c ( G (0) , Z ) −−−−→ . (5.4)We observe that Λ : L Z C c ( G (0) , Z ) −→ L Z C c ( G (0) , Z ) satisfiesΛ σ (Im ∂ ) ⊆ σ (Im ∂ ) , (5.5)where ∂ : C c (( G × c Z ) (1) , Z ) −→ C c (( G × c Z ) (0) , Z ) is given by (5.2) for the groupoid G × c Z . By Lemma 5.5, we onlyneed to show that Λ( σ (1 s ( U ) − r ( U ) )) ∈ Im ∂ , for U ∈ B co n ( G × c Z ) with n ∈ Z . By (3.3), we write U = F li =1 U i × { n i } with n i = n j for i = j . Then we haveΛ σ (1 s ( U ) − r ( U ) ) =Λ σ (1 F li =1 s ( U i ) ×{ n i } − F li =1 r ( U i ) ×{ n + n i } )= σ (1 F li =1 s ( U i ) ×{ n i +1 } − F li =1 r ( U i ) ×{ n + n i +1 } ) − σ (1 F li =1 s ( U i ) ×{ n i } − F li =1 r ( U i ) ×{ n + n i } )= σ (1 s ( U ′ ) − r ( U ′ ) ) − σ (1 s ( U ) − r ( U ) ) ∈ σ Im ∂ , where U ′ = F li =1 U i × { n i + 1 } . We have Σ σ (Im ∂ ) ⊆ Im ∂ ′ , (5.6)where ∂ ′ : C c ( G (1) , Z ) −→ C c ( G (0) , Z ) is given by (5.2) for the groupoid G . Observe that (5.6) follows from the followingequalities for U = F li =1 U i × { n i } ∈ B co ∗ ( G × c Z )Σ σ (1 s ( U ) − r ( U ) ) = Σ σ (1 F li =1 s ( U i ) ×{ n i } − F li =1 r ( U i ) ×{ n + n i } )= l X i =1 (1 s ( U i ) − r ( U i ) ) ∈ Im ∂ ′ . The statement that the following sequence of abelian groups is exact L Z C c ( G , Z ) /σ (Im ∂ ) Λ −−−−→ L Z C c ( G , Z ) /σ (Im ∂ ) Σ −−−−→ C c ( G , Z ) / Im ∂ ′ −−−−→ σ (Im ∂ ) −→ Im ∂ ′ is surjective.Recall that H gr0 ( G ) = C c (( G × c Z ) (0) , Z ) / Im ∂ and H ( G ) = C c ( G (0) , Z ) / Im ∂ ′ . Let e Λ : H gr0 ( G ) −→ H gr0 ( G ) be thecomposition H gr0 ( G ) σ −→ M Z C c ( G , Z ) /σ (Im ∂ ) Λ −→ M Z C c ( G , Z ) /σ (Im ∂ ) σ − −−→ H gr0 ( G )and e Σ : H gr0 ( G ) −→ H ( G ) be the composition H gr0 ( G ) σ −→ M Z C c ( G , Z ) /σ (Im ∂ ) Σ −→ C c ( G , Z ) / Im ∂ ′ . The theorem now follows from the following commutative diagram whose second row is an exact sequence. H gr0 ( G ) e Λ / / σ (cid:15) (cid:15) H gr0 ( G ) e Σ / / σ (cid:15) (cid:15) H ( G ) / / id (cid:15) (cid:15) L Z C c ( G , Z ) /σ (Im ∂ ) Λ / / L Z C c ( G , Z ) /σ (Im ∂ ) Σ / / C c ( G , Z ) / Im ∂ ′ / / . (cid:3) Graded homology and graded Grothendieck group of path algebras
In this section, we first recall the concept of Leavitt path algebra of a graph E and a basis for a Leavitt path algebra.We prove that a quotient of the diagonal of the Leavitt path algebra for the covering graph of E is isomorphic to thegraded Grothendieck group of the Leavitt path algebra L R ( E ) over a field R . This will be used to show that the for thegraph groupoid G E associated to an arbitrary graph E , we have H gr0 ( G E ) ∼ = K gr0 ( L R ( E )).We briefly recall the definition of a Leavitt path algebra and establish notation. For a comprehensive study of thesealgebras refer to [1].A directed graph E is a tuple ( E , E , r, s ), where E and E are sets and r, s are maps from E to E . We thinkof each e ∈ E as an edge pointing from s ( e ) to r ( e ). We use the convention that a (finite) path p in E is a sequence p = α α · · · α n of edges α i in E such that r ( α i ) = s ( α i +1 ) for 1 ≤ i ≤ n −
1. We define s ( p ) = s ( α ), and r ( p ) = r ( α n ).A directed graph E is said to be row-finite if for each vertex u ∈ E , there are at most finitely many edges in s − ( u ).A vertex u for which s − ( u ) is empty is called a sink , whereas u ∈ E is called an infinite emitter if s − ( u ) is infinite.If u ∈ E is neither a sink nor an infinite emitter, we call it a regular vertex . Definition 6.1.
Let E be a directed graph and R a unital ring. The Leavitt path algebra L R ( E ) of E is the R -algebragenerated by the set { v | v ∈ E } ∪ { e | e ∈ E } ∪ { e ∗ | e ∈ E } subject to the following relations:(1) uv = δ u,v v for every u, v ∈ E ;(2) s ( e ) e = er ( e ) = e for all e ∈ E ;(3) r ( e ) e ∗ = e ∗ = e ∗ s ( e ) for all e ∈ E ;(4) e ∗ f = δ e,f r ( e ) for all e, f ∈ E ; and(5) v = P e ∈ s − ( v ) ee ∗ for every regular vertex v ∈ E .Let Γ be a group with identity ε , and let w : E −→ Γ be a function. Extend w to vertices and ghost edges by defining w ( v ) = ε for v ∈ E and w ( e ∗ ) = w ( e ) − for e ∈ E . The relations in Definition 6.1 are homogeneous with respect to w and makes L R ( E ) a Γ-graded ring. The set of all finite sums of vertices in E is a set of graded local units for L R ( E )(see [1, Lemma 1.2.12]). Furthermore, L R ( E ) is unital if and only if E is finite.If p = α α · · · α n is a path in E of length n ≥
1, we define p ∗ = α ∗ n · · · α ∗ α ∗ . We have s ( p ∗ ) = r ( p ) and r ( p ∗ ) = s ( p ).For convention, we set v ∗ = v for v ∈ E . We observe that for paths p, q in E satisfying r ( p ) = r ( q ), pq ∗ = 0 in L R ( E ).Recall that the Leavitt path algebra L R ( E ) is spanned by the following set (cid:8) v, p, p ∗ , ηγ ∗ | v ∈ E , p, γ, and η are nontrivial paths in E with r ( γ ) = r ( η ) (cid:9) . In general, this set is not R -linearly independent.In [2] authors gave a basis for row-finite Leavitt path algebras over a field which can be extended to arbitrary graphsover commutative rings as well [3, Theorfem 2.7]. We need this result in the paper, thus we recall it here. For eachregular vertex v ∈ E , we fix an edge (called special edge ) starting at v . Lemma 6.2. [2, Theorem 1]
The following elements form a basis for the Leavitt path algebra L R ( E ) : (1) v , where v ∈ E ; (2) p, p ∗ , where p is a nontrivial path in E ; (3) qp ∗ with r ( p ) = r ( q ) , where p = α · · · α m and q = β · · · β n are nontrivial paths of E such that α m = β n , or α m = β n which is not special. (cid:3) For a Γ-graded ring A one can consider the abelian monoid of isomorphism classes of graded finitely generatedprojective modules denoted by V gr ( A ). For the precise definition of V gr ( A ) for a graded ring with graded local units,refer to [4, § T α : Gr - A −→ Gr - A restricts to the category of graded finitely generatedprojective modules. Thus the group Γ acts on V gr ( A ) which makes V gr ( A ) a Γ-module. The graded Grothendieck group, K gr0 ( A ), is defined as the group completion of V gr ( A ) which naturally inherits the Γ-module structure of V gr ( A ) andthus becomes a Z [Γ]-module. Here, Z [Γ] is a group ring.The monoid V gr ( L R ( E )) for a Γ-graded Leavitt path algebra of a graph E over a field R was studied in [4, § V gr ( L R ( E )) for an arbitrary graph E , an abelian monoid M gr E for an arbitrary graph E was defined in [4, § { a v ( γ ) | v ∈ E , γ ∈ Γ } are supplemented bygenerators b Z ( γ ), where γ ∈ Γ and Z runs through all nonempty finite subsets of s − ( u ) for infinite emitters u ∈ E ,subject to the relations(1) a v ( γ ) = P e ∈ s − ( v ) a r ( e ) ( w ( e ) − γ ) for all regular vertices v ∈ E and γ ∈ Γ;(2) a u ( γ ) = P e ∈ Z a r ( e ) ( w ( e ) − γ ) + b Z ( γ ) for all γ ∈ Γ, infinite emitters u ∈ E and nonempty finite subsets Z ⊆ s − ( u ); OMOLOGY OF ´ETALE GROUPOIDS 19 (3) b Z ( γ ) = P e ∈ Z \ Z a r ( e ) ( w ( e ) − γ ) + b Z ( γ ) for all γ ∈ Γ, infinite emitters u ∈ E and nonempty finite subsets Z ⊆ Z ⊆ s − ( u ).Recall that the group Γ acts on the monoid M gr E as follows. For any β ∈ Γ, β · a v ( γ ) = a v ( βγ ) and β · b Z ( γ ) = b Z ( βγ ) . (6.1)There is a Γ-module isomorphism V gr ( L R ( E )) ∼ = M gr E (see [4, Proposition 5.7]).Let Γ be a group and w : E −→ Γ a function. As in [16, § covering graph E of E with respect to w is given by E = { v α | v ∈ E and α ∈ Γ } , E = { e α | e ∈ E and α ∈ Γ } , (6.2) s ( e α ) = s ( e ) α , and r ( e α ) = r ( e ) w ( e ) − α . Example 6.3.
Let E be a graph and define w : E −→ Z by w ( e ) = 1 for all e ∈ E . Then E (sometimes denoted E × Z ) is given by E = (cid:8) v n | v ∈ E and n ∈ Z (cid:9) , E = (cid:8) e n | e ∈ E and n ∈ Z (cid:9) ,s ( e n ) = s ( e ) n , and r ( e n ) = r ( e ) n − . As examples, consider the following graphs E : ue f ! ! vg b b F : u e e e f q q Then E : . . . u e / / f ' ' PPPPPPPPP u e / / f ' ' ❖❖❖❖❖❖❖❖❖ u − e − / / f − ' ' ❖❖❖❖❖❖❖❖❖ · · · . . . v g ♥♥♥♥♥♥♥♥♥ v g ♦♦♦♦♦♦♦♦♦ v − g − ♦♦♦♦♦♦♦♦♦ · · · and F : . . . u f ! ! e : : u f e u − f − " " e − ; ; · · · For the rest of the section we work with Leavitt path algebras with integral coefficients. We denote by E ∗ the set ofall finite paths in E . Recall that the diagonal of the Leavitt path algebra L Z ( E ), denoted by D Z ( E ), is Z -span { αα ∗ | α ∈ E ∗ } , which is a commutative subalgebra of L Z ( E ). In particular, D Z ( E ) is an abelian group. Lemma 6.4.
For a graph E , the diagonal D Z ( E ) is the free abelian group generated by symbols { αα ∗ | α ∈ E ∗ } subjectto the relation αα ∗ = X e ∈ s − ( r ( α )) αee ∗ α ∗ , (6.3) for α ∈ E ∗ with r ( α ) a regular vertex of E .Proof. Denote by G the free abelian group generated by symbols { αα ∗ | α ∈ E ∗ } subject to the relation given in (6.3).We prove that G and D Z ( E ) are isomorphic as groups. We define a group homomorphism f : G −→ D Z ( E ) such that f ( αα ∗ ) = αα ∗ for the generators αα ∗ in G . Observe that f is well defined since it preserves the relation in G . Obviously, f is surjective. In order to show that f is injective, it suffices to define a group homomorphism g : D Z ( E ) −→ G such that gf = id G . Recall from Lemma 6.2 that there is a basis of normal forms for L Z ( E ). Since D Z ( E ) is a subset of L Z ( E ),any element in D Z ( E ) has the normal form P ni =1 m i α i α ∗ i such that α i = α i · · · α i li a path in E of length l i with α l i notspecial and m i is a nonzero integer. We define g : D Z ( E ) −→ G by g ( P ni =1 m i α i α ∗ i ) = P ni =1 m i α i α ∗ i with P ni =1 m i α i α ∗ i a normal form. Note that g ( x + y ) = g ( x ) + g ( y ) for two normal forms in D Z ( E ). To check that gf = id G , we onlyneed to show that gf ( αα ∗ ) = αα ∗ for any generator αα ∗ in G . We have the following two cases. If α = α · · · α l is apath in E of length l ≥ α l not special, then gf ( αα ∗ ) = g ( αα ∗ ) = αα ∗ . In the case l = 0, α is a vertex in E . If α = α · · · α l is a path in E of length l ≥ α l special, then we can write αα ∗ as the following normal form αα ∗ = α · · · α k α ∗ k · · · α ∗ − l − X j = k X { e ∈ s − ( r ( α j )) | e = α j +1 } α · · · α j ee ∗ α ∗ j · · · α ∗ , where 1 ≤ k ≤ l is the number such that α j is special for any j > k . Then we have gf ( αα ∗ ) = g ( αα ∗ )= α · · · α k α ∗ k · · · α ∗ − l − X j = k X { e ∈ s − ( r ( α j )) | e = α j +1 } α · · · α j ee ∗ α ∗ j · · · α ∗ = α · · · α l − α l α ∗ l α ∗ l − · · · α ∗ = αα ∗ , where the second last equality holds because we have α · · · α j α ∗ j · · · α ∗ = P e ∈ s − ( r ( α j )) α · · · α j ee ∗ α ∗ j · · · α ∗ for each j = k, k + 1 , · · · , l −
1. This finishes the proof. (cid:3)
Let E be a graph. We denote by E ∞ the set of infinite paths in E . Set X := E ∞ ∪ { µ ∈ E ∗ | r ( µ ) is not a regular vertex } . Let G E := { ( αx, | α | − | β | , βx ) | α, β ∈ E ∗ , x ∈ X, r ( α ) = r ( β ) = s ( x ) } . We view each ( x, k, y ) ∈ G E as a morphism with range x and source y . The formulas ( x, k, y )( y, l, z ) = ( x, k + l, z ) and( x, k, y ) − = ( y, − k, x ) define composition and inverse maps on G E making it a groupoid with G (0) E = { ( x, , x ) | x ∈ X } which we identify with the set X .Next, we describe a topology on G E . For µ ∈ E ∗ define Z ( µ ) = { µx | x ∈ X, r ( µ ) = s ( x ) } ⊆ X. For µ ∈ E ∗ and a finite F ⊆ s − ( r ( µ )), define Z ( µ \ F ) = Z ( µ ) \ [ α ∈ F Z ( µα ) . The sets Z ( µ \ F ) constitute a basis of compact open sets for a locally compact Hausdorff topology on X = G (0) E (see[31, Theorem 2.1]).For µ, ν ∈ E ∗ with r ( µ ) = r ( ν ), and for a finite F ⊆ E ∗ such that r ( µ ) = s ( α ) for α ∈ F , we define Z ( µ, ν ) = { ( µx, | µ | − | ν | , νx ) | x ∈ X, r ( µ ) = s ( x ) } , and then Z (( µ, ν ) \ F ) = Z ( µ, ν ) \ [ α ∈ F Z ( µα, να ) . The sets Z (( µ, ν ) \ F ) constitute a basis of compact open bisections for a topology under which G E is an ample groupoid.By [12, Example 3.2], the map π E : L R ( E ) −→ A R ( G E ) (6.4)with R a commutative ring with unit defined by π E ( µν ∗ − P α ∈ F µαα ∗ ν ∗ ) = 1 Z (( µ,ν ) \ F ) extends to an algebra isomor-phism. We observe that the isomorphism of algebras in (6.4) satisfies π E ( v ) = 1 Z ( v ) , π E ( e ) = 1 Z ( e,r ( e )) , π E ( e ∗ ) = 1 Z ( r ( e ) ,e ) , (6.5)for each v ∈ E and e ∈ E . Observe that the isomorphism π E in (6.4) restricts to an isomorphism D Z ( E ) ∼ = C c ( G (0) E , Z )when R = Z .If E is any graph, and w : E → Γ any function, we extend w to E ∗ by defining w ( v ) = 0 for v ∈ E , and w ( α · · · α n ) = w ( α ) · · · w ( α n ). We obtain from [21, Lemma 2.3] a continuous cocycle e w : G E −→ Γ satisfying e w ( αx, | α | − | β | , βx ) = w ( α ) w ( β ) − . Thus G E is Γ-graded with the grading map e w and thus we have the skew-product groupoid G E × e w Γ. There is a groupoidisomorphism ρ : G E −→ G E × e w Γ (see [4, § . (6.6)From now on, let Γ be an abelian group. We denote by D Z ( E ) the diagonal of the Leavitt path algebra L Z ( E ) ofthe covering graph E with respect to the map w : E → Γ (see (6.2)).In order to give D Z ( E ) a Γ-module structure, we define a right continuous Γ-action on the skew-product groupoid G E × e w Γ such that for g ∈ G E , α, β ∈ Γ ( g, α ) · β = ( g, αβ − ) . (6.7) OMOLOGY OF ´ETALE GROUPOIDS 21
Note that the above action is a continuous action, since Γ is an abelian group. Recall from subsection 5.1 that thereis an induced Γ-action on C c (( G E × e w Γ) (0) , Z ). Since we have D Z ( E ) ∼ = C c ( G (0) E , Z ) ∼ = C c (( G E × e w Γ) (0) , Z ), there is aΓ-action on D Z ( E ) such that ( α γ α ∗ γ ) · β = α γβ α ∗ γβ (6.8)for α ∈ E ∗ , β, γ ∈ Γ. Theorem 6.5.
Let E be an arbitrary graph, Γ an abelian group, R a field and w : E −→ Γ a function. Then there is a Γ -module isomorphism D Z ( E ) / h r ( α ) w ( α ) − γ − α γ α ∗ γ i α ∈ E ∗ ,γ ∈ Γ ∼ = K gr0 ( L R ( E )) . Proof.
We define a map f : D Z ( E ) −→ M gr E + such that f ( α γ α ∗ γ ) = a r ( α ) ( w ( α ) − γ ) for α ∈ E ∗ . In the case that α = v is a vetex, f ( v γ ) = a v ( γ ). By Lemma 6.4, themap f is well defined since f ( α γ α ∗ γ ) = a r ( α ) ( w ( α ) − γ ) = X e ∈ s − E ( r ( α )) a r ( e ) ( w ( e ) − w ( α ) − γ ) = f (cid:0) X e ∈ s − E ( r ( α )) α γ e w ( α ) − γ e ∗ w ( α ) − γ ) α ∗ γ (cid:1) which follows from the relation (1) in M gr E for each α ∈ E ∗ with r ( α ) a regular vertex. We observe that f ( r ( α ) w ( α ) − γ − α γ α ∗ γ ) = 0 for α ∈ E ∗ and γ ∈ Γ. Then there is an induced group homomorphism f : D Z ( E ) / h r ( α ) w ( α ) − γ − α γ α ∗ γ i α ∈ E ∗ ,γ ∈ Γ −→ M gr E + . (6.9)Next we define a monoid homomorphism g : M gr E −→ D Z ( E ) / h r ( α ) w ( α ) − γ − α γ α ∗ γ i α ∈ E ∗ ,γ ∈ Γ by g ( a v ( γ )) = v γ and g ( b Z ( γ ′ )) = u γ ′ − X e ∈ Z e γ ′ e ∗ γ ′ , for v ∈ E , Z ⊆ s − E ( u ) a nonzero finite subset for an infinite emitter u ∈ E and γ, γ ′ ∈ Γ. In order to check that g iswell defined, we need to show that the relations (1), (2), (3) of M gr E are satisfied. For (1), g ( a v ( γ )) = v γ = X e ∈ s − ( v ) e γ e ∗ γ = X e ∈ s − ( v ) r ( e ) w ( e ) − γ = g ( X e ∈ s − ( v ) a r ( e ) ( w ( e ) − γ )) , for regular vertex v ∈ E and γ ∈ Γ. For relation (2) of M gr E , we have g (cid:0) X e ∈ Z a r ( e ) ( w ( e ) − γ ) + b Z ( γ ) (cid:1) = X e ∈ Z r ( e ) w ( e ) − γ + ( u γ − X e ∈ Z e γ e ∗ γ ) = u γ = g ( a u ( γ )) , for u ∈ E an infinite emitter, Z ⊆ s − E ( u ) a nonzero finite subset and γ ∈ Γ. For (3), we have g (cid:0) X e ∈ Z \ Z a r ( e ) ( w ( e ) − γ ) + b Z ( γ ) (cid:1) = X e ∈ Z \ Z r ( e )( w ( e ) − γ ) + u γ − X e ∈ Z e γ e ∗ γ = u γ + (cid:0) X e ∈ Z \ Z e γ e ∗ γ − X e ∈ Z e γ e ∗ γ (cid:1) = u γ − X e ∈ Z e γ e ∗ γ = g ( b Z ( γ )) , for all γ ∈ Γ, infinite emitters u ∈ E and nonempty finite subsets Z ⊆ Z ⊆ s − ( u ). Then we have an induced grouphomomorphism g : M gr E + −→ D Z ( E ) / h r ( α ) w ( α ) − γ − α γ α ∗ γ i α ∈ E ∗ ,γ ∈ Γ . (6.10)Observe that f = g − as group homomorphisms. Indeed, gf ( α γ α ∗ γ ) = g ( a r ( α ) ( w ( α ) − γ )) = r ( α ) w ( α ) − γ = α γ α ∗ γ in D Z ( E ) / h r ( α ) w ( α ) − γ − α γ α ∗ γ i α ∈ E ∗ ,γ ∈ Γ for each α ∈ E ∗ and γ ∈ Γ . On the other hand, f g ( a v ( γ )) = f ( v γ ) = a v ( γ ) and f g ( b Z ( γ )) = f ( u γ − P e ∈ Z e γ e ∗ γ ) = a u ( γ ) − P e ∈ Z a r ( e ) ( w ( e ) − γ ) = b Z ( γ ). Comparing the actions given in (6.1) and(6.8), f is a Γ-module isomorphism. (cid:3) We are in a position to write the main theorem of this section.
Theorem 6.6.
Let E be an arbitrary graph, R a field and w : E −→ Z a function such that w ( e ) = 1 for each edge e ∈ E . Then there is a Z [ x, x − ] -module isomorphism (cid:0) H gr0 ( G E ) , H gr0 ( G E ) + (cid:1) ∼ = (cid:0) K gr0 ( L R ( E )) , K gr0 ( L R ( E )) + (cid:1) . Furthermore, If E is finite then, [1 G E ] [ L ( E )] . Proof.
Observe that by Lemma 5.5 for the groupoid G E we haveIm ∂ = Z -Span { s ( U ) − r ( U ) | U ∈ B co ∗ ( G E ) } = Z -Span { s ( U ) − r ( U ) | U = ⊔ ni =1 Z ( α i , β i ) ∈ B co ∗ ( G E ) for α i , β i ∈ E ∗ with r ( α i ) = r ( β i ) for each i } = Z -Span { Z ( β ) − Z ( α ) | α, β ∈ E ∗ with r ( α ) = r ( β ) } = Z -Span { Z ( r ( α )) − Z ( α ) | α ∈ E ∗ } . (6.11)Here the third equality holds since the intersection of Z ( α, β ) and Z ( α ′ , β ′ ) for α, β, α ′ , β ′ ∈ E ∗ with r ( α ) = r ( β ) and r ( α ′ ) = r ( β ′ ) is either empty or equal to one of them, and the source (or range) of ⊔ i Z ( α i , β i ) is ⊔ i Z ( β i ) (or ⊔ i Z ( β i )).Recall that π E in (6.4) restricts to an isomorphism D Z ( E ) ∼ = C c ( G (0) E , Z ) and sends r ( α ) − αα ∗ to 1 Z ( r ( α )) − Z ( α,α ) =1 Z ( r ( α )) − Z ( α ) . Then by (6.11), we have H ( G E ) = C c ( G (0) , Z ) / Im ∂ ∼ = D Z ( E ) / h r ( α ) − αα ∗ i α ∈ E ∗ . Similarly for E , wehave H ( G E ) ∼ = D Z ( E ) / h r ( α ) w ( α ) − γ − α γ α ∗ γ i α ∈ E ∗ ,γ ∈ Γ . (6.12)Observe that the Z -action on C c ( G (0) E × Z , Z ) is given by ( β · f )( x ) = f ( x · β ) for x ∈ G (0) E × Z , f ∈ C c ( G (0) E × Z , Z ) , and β ∈ Z , where x · β is the action given in (6.7). To show that the isomorphism given in (6.12) perserves the Z -modulestructure, it suffices to show that β · Z ( α ) ×{ γ } = 1 Z ( α ) ×{ γ + β } for α ∈ E ∗ , β, γ ∈ Z , which can be checked directly.Combining Theorem 6.5, H gr0 ( G E ) = H ( G E × e w Z ) ∼ = D Z ( E ) / h r ( α ) w ( α ) − γ − α γ α ∗ γ i α ∈ E ∗ ,γ ∈ Γ ∼ = K gr0 ( L R ( E )) , (6.13)are Z -module isomorphisms and thus are Z [ x, x − ]-module isomorphisms.Recall that V gr ( L R ( E )) ∼ = M gr E is the positive cone of K gr0 ( L R ( E )). We first show that the isomorphismΦ : K gr0 ( L R ( E )) −→ H gr0 ( G E ) , given in (6.13) carries K gr0 ( L R ( E )) + to H gr0 ( G E ) + . By (6.10), (6.5) and (6.6),Φ( a v ( γ )) = 1 Z ( v ) ×{ γ } ∈ H gr0 ( G E ) + and Φ( b Z ( γ ′ )) = ρ ( π E ( u γ ′ − X e ∈ Z e γ ′ e ∗ γ ′ ))= 1 Z ( u ) ×{ γ ′ } − X e ∈ Z Z ( e ) ×{ γ ′ } = 1 Z ( u ) \ ( ⊔ e ∈ Z Z ( e )) ×{ γ ′ } = 1 ⊔ e ∈ s − u ) \ Z Z ( e )) ×{ γ ′ } ∈ H gr0 ( G E ) + . Conversely we show that the isomorphism Φ − : H gr0 ( G E ) −→ K gr0 ( L R ( E )) carries H gr0 ( G E ) + to K gr0 ( L R ( E )) + . If[ h ] ∈ H gr0 ( G E ) + , then h ∈ C c ( G (0) E × Z , Z ) with h ( x ) ≥ x ∈ G (0) E × Z . We may write h = P i n i Z ( α i ) ×{ γ i } with n i ≥ Z ( α i ) × { γ i } ’s mutually disjoint. By (6.9) and (6.12), we haveΦ − ( h ) = X i n i f ( α iγ i α iγ i ∗ ) = X i n i a r ( α i ) ( w ( α i ) − γ i ) ∈ K gr0 ( L R ( E )) + . This completes the proof. (cid:3)
Theorem 6.6 allows us to use the graded homology as a capable invariant in symbolic dynamics. Recall thattwo-sided shift spaces X and Y are called eventually conjugate if X n is conjugate to Y n , for all sufficiently large n ([22, Definition 7.5.14]). For a two-sided shift X , we denote by X + the one-sided shift associated to X , namely, X + = { x [0 , ∞ ] | x ∈ X } (see [7] for a summary of concepts on symbolic dynamics). Theorem 6.7.
Let X and Y be two-sided shift of finite types. Then X is eventually conjugate to Y if and only ifthere is an order preserving Z [ x, x − ] -module isomorphism H gr0 ( G X + ) ∼ = H gr0 ( G Y + ) , where G X + and G Y + are groupoidsassociated to one-sided shift X + and Y + , respectively.Proof. Let E and F be the graphs such that X and Y are conjugate to X E and X F , respectively. Then X is eventuallyconjugate to Y if and only if there is an isomorphism between Krieger’s dimension groups, namely, (∆ A E , ∆ + A E , δ A E ) ∼ =(∆ A F , ∆ + A F , δ A F ), where A E and A F are the adjacency matrices of E and F (see [22, Theorem 7.5.8]). On the other hand,by [19, Theorem 3.11.7], ( K gr0 ( L R ( E ) , K gr0 ( L R ( E ) + ) ∼ = (∆ A E , ∆ + A E ) which respects the Z [ x, x − ]-module structure. Notethat the groupoid G X + associated to the one-sided shift X + is isomorphic to the graph groupoid G E . Combining thiswith Theorem 6.6 the result follows. (cid:3) OMOLOGY OF ´ETALE GROUPOIDS 23 Acknowledgements
This research was supported by the Australian Research Council grant DP160101481. The authors benefitted a greatdeal from the informative workshops in topological groupoids in Copenhagen in October 2017 and Faroe Islands in May2018. They would like to thank the organisers Kevin Brix, Toke Carlsen, Søren Eilers and Gunnar Restorff.
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