Tensor products of Leavitt path algebras

We compute the Hochschild homology of Leavitt path algebras over a field $k$. As an application, we show that $L_2$ and $L_2\otimes L_2$ have different Hochschild homologies, and so they are not Morita equivalent; in particular they are not isomorphic. Similarly, $L_\infty$ and $L_\infty\otimes L_\infty$ are distinguished by their Hochschild homologies and so they are not Morita equivalent either. By contrast, we show that $K$-theory cannot distinguish these algebras; we have $K_*(L_2)=K_*(L_2\otimes L_2)=0$ and $K_*(L_\infty)=K_*(L_\infty\otimes L_\infty)=K_*(k)$.


Introduction
Elliott's theorem [21] stating that O 2 ⊗ O 2 ∼ = O 2 plays an important role in the proof of the celebrated classification theorem of Kirchberg algebras in the UCT class, due to Kirchberg [14] and Phillips [19]. Recall that a Kirchberg algebra is a purely infinite, simple, nuclear and separable C*-algebra. The Kirchberg-Phillips theorem states that this class of simple C*-algebras is completely classified by its topological K-theory. The analogous question whether the algebras L 2 and L 2 ⊗ L 2 are isomorphic has remained open for some time. Here L 2 is the Leavitt algebra of type (1, 2) over a field k (see [17]), that is, the k-algebra with generators x 1 , x 2 , x * 1 , x * 2 and relations given by x * i x j = δ i,j and 2 i=1 x i x * i = 1. In this paper we obtain a negative answer to this question. Indeed, we analyze a much larger class of algebras, namely the tensor products of Leavitt path algebras of finite quivers, in terms of their Hochschild homology, and we prove that, for 1 ≤ n < m ≤ ∞, the tensor products E = n i=1 L(E i ) and F = m j=1 L(F j ) of Leavitt path algebras of non-acyclic finite quivers E i , F j , are distinguished by their Hochschild homologies (Theorem 5.1). Because Hochschild homology is Morita invariant, we conclude that E and F are not Morita equivalent for n < m. Since L 2 is the Leavitt path algebra of the graph with one vertex and two arrows, we obtain that L 2 ⊗ L 2 and L 2 are not Morita equivalent; in particular they are not isomorphic.
Recall that, by a theorem of Kirchberg [15], a simple, nuclear and separable C * -algebra A is purely infinite if and only if A ⊗ O ∞ ∼ = A. We also show that the analogue of Kirchberg's result is not true for Leavitt algebras. We prove in Proposition 5.3 that if E is a non-acyclic quiver, then L ∞ ⊗ L(E) and L(E) are not Morita equivalent, and also that L ∞ ⊗ L ∞ and L ∞ are not Morita equivalent.
Using the results in [5] we prove that the algebras L 2 and L 2 ⊗ L(F ), for F an arbitrary finite quiver, have trivial K-theory: all algebraic K-theory groups K i , i ∈ Z, vanish on them (this follows from Lemma 6.1 and Proposition 6.2). We also compute K * (L(F )) = K * (L ∞ ⊗ L(F )) and that K * (L ∞ ) = K * (L ∞ ⊗ L ∞ ) = K * (k) is the K-theory of the ground field (see Proposition 6.3 and Corollary 6.4). This implies in particular that, in contrast with the analytic situation, no classification result, in terms solely of K-theory, can be expected for a class of central, simple k-algebras, containing all purely infinite simple unital Leavitt path algebras, and closed under tensor products. It is worth mentioning that an important step towards a K-theoretic classification of purely infinite simple Leavitt path algebras of finite quivers has been achieved in [2].
We refer the reader to [3], [7] and [20] for the basics on Leavitt algebras, Leavitt path algebras and graph C*-algebras, and to [22] for a nice survey on the Kirchberg-Phillips Theorem.
Notations. We fix a field k; all vector spaces, tensor products and algebras are over k. If R and S are unital k-algebras, then by an (R, S)-bimodule we understand a left module over R ⊗ S op . By an R-bimodule we shall mean an (R, R) bimodule, that is, a left module over the enveloping algebra R e = R ⊗ R op . Hochschild homology of k-algebras is always taken over k; if M is an R-bimodule, we write HH n (R, M ) = Tor R e n (R, M ) for the Hochschild homology of R with coefficients in M ; we abbreviate HH n (R) = HH n (R, R).

Hochschild homology
Let k be a field, R a k-algebra and M an R-bimodule. The Hochschild homology HH * (R, M ) of R with coefficients in M was defined in the introduction; it is computed by the Hochschild complex HH(R, M ) which is given in degree n by It is equipped with the Hochschild boundary map b defined by b(a 0 ⊗ a 1 ⊗ · · · ⊗ a n ) = n−1 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 happen to be Z-graded, then HH(R, M ) splits into a direct sum of subcomplexes The homogeneous component of degree m of HH(R, M ) n is the linear subspace of HH(R, M ) n generated by all elementary tensors a 0 ⊗ · · · ⊗ a n with a i homogeneous and i |a i | = m. One of the first basic properties of the Hochschild complex is that it commutes with filtering colimits. Thus we have Let R i be a k-algebra and M i an R i -bimodule (i = 1, 2). The Künneth formula establishes a natural isomorphism ( [23, 9.4.1]) Another fundamental fact about Hochschild homology which we shall need is Morita invariance. Let R and S be Morita equivalent algebras, and let P ∈ R ⊗ S op − mod and Q ∈ S ⊗ R op − mod implement the Morita equivalence. Then ([23, Thm. 9.5.6]) Then no two of R, S ≤m and S are Morita equivalent.
Proof. By the Künneth formula, we have By the same argument, HH p (S ≤m ) is nonzero for p = m, and zero for p > m. Hence if n = m, R and S ≤m do not have the same Hochschild homology and therefore they cannot be Morita equivalent, by (2.2). Similarly, by Lemma 2.1, we have so that HH n (S) is nonzero for all n ≥ 1, and thus it cannot be Morita equivalent to either R or S ≤m .

Hochschild homology of crossed products
Let R be a unital algebra and G a group acting on R by algebra automorphisms. Form the crossed-product algebra S = R ⋊ G, and consider the Hochschild complex HH(S). For each conjugacy class ξ of G, the graded submodule HH ξ (S) ⊂ HH(S) generated in degree n by the elementary tensors a 0 ⋊g 0 ⊗· · ·⊗a n ⋊g n with g 0 · · · g n ∈ ξ is a subcomplex, and we have a direct sum decomposition HH(S) = ξ HH ξ (S). The following theorem of Lorenz describes the complex HH ξ (S) corresponding to the conjugacy class ξ = [g] of an element g ∈ G as hyperhomology over the centralizer subgroup Z g ⊂ G.
Theorem 3.1. [16]. Let R be a unital k-algebra, G a group acting on R by automorphisms, g ∈ G and Z g ⊂ G the centralizer subgoup. Let S = R ⋊ G be the crossed product algebra, and HH g (S) ⊂ HH(S) the subcomplex described above. Consider the R-submodule S g = R ⋊ g ⊂ S. Then there is a quasi-isomorphism In particular we have a spectral sequence p+q (S) Remark 3.2. Lorenz formulates his result in terms of the spectral sequence alone, but his proof shows that there is a quasi-isomorphism as stated above; an explicit formula is given for example in the proof of [11,Lemma 7.2].
Let A be a not necessarily unital k-algebra, writeÃ for its unitalization. Recall from [24] that A is called H-unital if the groups TorÃ n (k, A) vanish for all n ≥ 0. Wodzicki proved in [24] that A is H-unital if and only if for every embedding A ⊳ R of A as a two-sided ideal of a unital ring R, the map is a quasi-isomorphism.
Let R be a unital algebra, and φ : R → pRp a corner isomorphism. As in [6], we consider the skew Laurent polynomial algebra R[t + , t − , φ]; this is the R-algebra generated by elements t + and t − subject to the following relations.
The homogeneous component of degree n is given by Proof. If φ is an automorphism, then S = R ⋊ φ Z, the right hand side of (3.5) computes H(Z, HH(R, S m )), and the proposition becomes the particular case G = Z of Theorem 3.1. In the general case, let A be the colimit of the inductive system Note that φ induces an automorphismφ :

Hochschild homology of the Leavitt path algebra
Let E = (E 0 , E 1 , r, s) be a finite quiver and letÊ = (E 0 , E 1 ⊔ E * 1 , r, s) be the double of E, which is the quiver obtained from E by adding an arrow α * for each arrow α ∈ E 1 , going in the opposite direction. The Leavitt path algebra of E is the algebra L(E) with one generator for each arrow α ∈Ê 1 and one generator p i for each vertex i ∈ E 0 , subject to the following relations The algebra L = L(E) is equipped with a Z-grading. The grading is determined by |α| = 1, |α * | = −1, for α ∈ E 1 . Let L 0,n be the linear span of all the elements of the form γν * , where γ and ν are paths with r(γ) = r(ν) and |γ| = |ν| = n. By [7, proof of Theorem 5.3], we have L 0 = ∞ n=0 L 0,n . For each i in E 0 , and each n ∈ Z + , let us denote by P (n, i) the set of paths γ in E such that |γ| = n and r(γ) = i. The algebra L 0,0 is isomorphic to i∈E0 k. In general the algebra L 0,n is isomorphic to is a block diagonal map induced by the following identification in L(E) 0 : A matrix unit in a factor M |P (n,i)| (k), where i ∈ E 0 \ Sink(E), is a monomial of the form γν * , where γ and ν are paths of length n with r(γ) = r(ν) = i. Since i is not a sink, we can enlarge the paths γ and ν using the edges that i emits, obtaining paths of length n + 1, and the last relation in the definition of L(E) gives Assume E has no sources. For each i ∈ E 0 , choose an arrow α i such that r(α i ) = i. Consider the elements One checks that t − t + = 1. Thus, since |t ± | = ±1, the endomorphism is homogeneous of degree 0 with respect to the Z-grading. In particular it restricts to an endomorphism of L 0 . By [6, Lemma 2.4], we have Consider the matrix N ′ E = [n i,j ] ∈ M e0 Z given by n i,j = #{α ∈ E 1 : s(α) = i, r(α) = j} Let e ′ 0 = |Sink(E)|. We assume that E 0 is ordered so that the first e ′ 0 elements of E 0 correspond to its sinks. Accordingly, the first e ′ 0 rows of the matrix N ′ E are 0. Let N E be the matrix obtained by deleting these e ′ 0 rows. The matrix that enters the computation of the Hochschild homology of the Leavitt path algebra is By a slight abuse of notation, we will write 1 − N t E for this matrix. Note that by rotation of closed paths. We have: Proof. Let L = L(E), P = P (E) ⊂ L the path algebra of E and W m ⊂ P be the subspace generated by all paths of length m. For each fixed n ≥ 1, and m ∈ Z, consider the following L 0,n -bimodule Similarly if m < 0, then Next, by (4.1), we have Here r(i, n) = max{r ≤ n : P (r, i) = ∅} Now note that, because L 0,n is a product of matrix algebras, it is separable, and thus  iii) There exist m > 0 such that m HH 0 (L(E)) and m HH 1 (L(E)) are both nonzero.
Proof. We first reduce to the case where the graph does not have sources. By the proof of [5, Theorem 6.3], there is a finite complete subgraph F of E such that F has no sources, F contains all the non-trivial closed paths of E, Sink(F ) = Sink(E), and L(F ) is a full corner in L(E) with respect to the homogeneous idempotent v∈F 0 p v . It follows that HH * (L(E)) and HH * (L(F )) are graded-isomorphic. Therefore we can assume that E has no sources.
The first two assertions are already part of Theorem 4.4. For the last assertion, let α be a primitive closed path in E, and let m = |α|. Let σ be the cyclic permutation; then {σ i α : i = 0, . . . , m − 1} is a linearly independent set. Hence N (α) = m−1 i=0 σ i α is a nonzero element of V σ m = m HH 1 (L(E)). Since on the other hand N vanishes on the image of 1 − σ : V m → V m , it also follows that the class of α in m HH 0 (L(E)) is nonzero.

Applications
Theorem 5.1. Let E 1 , . . . , E n and F 1 , . . . , F m be finite quivers. Assume that n = m and that each of the E i and the F j has at least one non-trivial closed path. Then the algebras L(E 1 )⊗ · · ·⊗ L(E n ) and L(F 1 )⊗ · · ·⊗ L(F m ) are not Morita equivalent.
Example 5.2. It follows from Theorem 5.1 that L 2 and L 2 ⊗ k L 2 are not Morita equivalent. There is another way of proving this, due to Jason Bell and George Bergman [8]. By Theorem 3.3 of [9], l.gl.dim L 2 ≤ 1. Using a module-theoretic construction, Bell and Bergman show that l.gl.dim(L 2 ⊗ k L 2 ) ≥ 2, which forces L 2 and L 2 ⊗ k L 2 to be not Morita equivalent. Bergman then asked Warren Dicks whether general results were known about global dimensions of tensor products and was pointed to Proposition 10(2) of [12], which is an immediate consequence of Theorem XI.3.1 of [10], and says that if k is a field and R and S are k-algebras, then l.gl.dim R + w.gl.dim S ≤ l.gl.dim(R ⊗ k S). Consequently, if l.gl.dim R < ∞ and 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 Morita equivalent. To see that w.gl.dim L 2 > 0, write x 1 , x 2 , x * 1 , x * 2 for the usual generators of L 2 and use normal-form arguments to show that {a ∈ L 2 | ax 1 = a + 1} = ∅ and {b ∈ L 2 | x 1 b = b} = {0}. Hence, in L 2 , x 1 − 1 does not have a left inverse and is not a left zerodivisor (or see [4]) ; thus, w.gl.dim L 2 > 0.
We denote by L ∞ the unital algebra presented by generators x 1 , x * 1 , x 2 , x * 2 , . . . and relations x * i x j = δ i,j 1. Proof. Let C n be the algebra presented by generators x 1 , x * 1 , . . . , x n , x * n and relations x * i x j = δ i,j 1, for 1 ≤ i, j ≤ n. Then  It would be interesting to know the answer to the following question: Question 5.7. Is there a unital homomorphism φ : Observe that, to build a unital homomorphism φ : L 2 ⊗ L 2 → L 2 , it is enough to exhibit a non-zero homomorphism ψ : L 2 ⊗ L 2 → L 2 , because eL 2 e ∼ = L 2 for every non-zero idempotent e in L 2 .

K-theory
To conclude the paper we note that algebraic K-theory cannot distinguish between L 2 and L 2 ⊗ L 2 or between L ∞ and L ∞ ⊗ L ∞ . For this we need a lemma, which might be of independent interest. Recall that a unital ring R is said to be regular supercoherent in case all the polynomial rings R[t 1 , . . . , t n ] are regular coherent in the sense of [13]. Proof. Let P (E) be the usual path algebra of E. It was observed in the proof of [3,Lemma 7.4] that the algebra P (E)[t] is regular coherent. The same proof gives that all the polynomial algebras P (E)[t 1 , . . . , t n ] are regular coherent. This shows that P (E) is regular supercoherent. By [3,Proposition 4.1], the universal localization P (E) → L(E) = Σ −1 P (E) is flat on the left. It follows that L(E) is left regular supercoherent (see [5, page 23]). Since L(E) ⊗ k[t 1 , . . . , t n ] admits an involution, it follows that L(E) is regular supercoherent. Proposition 6.2. Let R be regular supercoherent. Then the algebraic K-theories of L 2 and of L 2 ⊗ R are both trivial.
Proof. Let E be the quiver with one vertex and two arrows. Then L 2 ∼ = L(E), and we have L 2 ⊗ R = L R (E). Applying [5, Theorem 7.6] we obtain that K * (L R (E)) = K * (L(E)) = 0. The result follows.
We finally obtain a K-absorbing result for Leavitt path algebras of finite graphs, indeed for any regular supercoherent algebra. Proposition 6.3. Let R be a regular supercoherent algebra. Then the natural inclusion R → R ⊗ L ∞ induces an isomorphism K i (R) → K i (R ⊗ L ∞ ) for all i ∈ Z. Proof. Adopting the notation used in the proof of Proposition 5.3, we see that it is enough to show that the natural 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 regular supercoherent the K-theory of R ⊗ L(E n ) ∼ = L R (E n ) can be computed by using [5,Theorem 7.6]. By the explicit form of the quiver E n , we thus obtain 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 as R → Rv ⊕ Rw → L R (E n ) .
The first map induces the diagonal homomorphism K i (R) → K i (R)⊕K i (R) sending x to (x, x). The second map induces the natural surjection K i (R) ⊕ K i (R) → (K i (R) ⊕ K i (R))/(−n, 1 − n)K i (R).
Therefore the natural homomorphism R → L R (E n ) induces an isomorphism This concludes the proof.
Proof. A first application of Proposition 6.3 gives K * (k) = K * (L ∞ ). A second application shows that for E n as in the proof above, the inclusion L(E n ) → L(E n )⊗ L ∞ induces a K-theory isomorphism; passing to the limit, we obtain the corollary.