Locally Acyclic Cluster Algebras

• Locally acyclic cluster algebras are commutative algebras defined via seeds that admit an acyclic cover, ensuring desirable properties like finite generation and normality.
• They unify algebraic, geometric, and homological characteristics across diverse structures such as Grassmannians, marked surfaces, and positroid varieties.
• These algebras coincide with their upper cluster algebras and support computational methods for divisor class group analysis and factoriality testing.

A locally acyclic cluster algebra is a commutative algebra defined via the combinatorial data of a seed, but which admits a finite affine cover by acyclic cluster localizations—cluster algebras associated to seeds with no oriented cycles in the mutable part of the quiver. This property, introduced by Muller, generalizes the acyclic case of cluster algebras and is central in unifying favorable algebraic, geometric, and homological properties across important families, including coordinate rings of Grassmannians, marked surface cluster algebras (with at least two boundary marked points per component), and positroid varieties. Locally acyclic cluster algebras inherit finite generation, normality, and other robust geometric properties from the acyclic case, and provide the most general known framework in which the cluster algebra coincides with its upper cluster algebra.

1. Definitions and Structural Properties

A cluster algebra $A$ over a commutative ring $D$ is constructed from an initial seed $(Q,\mathbf{x})$, consisting of a finite quiver $Q$ (without loops or 2-cycles) on $n$ vertices (partitioned into mutable and frozen) and a bijection $\mathbf{x} = \{x_1,\dots,x_n\}$ from vertices to algebraically independent generators of an ambient field of rational functions. Mutation at a mutable vertex replaces a cluster variable via the exchange relation

$$x_k' = \frac{\prod_{i \rightarrow k} x_i + \prod_{k \rightarrow j} x_j}{x_k},$$

while the quiver $Q$ is mutated according to Fomin–Zelevinsky's rule. The cluster algebra $A(Q,\mathbf{x})$ is generated by all variables arising under sequences of mutations, together with inverses for the frozen variables.

A seed is acyclic if its mutable part has no directed cycles. A cluster algebra is acyclic if it has an acyclic seed; acyclic cluster algebras are finitely generated, normal, Cohen–Macaulay, and are (local) complete intersections. The upper cluster algebra

$$U(Q, \mathbf{x}) = \bigcap_{\text{clusters } \mathbf{y}} D[\mathbf{y}_1^{\pm1},\dots,\mathbf{y}_n^{\pm1}]$$

contains $A(Q,\mathbf{x})$, but equality need not hold in general.

A cluster algebra $A$ is locally acyclic if $\mathrm{Spec}\,A$ admits a finite cover by open subvarieties isomorphic to spectra of acyclic cluster algebras—i.e., by cluster localizations at sets of mutable variables that make the localized quiver acyclic. Alternatively, there must exist finitely many localizations $A_i$, each acyclic, such that their spectra cover $\mathrm{Spec}\,A$ (Muller, 2011, Muller, 2013, Muller et al., 2014, Benito et al., 2014).

Key results:

• Every acyclic cluster algebra is locally acyclic.
• Every locally acyclic cluster algebra coincides with its upper cluster algebra ($A=U$) and is finitely generated, normal, and a local complete intersection (Muller, 2013, Benito et al., 2014).
• Locally acyclic cluster algebras form a robust class, encompassing algebras associated to Grassmannians, marked surfaces with $\geq2$ boundary marked points, double Bruhat cells, and positroid varieties (Muller et al., 2014, Muller, 2011, Benito et al., 2014).

2. Local Acyclicity, Cluster Localizations, and Covering Criteria

Local acyclicity is formalized in terms of cluster localizations: freezings at subsets of mutable variables, which, when invertible, yield cluster algebras whose spectra correspond to open subschemes covering the original variety. An algebra is locally acyclic if these covers are by acyclic charts.

An efficient operational test relies on covering pairs: if a pair of cluster variables $(x_i, x_j)$ is such that there is an arrow $i \rightarrow j$ not on any bi-infinite path in the mutable subquiver, then $(x_i, x_j)$ generate the unit ideal, and the open sets $D(x_i), D(x_j)$ cover $\mathrm{Spec}\,A$; freezing at either variable reduces cycles. The Banff algorithm exploits this to construct acyclic covers inductively (Muller, 2011, Mills, 2018).

Table: Characterizations of Local Acyclicity

Characterization	Description
Existence of acyclic cover	$\mathrm{Spec}\,A$ admits finite cover by cluster localizations that are acyclic
Existence of covering pairs	Every mutable cycle can be broken via inverting variables in a covering pair
Banff algorithm terminates with acyclic seeds	Iterative covering by freezing along covering pairs successfully yields an acyclic chart cover

Locally acyclic cluster algebras are preserved under mutation-finite operations not introducing obstruction classes ($\mathbb{X}_7$ exception; certain once-punctured closed surfaces), and all acyclic, tree, or finite type charts guarantee local acyclicity (Mills, 2018).

3. Algebraic and Geometric Properties

Locally acyclic cluster algebras inherit improved algebraic properties over general cluster algebras:

• Finiteness: $A$ is finitely generated over the base ring $D$ (Muller, 2011, Benito et al., 2014).
• Normality and complete intersection: $A$ is normal and locally a complete intersection, often Gorenstein (Muller et al., 2014, Benito et al., 2014).
• Equality with the upper cluster algebra: $A=U$ (Muller, 2013, Benito et al., 2014).
• Krull domain structure: $A$ is a Krull domain, and divisor class groups can be computed via explicit Laurent intersection techniques (Elsener et al., 2017, Pompili et al., 12 Jan 2026).
• Regularity in full rank: If the extended exchange matrix has full rank, $A$ is regular, and the associated variety is smooth (Muller, 2011, Lam et al., 2016).
• Singularity class: $A$ has canonical singularities in characteristic zero, and is strongly F-regular in characteristic $p>0$ (Benito et al., 2014).

Canonical forms (log-volume, GSV forms) generate the canonical module and induce the curious Lefschetz property on cohomology for full-rank (Louise) cluster varieties (Lam et al., 2016).

4. Connections to Combinatorics, Topology, and Cohomology

The stratification and cohomological structure of locally acyclic cluster varieties are described in terms of the independent set complex of the undirected mutable subquiver:

• Stratification: The cluster variety is stratified by vanishing loci of cluster variables corresponding to independent sets (anticliques). Each stratum is (up to torsor structure) a product of an affine space and a torus, depending on the size of the independent set (Lam et al., 2021).
• Cohomology: Mixed Hodge numbers are computed via a Gysin spectral sequence, whose $E_1$ page is controlled by the topology of independence complexes from quiver combinatorics (Lam et al., 2016, Lam et al., 2021).
• Curious Lefschetz: Full-rank Louise cluster varieties exhibit the curious Lefschetz property: cohomology is mixed Tate and split over $\mathbb{Q}$, with hard Lefschetz-type isomorphisms induced by powers of a GSV form (Lam et al., 2016).
• Point counting: The number of points over finite fields is a polynomial in $q$ and $q-1$, and coincides with (specializations of) the Poincaré–Deligne generating function for cohomology (Lam et al., 2016, Lam et al., 2021).

5. Computations and Factoriality

Locally acyclic cluster algebras are finite Laurent intersection rings (FLIRs), enabling computational approaches to divisor class group and unique factorization:

• Divisor class group: The class group of a locally acyclic cluster algebra is a finitely generated free abelian group whose rank and generators can be computed algorithmically via the configuration of charts and factorization data (Elsener et al., 2017, Pompili et al., 12 Jan 2026).
• Factoriality: $A$ is a UFD precisely when its class group is trivial. Explicit rank formulas connect factoriality to the irreducibility and pairwise coprimality of exchange polynomials in the initial seed (e.g., simply-laced Dynkin types) (Elsener et al., 2017).
• Algorithms: Algorithms for class-group computation, factoriality testing, and enumeration of all factorizations are available for FLIRs, enabling practical investigations for explicit seeds and models (Pompili et al., 12 Jan 2026).

Table: Factorization Data for Locally Acyclic Cluster Algebras

Property	Criterion/Description
Krull domain	Always holds (Elsener et al., 2017, Pompili et al., 12 Jan 2026)
UFD	If and only if class group is trivial; explicit combinatorics from seed/FLIR structure
Computable algorithms	Chart data + multivariate factorization; linear algebra on divisor relations

6. Examples and Applications

Prominent classes of cluster algebras that are locally acyclic include:

• Cluster algebras of Grassmannians: All coordinate rings $\mathcal{O}(\mathrm{Gr}(k,n))$ carry a locally acyclic cluster structure, arising as specializations from Postnikov's plabic graphs and alternating strand diagrams. More generally, open positroid varieties carry locally acyclic (Louise) cluster algebras (Muller et al., 2014).
• Cluster algebras of marked surfaces: For any marked surface with at least two boundary marked points per component, the associated cluster algebra is locally acyclic—even when not acyclic globally (Muller, 2011, Muller et al., 2014).
• Double Bruhat cells and positroid varieties: These cases generalize the above and inherit full locally acyclic structure via the geometry of the corresponding charts (Muller et al., 2014).
• Mutation-finite quivers: All but the explicit exceptions (once-punctured closed surfaces, $\mathbb X_7$) yield locally acyclic algebras (Mills, 2018).

Non-examples include the Markov quiver, closed surfaces with a single marked point, and certain exceptional classes, which demonstrate pathologies such as infinite generation, non-noetherianity, and failure of $A=U$ (Benito et al., 2014, Muller, 2011, Mills, 2018).

7. Relationship to Combinatorial Dynamics and Recent Developments

While initial conjectures posited max green sequences as equivalent to $A=U$ and local acyclicity, counterexamples demonstrated in (Mills, 2018) refute this: there exist locally acyclic cluster algebras and $A = U$ cases for which no maximal green sequence exists (e.g., special rank 4 quiver $Q_{ce}$). The correct unifying invariant is the existence of a green-to-red sequence, which matches local acyclicity and $A=U$ for an open set $$O \subset \mathrm{Spec}\,\mathbb Z[y_1^{\pm1},\ldots,y_n^{\pm1}]$$. This reframing connects cluster dynamics, structure theory, and algebraic geometry.

Current computational advances (Pompili et al., 12 Jan 2026) leverage the FLIR framework for locally acyclic cluster algebras, giving divisorial and factorization algorithms and broadening explicit investigation.

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References: (Muller, 2011, Muller, 2013, Muller et al., 2014, Benito et al., 2014, Lam et al., 2016, Elsener et al., 2017, Mills, 2018, Lam et al., 2021, Pompili et al., 12 Jan 2026)