Algebraic bivariant K -theory and Leavitt path algebras
aa r X i v : . [ m a t h . K T ] A ug ALGEBRAIC BIVARIANT K -THEORY AND LEAVITT PATHALGEBRAS GUILLERMO CORTI ˜NAS AND DIEGO MONTEROA bstract . We investigate to what extent homotopy invariant, excisive and ma-trix stable homology theories help one distinguish between the Leavitt path al-gebras L ( E ) and L ( F ) of graphs E and F over a commutative ground ring ℓ . Weapproach this by studying the structure of such algebras under bivariant alge-braic K -theory kk , which is the universal homology theory with the propertiesabove. We show that under very mild assumptions on ℓ , for a graph E withfinitely many vertices and reduced incidence matrix A E , the structure of L ( E ) in kk depends only on the groups Coker( I − A E ) and Coker( I − A tE ). We also provethat for Leavitt path algebras, kk has several properties similar to those that Kas-parov’s bivariant K -theory has for C ∗ -graph algebras, including analogues of theUniversal coe ffi cient and K¨unneth theorems of Rosenberg and Schochet.
1. I ntroduction
This article is the first part of a two part project motivated by the classificationproblem for Leavitt path algebras [3]. We consider homological invariants of suchalgebras; we investigate to what extent they help one distinguish between them. Inthis first part we investigate general graphs and their algebras over general com-mutative ground rings; the second part [6] focuses mostly on purely infinite simpleunital algebras over a field. We fix a commutative ring ℓ and write L ( E ) for theLeavitt path algebra of a graph E over ℓ . Here a homology theory of the categoryAlg ℓ of algebras is simply a functor X : Alg ℓ → T with values in some triangu-lated category T . If S is a set and A ∈ Alg ℓ , we write M S A for the algebra ofthose matrices M : S × S → A which are finitely supported. We call a homol-ogy theory X S -stable if for s ∈ S and A ∈ Alg ℓ , the inclusion ι s : A → M S A , ι s ( a ) = ǫ s , s ⊗ a induces an isomorphism X ( ι s ). Write E and E for the sets of ver-tices and edges of the graph E . We call X E -stable if it is E ⊔ E ⊔ N -stable. Thusif E and E are both countable, E -stability is the same as stability with respectto M ∞ = M N . We are interested in those homology theories which are excisive,(polynomially) homotopy invariant and E -stable. For example Weibel’s homotopyalgebraic K -theory KH has all these properties and, if ℓ is either Z or a field, then KH ∗ ( L ( E )) = K ∗ ( L ( E )) is Quillen’s K -theory. There is also a universal homologytheory with all the above properties, j : Alg ℓ → kk ([7],[11]); this is the bivariant K -theory of the title. For two algebras A , B ∈ Alg ℓ , the statement j ( A ) (cid:27) j ( B ) isequivalent the statement that X ( A ) (cid:27) X ( B ) for any excisive, homotopy invariant Both authors were supported by CONICET and by grants UBACyT 20021030100481BA andPICT 2013-0454. Corti˜nas research was supported by grant MTM2015-65764-C3-1-P (Feder funds). and E -stable homology theory X . Let Ω : kk → kk be the inverse suspension; if A , B ∈ Alg ℓ , put kk n ( A , B ) = hom kk ( j ( A ) , Ω n j ( B )) , kk ( A , B ) = kk ( A , B ) . By [7, Theorem 8.2.1], setting the first variable equal to the ground ring we recoverWeibel’s homotopy algebraic K -theory KH [16]: kk n ( ℓ, B ) = KH n ( B ) . Set KH n ( B ) : = kk − n ( B , ℓ ) . Recall that a vertex v ∈ E is regular if it emits a nonzero finite number of edgesand that it is singular otherwise. Write reg( E ) and sing( E ) for the sets of regularand of singular edges. Let A E ∈ Z reg( E ) × E be the matrix whose ( v , w ) entry is thenumber of edges from v to w and let I ∈ Z E × reg( E ) be the matrix that results fromthe identity matrix upon removing the columns corresponding to singular vertices.It follows from [4] that if KH ( ℓ ) = Z , KH − ( ℓ ) = E is finite, then for thereduced incidence matrix A E we have KH ( L ( E )) = Coker( I − A tE ) . (1.1)We show here (see Section 6) that, abusing notation, and writing I for I t , KH ( L ( E )) = Coker( I − A E ) . (1.2)For n ≥
0, let L n be the Leavitt path algebra of the graph with one vertex and n loops. Thus L = ℓ and L = ℓ [ t , t − ]. We prove the following structure theorem. Theorem 1.3.
Assume that KH ( ℓ ) = Z and KH − ( ℓ ) =
0. Let E be a graph suchthat E is finite. Let d , . . . , d n , d i \ d i + be the invariant factors of the torsion group τ ( E ) = tors( K ( L ( E )) , s = E ) and r = rk( KH ( L ( E )) . Let j : Alg ℓ → kk bethe universal excisive, homotopy invariant, E-stable homology theory. Thenj ( L ( E )) (cid:27) j ( L s ⊕ L r ⊕ n M i = L d i + )In particular, any unital Leavitt path algebra with trivial KH is zero in kk . Forexample both L and its Cuntz splice L − ([12]) are zero in kk . We also have the fol-lowing corollary; here, and in any other statement which involves the image under j of the Leavitt path algebras of finitely many graphs E , . . . , E n , j is understood torefer to the ⊔ ni = E i -stable j . Corollary 1.4.
Let ℓ be as in Theorem 1.3. The following are equivalent for graphsE and F with finitely many vertices. i) j ( L ( E )) (cid:27) j ( L ( F )) . ii) KH ( L ( E )) (cid:27) KH ( L ( F )) and KH ( L ( E )) (cid:27) KH ( L ( F )) . iii) KH ( L ( E )) (cid:27) KH ( L ( F )) and E ) = F ) .Proof. It is not hard to check, using (1.1) and (1.2) (see Lemma 6.7) that the groups KH ( L ( E )) and KH ( L ( E )) have isomorphic torsion subgroups and that E ) = rk KH ( L ( E )) − rk KH ( L ( E )) . (1.5) LGEBRAIC BIVARIANT K -THEORY AND LEAVITT PATH ALGEBRAS 3 The corollary is immediate from this and the theorem above. (cid:3)
To put the above result in perspective, let us recall that E. Ruiz and M. Tomfordehave shown in [14] that if ℓ is a field, L ( E ) and L ( F ) are simple and both E and F have infinite emitters, then condition iii) of the corollary holds if and only if L ( E )and L ( F ) are Morita equivalent. Our result applies far more generally, but it is inprinciple weaker, since kk -isomorphic algebras need not be Morita equivalent. Forexample L is not Morita equivalent to the 0 ring. Observe also that the identity(1.5) helps us replace the graphic condition about K -theoreticor homological condition about KH .By (1.2) and [8, Theorem 5.3], when E is finite and regular KH ( L ( E )) is iso-morphic to the group of extensions of the C ∗ -algebra of E by the algebra of com-pact operators. We shall see presently that KH ( L ( E )) is also related to algebraextensions 0 → M ∞ → E → L ( E ) → . One can form an abelian monoid of homotopy classes of such extensions (see Sec-tion 2); we write E xt ( L ( E )) for its group completion. When E is finite and E is countable, there is a natural map We show in Proposition 6.5 that, under theassumptions of Theorem 1.3, if in addition E is countable and E has no sources,then there is a natural surjection E xt ( L ( E )) ։ KH ( L ( E )) . (1.6)As another similarity with the operator algebra case, we prove (Corollary 7.20) thatif ℓ and E are as in Theorem 1.3 and R ∈ Alg ℓ , then there is a short exact sequence0 → Ext Z ( KH ( L ( E )) , KH n + ( R )) → kk n ( L ( E ) , R ) [ KH ,γ ∗ KH ] −→ Hom( KH ( L ( E )) , KH n ( R )) ⊕ Hom(Ker( I − A tE ) , KH n + ( R )) → . (1.7)Observe that, for operator algebraic K -theory, K top1 ( C ∗ ( E )) = Ker( I − A tE ), so substi-tuting K top and KK for KH and kk in (1.7) one obtains the usual UCT of [13, Theo-rem 1.17]. Moreover, in Proposition 7.23 we also prove an analogue of the K ¨unneththeorem of [13, Theorem 1.18].Up to here in this introduction we have only discussed results that hold for E with finitely many vertices and for ℓ such that KH ( ℓ ) = Z and KH − ℓ =
0. Withno hypothesis on ℓ we show that if E and F have finitely many vertices and θ ∈ kk ( L ( E ) , L ( F )) then θ is an isomorphism ⇐⇒ KH ( θ ) and KH ( θ ) are isomorphisms. (1.8)It is however not true that unital Leavitt path algebras with isomorphic KH and KH are kk -isomorphic, even when ℓ is a field (see Remark 5.11). Thus in viewof Corollary 1.4, the pair ( KH , KH ) is a better invariant of Leavitt path algebrasthan the pair ( KH , KH ).Next let ℓ and E be arbitrary and let R ∈ Alg ℓ . If I is a set, write R ( I ) = M i ∈ I R GUILLERMO CORTI ˜NAS AND DIEGO MONTERO for the algebra of finitely supported functions I → R . Let X : Alg ℓ → T bean excisive, homotopy invariant, E -stable homology theory. Further assume thatdirect sums of at most E summands exist in T and that for any family of algebras { R i : i ∈ I } the natural map M i ∈ I X ( R i ) → X ( M i ∈ I R i )is an isomorphism if I ≤ E . We prove in Theorem 5.4 that there is a distin-guished triangle in T of the following form X ( R ) (reg( E )) I − A tE / / X ( R ) ( E ) / / X ( L ( E ) ⊗ R ) . (1.9)This applies, in particular, when we take X = KH , generalizing [4, Theorem 8.4].Thus we get a long exact sequence KH n + ( L ( E ) ⊗ R ) ( E ) → KH n ( R ) (reg( E )) I − A tE −→ KH n ( R ) ( E ) → KH n ( L ( E ) ⊗ R ) (1.10)When R is regular supercoherent we may substitute K for KH in (1.10), general-izing [4, Theorem 7.6] (see Example 5.5). Infinite direct sums are not known toexist in kk ; however finite direct sums do exist, and j does commute with them.Hence when E is finite and ℓ is arbitrary, we may take X = j above to obtain adistinguished triangle j ( R ) reg( E ) I − A tE / / j ( R ) E / / j ( L ( E ) ⊗ R ) . (1.11)This triangle is the basic tool we use to establish all the results on unital Leavittpath algebras mentioned above.The rest of this article is organized as follows. Some notations used throughoutthe paper (in particular pertaining matrix algebras) are explained at the end of thisIntroduction. In Section 2 we recall some basic notions about algebraic homotopy,prove some elementary lemmas about it, and use them to define, for every pair ofalgebras A and R with R unital, a group E xt ( A , R ) of virtual homotopy classes ofextensions of A by M ∞ R . In Section 3 we recall some basic properties of kk andquasi-homomorphisms. Also, we prove in Proposition 3.12 that if ι i : A i → M S i A i ( i ∈ I ) are corner inclusions, S is an infinite set with S ≥ ⊔ i S i ) and j : Alg ℓ → kk is the universal excisive, homotopy invariant and M S -stable homology theory,then the direct sum ⊕ i ι i induces an isomorphism in M S -stable kk , even if I -directsums might not exist in kk . Section 4 is devoted to the characterization of theimage under j : Alg ℓ → kk of the Cohn path algebra C ( E ) of a graph E . The latteris related to Leavitt path algebra L ( E ) by means of an exact sequence0 → K ( E ) → C ( E ) → L ( E ) → , (1.12)where K ( E ) is a direct sum of matrix algebras. The algebra C ( E ) receives a canon-ical homomorphism φ : ℓ ( E ) → C ( E ). We prove in Theorem 4.2 that the universal LGEBRAIC BIVARIANT K -THEORY AND LEAVITT PATH ALGEBRAS 5 excisive, homotopy invariant, E -stable homology theory j maps φ to an isomor-phism j ( ℓ ( E ) ) (cid:27) j ( C ( E )) . (1.13)The proof uses quasi-homomorphisms, much in the spirit of Cuntz’ proof of Bottperiodicity for C ∗ -algebra K -theory. As a corollary we obtain that if KH ( ℓ ) = Z and E and F are graphs, then for the universal excisive, homotopy invariant, E ⊔ F -stable homology theory j we have (Corollary 4.3) j ( C ( E )) (cid:27) j ( C ( F )) ⇐⇒ K ( C ( E )) (cid:27) K ( C ( F )) ⇐⇒ E = F . (1.14)In Section 5 we use Proposition 3.12 to prove that j ( ℓ (reg( E )) ) (cid:27) j ( K ( E )). Puttingthis together with (1.14) we get that, for arbitrary ℓ and E , the kk -triangle inducedby (1.12) is isomorphic to one of the form j ( ℓ (reg( E )) ) f → j ( ℓ ( E ) ) → j ( L ( E )) . We show in Proposition 5.2 that for each pair ( v , w ) ∈ E × reg( E ) the compos-ite π v f i w : ℓ → ℓ induced by the inclusion at the w -summand and the projectiononto the v -summand is multiplication by the ( v , w )-entry of I − A tE . We use thisto prove (1.9) (Theorem 5.4). The exact sequence (1.10), the fact that K can besubstituted for KH when R is regular supercoherent, as well as triangle (1.9), arededuced in Example 5.5. The equivalence (1.8) is proved in Propostion 5.10. Be-ginning in Section 6 we work under the Standing assumptions 6.1, which are that KH − ℓ = KH ( ℓ ) = Z . The surjection (1.6) is established in Proposition6.5. The fact that KH ( L ( E )) and KH ( L ( E )) have isomorphic torsion subgroupsand the identity (1.5) are proved in Lemma 6.7. Theorem 1.3 and Corollary 1.4 areTheorem 6.10 and Corollary 6.11. In Section 7 we introduce a descending filtration { kk ( L ( E ) , R ) i : 0 ≤ i ≤ } on kk ( L ( E ) , R ) for every algebra R and every unital Leav-itt path algebra L ( E ) and compute the slices kk ( L ( E ) , R ) i / kk ( L ( E ) , R ) i + (Theorem7.12). We use this to prove the universal coe ffi cient theorem (1.7) in Corollary 7.20and the K ¨unneth theorem in Proposition 7.23. Notation . A commutative ground ring ℓ is fixed throughout the paper. Allalgebras, modules and tensor products are over ℓ . If A is an algebra and X ⊂ A a subset, we write span( X ) and h X i for the ℓ -submodule and the two-sided idealgenerated by X . At the beginning of this Introduction we introduced, for a set S and an algebra A , the algebra M S A of finitely supported S × S -matrices. We write M S = M S ℓ and ǫ s , t ∈ M S for the matrix whose only nonzero entry is a 1 at the( s , t )-spot ( s , t ∈ S ). We also consider the algebra Γ S ( R ) : = { A : S × S → R | A i , ∗ , A ∗ , i < ∞} of those matrices which have finitely many nonzero coe ffi cients in each row andcolumn. If S = n < ∞ , then Γ S = M S = M n is the usual matrix algebra. We usespecial notation for the case S = N ; we write M ∞ for M N and Γ for Γ N . Observethat M ∞ R is an ideal of Γ ( R ). Put Σ ( R ) = Γ ( R ) / M ∞ R . (1.16) GUILLERMO CORTI ˜NAS AND DIEGO MONTERO
The algebras Γ ( R ) and Σ ( R ) are Wagoner’s cone and suspension algebras [15]. A ∗ -algebra is an algebra R equipped with an involutive algebra homomorphism R → R op . For example ℓ is a ∗ -algebra with trivial involution. If R is a ∗ -algebra, theconjugate matricial transpose makes both Γ S ( R ) and M S R into ∗ -algebras.2. H omotopy and extensions Let ℓ be a commutative ring. Let Alg ℓ be the category of associative, not neces-sarily unital algebras over ℓ . If B ∈ Alg ℓ , we write ev i : B [ t ] → B , ev i ( f ) = f ( i ), i = , φ , φ : A → B be two algebra homo-morphisms; an elementary homotopy from φ to φ is an algebra homomorphism H : A → B [ t ] such that ev H = φ and ev H = φ . We say that two algebrahomomorphisms φ, ψ : A → B are homotopic , and write φ ≈ ψ , if for some n ≥ φ = φ , . . . , φ n = ψ such that for each 0 ≤ i ≤ n − φ i to φ i + . We write[ A , B ] = hom Alg ℓ ( A , B ) / ≈ for the set of homotopy classes of homomorphisms A → B . Lemma 2.1.
Let A be a ring. Then the maps ι , ι ′ : A → M A, ι ( a ) = ǫ , ⊗ a, ι ′ ( a ) = ǫ , ⊗ a are homotopic.Proof. Let R = ˜ A be the unitalization. Consider the element U ( t ) = " (1 − t ) ( t − t ) t (1 − t ) ∈ GL R [ t ] . Let ad( U ( t )) : R [ t ] → R [ t ] be the conjugation map. Then H = ad( U ( t )) ι : A → M A [ t ], satisfies ev H = ι , ev H = ι ′ . (cid:3) Let A and R be algebras, φ, ψ ∈ hom Alg ℓ ( A , R ) and ι : R → M R , as in Lemma2.1. We say that φ and ψ are M -homotopic , and write φ ≈ M ψ , if ι φ ≈ ι φ . Put[ A , R ] M = hom Alg ℓ ( A , R ) / ≈ M . Let C be an algebra, A , B ⊂ C subalgebras and inc A : A → C , inc B : B → C theinclusion maps. Let x , y ∈ C such that yAx ⊂ B and axya ′ = aa ′ for all a , a ′ ∈ A .Then ad( y , x ) : A → B , ad( y , x )( a ) = yax (2.2)is a homomorphism of algebras, and we have the following. Lemma 2.3.
Let A , B , C and x , y be as above. Then inc B ad( y , x ) ≈ M inc A . Ifmoreover A = B and yA , Ax ⊂ A, then ad( y , x ) ≈ M id A .Proof. Consider the diagonal matrices ¯ y = diag( y , , ¯ x = diag( x , ∈ M ˜ C . Onechecks that a ¯ x ¯ ya ′ = aa ′ for all a , a ′ ∈ M A . Hence φ : = ad(¯ y , ¯ x ) : M A → M C isa homomorphism. Moreover we have φι = ι inc B ad( y , x ) and φι ′ = ι ′ inc A . Thusapplying Lemma 2.1 twice, we get ι inc B ad( y , x ) ≈ ι ′ inc A ≈ ι inc A . LGEBRAIC BIVARIANT K -THEORY AND LEAVITT PATH ALGEBRAS 7 This proves the first assertion. Under the hypothesis of the second assertion, φ maps M A → M A , and we have φι = ι ad( y , x ) and φι ′ = ι ′ . The proof isimmediate from this using Lemma 2.1. (cid:3) A C -algebra is a unital algebra R together with a unital algebra homomorphismfrom the Cohn algebra C to R . Equivalently, R is a unital algebra together withelements x , x , y , y ∈ R satisfying y i x j = δ i , j .If R is a C -algebra the map ⊞ : R ⊕ R → R , a ⊞ b = x ay + x ay (2.4)is an algebra homomorphism. An infinite C -algebra is a C -algebra together withan endomorphism φ : R → R such that for all a ∈ R we have a ⊞ φ ( a ) = φ ( a ) . In the following lemma and elsewhere, if M is an abelian monoid, we write M + for the group completion. Lemma 2.5.
Let A be an algebra, R = ( R , x , x , y , y ) a C -algebra, and B ⊳ R an ideal. Then (2.4) induces an operation in [ A , B ] M which makes it into anabelian monoid whose neutral element is the zero homomorphism. If furthermoreR is an infinite C -algebra, then [ A , R ] + M = .Proof. By Lemma 2.3, the homomorphisms B → B , b x i by i ( i = ,
1) are M -homotopic to the identity. Hence to prove the first assertion, it su ffi ces to showthat (2.4) associative and commutative up to M -homotopy. This is straightforwardfrom Lemma 2.3, since all diagrams involved commute up to a map of the form(2.2). The second assertion is clear. (cid:3) Example 2.6.
Any purely infinite simple unital algebra is a C -algebra, by [2,Proposition 1.5]. Example 2.7. If R is a unital algebra, its cone Γ ( R ) is an infinite C -algebra ([15])and Σ ( R ) is a C -algebra. For every algebra R , Γ ( R ) ⊳ Γ ( ˜ R ) and Σ ( R ) ⊳ Σ ( ˜ R ). Bydefinition, we have an exact sequence0 → M ∞ R → Γ ( R ) → Σ ( R ) → . (2.8) Lemma 2.9.
Let R be a unital algebra and let E be an algebra containing M ∞ R asan ideal. Then there exists a unique algebra homomorphism ψ = ψ E : E → Γ ( R ) which restricts to the identity on M ∞ R.Proof. If a ∈ E then for each i , j ∈ N there is a unique element a i , j ∈ R such that( ǫ i , i ⊗ a ( ǫ j , j ⊗ = ǫ i , j ⊗ a i , j . One checks that ψ : E → Γ ( R ), ψ ( a ) = ( a i , j ) satisfiesthe requirements of the lemma. (cid:3) It follows from Lemma 2.9 that if R is unital then every exact sequence of alge-bras 0 → M ∞ R → E → A → GUILLERMO CORTI ˜NAS AND DIEGO MONTERO induces a homomorphism ψ : A → Σ ( R ) and that (2.10) is isomorphic to thepullback along ψ of (2.8). Hence we may regard [ A , Σ ( R )] M as the abelian monoidof homotopy classes of all sequences (2.10). Put E xt ( A , R ) = [ A , Σ ( R )] + M , E xt ( A ) = E xt ( A , ℓ ) . (2.11)Observe that, by Lemma 2.5, any sequence (2.10) which is split by an algebrahomomorphism A → E maps to zero in E xt ( A , R ).3. A lgebraic bivariant K - theory Let T be a triangulated category and Ω the inverse suspension functor of T . A homology theory with values in T is a functor X : Alg ℓ → T . An extension ofalgebras is a short exact sequence of algebra homomorphisms( E ) : 0 → A → B → C → ℓ -linearly split. We write E for the class of all extensions. An excisivehomology theory for ℓ -algebras with values in T consists of a functor X : Alg ℓ →T , together with a collection { ∂ E : E ∈ E} of maps ∂ XE = ∂ E ∈ hom T ( Ω X ( C ) , X ( A ))satisfying the compatibility conditions of [7, Section 6.6]. Observe that if X :Alg ℓ → T is excisive and A , B ∈ Alg ℓ , then the canonical map X ( A ) ⊕ X ( B ) → X ( A ⊕ B ) is an isomorphism. Let I be a set. We say that a homology theory X : Alg ℓ → T is I-additive if first of all direct sums of cardinality ≤ I exist in T and second of all the map M j ∈ J X ( A j ) → X ( M j ∈ J A j )is an isomorphism for any family of algebras { A j : j ∈ J } ⊂ Alg ℓ with J ≤ I .We say that the functor X : Alg ℓ → T is homotopy invariant if for every A ∈ Alg ℓ , X maps the inclusion A ⊂ A [ t ] to an isomorphism.Let S be a set, s ∈ S and let ι s : A → M S A , ι s ( a ) = ǫ s , s ⊗ a ( A ∈ Alg ℓ ) . (3.2)Call X M S -stable if for every A ∈ Alg ℓ , it maps ι s : A → M S A to an iso-morphism. This definition is independent of the element s ∈ S , by the argu-ment of [5, Lemma 2.2.4]. One can further show, using [5, Proposition 2.2.6]and [11, Example 5.2.6] that if S is infinite and X is M S -stable, and T is a set suchthat T ≤ S , then X is T -stable. Definition 3.3.
Let A , B ∈ Alg ℓ . A quasi-homomorphism from A to B is a pair ofhomomorphisms φ, ψ : A → D ∈ Alg ℓ , where D contains B as an ideal, such that φ ( a ) − ψ ( a ) ∈ B ( a ∈ A ) . We use the notation ( φ, ψ ) : A → D ⊲ B . Two algebra homomorphisms φ, ψ : A → B are said to be orthogonal , in sym-bols φ ⊥ ψ , if φ ( x ) ψ ( y ) = = ψ ( x ) φ ( y ) ( x , y ∈ A ). In this case, we will write φ ⊥ ψ .If φ ⊥ ψ then φ + ψ is an algebra homomorphism. LGEBRAIC BIVARIANT K -THEORY AND LEAVITT PATH ALGEBRAS 9 Proposition 3.4. ( [9, Proposition 3.3] ) Let X : Alg ℓ → τ be an excisive homologytheory and let ( φ, ψ ) : A → D ⊲ B be a quasi-homomorphism. Then, there is aninduced map X ( φ, ψ ) : X ( A ) → X ( B ) which satisfies the following naturality conditions: (1) X ( φ, = X ( φ ) . (2) X ( φ, ψ ) = − X ( ψ, φ ) . (3) If ( φ , ψ ) and ( φ , ψ ) are quasi-homomorphisms A → D ⊲ B with φ ⊥ φ and ψ ⊥ ψ , then ( φ + φ , ψ + ψ ) is a quasi-homomorphism andX ( φ + φ , ψ + ψ ) = X ( φ , ψ ) + X ( φ , ψ ) . (4) X ( φ, φ ) = . (5) If α : C → A is an ℓ -algebra homomorphism, thenX ( φα, ψα ) = X ( φ, ψ ) X ( α ) . (6) If η : D → D ′ is an ℓ -algebra homomorphism which maps B into an idealB ′ ⊳ D ′ , then X ( ηφ, ηψ ) = X ( η | B ) X ( φ, ψ ) . (7) Let H = ( H + , H − ) : A → D [ t ] ⊲ B [ t ] with ev ◦ H = ( φ + , φ − ) and ev ◦ H = ( ψ + , ψ − ) . If, in addition, X is homotopy invariant thenX ( φ + , φ − ) = X ( ψ + , ψ − ) . (8) Let ( ψ, ̺ ) be another quasi-homomorphism A → D ⊲ B. Then ( φ, ̺ ) is aquasi-homomorphism andX ( φ, ̺ ) = X ( φ, ψ ) + X ( ψ, ̺ ) . The excisive homology theories form a category, where a homomorphism be-tween the theories X : Alg ℓ → T and Y : Alg ℓ → U is a triangulated functor G : T → U such that GX = Y and such that for every extension (3.1) in E , thenatural isomorphism φ : G ( Ω X ( C )) → Ω Y ( C ) makes the following into a commu-tative diagram G ( Ω X ( C )) G ( ∂ XE ) / / φ (cid:15) (cid:15) Y ( A ) Ω Y ( C ) . ∂ YE ssssssssss In [7] a functor j : Alg ℓ → kk was defined which is an initial object in the fullsubcategory of those excisive homology theories which are homotopy invariant and M ∞ -stable. It was shown in [11] that, for any fixed infinite set S , by a slight vari-ation of the construction of [7] one obtains an initial object in the full subcategoryof those excisive and homotopy invariant homology theories which are M S -stable.Starting in the next section we shall fix S and use j and kk for the universal exci-sive, homotopy invariant and S -stable homology theory and its target triangulatedcategory. Moreover, we shall often omit j from our notation, and say, for example, that an algebra homomorphism is an isomorphism in kk or that a diagram in Alg ℓ commutes in kk or that a sequence of algebra maps A → B → C is a triangle in kk to mean that j applied to the corresponding morphism, diagramor sequence is an isomorphism, a commutative diagram or a distinguished trian-gle. Also, since as explained above, in kk the corner inclusion ι s : A → M S A isindependent of s , we shall simply write ι for j ( ι s ).The loop functor Ω in kk and its inverse have a concrete description as follows.Let Ω = t ( t − ℓ [ t ], Ω − = ( t − ℓ [ t , t − ]. For A ∈ Alg ℓ we have Ω ± j ( A ) = j ( Ω ± ⊗ A ) . (3.5) Example 3.6.
Let S be an infinite set and j : Alg ℓ → kk the universal homotopyinvariant, excisive and M S -stable homology theory. If R ∈ Alg ℓ , then the func-tor j (( − ) ⊗ R ) : Alg ℓ → kk is again a homotopy invariant, M S -stable, excisivehomology theory. Hence it gives rise to a triangulated functor kk → kk . In particu-lar, triangles in kk are preserved by tensor products. Moreover, the tensor productinduces a “cup product” ∪ : kk ( A , B ) ⊗ kk ( R , S ) → kk ( A ⊗ R , B ⊗ S ) , ξ ∪ η = ( B ⊗ η ) ◦ ( ξ ⊗ R ) . For A , B ∈ Alg ℓ and n ∈ Z , set kk n ( A , B ) = hom kk ( j ( A ) , Ω n j ( B )) , kk ( A , B ) = kk ( A , B ) . (3.7)The groups kk ∗ ( A , B ) are the bivariant K-theory groups of the pair ( A , B ). Set-ting A = ℓ in (3.7) we recover the homotopy algebraic K -groups of Weibel [16];there is a natural isomorphism ([7, Theorem 8.2.1], [11, Theorem 5.2.20]) kk ∗ ( ℓ, B ) ∼ −→ KH ∗ ( B ) ( B ∈ Alg ℓ ) . (3.8) Remark . Even though KH is I -additive for every set I , the universal functor j : Alg ℓ → kk is not known to be infinitely additive. Lemma 3.10.
Let { A i : i ∈ I } ⊂ Alg ℓ be a family of algebras, A = L i ∈ I A i , T aset, j : I → T a function and v ∈ T . Then the homomorphism ι j : A → M T A , ι j ( X i a i ) = X i ∈ I ǫ j ( i ) , j ( i ) ⊗ a i is homotopic to ι v .Proof. Because ( M T A )[ x ] = L i ∈ I ( M T A i [ x ]), we may assume that I has a singleelement, in which case the lemma follows using a rotational homotopy, as in theproof of Lemma 2.1. (cid:3) Lemma 3.11.
Let { S i : i ∈ I } be a family of sets, σ i : S i → S i an injectivemap, ( σ i ) ∗ : M S i → M S i , ( σ i ) ∗ ( ǫ s , t ) = ǫ σ i ( s ) ,σ i ( t ) the induced homomorphism,D = L i ∈ I M S i , and σ ∗ = L i ∈ I ( σ i ) ∗ : D → D. If X : Alg ℓ → T is M -invariant,then X ( σ ∗ ) is the identity map. LGEBRAIC BIVARIANT K -THEORY AND LEAVITT PATH ALGEBRAS 11 Proof.
The map σ i induces an ℓ -module homomorphism ℓ ( S i ) → ℓ ( S i ) whose ma-trix [ σ i ] is an element of the ring Γ S i of 1.15. Let [ σ i ] ∗ be the transpose ma-trix; we have [ σ i ] ∗ [ σ i ] =
1, and ( σ i ) ∗ ( a ) = [ σ i ] a [ σ ∗ i ] ( a ∈ M S i ). Hence for[ σ ] = L i ∈ I [ σ i ] ∈ R = L i ∈ I Γ S i , we have σ ∗ ( a ) = [ σ ] a [ σ ] ∗ . Since D ⊳ R , X ( σ ∗ )is the identity by [5, Proposition 2.2.6]. (cid:3) Proposition 3.12.
Let { S i : i ∈ I } be a family of sets, v i ∈ S i and S = ` i ∈ I S i .Let f = L i ∈ I ι v i : ℓ ( I ) → ⊕ i ∈ I M S i . Let T be an infinite set with T ≥ S . Let j :Alg ℓ → kk be the universal excisive, homotopy invariant and M T -stable homologytheory. Then j ( f ) is an isomorphism.Proof. Put D = L i ∈ I M S i . Let inc : D → M S ℓ ( I ) be the inclusion. By Lemma3.10, the composite inc f equals the canonical inclusion ι in kk . Next let g = ( M S f ) inc : D → M S D . We have g ( ǫ α,β ) = ǫ α,β ⊗ ǫ v i , v i ( α, β ∈ S i ). For each i ∈ I extend the coordinate permutation map S i × { v i } → { v i } × S i , to a bijection σ i : S × S i → S × S i , and let ( σ i ) ∗ be the induced automorphism of M S M S i (cid:27) M S × S i .Consider the automorphism σ ∗ = L i ∈ I ( σ i ) ∗ : M S D → M S D ; by Lemmas 3.10and 3.11, ι = j ( σ ∗ g ) = j ( g ). From what we have just seen and Example 3.6, in kk the following diagram commutes and its horizontal arrows are isomorphisms. ℓ ( I ) f (cid:15) (cid:15) ι / / M S ℓ ( I ) M S f (cid:15) (cid:15) M S ι / / M S M S ℓ ( I ) M S M S f (cid:15) (cid:15) D ι / / inc ; ; ①①①①①①①①① M S D M S ι / / M S inc rrrrrrrrrr M S M S D It follows that M S f and f are isomorphisms in kk . (cid:3)
4. C ohn algebras and kk A directed graph is a quadruple E = ( E , E , r , s ) where E and E are the setsof vertices and edges, and r and s are the range and source functions E → E .We call E finite if both E and E are finite. A vertex v ∈ E is a sink if s − ( v ) = ∅ and is an infinite emitter if s − ( v ) is infinite. A vertex v is singular if it is either asink or an infinite emitter; we call v regular if it is not singular. A vertex v ∈ E isa source if r − ( v ) = ∅ . We write sink( E ), inf( E ) and sour( E ) for the sets of sinks,infinite emitters, and sources, and sing( E ) and reg( E ) for those of singular and ofregular vertices.A finite path µ in a graph E is a sequence of edges µ = e . . . e n such that r ( e i ) = s ( e i + ) for i = , . . . , n −
1. In this case | µ | : = n is the length of µ . We viewthe vertices of E as paths of length 0. Write P ( E ) for the set of all finite paths in E .The range and source functions r , s extend to P ( E ) → E in the obvious way. Anedge f is an exit for a path µ = e . . . e n if there exist i such that s ( f ) = s ( e i ) and f , e i . A path µ = e . . . e n with n ≥ closed path at v if s ( e ) = r ( e n ) = v . Aclosed path µ = e . . . e n at v is a cycle at v if s ( e j ) , s ( e i ) for i , j .The Cohn path algebra C ( E ) of a graph E is the quotient of the free associative ℓ -algebra generated by the set E ∪ E ∪ { e ∗ | e ∈ E } , subject to the relations:(V) v · w = δ v , w v . (E1) s ( e ) · e = e = e · r ( e ).(E2) r ( e ) · e ∗ = e ∗ = e ∗ · s ( e ).(CK1) e ∗ · f = δ e , f r ( e ).The algebra C ( E ) is in fact a ∗ -algebra; it is equipped with an involution ∗ : C ( E ) → C ( E ) op which fixes vertices and maps e e ∗ ( e ∈ Q ). Condition V says that thevertices of E are orthogonal idempotents in C ( E ). Hence the subspace generatedby E is a subalgebra of C ( E ), isomorphic to the algebra ℓ ( E ) finitely supportedfunctions E → ℓ . For v ∈ E , let χ v ∈ ℓ ( E ) be the characteristic function of { v } .We have a monomorphism ϕ : ℓ ( E ) → C ( E ) , ϕ ( χ v ) = v . (4.1)Observe that if E is finite, then ℓ ( E ) = ℓ E is the algebra of all functions E → ℓ .We shall say that a homology theory is E-stable if it is stable with respect to aset of cardinality E ` E ` N ).The main result of this section is the following theorem. Theorem 4.2.
Let ϕ be the algebra homomorphism (4.1) and let j : Alg ℓ → kkbe the universal excisive, homotopy invariant and E-stable homology theory. Thenj ( ϕ ) is an isomorphism. Corollary 4.3.
Let E and F be graphs and j : Alg ℓ → kk the universal excisive,homotopy invariant and E ⊔ F-stable homology theory. Assume that KH ( ℓ ) (cid:27) Z .Then C ( E ) and C ( F ) are isomorphic in kk if and only if E = F .Proof. By Theorem 4.2, C ( E ) and C ( F ) are isomorphic in kk if and only if ℓ ( E ) and ℓ ( F ) are. If E = F then ℓ ( E ) and ℓ ( F ) are isomorphic in Alg ℓ , and thereforealso in kk . Assume conversely that ℓ ( E ) and ℓ ( F ) are isomorphic in kk . Then inview of (3.8) and of the hypothesis that KH ( ℓ ) (cid:27) Z , we have E = F . (cid:3) The proof of Theorem 4.2 is organized in four parts, with three lemmas inter-spersed. First we need some preliminaries.Associate an element m v ∈ C ( E ) to each v ∈ E \ inf( E ) as follows m v = P e ∈ s − ( v ) ee ∗ if v ∈ reg( E )0 if v ∈ sour( E ) . Observe that m v satisfies the following identities: m v = m ∗ v , m v = m v , m v w = δ w , v m v , m v e = δ v , s ( e ) e ( w ∈ E , e ∈ E ) . (4.4)Let C m ( E ) be the ∗ -algebra obtained from C ( E ) by formally adjoining an element m v for each v ∈ inf( E ) subject to the identities (4.4). We have a canonical ∗ -homomorphism can : C ( E ) → C m ( E ) . (4.5)Let P = P ( E ). For v ∈ E , set P v = { µ ∈ P ( E ) | r ( µ ) = v } , P v = { µ ∈ P | s ( µ ) = v } . (4.6) LGEBRAIC BIVARIANT K -THEORY AND LEAVITT PATH ALGEBRAS 13 Let Γ P be the ring introduced in 1.15. Using the notation (4.6) in the summationindexes, define a ∗ -homomorphism ρ : C m ( E ) → Γ P , (4.7) ρ ( v ) = X α ∈P v ǫ α,α , ρ ( e ) = X α ∈P r ( e ) ǫ e α,α , ( v ∈ E , e ∈ E ) ρ ( m w ) = X α ∈P w , | α |≥ ǫ α,α ( w ∈ inf( E )) . Lemma 4.8.
The maps (4.5) and (4.7) are monomorphisms.Proof.
It is well-known that the set B = { αβ ∗ | α, β ∈ P , r ( α ) = r ( β ) } is a basis of C ( E ) ([1, Proposition 1.5.6]). Set B = { α m v β ∗ | α, β ∈ P v , v ∈ inf( E ) } . It follows from (4.4) that B = B ∪ B generates C m ( E ) as an ℓ -module. It is clearthat ρ is injective on B ; hence it su ffi ces to show that the set ρ ( B ) ⊂ Γ P is ℓ -linearlyindependent. Let F ⊂ B be a finite set and c : F → ℓ \ { } a function such that X x ∈F c x x = . Let Q = { ( α, β ) ∈ P | r ( α ) = r ( β ) } ; give Q a partial order by setting ( α, β ) ≥ ( α ′ , β ′ ) if and only if there exists θ ∈ P r ( α ) such that α ′ = αθ , β ′ = βθ . Let f : B → Q , f ( αβ ∗ ) = ( α, β ), f ( α m v β ∗ ) = ( α, β ). Assume that F , ∅ . Then f ( F )has a maximal element ( α, β ). If αβ ∗ ∈ F , then ρ ( αβ ∗ ) is the only matrix in ρ ( F )whose ( α, β ) entry is nonzero. Thus c αβ ∗ =
0, a contradiction. Hence v = r ( α ) ∈ inf( E ), αβ ∗ < F and α m v β ∗ ∈ F . Then f ( F \ { α m v β ∗ } ) contains only finitely manyelements of the form ( α e , β e ) with e ∈ s − ( v ). However ρ ( α m v β ∗ ) α e ,β e = e ∈ s − ( v ). Thus c α m v β ∗ =
0, which again is a contradiction. Hence F mustbe empty; this concludes the proof. (cid:3) Remark . By Lemma 4.8 we may identify C m ( E ) with its image in Γ P . Underthis identification, the formula m v = X e ∈ s − ( v ) ee ∗ holds for every v ∈ E . Proof of Theorem 4.2, part I . Set C m ( E ) ∋ q v = v − m v ( v ∈ E ) . (4.10)Consider the following ideals of C m ( E ) K ( E ) = h q v | v ∈ reg( E ) i ⊂ ˆ K ( E ) = h q v | v ∈ E i . (4.11)One checks, using [1, Proposition 1.5.11] that the maps M P v → ˆ K ( E ) , ǫ α,β α q v β ∗ ( v ∈ E ) assemble to an isomorphism M v ∈ E M P v ∼ −→ ˆ K ( E ) . (4.12)Observe that (4.12) restricts to an isomorphism M v ∈ reg( E ) M P v ∼ −→ K ( E ) . (4.13)Let ˆ ι : ℓ ( E ) → ˆ K ( E ) be the homomorphism that sends the canonical basis element χ v to q v and let ξ : C ( E ) → C m ( E ) be the ∗ -homomorphism determined by ξ ( v ) = m v , ξ ( e ) = em r ( e ) . One checks that (can , ξ ) is a quasi-homomorphism C ( E ) → C m ( E ) ⊲ ˆ K ( E ). Fromthe equality can ϕ = ξϕ + ˆ ι and items (1), (3), (4) and (5) of Proposition 3.4, itfollows that j (can , ξ ) j ( ϕ ) = j (can ϕ, ξϕ ) = j ( ξϕ + ˆ ι, ξϕ ) = j ( ξϕ, ξϕ ) + j (ˆ ι, = j (ˆ ι ) . By Proposition 3.12, ˆ ι is an isomorphism in kk . Hence j (ˆ ι ) − j (can , ξ ) j ( ϕ ) = j ( ℓ ( E ) . It remains to show that j ( ϕ ) j (ˆ ι ) − j (can , ξ ) = j ( C ( E )) . (4.14)Let P = P ( E ); consider the algebra M P of finite matrices indexed by P . Let ˆ ϕ :ˆ K ( E ) → M P C ( E ) be the homomorphism that sends α q v β ∗ to ǫ α,β ⊗ v , where ǫ α,β isthe matrix unit. We shall need a twisted version ˆ ι τ of ˆ ι ; this is the ∗ -homomorphismˆ ι τ : C ( E ) → M P C ( E ) , ˆ ι τ ( v ) = ǫ v , v , ˆ ι τ ( e ) = ǫ s ( e ) , r ( e ) ⊗ e ( v ∈ E , e ∈ E ) . (4.15)We have a commutative diagram ℓ ( E ) ϕ (cid:15) (cid:15) ˆ ι / / ˆ K ( E ) ˆ ϕ (cid:15) (cid:15) C ( E ) ˆ ι τ / / M P C ( E ) (4.16) Lemma 4.17.
Let α ∈ P and let ι α : C ( E ) → M P C ( E ) as in (3.2) . Then ι α and themap ˆ ι τ of (4.15) induce the same isomorphism in kk.Proof. Because j is E -stable, it is M P -stable, whence ι α is an isomorphism anddoes not depend on α . Hence we may and do assume that α = w ∈ E . Because j is homotopy invariant, it is enough to find a polynomial homotopy between ι w andˆ ι τ . For each v ∈ E \{ w } set A v = [(1 − t ) ǫ w , w + ( t − t ) ǫ w , v + t ǫ v , w + (1 − t ) ǫ v , v ] ⊗ v , B v = [(1 − t ) ǫ w , w + (2 t − t ) ǫ w , v − t ǫ v , w + (1 − t ) ǫ v , v ] ⊗ v , A w = ǫ w , w ⊗ w = B w . LGEBRAIC BIVARIANT K -THEORY AND LEAVITT PATH ALGEBRAS 15 The desired homotopy is the homomorphism H : C ( E ) → M P C ( E )[ t ] defined by H ( v ) = A v ( ǫ v , v ⊗ v ) B v , H ( e ) = A s ( e ) ( ǫ s ( e ) , r ( e ) ⊗ e ) B r ( e ) , H ( e ∗ ) = A r ( e ) ( ǫ r ( e ) , s ( e ) ⊗ e ∗ ) B s ( e ) . (cid:3) Proof of Theorem 4.2, part II . Let M P C ( E ) ⊃ A = span { ǫ γ,δ ⊗ αβ ∗ | s ( α ) = r ( γ ) , s ( β ) = r ( δ ) , r ( α ) = r ( β ) } . One checks that A is a subalgebra containing the images of both ˆ ι τ and ˆ ϕ . Fromthe commutative diagram 4.16 we obtain, by corestriction, another commutativediagram ℓ ( E ) ϕ (cid:15) (cid:15) ˆ ι / / ˆ K ( E ) ˆ ϕ (cid:15) (cid:15) C ( E ) / / A (4.18)By Lemma 4.17, the bottom arrow of (4.18) is a monomorphism in kk . We shallabuse notation and write ˆ ι τ for the latter map.Let e C m ( E ) be the unitalization; put R = Γ P e C m ( E ). Consider the homomorphism ρ ′ = ρ ⊗ C ( E ) → R . One checks that the subalgebra A ⊂ R is closed under bothleft and right multiplication by elements in the image of ρ ′ . We can thus form thesemi-direct product C m ( E ) ⋉ A = C m ( E ) ⋉ ρ ′ A . As an ℓ -module, C m ( E ) ⋉ A is just C m ( E ) ⊕ A . Multiplication is defined by the rule( r , x ) · ( s , y ) = ( rs , ρ ′ ( r ) x + y ρ ′ ( s ) + xy ) . Let J be the ideal in C m ( E ) ⋉ A generated by the elements ( α q v β ∗ , − ǫ α,β ⊗ v ) with v = r ( α ) = r ( β ). One checks that J = span { ( α q v β ∗ , − ǫ α,β ⊗ v ) : v = r ( α ) = r ( β ) } . Set D = ( C m ( E ) ⋉ A ) / J . Lemma 4.19.
The composite of the inclusion and projection maps A = ⋊ A ⊂ C m ( E ) ⋉ A → D is injective.Proof.
It follows from (4.12) that there is an injective homomorphism j : ˆ K ( E ) → A , j ( α q v β ∗ ) = ǫ α,β ⊗ v ( r ( α ) = r ( β ) = v ) . Let inc : ˆ K ( E ) → C m ( E ) be the inclusion. Observe that J is the image of the mapinc ⋊ ( − j ) : ˆ K ( E ) → C m ( E ) ⋊ A . In particular, the projection π : C m ( E ) ⋉ A → C m ( E )is injective on J . It follows that J ∩ (0 ⋊ A ) =
0; this finishes the proof. (cid:3)
Proof of Theorem 4.2, part III.
By Lemma 4.19, we may regard A as an ideal of D .Let Υ : C m ( E ) → D be the composite of the inclusion C m ( E ) ⊂ C m ( E ) ⋊ A and theprojection C m ( E ) ⋊ A → D . We may embed diagram (4.18) into a commutativediagram ℓ ( E ) ϕ (cid:15) (cid:15) ˆ ι / / ˆ K ( E ) ˆ ϕ (cid:15) (cid:15) / / C m ( E ) Υ (cid:15) (cid:15) C ( E ) ˆ ι τ / / A / / D (4.20)Let ψ = Υ can, ψ = Υ ξ . Note that ψ ⊥ ˆ ι τ , so ψ / = ψ + ˆ ι τ is an algebrahomomorphism. We have quasi-homomorphisms( ψ , ψ ) , ( ψ , ψ / ) , ( ψ / , ψ ) : C ( E ) → D ⊲ A . Lemma 4.21.
The quasi-homorphism ( ψ , ψ / ) induces the zero map in kk.Proof. Let H + : C ( E ) → D [ t ] be the algebra homorphism determined by setting H + ( v ) = ( v , , H + ( e ) = ( em r ( e ) , + (1 − t )(0 , ǫ s ( e ) , r ( e ) ⊗ e ) + t (0 , ǫ e , r ( e ) ⊗ r ( e )) H + ( e ∗ ) = ( m r ( e ) e ∗ , + (1 − t )(0 , ǫ r ( e ) , s ( e ) ⊗ e ∗ ) + (2 t − t )(0 , ǫ r ( e ) , e ⊗ r ( e ))for v ∈ E and e ∈ E . It is a matter of calculation to show that H + a homotopybetween ψ and ψ / , and that ( H + , ψ / ) : C ( E ) → D [ t ] ⊲ A [ t ] is a homotopybetween ( ψ , ψ / ) and ( ψ / , ψ / ). Hence by item (7) of Proposition 3.4, we obtain j ( ψ , ψ / ) = j ( ψ / , ψ / ) = (cid:3) Proof of Theorem 4.2, conclusion . Using the commutativity of diagram (4.20) anditems (6), (8) and (1) of Proposition 3.4 and Lemma 4.21 we have j ( ˆ ϕ ) j (can , ξ ) = j ( ψ , ψ ) = j ( ψ , ψ / ) + j ( ψ / , ψ ) = j (ˆ ι τ ) . On the other hand j ( ˆ ϕ ) j (can , ξ ) = j (ˆ ι τ ) j ( ϕ ) j (ˆ ι ) − j (can , ξ ) . Hence j (ˆ ι τ ) = j (ˆ ι τ ) j ( ϕ ) j (ˆ ι ) − j (1 , ξ )Since j (ˆ ι τ ) is a monomorphism, this implies that1 j ( C ( E )) = j ( ϕ ) j (ˆ ι ) − j (1 , ξ ) . This finishes the proof. (cid:3)
5. T he L eavitt path algebra and a fundamental triangle Let E be a graph; for v ∈ E let q v ∈ C ( E ) be the element (4.10). The Leavittpath algebra L ( E ) is the quotient of C ( E ) modulo the relation( CK q v = v ∈ reg( E )) . In other words, for the ideal K ( E ) ⊳ C ( E ) of (4.11), we have a short exact sequence0 → K ( E ) → C ( E ) → L ( E ) → . (5.1)It follows from [1, Proposition 1.5.11] that the sequence (5.1) is ℓ -linearly split,and is thus an algebra extension in the sense of Section 3. LGEBRAIC BIVARIANT K -THEORY AND LEAVITT PATH ALGEBRAS 17 The adjacency matrix A ′ E ∈ Z (( E \ inf( E )) × E ) is the matrix whose entries are givenby ( A ′ E ) v , w = { e ∈ E : s ( e ) = v and r ( e ) = w } . The reduced adjacency matrix is the matrix A E ∈ Z (reg( E )) × E ) which results from A E upon removing the rows corresponding to sinks. We also consider the matrix I ∈ Z ( E × reg( E )) , I v , w = δ v , w . Proposition 5.2.
Let j : Alg ℓ → kk be as in Theorem 4.2. i) There is a distinguished triangle in kkj ( ℓ (reg( E )) ) f / / j ( ℓ ( E ) ) / / j ( L ( E )) . (5.3)ii) Let χ v : ℓ → ℓ (reg( E )) be the inclusion in the v-summand and let c v ∈ Z ( E ) ×{ v } bethe v-column of the matrix I − A tE ( v ∈ reg( E )) . Under the isomorphism (3.8) , thecomposite f j ( χ v ) corresponds to the map ⊗ c v : KH ( ℓ ) → KH ( ℓ ) ⊗ Z ( E ) Proof.
Consider the map q : ℓ (reg( E )) → K ( E ), q ( χ v ) = q v . In view of (4.13), j ( q ) is an isomorphism by Proposition 3.12. By Theorem 4.2, the map j ( φ ) is anisomorphism. Hence the kk -triangle induced by (5.1) is isomorphic to the triangle(5.3) where for the inclusion inc : K ( E ) ⊂ C ( E ), we have f = j ( φ ) − j (inc) j ( q ).This proves i). To prove ii), fix v ∈ reg( E ) and consider the elements q v , m v and ee ∗ ∈ C ( E ) ( e ∈ E , s ( e ) = v ). As the latter elements are idempotent, we regardthem as homomorphisms ℓ → C ( E ). In particular, q v = inc q χ v . Because q v ⊥ m v and v = q v + m v , j ( q v ) = j ( v ) − j ( m v ). On the other hand, by (CK1), j ( m v ) = P s ( e ) = v j ( r ( e )). Summing up, q v = j ( v ) − P s ( e ) = v j ( r ( e )); this proves ii). (cid:3) Theorem 5.4.
Let X : Alg ℓ → T be an excisive, homotopy invariant, E-stable andE -additive homology theory and let R ∈ Alg ℓ . Then (5.3) induces a triangle in T X ( R ) (reg( E )) I − A tE / / X ( R ) ( E ) / / X ( L ( E ) ⊗ R ) . Proof.
Tensoring the triangle (5.3) by R yields another triangle in kk , by Exam-ple 3.6. By the universal property of j , applying X to the latter triangle gives adistinguished triangle in T . Now apply Proposition 5.2 (ii) and the E -additivityhypothesis on X to finish the proof. (cid:3) Example 5.5.
Theorem 5.4 applies to X = KH and arbitrary E , generalizing[4, Theorem 8.4] from the row-finite to the general case. Recall a ring A is K n -regular if for every m ≥
1, the inclusion A → A [ t , . . . , t m ] induces an isomorphism K n ( R ) → K n ( R [ t , . . . , t m ]). We call A K-regular if it is K n -regular for all n . By[16, Proposition 1.5], the canonical map K ( A ) → KH ( A ) is a weak equivalencewhen A is K -regular. For example, if R is an ℓ -algebra which is a regular superco-herent ring, then L ( E ) ⊗ R is K -regular (by the argument of [4, page 23]), so we may replace KH by K to obtain the following triangle in the homotopy category ofspectra which generalizes [4, Theorem 7.6] K ( R ) (reg( E )) I − A tE / / K ( R ) ( E ) / / K ( L ( E ) ⊗ R ) . In particular this applies when R = ℓ is a field. When E is finite and ℓ is arbitrary,Theorem 5.4 also applies to the universal homology theory j : Alg ℓ → kk ofTheorem 4.2. In particular, if E < ∞ we have a triangle in kkj ( ℓ reg( E ) ) I − A tE / / j ( ℓ E ) / / j ( L ( E )) . (5.6)In particular L ( E ) belongs to the bootstrap category of [7, Section 8.3] whenever E is finite, or equivalently, when L ( E ) is unital [1, Lemma 1.2.12]. Remark . When E is finite, we can also fit L ( E ) into a kk -triangle associated toa matrix with entries in { , } . Let B ′ E ∈ { , } ( E ` sink( E )) × ( E ` sink( E )) ,( B ′ E ) x , y = δ r ( x ) , s ( y ) x , y ∈ E δ r ( x ) , y x ∈ E , y ∈ sink( E )0 x ∈ sink( E )The matrix B ′ E = A ′ E ′ is the incidence matrix of the maximal out-split graph E ′ of[1, Definition 6.3.23]. Since by [1, Proposition 6.3.25], L ( E ) (cid:27) L ( E ′ ) in Alg ℓ , (5.6)gives a triangle j ( ℓ E ) I − B tE / / j ( ℓ E ` sink( E ) ) / / j ( L ( E )) . Here I , B tE ∈ ( E ` sink( E )) × E are obtained from the identity matrix and from( B ′ E ) t by removing the columns corresponding to sinks. Remark . In [7], a functor j ′ : Alg ℓ → kk ′ was constructed that is universalfor those homotopy invariant and M ∞ -stable homology theories which are excisivewith respect to all, not just the linearly split short exact sequences of algebras (3.1).The suspension functor in kk ′ is induced by Wagoner’s suspension (1.16); we have Ω − j = j Σ . The universal property of j implies that there is a triangulated functor F : kk → kk ′ such that j ′ = F j , and it follows from [7, Theorem 8.2.1] that F : KH n ( R ) = kk n ( ℓ, R ) → kk ′ n ( ℓ, R ) is an isomorphism for all n ∈ Z and R ∈ Alg ℓ .Note that when E is finite and E is countable, Theorem 5.4 applies to X = j ′ . Itfollows that F n : kk ( L ( E ) , R ) → kk ′ n ( L ( E ) , R ) is an isomorphism for all n ∈ Z and R ∈ Alg ℓ . In particular, if R is unital, E is countable and E is finite, then for the E xt -group we have a natural map E xt ( L ( E ) , R ) → kk − ( L ( E ) , R ) . Convention 5.9.
From now on, every statement about the image under j of theCohn or Leavitt path algebras of finitely many graphs E , . . . , E n will refer to the ⊔ ni = E i -stable, homotopy invariant, excisive homology theory j : Alg ℓ → kk . LGEBRAIC BIVARIANT K -THEORY AND LEAVITT PATH ALGEBRAS 19 Proposition 5.10.
Let E and F be graphs and θ ∈ kk ( L ( E ) , L ( F )) . Assume thatE and F are finite and that KH i ( θ ) is an isomorphism for i = , . Then θ is anisomorphism. In particular KH n ( θ ) is an isomorphism for all n ∈ Z .Proof. The map θ induces a natural transformation θ A : kk ( A , L ( E )) → kk ( A , L ( F ))( A ∈ Alg ℓ ). Our hypothesis that KH i ( θ ) is an isomorphism for i = , θ Ω − i j ( ℓ ) is an isomorphism. Since F is finite by assumption, this implies that also θ Ω − i j ( ℓ F ) and θ Ω − i j ( ℓ reg( F ) ) are isomorphisms. Hence applying θ : kk ( − , L ( E )) → kk ( − , L ( F )) to the triangle j ( ℓ reg( F ) ) I − A tF / / j ( ℓ F ) / / j ( L ( F ))and using the five lemma, we obtain that θ L ( F ) is an isomorphism. In particular thereis an element µ ∈ kk ( L ( F ) , L ( E )) such that µθ = L ( F ) . Our hypothesis implies that KH i ( µ ) must be an isomorphism for i = ,
1. Hence reversing the role of E and F in the previous argument shows that µ has a left inverse. It follows that θ is anisomorphism. (cid:3) Remark . The conclusion of Proposition 5.10 does not follow if we only as-sume that there are group isomorphisms θ i : KH i ( L ( E )) ∼ −→ KH i ( L ( F )) ( i = , ℓ = Q , K ( L ) = K ( L ) = Z and K ( L ) = Q ∗ (cid:27) Z / Z ⊕ Z ( N ) (cid:27) K ( L ). However L and L are not isomorphic in kk , since they have di ff erentperiodic cyclic homology: HP ( L ) = HP ( L ) = Q .6. A structure theorem for L eavitt path algebras in kk Standing assumptions 6.1.
From here on, we shall assume that the commutativebase ring ℓ satisfies the following conditions.i) KH − ( ℓ ) = Z = K ( Z ) = KH ( Z ) → KH ( ℓ ) is an isomorphism.Moreover, all graphs considered henceforth are assumed to have finitely many ver-tices. In particular, all Leavitt path algebras will be unital. Remark . Any regular supercoherent ground ring ℓ satisfies standing assumptioni), and moreover any Leavitt path algebra over ℓ is K -regular. Hence all statementsof this section are valid for regular supercoherent ℓ satisfying standing assumptionii), with K substituted for KH . In particular, this applies when ℓ = Z or any field. Definition 6.3.
Let L ( E ) the Leavitt path algebra associated to the graph E . Put KH ( L ( E )) = kk − ( L ( E ) , ℓ ) . It follows from (5.6) and the standing assumptions that, abusing notation, and writ-ing I for I t , KH ( L ( E )) (cid:27) Coker( I − A E : Z E → Z reg( E ) ) . (6.4) Proposition 6.5. (Compare [8, Theorem 5.3] .) Let E be a graph with finitely manyvertices, such that E is countable and sour( E ) = ∅ . Then the natural map ofRemark 5.8 is a surjection E xt ( L ( E )) ։ KH ( L ( E )) . (6.6) Proof.
Our hypothesis on E imply that, with the notation of(4.6), we have P v = N for all v ∈ E . Hence by (4.13), K ( E ) (cid:27) M ∞ ℓ reg( E ) , and (5.1) is an extensionof L ( E ) by M ∞ ℓ reg( E ) . Let ψ : L ( E ) → Σ ( ℓ ) reg( E ) be its classifying map and for v ∈ reg( E ) let π v : Σ ( ℓ ) reg( E ) → Σ ( ℓ ) be the projection, and put ψ v = π v ψ . With thenotation of Remark 5.8 we have a triangle in kk ′ j ( ℓ E ) → j ( L ( E )) ψ −→ j ( Σ ( ℓ ) reg( E ) ) → j ( Σ ( ℓ ) E ) . Applying kk ′ ( − , Σ ( ℓ )) to it and using Remark 5.8 we see that KH ( L ( E )) is gener-ated by the kk -classes of the ψ v ; since these are in the image of (6.6), it follows thatthe latter map is surjective. (cid:3) Lemma 6.7. i) The groups KH ( L ( E )) and KH ( L ( E )) have isomorphic torsion subgroups. ii) E ) = rk( KH ( L ( E )) − rk( KH ( L ( E )) .Proof. Let D = diag( d , . . . , d n , , . . . , ∈ Z E × reg( E ) , d i ≥ d i \ d i + be the Smithnormal form of I − A tE . Then D t is the Smith normal form of I − A E , whencetors KH ( L ( E )) = n M i = Z / d i = tors KH ( L ( E )) . (6.8)Similarly,rk KH ( L ( E )) − rk KH ( L ( E )) = ( E − rk( I − A E )) − ( E ) − rk( I − A E ))) = E ) . (cid:3) We shall write τ ( E ) = tors KH ( L ( E )) . For 0 ≤ n ≤ ∞ , let R n be the graph with exactly one vertex and n loops and let L n = L ( R n ). Thus L = ℓ , L = ℓ [ t , t − ] is the algebra of Laurent polynomials andfor 2 ≤ n < ∞ , L n = L (1 , n ) is the Leavitt algebra of [10]. By (5.6), j ( L ∞ ) (cid:27) j ( L )and we have a distinguished triangle in kkj ( ℓ ) n − −→ j ( ℓ ) −→ j ( L n ) ( n ≥ . (6.9) Theorem 6.10.
Let E be a graph with finitely many vertices. Assume that ℓ satisfiesthe standing assumptions 6.1. Let d , . . . , d n , d i \ d i + be the invariant factors of thefinite abelian group τ ( E ) , s = E ) and r = rk( KH ( L ( E )) . Let j : Alg ℓ → kkbe the universal excisive, homotopy invariant, E-stable homology theory. Thenj ( L ( E )) (cid:27) j ( L s ⊕ L r ⊕ n M i = L d i + ) . Proof.
Let D = diag( d , . . . , d n , , . . . , ∈ Z E × reg( E ) . Then there are P ∈ GL E Z , Q ∈ GL E ) Z such that P ( I − A tE ) Q = D where D : = diag ( d , . . . , d r , , . . . , LGEBRAIC BIVARIANT K -THEORY AND LEAVITT PATH ALGEBRAS 21 Hence we have the following commutative square in kk with vertical isomorphisms j ( ℓ reg( E ) ) Q − (cid:15) (cid:15) I − A tE / / j ( ℓ E ) P (cid:15) (cid:15) j ( ℓ reg( E ) ) D / / j ( ℓ E )Hence both rows have isomorphic cones. By (5.6), the cone of the top row is L ( E );by (6.9) and Lemma 6.7 that of the bottom row is L s ⊕ L r ⊕ L ni = L d i + . (cid:3) Corollary 6.11.
The following are equivalent for graphs E and F with finitelymany vertices. i) j ( L ( E )) (cid:27) j ( L ( F )) . ii) KH ( L ( E )) (cid:27) KH ( L ( F )) and KH ( L ( E )) (cid:27) KH ( L ( F )) . iii) KH ( L ( E )) (cid:27) KH ( L ( F )) and E ) = F ) .Proof. Immediate from Lemma 6.7 and Theorem 6.10. (cid:3)
Remark . Let E and F be as in Corollary 6.11. Assume in addition that ℓ is afield, that L ( E ) and L ( F ) are simple and that inf( E ) , ∅ , inf( F ). In [14, Theorem7.4], E. Ruiz and M. Tomforde show that under these assumptions condition iii)of Corollary 6.11 is equivalent to the existence of a Morita equivalence between L ( E ) and L ( F ). It follows that for such E and F , the algebras L ( E ) and L ( F ) areisomorphic in kk if and only if they are Morita equivalent. Ruiz and Tomforde showalso that under the additional assumption that the group of invertible elements U ( ℓ )has no free quotients, the condition that E ) = F ) in iii) can be replacedby the condition that K ( L ( E )) (cid:27) K ( L ( F )). The additional assumption guaranteesthat rk( K ( L ( E ))) = rk(Ker(1 − A tE )) = rk( KH ( L ( E )) whenever E < ∞ , so that E ) = rk( K ( L ( E )) − rk( K ( L ( E ))).7. A canonical filtration in kk(L(E),R) Let ℓ be a ground ring satisfying the Standing assumptions 6.1, let E be a graphwith finitely many vertices, L ( E ) its Leavitt path algebra over ℓ , and n ∈ Z . Itfollows from (5.3) that we have an exact sequence0 → KH n ( ℓ ) ⊗ KH ( L ( E )) −→ KH n ( L ( E )) → Ker(( I − A tE ) ⊗ KH n − ( ℓ )) → . (7.1) Lemma 7.2.
The map KH n ( ℓ ) ⊗ KH ( L ( E )) −→ KH n ( L ( E )) of (7.1) is the cupproduct map of Example 3.6.Proof. Because by assumption 6.1 (ii), KH ( ℓ ) = Z , for any finite set X , the cupproduct of Example 3.6 gives an isomorphism ∪ : KH n ( ℓ ) ⊗ KH ( ℓ X ) ∼ −→ KH n ( ℓ X ) . (7.3) Hence by (5.3) we have a commutative diagram with exact rows KH n ( ℓ reg( E ) ) I − A tE / / KH n ( ℓ E ) / / KH n ( L ( E )) KH n ( ℓ ) ⊗ KH ( ℓ reg( E ) ) ∪ O O I − A tE / / KH n ( ℓ ) ⊗ KH ( ℓ E ) ∪ O O / / KH n ( ℓ ) ⊗ KH ( L ( E )) . ∪ O O (cid:3) Let R be an algebra and n ∈ Z . Consider the map KH n : kk ( L ( E ) , R ) → Hom Z ( KH n ( L ( E )) , KH n ( R )) . (7.4)Define a descending filtration { kk ( L ( E ) , R ) i | ≤ i ≤ } on kk ( L ( E ) , R ) as follows.Let kk ( L ( E ) , R ) = kk ( L ( E ) , R ) , kk ( L ( E ) , R ) = Ker KH , (7.5) kk ( L ( E ) , R ) = (Ker KH ) ∩ kk ( L ( E ) , R ) . (7.6)It follows from the definition of kk ( L ( E ) , R ) and kk ( L ( E ) , R ) that KH induces acanonical homomorphism kk ( L ( E ) , R ) / kk ( L ( E ) , R ) → hom( KH ( L ( E )) , KH ( R )) . (7.7)Let ξ ∈ kk ( L ( E ) , R ) ; by Lemma 7.2, KH ( ξ ) vanishes on the image of KH ( ℓ ) ( E ) ,whence it induces a map Ker( I − A tE ) → KH ( R ). Thus we have a map kk ( L ( E ) , R ) / kk ( L ( E ) , R ) → hom(Ker( I − A tE ) , KH ( R )) . (7.8)Let ξ ∈ kk ( L ( E ) , R ) ; embed ξ into a distinguished triangle C ξ → L ( E ) ξ −→ R . (7.9)We have an extension of abelian groups( E ( ξ )) 0 → KH ( R ) → K ( C ξ ) → KH ( L ( E )) → . (7.10)Let kk ( L ( E ) , R ) → Ext Z ( KH ( L ( E )) , KH ( R )) , ξ [ E ( ξ )] . (7.11) Theorem 7.12.
Let E be a graph with finitely many vertices, ℓ a ring satisfying theStanding assumptions 6.1, L ( E ) the Leavitt path algebra over ℓ and R an ℓ -algebra.Then the maps (7.7) , (7.8) and (7.11) are isomorphisms.Proof. Observe that if B is an algebra and X a finite set, then the isomorphism(3.8) induces an isomorphism kk n ( ℓ X , B ) ∼ −→ hom( Z X , KH n ( B )). Using this andapplying kk ( − , R ) to the triangle (5.6) we obtain an exact sequenceHom( Z E , KH ( R )) → Hom( Z reg( E ) , KH ( R ))) → kk ( L ( E ) , R ) → Hom( Z E , KH ( R )) → Hom( Z reg( E ) , KH ( R )) . (7.13)Since 0 → Ker( I − A tE ) → Z reg( E ) → Z E → KH ( L ( E )) → LGEBRAIC BIVARIANT K -THEORY AND LEAVITT PATH ALGEBRAS 23 is a free Z -module resolution, the kernel of the last map in (7.13) isHom( KH ( L ( E )) , KH ( R )), and it is straightforward to check that the induced sur-jection kk ( L ( E ) , R ) ։ Hom( KH ( L ( E )) , KH ( R ))is precisely the map KH of (7.4). Hence the cokernel of the first map in (7.13) is kk ( L ( E ) , R ) , and again because (7.14) is a free resolution, we have a short exactsequence0 → Ext Z ( KH ( L ( E ) , KH ( R )) → kk ( L ( E ) , R ) → Hom(Ker( I − A tE ) , KH ( R )) → . (7.15)It is again straightforward to check that the surjective map from kk ( L ( E ) , R ) in(7.15) is (7.8). Hence by (7.15) we have an isomorphism kk ( L ( E ) , R ) ∼ −→ Ext Z ( KH ( L ( E ) , KH ( R )) (7.16)It remains to show that the above isomorphism agrees with (7.11).Let ξ ∈ kk ( L ( E ) , R ) and let ∂ : j ( L ( E )) → Ω − j ( ℓ ) reg( E ) be the boundary map in(5.6). Because KH ( ξ ) =
0, there is an element ˆ ξ ∈ kk ( ℓ reg( E ) , R ) such that ξ = ˆ ξ∂ .Hence because kk is triangulated, there exists θ ∈ kk ( ℓ E , C ξ ) such that we have amap of distinguished triangles j ( ℓ ) reg( E ) Ω j ˆ ξ (cid:15) (cid:15) / / j ( ℓ ) E θ (cid:15) (cid:15) / / j ( L ( E )) ∂ / / Ω − j ( ℓ ) reg( E )ˆ ξ (cid:15) (cid:15) Ω j ( R ) / / C ξ / / j ( L ( E )) ξ / / j ( R )Applying the functor kk ( ℓ, − ) and using that KH ( ξ ) =
0, we obtain a map ofextensionsKer( I − A tE ) (cid:15) (cid:15) / / Z reg( E )ˆ ξ (cid:15) (cid:15) / / Z E θ (cid:15) (cid:15) / / K ( L ( E )) / / / / K ( R ) / / K ( C ξ ) / / K ( L ( E )) / / ξ to the class [ ˆ ξ ] of ˆ ξ modulo the image ofHom( Z E , KH ( R )). It is clear from (7.17) that [ ˆ ξ ] = [ C ξ ]. (cid:3) Corollary 7.18.
Let ξ ∈ kk ( L ( E ) , R ) and let C ξ be as in (7.9) . Then ξ = if andonly if KH ( ξ ) = KH ( ξ ) = and the extension (7.10) is split. In the next corollary we shall use the fact that, since Ker( I − A tE ) is a free abeliangroup, the canonical surjection KH ( L ( E )) → Ker( I − A tE ) admits a section γ : Ker( I − A tE ) → KH ( L ( E )) . (7.19)The map γ induces a natural transformation γ ∗ : Hom( KH ( L ( E )) , − ) → Hom(Ker( I − A tE ) , − )) . Corollary 7.20. (UCT) For every n ∈ Z we have an exact sequence → Ext Z ( KH ( L ( E )) , KH n + ( R )) → kk n ( L ( E ) , R ) [ KH ,γ ∗ KH ] −→ Hom( KH ( L ( E )) , KH n ( R )) ⊕ Hom(Ker( I − A tE ) , KH n + ( R )) → . Proof.
In view of (3.5) we may assume that n =
0. By Theorem 7.12 the map KH : kk ( L ( E ) , R ) → hom( KH ( L ( E )) , KH ( R )) is a surjection; by definition, itskernel is kk ( L ( E ) , R ) , and γ ∗ KH induces the map (7.8), which is surjective byTheorem 7.12. Hence [ KH , γ ∗ KH ] is surjective, and its kernel is by definition kk ( L ( E ) , R ) , which, again by Theorem 7.12, is Ext Z ( KH ( L ( E )) , KH ( R )). (cid:3) Lemma 7.21.
Let E be a graph and R an algebra. Assume that E < ∞ . Thenthe composition map induces an isomorphismKH ( L ( E )) ⊗ KH ( R ) ∼ −→ kk ( L ( E ) , R ) Proof.
By our Standing assumptions, KH − ℓ =
0; it follows from this that KH ( L ( E )) = kk − ( L ( E ) , ℓ ) and that the composition map lands in kk ( L ( E ) , R ) .In particular, writing ∨ for the dual group, we have KH ( L ( E )) / kk − ( L ( E ) , ℓ ) = Ker( I − A tE ) ∨ ; since the latter is free, tensoring with KH ( R ) we obtain the top exactsequence of the commutative diagram below; the bottow row is exact by Theorem7.12.Ext Z ( τ ( E ) , Z ) ⊗ KH ( R ) (cid:15) (cid:15) / / KH ( LE ) ⊗ KH ( R ) (cid:15) (cid:15) / / / / Ker( I − A tE ) ∨ ⊗ KH ( R ) (cid:15) (cid:15) Ext Z ( τ ( E ) , KH ( R )) / / kk ( L ( E ) , R ) / / / / Hom Z (Ker( I − A tE ) , KH ( R )) . One checks, using the fact that for a free, finitely generated group L , L ∨ ⊗ ( − ) (cid:27) Hom Z ( L , − ), that the vertical arrows on the right and left are isomorphisms; itfollows that the vertical arrow at the middle is an isomorphism as well. (cid:3) Lemma 7.22.
Let E and R be as in Lemma 7.21. There is an exact sequence
Ker( I − A E ) ⊗ KH ( R ) ֒ → Hom( KH ( L ( E )) , KH ( R )) ։ Tor Z ( KH ( L ( E )) , KH ( R )) . Proof.
It follows from (5.6) that we have a free Z -module resolution0 → Ker( I − A E ) → ( Z E ) ∨ → ( Z reg( E ) ) ∨ → KH ( L ( E )) → . Now tensor by KH ( R ) and observe thatKer(( I − A E ) ⊗ id KH ( R ) ) = hom( KH ( L ( E )) , KH ( R )) . (cid:3) Proposition 7.23. (K ¨unneth theorem) Let L ( E ) and R be as in Theorem 7.12 andn ∈ Z . Then there is an exact sequence → KH ( L ( E )) ⊗ KH n + ( R ) ⊕ Ker( I − A E ) ⊗ KH n ( R ) → kk ( L ( E ) , R ) → Tor Z ( KH ( L ( E )) , KH n ( R )) → . LGEBRAIC BIVARIANT K -THEORY AND LEAVITT PATH ALGEBRAS 25 Proof.
It su ffi ces to prove the proposition for n =
0. By Theorem 7.12 we havea canonical surjection π : kk ( L ( E ) , R ) → Hom( KH ( L ( E )) , KH ( R )). By Lemma7.22 we have an inclusioninc : Ker( I − A E ) ⊗ KH ( R ) ⊂ Hom( KH ( L ( E )) , KH ( R )) . (7.24)Let Q = π − (Ker( I − A E ) ⊗ KH ( R )); by Lemmas 7.21 and 7.22 we have exactsequences 0 → Q → kk ( L ( E ) , R ) → Tor Z ( KH ( L ( E )) , KH ( R )) → → KH ( L ( E )) ⊗ KH ( R ) → Q → Ker( I − A E ) ⊗ KH ( R ) → . (7.25)We have to show that the second sequence above splits. Let θ : Ker( I − A E ) → KH ( L ( E )) be a section of the canonical projection. One checks that for inc as in(7.24), the composite θ ′ : Ker( I − A E ) ⊗ KH ( R ) θ ⊗ id −→ KH ( L ( E )) ⊗ KH ( R ) ◦ −→ kk ( L ( E ) , R )satisfies πθ ′ = inc. It follows that the sequence (7.25) splits, completing the proof. (cid:3) Remark . The key property of the algebra B = L ( E ) that we have used in thissection is that for some m , n ∈ N and M ∈ Z m × n we have a distinguished triangle in kk j ( ℓ ) n M −→ j ( ℓ ) m → j ( B ) . All the results and proofs in this section apply to any algebra B with the aboveproperty, substituting M for I − A tE , and assuming of course that ℓ satisfies theStanding assumptions 6.1. However one can show, using the Smith normal formof M , that any such B is kk -isomorphic to the sum of Leavitt path algebra and anumber of copies of the suspension Ω − ℓ .R eferences [1] Gene Abrams, Pere Ara, and Mercedes Siles Molina, Leavitt path algebras , Lecture Notes inMath., vol. 2008, Springer, 2017. ↑
13, 16, 18[2] P. Ara, K. Goodearl, and E. Pardo, K of purely infinite simple regular rings , K-theory (2002), no. 1, 69–100. ↑ Flow invariants in the clas-sification of Leavitt path algebras , J. Algebra (2011), 202–231. MR2785945 ↑ K-theory of Leavitt path algebras ,M¨unster J. Math. (2009), 5–33. MR2545605 ↑
2, 4, 17, 18[5] Guillermo Corti˜nas,
Algebraic v. topological K-theory: a friendly match , Topics in algebraicand topological K -theory, Lecture Notes in Math., vol. 2008, Springer, Berlin, 2011, pp. 103–165. MR2762555 ↑
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Homotopy classification of Leavitt path algebras ,available at arXiv:1806.09242 . ↑ Bivariant algebraic K-theory , J. Reine Angew. Math. (2007), 71–123. MR2359851 ↑
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A class of C ∗ -algebras and topological Markov chains ,Inventiones Mathematicae. ↑
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Topological and bivariant K-theory ,Oberwolfach Seminars, vol. 36, Birkh¨auser Verlag, Basel, 2007. MR2340673 ↑ [10] W. G. Leavitt, The module type of a ring , Trans. Amer. Math. Soc. (1962), 113–130.MR0132764 ↑ Bivariant algebraic K-theory categories and a spectrum for G-equivariant bivariant algebraic K-theory , PhD thesis, Buenos Aires, 2017. ↑
1, 8, 9, 10[12] Mikael Rørdam,
Classification of Cuntz-Krieger algebras , K-theory (1995), no. 1, 31–58. ↑ The K¨unneth theorem and the universal coef-ficient theorem for Kasparov’s generalized K-functor , Duke Math. J. (1987), 431–474.MR0894590 ↑ Classification of unital simple Leavitt path algebras of infinitegraphs , J. Algebra (2013), 45–83. MR3045151 ↑
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Delooping classifying spaces in algebraic K-theory , Topology (1972), 349–370. MR0354816 ↑
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Homotopy algebraic K-theory , Algebraic K -theory and algebraic numbertheory (Honolulu, HI, 1987), Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989,pp. 461–488. MR991991 ↑
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E-mail address : [email protected], [email protected] URL : http://mate.dm.uba.ar/˜gcorti D ep . M atem ´ atica -IMAS, FCE y N-UBA, C iudad U niversitaria P ab
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