A=U for Locally Acyclic Cluster Algebras

This note presents a self-contained proof that acyclic and locally acyclic cluster algebras coincide with their upper cluster algebras.


Introduction
Cluster algebras are commutative domains with distinguished generators (cluster variables) and certain combinatorial identities between them (mutation relations). They were introduced in [FZ02] to study dual canonical basis for Lie groups, and have since been discovered in a remarkable range of applications (see [FZ03,GSV10,Kel11] for surveys).
1.1. Upper cluster algebras. Each cluster algebra A also determines an upper cluster algebra U, with A ⊆ U. Defined as an interection of Laurent rings, the upper cluster algebra is more natural than the cluster algebra from a geometric perspective.
Upper cluster algebras were introduced in the seminal paper [BFZ05], where the authors sought to show that certain rings are cluster algebras. 1 They instead proved that these rings are upper cluster algebras. Naturally, they also asked when A ⊆ U is equality.
To this end, they introduced acyclic cluster algebras: a class of elementary cluster algebras which have since proven to be particularly easy to work with. They also defined a totally coprime condition; a more technical condition that can depend on the coefficients of the cluster algebra. Using both properties, they were able to trap A and U between two simpler algebras (the lower bound and the upper bound ) and 'close the gap' between the bounds to show that A = U.
Theorem 1.1.1. [BFZ05] If A is a totally coprime, acyclic cluster algebra, then A = U.
1.2. Locally acyclic cluster algebras. The current paper's author became interested in studying when A = U for cluster algebras coming from marked surfaces. 2 Fortuitously, the geometric techniques developed for that problem (cluster localization and covers) generalized beyond marked surfaces. In [Mul13], locally acyclic cluster algebras were defined to capture those cluster algebras for which these techniques could show A = U.
Unfortunately, the proof that A = U for locally acyclic cluster algebras which appears in [Mul13, Theorem 4.1] depends an incorrectly stated version of Theorem 1.1.1; specifically, the totally coprime hypothesis was omitted. This has led to some confusion about whether A = U for locally acyclic cluster algebras. 3 Thankfully, the techniques of [Mul13] can be used to show that A = U for locally acyclic cluster algebras without assuming totally coprime or using Theorem 1.1.1; 4 this was alluded to in [Mul13, Remark 6.7] but not shown. Since acyclic cluster algebras are locally acyclic, this shows that the totally coprime assumption may be removed from Theorem 1.1.1.
Theorem 1.2.1. If A is an acyclic or locally acyclic cluster algebra, then A = U.
The purpose of this note is to present an elementary proof of this fact, which assumes nothing except the Laurent phenomenon. This not only resolves the dependency error in [Mul13], but serves as a short and self-contained introduction to the techniques and effectiveness of cluster localization and locally acyclic cluster algebras. The reader might find this a more straight-forward and accessible motivation for the study of locally acyclic cluster algebras than [Mul13], which relies heavily on geometric intuition and techniques.
We also work in the setting of cluster algebras with normalized coefficients, which are a bit more general than the cluster algebras with geometric coefficients studied in [Mul13].

Cluster algebra recollections
We recall the definition of cluster algebras with normalized coefficients. This generalizes cluster algebras with geometric coefficients; see [FZ07] for the appropriate correspondence.
2.1. Seeds and mutation. Let P be a semifield : a torsion-free abelian group (written multiplicatively) equipped with an auxillary addition ⊕ which is commutative, associative and distributive over multiplication. Its integral group ring ZP will be the coefficient ring of the cluster algebra.
Let F be a field which contains ZP. A seed of rank n in F is a triple (x, y, B) consisting of three parts.
• The cluster: x = {x 1 , x 2 , ..., x n } is an n-tuple in F which freely generate F as a field over the fraction field of ZP, • The coefficients: y = {y 1 , y 2 , ..., y n } is an n-tuple in P, and • The exchange matrix: B is an integral, skew-symmetrizable 5 n × n matrix. A seed (x, y, B) may be mutated at an index 1 ≤ k ≤ n, to produce a new seed (µ k (x), µ k (y), µ k (B)) defined as follows.
• µ(x) := {x 1 , x 2 , ..., Mutating at the same index twice in a row returns to the original seed. Permuting the indices {1, 2, ..., n} induces a new seed in the obvious way. Two seeds are mutationequivalent if they are related by a sequence of mutations and permutations.
2.2. Cluster algebras. Given a seed (x, y, B), the union of all the clusters which appear in mutation-equivalent seeds defines a set of cluster variables in the embedding field F . The cluster algebra A(x, y, B) determined by (x, y, B) is the subring of F generated by ZP and the cluster variables. The cluster algebra only depends on the mutationequivalence class of the initial seed, and so the initial seed (x, y, B) will often be omitted from the notation. We A fundamental property of cluster algebras is the Laurent phenonemon, which states that the localization of A at a cluster x is the ring of Laurent polynomials in x over ZP.
Every cluster in A defines such an inclusion. Define the upper cluster algebra U of A to be the intersection of each of these Laurent rings, taken inside the common fraction field F .

Cluster localization and covers
This section reviews the basic algebraic techniques of cluster localization and covers.
3.1. Freezing. 'Freezing' a cluster variable in a seed promotes one or more cluster variables to the coefficient ring. The name is motivated by the case of geometric seeds, in which the generators of P are regarded as 'frozen' cluster variables.
Let (x, y, B) be a seed of rank n over a coefficient ring ZP. Define the freezing of (x, y, B) at x n to be the seed (x † , y † , B † ) of rank n − 1 defined as follows.
• P † := P × Z, the direct product of P with a free abelian group Z, whose generator will be denoted x n . The auxilliary addition is defined by . Identify x i with its image under this isomorphism, and let x † := {x 1 , x 2 , ..., x n−1 }. • y † i := y i x Bin n and y † := {y † 1 , y † 2 , ..., y † n }. • Let B † be the submatrix of B obtained by deleting the nth row and column.
Freezing an arbitrary variable x i ⊂ x is defined by conjugating the above construction by any permutation that sends x i to x n . The following proposition is straightforward.
Proposition 3.1.1. For x i = x j in some seed (x, y, B), freezing at x i commutes with mutating at x j .
Since every mutation of a freezing (x † , y † , B † ) is the freezing of a mutation at some x j = x i , the exchange graph 6 of (x † , y † , B † ) is the subgraph of the exchange graph of (x, y, B) which contains the initial seed and avoids mutating at x i .
Given a subset S ⊂ x, the freezing of (x, y, B) at S is the iterated freezing of each variable in S in any order. As above, freezing multiple variables S commutes with mutation away from S.
3.2. Cluster localization. The effect of freezing a seed on the corresponding cluster algebras and upper cluster algebras is given by the following sequence of containments.
x be a set of variables in a seed (x, y, B), with freezing (x † , y † , B † ). Let A and U be the cluster algebra and upper cluster algebra of (x, y, B) and let A † and U † be the cluster algebra and upper cluster algebra of (x † , y † , B † ). Then there are inclusions in F Since the cluster variables of A contain the cluster variables of A † , this implies the first inclusion.
Since the clusters of A contain the clusters of A † , there is a containment U ⊂ U † . Furthermore, (x i1 x i2 · · · x i k ) −1 ∈ P † ⊂ U † , which implies the last inclusion.
The middle inclusion follows from the Laurent phenomenon.
Each of these inclusions is fairly interesting; but for the purposes at hand, we are most interested in when the first inclusion is equality.
This is the most natural way a cluster structure can descend to a cluster structure on a localization; hence the name. Whenever A(x † , y † , B † ) = U(x † , y † , B † ), each inclusion in Lemma 3.2.1 is an equality, and so A(x † , y † , B † ) is a cluster localization of A(x, y, B).
Cluster localizations are transitive; if A † is a cluster localization of A, and A ‡ is a cluster localization of A † , then A ‡ is a cluster localization of A.
3.3. Covers. For a cluster algebra A, a set {A i } i∈I of cluster localizations of A is called a cover if, for every prime ideal P in A, there is some Covers are transitive; that is, if {A i } i∈I is a cover of A, and {A ij } j∈Ji is a cover of A i , then i∈I {A ij } j∈Ji is a cover of A. However, because there is no 'geometric intersection' for cluster localizations, there is no notion of a common refinement of two covers.
The following is a useful property of covers.
The exchange graph of A is the infinite graph with a vertex for each seed (up to permutation) and an edge for each mutation between seeds.
Hence, no prime A-ideals contain J, and so J = A. In particular, 1 ∈ J, and so a ∈ A.
The equality A = U may be checked locally in a cover.
Proof. Lemma 3.2.1 implies that U ⊂ U i for all i, and so U ⊆ i∈I U i . Then

Acyclic and locally acyclic cluster algebras
This section shows that acyclic cluster algebras admit covers by isolated cluster algebras.

Isolated cluster algebras.
A cluster algebra is isolated if the exchange matrix of any seed (equivalently, every seed) is the zero matrix. 7 Proposition 4.1.1. Let A be an isolated cluster algebra. Then A = U. 8 Proof. Let (x, y, B) be a seed for A. Let x = {x 1 , x 2 , ..., x n } and let x ′ i := P i x −1 i denote the mutation of x i in the initial seed. Since B = 0, each of the P i ∈ ZP, and mutation does not change y or B. Hence, mutating at distinct indices is order-independent. It follows that {x 1 , x 2 , ..., x n , x ′ 1 , x ′ 2 , ..., x ′ n } is the complete set of cluster variables in A, and the clusters are of the form {x i |i ∈ I} {x ′ i | i ∈ I} for any subset I ⊂ {1, 2, ..., n}.
7 For a seed of geometric type defined by a quiver, being isolated is equivalent to having no arrows between mutable vertices; this is the origin of the term 'isolated'. 8 This is a straightforward generalization of the statement and proof of [BFZ05, Lemma 6.2].
Remark 4.1.2. This proves stronger statement: the cluster monomials are a ZP-basis for A and U. This has been proven for all skew-symmetric cluster algebras of finite-type in [CI11], for which isolated cluster algebras are a special case.
Corollary 4.1.3. If A admits a cover by isolated cluster algebras, then A = U.
Proof. This is an immediate corollary of Proposition 4.1.1 and Lemma 3.3.2.
4.2. Acyclic cluster algebras. Isolated cluster algebras are too simple to be interesting by themselves, but the key idea of [Mul13] is that many interesting cluster algebras can be covered by isolated cluster algebras.
A cluster algebra is acyclic if it has an acyclic seed : a seed with an exchange matrix B, such that there is no sequence of indices i 1 , i 2 , ..., i ℓ ∈ {1, 2, ..., n} with B ij+1ij > 0 for all j and i ℓ = i 1 . 9 Acyclic cluster algebras form a class of well-behaved cluster algebras which contain many notable examples, including all finite-type cluster algebras.
Proposition 4.2.1. If A is acyclic, then it admits a cover by isolated cluster algebras.
Proof. The proof will be by induction on the rank n of A. If acyclic A has rank n ≤ 1, then A is isolated, and trivially has a cover by isolated cluster algebras.
Assume that every acyclic cluster algebra of rank < n admits a cover by isolated cluster algebras. Let A be an acyclic cluster algebra of rank n, and let (x, y, B) be an acyclic seed. There must be some index i ∈ {1, 2, ..., n} which is a sink ; that is, B ji ≤ 0 for all j. Otherwise, it would be possible to create arbitrarily long sequences of indices i 1 , i 2 , ..., i ℓ with B ij+1ij > 0; by finiteness, at least one such sequence will have i ℓ = i 1 .
If A is isolated, then it trivially has a cover by isolated cluster algebras. If A is not isolated, then there must be indices i and j such that i is a sink and B ji < 0. The mutation relation at i is then Since yi yi⊕1 is invertible in ZP, this can be rewritten.
Since x j appears in the right-hand product, x i and x j generate the trivial A-ideal. Let A i and A j denote the freezings of A at the indices i and j, respectively. The freezing of an acyclic seed is an acyclic seed, and so A i and A j are acyclic cluster algebras of rank n − 1. By the inductive hypothesis, they admit covers by isolated cluster algebras, and by Corollary 4.1.3, A i = U i and A j = U j . By Lemma 3.2.1, these are cluster localizations Let P be a prime A-ideal. Since x i and x j generate the trivial A-ideal, P cannot contain both elements. If x i ∈ P , then A i P A i . If x j ∈ P , then A j P A j . Hence, A i and A j cover A. The union of the covers of A i and A j by isolated cluster algebras defines a cover A by isolated cluster algebras.
The main result then follows directly from Corollary 4.1.3.
Corollary 4.2.2. If A is acyclic, then A = U.
9 For a seed of geometric type defined by a quiver, being acyclic is equivalent to having no directed cycles; this is the origin of the term 'acyclic'. Not every seed of an acyclic cluster algebra is an acyclic seed.

4.3.
Locally acyclic cluster algebras. Many non-acyclic cluster algebras also admit covers by isolated cluster algebras. As an example, every triangulable marked surface determines a cluster algebra (see [GSV05,FG06,FST08]), and these cluster algebras are locally acyclic as long as there are at least two marked points on the boundary [Mul13, Theorem 10.6]. Since this class of cluster algebras is interesting in its own right, we give it a name. Definition 4.3.1. [Mul13] A cluster algebra is locally acyclic if it admits a cover by isolated cluster algebras.
Since isolated cluster algebras are acyclic, and every acyclic cluster algebra admits a cover by isolated cluster algebras, this condition is equivalent to admitting a cover by acyclic cluster algebras; hence the name. By Corollary 4.1.3, every locally acyclic cluster algebra has A = U. More generally, locally acyclic cluster algebras possess any property of isolated cluster algebras which is 'geometrically local'. For example, they are integrally closed, finitely generated and locally a complete intersection; see [Mul13] for details.