Tensor products of Leavitt path algebras

TENSOR PR ODUCTS OF LEA VITT P A TH ALGEBRAS PERE ARA AND GUILLERMO COR TI ˜ NAS Abstract. W e compute the Hochsc hild homology o f Lea vitt path alge bras o v er a field k . As an application, we sho w that L 2 and L 2 ⊗ L 2 ha v e differen t Hochsc hild homologies, and so they are not Mori ta equiv alen t; in particular they are not isomorphic. Similarly , L ∞ and L ∞ ⊗ L ∞ are distinguished by their Hochsc hild homologies and so they are not Morita equiv alen t either. By con tr ast, we sho w that K -theory cannot distinguish these algebras; we ha ve K ∗ ( L 2 ) = K ∗ ( L 2 ⊗ L 2 ) = 0 and K ∗ ( L ∞ ) = K ∗ ( L ∞ ⊗ L ∞ ) = K ∗ ( k ). 1. Introduction Elliott’s theorem [21] stating that O 2 ⊗ O 2 ∼ = O 2 plays an imp ortant role in the pro of of the celebrated cla ssification theorem of Kirchberg algebras in the UCT class, due to K irch b erg [14] and Phillips [1 9]. Recall that a K ir ch b erg algebr a is a purely infinite, simple, nuclear a nd s eparable C* - algebra . The Kirch b er g-Phillips theorem states that this class of simple C*-a lg ebras is completely classified by its topolog ical K -theory . The analogous question whether the algebr as L 2 and L 2 ⊗ L 2 are isomorphic has re mained op en for some time. Here L 2 is the Leavitt algebra of t ype (1 , 2) ov er a field k (see [1 7]), that is, the k -alg ebra with ge ne r ators x 1 , x 2 , x ∗ 1 , x ∗ 2 and relations given by x ∗ i x j = δ i,j and P 2 i =1 x i x ∗ i = 1. In this pap er we o btain a nega tive answer to this que s tion. Indeed, we analy ze a m uch larg er class of algebra s, namely the tens or pro ducts of L e avitt path a lgebras of finite quivers, in ter ms of their Ho chsc hild homology , and we pro ve that, for 1 ≤ n < m ≤ ∞ , the tenso r pro ducts E = N n i =1 L ( E i ) and F = N m j =1 L ( F j ) o f Leavitt path algebra s of non-acyclic finite quivers E i , F j , are distinguished b y their Ho chsc hild homologies (Theo rem 5.1). Because Hochsch ild homology is Morita inv ariant, we conclude tha t E and F are no t Morita e q uiv alent for n < m . Since L 2 is the Leavitt pa th algebra of the g raph with one vertex a nd tw o ar rows, we obtain that L 2 ⊗ L 2 and L 2 are not Morita equiv a lent; in particula r they ar e not isomorphic. Recall that, by a theor e m of Kir ch b erg [15], a s imple, nuclear and separable C ∗ -algebra A is purely infinite if and only if A ⊗ O ∞ ∼ = A . W e also s how that the ana logue of Kir chberg’s result is not true for Le avitt algebr as. W e prove in Prop ositio n 5.3 that if E is a non- acyclic quiver, then L ∞ ⊗ L ( E ) a nd L ( E ) a r e not Morita equiv alent, and als o that L ∞ ⊗ L ∞ and L ∞ are not Morita equiv alent. Date : No v em ber 7, 2018. The first author was partiall y supp orted by DGI MICI IN-FEDER MTM2008-06201-C02-01, and b y the Comissi onat p er Unive rsitats i Recerca de l a Generalitat de Cataluny a. The second named author was supp orted by CONICET and partially supported by grants PIP 112-200801- 00900, UBACyTs X051 and 2002010010038 6, and MTM2007-64074. 1 2 PERE ARA AND GUILLERMO COR TI ˜ NAS Using the results in [5] we prove that the algebras L 2 and L 2 ⊗ L ( F ), for F an arbitrar y finite quiver, hav e trivia l K -theory: all a lgebraic K -theory gr oups K i , i ∈ Z , v anish on them (this follows from Lemma 6.1 and Prop osition 6.2). W e also compute K ∗ ( L ( F )) = K ∗ ( L ∞ ⊗ L ( F )) and that K ∗ ( L ∞ ) = K ∗ ( L ∞ ⊗ L ∞ ) = K ∗ ( k ) is the K -theory o f the gr ound field (see Pr op osition 6.3 and Corolla ry 6.4). This implies in particula r that, in contrast with the analytic s itua tion, no clas sification result, in terms so le ly of K -theory , ca n b e exp ected for a c la ss o f central, simple k -algebr as, c o ntaining all purely infinite simple unital L e avitt path alg ebras, and closed under tensor pro ducts. It is worth mentioning tha t an imp or ta nt step to- wards a K -theoretic classification of purely infinite s imple Leavitt pa th alg ebras of finite quivers has b een achiev ed in [2]. W e refer the reader to [3], [7] a nd [20] for the basics on Leavitt a lgebras, Leavitt path algebras and graph C*-algebra s, and to [22] for a nice survey on the Kirch be r g- Phillips Theorem. Notations. W e fix a field k ; all vector spaces, tensor pro ducts a nd algebr as are ov e r k . If R and S are unital k - a lgebras , then b y an ( R, S )-bimo dule we under stand a left mo dule over R ⊗ S op . By a n R -bimo dule we shall mean a n ( R , R ) bimo dule, that is, a left mo dule ov er the env e loping alg ebra R e = R ⊗ R op . Hochschild ho mology of k -algebr as is alwa ys taken over k ; if M is an R -bimo dule, we write H H n ( R, M ) = T or R e n ( R, M ) for the Ho chschild homology of R with c o efficien ts in M ; we abbreviate H H n ( R ) = H H n ( R, R ). 2. Hochschil d homol ogy Let k b e a field, R a k -alg ebra and M an R -bimo dule. The Ho chschild homol- ogy H H ∗ ( R, M ) of R w ith co efficien ts in M w a s defined in the intro duction; it is computed b y the Ho chschild c omplex H H ( R, M ) which is given in degree n by H H ( R , M ) n = M ⊗ R ⊗ n It is equipp ed with the Hochschild b oundary map b defined by b ( a 0 ⊗ a 1 ⊗ · · · ⊗ a n ) = n − 1 X i =0 ( − 1) i a 0 ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a n + ( − 1) n a n a 0 ⊗ · · · ⊗ a n − 1 If R and M happ en to b e Z -gr aded, then H H ( R, M ) splits into a direct sum of sub c omplexes H H ( R , M ) = M m ∈ Z m H H ( R , M ) The ho mogeneous compo nent of degree m of H H ( R , M ) n is the linear subspace of H H ( R , M ) n generated by all elemen tary tenso rs a 0 ⊗ · · · ⊗ a n with a i homogeneous and P i | a i | = m . One of the first basic prop erties of the Ho chsc hild complex is that it commutes with filtering colimits. Thus we hav e Lemma 2. 1. L et I b e a filter e d or der e d set and let { ( R i , M i ) : i ∈ I } b e a dir e cte d system of p airs ( R i , M i ) c onsisting of an algebr a R i and and an R i -bimo dule M i , with algebr a maps R i → R j and R i -bimo dule m aps M i → M j for e ach i ≤ j . L et ( R, M ) = colim i ( R i , M i ) . Then H H n ( R, M ) = colim i H H n ( R i , M i ) ( n ≥ 0 ). TENSOR PRODUCT S OF LEA VITT P A TH ALGEBRAS 3 Let R i be a k -algebra a nd M i an R i - bimo dule ( i = 1 , 2). T he K ¨ unneth formula establishes a natura l is o morphism ([23, 9.4 .1]) H H n ( R 1 ⊗ R 2 , M 1 ⊗ M 2 ) ∼ = n M p =0 H H p ( R 1 , M 1 ) ⊗ H H n − p ( R 2 , M 2 ) Another fundamental fact ab out Ho chsc hild homo lo gy which w e shall need is Mo rita inv ariance. Let R and S b e Morita equiv alent alg ebras, and let P ∈ R ⊗ S op − mo d and Q ∈ S ⊗ R op − mo d implement the Morita eq uiv a lence. Then ([23, Thm. 9.5.6]) (2.2) H H n ( R, M ) = H H n ( S, Q ⊗ R M ⊗ R P ) Lemma 2.3 . L et R 1 , . . . , R n and S 1 , . . . , S m , . . . b e a finite and an infinite se qu en c e of algebr as, and let R = N n i =1 R i , S ≤ m = N m j =1 S j and S = N ∞ j =1 S j . Assume: (1) H H q ( R i ) 6 = 0 6 = H H q ( S j ) ( q = 0 , 1 ), ( 1 ≤ i ≤ n ), ( 1 ≤ j ). (2) H H p ( R i ) = H H p ( S j ) = 0 for p ≥ 2 , 1 ≤ i ≤ n , 1 ≤ j . (3) n 6 = m . Then n o two of R , S ≤ m and S ar e Morita e quivalent. Pr o of. By the K ¨ unneth formula, we hav e H H n ( R ) = n O i =1 H H 1 ( R i ) 6 = 0 , H H p ( R ) = 0 p > n By the same argument, H H p ( S ≤ m ) is nonzero for p = m , and ze ro for p > m . Hence if n 6 = m , R and S ≤ m do not hav e the sa me Ho chsc hild homology a nd therefore they cannot b e Mo rita equiv a lent, by (2.2). Similar ly , b y Lemma 2.1, we hav e H H n ( S ) = M J ⊂ N , | J | = n ( O j ∈ J H H 1 ( S j )) ⊗ ( O j / ∈ J H H 0 ( S j )) , so that H H n ( S ) is nonzero for all n ≥ 1, and thu s it c a nnot be Morita equiv alent to either R o r S ≤ m .  3. Hochschil d homology of crossed products Let R be a unital algebra and G a group acting on R by alg ebra a utomorphisms. F o rm the cros sed-pro duct a lgebra S = R ⋊ G , and consider the Ho chsc hild co mplex H H ( S ). F or ea ch conjugacy class ξ of G , the grade d submo dule H H ξ ( S ) ⊂ H H ( S ) generated in degree n by the e le men tary tensors a 0 ⋊ g 0 ⊗ · · · ⊗ a n ⋊ g n with g 0 · · · g n ∈ ξ is a sub co mplex, and we hav e a direct s um deco mpo sition H H ( S ) = L ξ H H ξ ( S ). The following theorem of Lorenz describ es the complex H H ξ ( S ) corresp onding to the co njugacy clas s ξ = [ g ] of a n element g ∈ G as hyperhomolo gy ov e r the centralizer subg roup Z g ⊂ G . Theorem 3.1. [16] . L et R b e a unital k -algebr a, G a gr oup acting on R by au- tomorphisms, g ∈ G and Z g ⊂ G the c entr alizer su b goup. L et S = R ⋊ G b e the cr osse d pr o duct algebr a, and H H h g i ( S ) ⊂ H H ( S ) the sub c omplex describ e d ab ove. Consider the R -submo dule S g = R ⋊ g ⊂ S . Then ther e is a qu asi-isomorph ism H H [ g ] ( S ) ∼ → H ( Z g , H H ( R , S g )) In p articular we have a s p e ctr al se qu en c e E 2 p,q = H p ( Z g , H H q ( R, S g )) ⇒ H H [ g ] p + q ( S ) 4 PERE ARA AND GUILLERMO COR TI ˜ NAS R emark 3.2 . Lore nz formulates his result in terms of the sp ectral sequence alone, but his pro of shows that there is a quasi-isomo rphism as s ta ted ab ove; an explicit formula is g iven for exa mple in the pro of of [11, Le mma 7.2]. Let A b e a not necessar ily unital k -algebra, write ˜ A for its unitalization. Recall from [24] that A is called H -un ital if the gro ups T or ˜ A n ( k , A ) v anish for a ll n ≥ 0. W o dz icki prov ed in [24] that A is H -unital if and only if for ev ery embedding A ⊳ R of A as a tw o -sided ideal of a unital ring R , the map H H ( A ) → H H ( R : A ) = ker ( H H ( R ) → H H ( R/ A )) is a quasi-iso morphism. Lemma 3.3. The or em 3.1 st il l holds if the c ondition t hat R b e unital is r eplac e d by the c ondition that it b e H - unital. Pr o of. F o llows from Theorem 3.1 and the fact, proved in [11, P rop. A.6.5], that R ⋊ G is H -unital if R is.  Let R b e a unital algebra , and φ : R → pR p a co r ner isomor phism. As in [6], we consider the skew Laur ent p olynomial algebra R [ t + , t − , φ ]; this is the R -algebr a generated by elements t + and t − sub ject to the following r elations. t + a = φ ( a ) t + at − = t − φ ( a ) t − t + = 1 t + t − = p Observe that the a lgebra S = R [ t + , t − , φ ] is Z -graded by deg( r ) = 0 , deg ( t ± ) = ± 1. The homogeneous comp onent o f degree n is given b y R [ t + , t − , φ ] n =    t − n − R n < 0 R n = 0 Rt n + n > 0 Prop ositio n 3 .4. L et R b e a unital ring, φ : R → pR p a c orner isomorphi sm, and S = R [ t + , t − , φ ] . Consider t he weight de c omp osition H H ( S ) = L m ∈ Z m H H ( S ) . Ther e is a quasi-isomorphi sm (3.5) m H H ( S ) ∼ → Co ne(1 − φ : H H ( R, S m ) → H H ( R , S m )) Pr o of. If φ is an automorphism, then S = R ⋊ φ Z , the right hand side o f (3.5) computes H ( Z , H H ( R, S m )), and the pr op osition bec o mes the pa rticular case G = Z of Theorem 3.1. In the g eneral c ase, let A be the c o limit o f the inductive system R φ / / R φ / / R φ / / . . . Note that φ induces an automo r phism ˆ φ : A → A . Now A is H -unita l, sinc e it is a filtering co limit of unital algebras, and thus the asser tion of the pro po sition is true for the pair ( A, ˆ φ ), by Lemma 3.3. Hence it suffices to show that for B = A ⋊ ˆ φ Z the maps H H ( S ) → H H ( B ) a nd Cone(1 − φ : H H ( R , S m ) → H H ( R , S m )) → Cone(1 − φ : H H ( A, B m ) → H H ( A, B m )) ( m ∈ Z ) are quasi-is omorphisms. The analogo us prop erty for K -theory is shown in the cour se of the thir d step of the pro of of [5, Thm. 3.6]. Since the pro of in lo c. cit. uses only that K -theory commutes with filter ing colimits and is ma trix inv a r iant on those ring s for which it satisfies excision, it applies verbatim to Hochschild homology . This concludes the pr o of.  TENSOR PRODUCT S OF LEA VITT P A TH ALGEBRAS 5 4. Hochschil d homology of the Lea vitt p a th algebra Let E = ( E 0 , E 1 , r , s ) b e a finite quiver a nd let ˆ E = ( E 0 , E 1 ⊔ E ∗ 1 , r , s ) b e the double of E , which is the quiver obtained fr o m E by adding an a rrow α ∗ for each arrow α ∈ E 1 , going in the o pp o site direction. The L e avitt p ath algebr a of E is the algebra L ( E ) with one g enerator for ea ch a rrow α ∈ ˆ E 1 and one g enerator p i for each vertex i ∈ E 0 , sub ject to the following relations p i p j = δ i,j p i , ( i, j ∈ E 0 ) p s ( α ) α = α = αp r ( α ) , ( α ∈ ˆ E 1 ) α ∗ β = δ α,β p r ( α ) , ( α, β ∈ E 1 ) p i = X α ∈ E 1 ,s ( α )= i αα ∗ , ( i ∈ E 0 \ Sink( E )) The algebra L = L ( E ) is equipp ed with a Z -g rading. The gr ading is determined by | α | = 1, | α ∗ | = − 1, for α ∈ E 1 . Let L 0 ,n be the linea r spa n of all the elements of the for m γ ν ∗ , wher e γ and ν ar e paths with r ( γ ) = r ( ν ) and | γ | = | ν | = n . By [7, pro o f of Theorem 5 .3], we hav e L 0 = S ∞ n =0 L 0 ,n . F or each i in E 0 , and each n ∈ Z + , let us denote by P ( n, i ) the set of paths γ in E suc h that | γ | = n and r ( γ ) = i . The algebr a L 0 , 0 is is omorphic to Q i ∈ E 0 k . In genera l the algebra L 0 ,n is isomorphic to (4.1)  n − 1 Y m =0 ( Y i ∈ Sink( E ) M | P ( m, i ) | ( k ))  ×  Y i ∈ E 0 M | P ( n, i ) | ( k )  . The transitio n ho mo morphism L 0 ,n → L 0 ,n +1 is the identit y o n the factor s Y i ∈ Sink( E ) M | P ( m, i ) | ( k ) , for 0 ≤ m ≤ n − 1, and also on the factor Y i ∈ Sink( E ) M | P ( n, i ) | ( k ) of the last term o f the display ed fo r mula. The transitio n ho mo morphism Y i ∈ E 0 \ Sink( E ) M | P ( n, i ) | ( k ) → Y i ∈ E 0 M | P ( n +1 ,i ) | ( k ) is a blo ck diagona l ma p induced by the following iden tification in L ( E ) 0 : A ma trix unit in a factor M | P ( n, i ) | ( k ), where i ∈ E 0 \ Sink( E ), is a monomia l of the form γ ν ∗ , where γ and ν ar e paths o f length n with r ( γ ) = r ( ν ) = i . Since i is not a sink, we ca n enla rge the paths γ and ν using the edges that i emits, obta ining pa ths o f length n + 1, and the last relatio n in the definition of L ( E ) giv es γ ν ∗ = X { α ∈ E 1 | s ( α )= i } ( γ α )( ν α ) ∗ . Assume E has no source s. F o r each i ∈ E 0 , choo s e an ar row α i such that r ( α i ) = i . Cons ide r the elements t + = X i ∈ E 0 α i , t − = t ∗ + 6 PERE ARA AND GUILLERMO COR TI ˜ NAS One chec ks that t − t + = 1. Thus, since | t ± | = ± 1, the endomorphism (4.2) φ : L → L, φ ( x ) = t + xt − is homog eneous o f degree 0 with resp ect to the Z -grading. In particular it res tricts to an endomorphism of L 0 . By [6, Lemma 2 .4], we hav e (4.3) L = L 0 [ t + , t − , φ ] . Consider the matr ix N ′ E = [ n i,j ] ∈ M e 0 Z given by n i,j = # { α ∈ E 1 : s ( α ) = i, r ( α ) = j } Let e ′ 0 = | Sink( E ) | . W e assume that E 0 is ordered so that the first e ′ 0 elements of E 0 corres p o nd to its sinks. Accordingly , the fir s t e ′ 0 rows of the matrix N ′ E are 0. Let N E be the matrix obta ined by deleting these e ′ 0 rows. The matrix that enters the computation of the Ho chsch ild homolo g y of the Leavitt path a lgebra is  0 1 e 0 − e ′ 0  − N t E : Z e 0 − e ′ 0 − → Z e 0 . By a slig ht a bus e of notation, w e will write 1 − N t E for this matrix. Note that 1 − N t E ∈ M e 0 × ( e 0 − e ′ 0 ) ( Z ). Of course N E = N ′ E in case E has no sinks . Theorem 4.4. L et E b e a finite qu iver without sour c es, and let N = N E . F or e ach i ∈ E 0 \ Sink( E ) , and m ≥ 1 , let V i,m b e the ve ctor sp ac e gener ate d by al l close d p aths c of length m with s ( c ) = r ( c ) = i . L et Z = < σ > act on V m = M i ∈ E 0 \ Sink( E ) V i,m by r otation of close d p aths. We have: m H H n ( L ( E )) =            coker (1 − σ : V | m | → V | m | ) n = 0 , m 6 = 0 coker (1 − N t ) n = m = 0 ker (1 − σ : V | m | → V | m | ) n = 1 , m 6 = 0 ker (1 − N t ) n = 1 , m = 0 0 n / ∈ { 0 , 1 } Pr o of. Let L = L ( E ), P = P ( E ) ⊂ L the path a lgebra of E and W m ⊂ P b e the subspace gener ated by all paths of length m . F o r each fixed n ≥ 1, and m ∈ Z , consider the following L 0 ,n -bimo dule L m,n =  L 0 ,n W m L 0 ,n m > 0 L 0 ,n W ∗ − m L 0 ,n m < 0 W r ite L = L ( E ), a nd let m L b e the homog eneous part of degr ee m ; we hav e m L = [ n ≥ 1 L m,n If m is p ositive, then there is a basis of L m,n consisting of the pro ducts αθβ ∗ where each o f α , β and θ is a path in E , r ( α ) = s ( θ ), r ( β ) = r ( θ ), | α | = | β | = n and | θ | = m . Hence the formula π ( αθ β ∗ ) =  θ if α = β 0 else defines a surjective linear map L m,n → V m . One ch ecks that π induces an isomor - phism H H 0 ( L 0 ,n , L m,n ) ∼ = V m ( m > 0) TENSOR PRODUCT S OF LEA VITT P A TH ALGEBRAS 7 Similarly if m < 0, then H H 0 ( L 0 ,n , L m,n ) = V ∗ | m | ∼ = V − m . Next, by (4.1), w e hav e H H 0 ( L 0 ,n ) = k [ E \ Sink( E )] ⊕ M i ∈ Sink( E ) k r ( i,n ) Here r ( i, n ) = max { r ≤ n : P ( r, i ) 6 = ∅ } Now note that, because L 0 ,n is a pro duct of matr ix algebra s, it is separable, and thus H H 1 ( L 0 ,n , M ) = 0 for a n y bimo dule M . As observed in (4.3), for the auto morphism (4.2), we hav e L = L 0 [ t + , t − , φ ]. Hence in view of Pr op osition 3 .4 and Lemma 2.1, it o nly re ma ins to identif y the ma ps H H 0 ( L 0 ,n , L m,n ) → H H 0 ( L 0 ,n +1 , L m,n +1 ) induced by inclus io n a nd by the homomorphism φ . One chec ks that for m 6 = 0, these ar e resp ectively the cyclic p er m utation and the identit y V | m | → V | m | . The case m = 0 is de a lt with in the sa me w ay as in [5 , Pro of of Theor em 5 .10].  Corollary 4.5 . L et E b e a finit e quiver with at le ast one non-trivial close d p ath. i) H H n ( L ( E )) = 0 for n / ∈ { 0 , 1 } . ii) m H H ∗ ( L ( E )) ∼ = − m H H ∗ ( L ( E )) ( m ∈ Z ). iii) Ther e ex ist m > 0 such that m H H 0 ( L ( E )) and m H H 1 ( L ( E )) ar e b oth nonzer o. Pr o of. W e first reduce to the case wher e the gra ph do es no t have sources. By the pro of of [5 , Theorem 6 .3], there is a finite complete subg raph F of E such that F ha s no sour ces, F contains all the non-trivia l closed paths of E , Sink ( F ) = Sink( E ), and L ( F ) is a full c o rner in L ( E ) with resp ect to the homogeneo us idemp otent P v ∈ F 0 p v . It follows that H H ∗ ( L ( E )) and H H ∗ ( L ( F )) are graded-isomor phic. Therefore we can as sume that E has no s o urces. The first tw o asse rtions a r e already part of Theorem 4.4. F or the last as sertion, let α b e a primitive closed path in E , and let m = | α | . Let σ b e the cyclic per mut ation; then { σ i α : i = 0 , . . . , m − 1 } is a linearly indep endent se t. Hence N ( α ) = P m − 1 i =0 σ i α is a nonzero element o f V σ m = m H H 1 ( L ( E )). Since o n the other hand N v anis hes on the imag e of 1 − σ : V m → V m , it also follo ws that the class of α in m H H 0 ( L ( E )) is nonzer o.  5. Applica tions Theorem 5. 1. L et E 1 , . . . , E n and F 1 , . . . , F m b e finite quivers. Assu me that n 6 = m and that e ach of the E i and the F j has at le ast one non- t rivial close d p ath. The n the algebr as L ( E 1 ) ⊗ · · · ⊗ L ( E n ) and L ( F 1 ) ⊗ · · · ⊗ L ( F m ) ar e not Morita e qu ivalent. Pr o of. Immediate from Lemma 2.3 a nd Corollar y 4.5(iii).  Example 5.2 . It follows fr o m Theore m 5 .1 that L 2 and L 2 ⊗ k L 2 are not Morita equiv alent. There is another way o f proving this, due to J ason Bell and Geor ge Bergman [8]. By Theorem 3.3 of [9], l . gl . dim L 2 ≤ 1. Using a mo dule-theor etic construction, Bell and B e rgman show that l . g l . dim(L 2 ⊗ k L 2 ) ≥ 2 , whic h forces L 2 and L 2 ⊗ k L 2 to b e not Morita equiv alent. Bergman then asked W arren Dic ks whether genera l r e s ults were known a bo ut global dimensions of tenso r pro ducts and w a s pointed to Pr op osition 10(2) o f [12], whic h is an immediate consequenc e of Theorem XI.3.1 of [10 ], a nd says that if k is a field and R and S ar e k -algebr as, then 8 PERE ARA AND GUILLERMO COR TI ˜ NAS l . gl . dim R + w . gl . dim S ≤ l . gl . dim( R ⊗ k S ) . Co nsequently , if l . gl . dim R < ∞ a nd w . gl . dim S > 0 , then l . gl . dim R < l . gl . dim( R ⊗ k S ); in particular, R and R ⊗ k S are then not Mo rita equiv alent. T o see that w . gl . dim L 2 > 0, write x 1 , x 2 , x ∗ 1 , x ∗ 2 for the usual g e nerators of L 2 and use norma l-form ar guments to show that { a ∈ L 2 | ax 1 = a + 1 } = ∅ a nd { b ∈ L 2 | x 1 b = b } = { 0 } . Hence, in L 2 , x 1 − 1 do e s no t hav e a left inv er se and is not a left zero diviso r (or see [4 ]) ; thus, w . gl . dim L 2 > 0. W e denote by L ∞ the unital algebr a pre s ent ed by generato rs x 1 , x ∗ 1 , x 2 , x ∗ 2 , . . . and relations x ∗ i x j = δ i,j 1. Prop ositio n 5. 3. L et E b e any finite quiver having at le ast one n on- trivial close d p ath. Then L ∞ ⊗ L ( E ) and L ( E ) ar e not Morita e quivalent. Similarly L ∞ ⊗ L ∞ and L ∞ ar e not Morita e quivalent. Pr o of. Let C n be the algebra prese n ted by generator s x 1 , x ∗ 1 , . . . , x n , x ∗ n and rela- tions x ∗ i x j = δ i,j 1, for 1 ≤ i, j ≤ n . Then (5.4) L ∞ = lim − → C n , and C n ∼ = L ( E n ), wher e E n is the g raph having tw o vertices v , w and 2 n arr ows e 1 , . . . , e n , f 1 , . . . , f n , with s ( e i ) = r ( e i ) = v = s ( f i ) and r ( f i ) = w for 1 ≤ i ≤ n . (The isomorphism C n → L ( E n ) is obtained by s e nding x i to e i + f i and x ∗ i to e ∗ i + f ∗ i .) It follows from Theo rem 4.4 a nd (5.4) tha t the formulas in Theor e m 4 .4 for m H H n ( L ∞ ), m 6 = 0, hold taking as V i,m the vector space g e ne r ated by all the words in x 1 , x 2 , . . . of length m , and that 0 H H 0 ( L ∞ ) = k and 0 H H n ( L ∞ ) = 0 fo r n ≥ 1. As b efore, Le mma 2 .3 gives the r esult.  Theorem 5 .5. L et E 1 , . . . , E n and F 1 , . . . , F m , . . . b e a finite and an infin it e se- quenc e of quivers. A ssume that the numb er of indic es i su ch that F i has at le ast one non-trivial close d p ath is infinite. Then t he algebr as L ( E 1 ) ⊗ · · · ⊗ L ( E n ) and N ∞ i =1 L ( F i ) ar e not Morita e quivalent. Pr o of. Immediate from Lemma 2.3 a nd Corollar y 4.5(iii).  Example 5.6 . Let L ( ∞ ) = N ∞ i =1 L 2 , and let E b e any quiver having at le a st one non-trivial closed path. Then L ( ∞ ) ⊗ L ( E ) and L ( E ) ar e not Morita equiv alent. It would b e in teresting to know the answer to the following ques tion: Question 5.7 . Is there a unital homomorphism φ : L 2 ⊗ L 2 → L 2 ? Observe that, to build a unital homomor phism φ : L 2 ⊗ L 2 → L 2 , it is enough to exhibit a n on-zer o homo mo rphism ψ : L 2 ⊗ L 2 → L 2 , b eca us e eL 2 e ∼ = L 2 for every non-zero idemp otent e in L 2 . 6. K -theor y T o co nc lude the pap er we note that algebr aic K -theor y cannot dis tinguish b e- t ween L 2 and L 2 ⊗ L 2 or be tw een L ∞ and L ∞ ⊗ L ∞ . F o r this we need a lemma, which might b e of indep endent interest. Recall that a unital ring R is sa id to be r e gular sup er c oher ent in cas e all the p o lynomial r ings R [ t 1 , . . . , t n ] a re regula r coherent in the s ense of [13]. Lemma 6. 1. L et E b e a fi n ite gr aph. Then L ( E ) is r e gular sup er c oher ent . TENSOR PRODUCT S OF LEA VITT P A TH ALGEBRAS 9 Pr o of. Let P ( E ) b e the usua l path algebra of E . It was o bserved in the pro o f of [3, Lemma 7.4 ] that the algebra P ( E )[ t ] is r egular co herent. The same pro of gives that all the po lynomial a lgebras P ( E )[ t 1 , . . . , t n ] a re regular coherent. This shows that P ( E ) is re g ular sup ercoherent. By [3, Pr op osition 4.1], the universal lo calization P ( E ) → L ( E ) = Σ − 1 P ( E ) is flat on the left. It follows that L ( E ) is left reg ular sup e rcoherent (see [5, page 23 ]). Since L ( E ) ⊗ k [ t 1 , . . . , t n ] a dmits an in volution, it follows that L ( E ) is r egular sup ercoher ent.  Prop ositio n 6.2. L et R b e r e gular sup er c oher ent. Then the algebr aic K -the ories of L 2 and of L 2 ⊗ R ar e b oth trivial. Pr o of. Let E b e the quiver with one vertex and tw o ar rows. Then L 2 ∼ = L ( E ), and we hav e L 2 ⊗ R = L R ( E ) . Applying [5, Theor em 7 .6] we obtain that K ∗ ( L R ( E )) = K ∗ ( L ( E )) = 0. The result follows.  W e finally obtain a K -absorbing r esult for Leavitt pa th algebras o f finite gra phs, indeed for any r e gular sup ercoherent algebra. Prop ositio n 6.3. L et R b e a r e gu lar sup er c oher ent algebr a. The n the n atur al inclusion R → R ⊗ L ∞ induc es an isomorphism K i ( R ) → K i ( R ⊗ L ∞ ) for al l i ∈ Z . Pr o of. Adopting the notation used in the pro of o f P rop osition 5.3, we se e that it is eno ugh to show that the na tur al map R → R ⊗ L ( E n ) induces isomorphisms K i ( R ) → K i ( R ⊗ L ( E n )) for all i ∈ Z and all n ≥ 1 . Since R is regula r sup ercoher e nt the K -theory o f R ⊗ L ( E n ) ∼ = L R ( E n ) ca n b e computed by using [5, Theo rem 7.6 ]. By the explicit for m of the quiver E n , w e thus o bta in that K i ( R ⊗ L ( E n )) ∼ = ( K i ( R ) ⊕ K i ( R )) / ( − n, 1 − n ) K i ( R ) . The natural map R → L R ( E n ) factors a s R → R v ⊕ Rw → L R ( E n ) . The first map induces the diag onal homomorphism K i ( R ) → K i ( R ) ⊕ K i ( R ) sending x to ( x, x ). The second map induces the na tural surjection K i ( R ) ⊕ K i ( R ) → ( K i ( R ) ⊕ K i ( R )) / ( − n, 1 − n ) K i ( R ) . Therefore the natural ho momorphism R → L R ( E n ) induces an iso morphism K i ( R ) ∼ − → K i ( L R ( E n )) . This concludes the pro o f.  Corollary 6.4. The natura l maps k → L ∞ → L ∞ ⊗ L ∞ induc e K -the ory isomor- phisms K ∗ ( k ) = K ∗ ( L ∞ ) = K ∗ ( L ∞ ⊗ L ∞ ) . Pr o of. A fir st applica tion o f Pro po sition 6.3 giv es K ∗ ( k ) = K ∗ ( L ∞ ). A second application shows that for E n as in the pro of above, the inclusion L ( E n ) → L ( E n ) ⊗ L ∞ induces a K - theo ry iso morphism; pa s sing to the limit, w e o btain the corolla ry .  A cknow le dgement. 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