A categorification of acyclic principal coefficient cluster algebras
AA CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENTCLUSTER ALGEBRAS
MATTHEW PRESSLAND
Abstract.
In earlier work, the author introduced a method for constructing a Frobenius cat-egorification of a cluster algebra with frozen variables, requiring as input a suitable candidatefor the endomorphism algebra of a cluster-tilting object in such a category. In this paper, weconstruct such candidates in the case of cluster algebras with ‘polarised’ principal coefficients,and obtain Frobenius categorifications in the acyclic case. Since cluster algebras with principalcoefficients are obtained from those with polarised principal coefficients by setting half of thefrozen variables to 1, our categories also indirectly model principal coefficient cluster algebras, forwhich no Frobenius categorification exists in general. Moreover, partial stabilisation provides anextriangulated category, in the sense of Nakaoka and Palu, that may more directly model theprincipal coefficient cluster algebra. Many of the intermediate results remain valid without theacyclicity assumption, as we will indicate. Along the way, we establish a Frobenius version ofKeller’s result that the Ginzburg dg-algebra of a quiver with potential is bimodule 3-Calabi–Yau. Introduction
Cluster algebras, introduced by Fomin–Zelevinsky [17], are combinatorially defined algebraswith applications to many areas of mathematics, and have been the subject of intense study; seeKeller [29] for a survey of connections between cluster algebras and the representation theory ofassociative algebras, and the references therein for applications to other fields.A key obstruction to studying cluster algebras is their recursive definition—one is given someinitial data (a seed), and constructs the cluster algebra inductively via sequences of mutations ofthis seed. Typically one may obtain an infinite number of seeds in this way, and so the output isnot easily controllable. To gain a better understanding of the cluster algebra, it has been fruitfulto construct categorical models, which allow the combinatorics to be understood in a more globalway. In such a model, given by an exact or triangulated category C , the clusters are replaced bycluster-tilting objects T , defined by the property thatadd T = { X ∈ C : Ext C ( T, X ) = 0 } = { X ∈ C : Ext C ( X, T ) = 0 } . When C is 2-Calabi–Yau in a suitable sense, these objects may be mutated via a process analogousto that of the mutation of seeds [27]. Categorification allows one to give clean, conceptual proofs ofmany key statements for any cluster algebra admitting such a categorical model, such as ‘clusterdetermines seed’ [10], linear independence of cluster monomials [12], sign coherence of c-vectors[40], and so on.We consider skew-symmetric cluster algebras of geometric type, so that a seed is given by thedata of a collection of cluster variables forming the vertices of a quiver. Mutation of seeds replacesa single cluster variable by a different one, and alters the quiver by Fomin–Zelevinsky mutation(see for example [29, §3.2]). A subset of the variables may be frozen, indicating that they may notbe mutated, and thus occur in every seed. We use the terminology ‘ice quiver’ to refer to a quiverwith a specified ‘frozen’ subquiver; from the point of view of cluster algebras, the only relevantadditional data is the set of frozen vertices, since arrows between these play no role in the clusterstructure, but we will want to consider such arrows later. Date : October 1, 2018.2010
Mathematics Subject Classification.
Key words and phrases. cluster algebra, cluster-tilting object, principal coefficients, Calabi–Yau algebra, Frobeniuscategory, Jacobian algebra, quiver with potential. a r X i v : . [ m a t h . R T ] O c t MATTHEW PRESSLAND
For cluster algebras without frozen variables, categorical models, known as cluster categories,have been constructed in great generality, beginning with Buan–Marsh–Reineke–Reiten–Todorov’sconstruction [9] for the case of acyclic quivers, later generalised by Amiot [1] to allow for theexistence of cycles. Unfortunately, most cluster algebras appearing naturally in other contexts,such as the cluster structures on the coordinate rings of partial flag varieties [21] and their doubleBruhat cells [4], do have frozen variables, which we would like to capture in a categorical model.This can be achieved by replacing cluster categories, which are 2-Calabi–Yau triangulatedcategories, by stably 2-Calabi–Yau Frobenius categories. A Frobenius category is, by definition, anexact category with enough projective objects and enough injective objects, such that these twoclasses of objects coincide. It is the indecomposable projective-injective objects, which necessarilyappear as summands of any cluster-tilting object, that will model the frozen variables. The factthat setting all frozen variables to 1 in a cluster algebra produces a new cluster algebra withoutfrozen variables corresponds to the fact that the stable category of a Frobenius category, givenby taking the quotient by the ideal of morphisms factoring through a projective object (‘settingthe projective objects to 0’) produces a triangulated category [26, §I.2], which we require to be2-Calabi–Yau.Such Frobenius models for cluster algebras with frozen variables have been constructed spo-radically for families of examples, often geometric in nature. For example, Geiß–Leclerc–Schröer[21] construct cluster structures on coordinate rings of open cells in partial flag varieties using aFrobenius model. They then combinatorially lift such structures to the homogeneous coordinatering of the whole flag variety. Frobenius categorifications of the resulting structures have beenobtained in the case of Grassmannians of planes by Demonet–Luo [15], all Grassmannians byJensen–King–Su [28], and in general by Demonet–Iyama [14]. However, these constructions areall somewhat specialised to the case at hand, and depend to some extent on using the geometryof the partial flag varieties to gain some insight into the global structure of the cluster algebra,before constructing the categorification. Nájera Chávez [33] has constructed Frobenius models forfinite-type cluster algebras with universal coefficients, in this case using the finiteness to control theglobal structure.By contrast, the constructions of cluster categories for cluster algebras without frozen variables byBuan–Marsh–Reineke–Reiten–Todorov and Amiot did not depend on such global information, butinstead start from the data of a single seed, as in the original definition of a cluster algebra, possiblyenhanced by some additional (but still local) data, such as a potential on the quiver in Amiot’s case.In earlier work [37], the author introduced a similar framework for constructing Frobenius models ofa cluster algebra, starting again from the data of an initial seed. Starting from the quiver Q of thisseed (some of the vertices of which are frozen), one attempts to find a Noetherian algebra A suchthat the Gabriel quiver of A agrees with Q up to the addition of arrows between frozen vertices, thequotient of A by paths passing through these vertices is finite-dimensional, and, most importantly, A is bimodule internally 3-Calabi–Yau with respect to these vertices [37, Def. 2.4]. Such an algebra A then determines a candidate Frobenius model of the cluster algebra [37, Thm. 4.1, Thm. 4.10].Passing from Q to A requires (mostly necessarily [37, Rem. 4.11]) the choice of a great dealof extra data, satisfying restrictive conditions. Thus it was not clear from the results of [37] howrealistic it would be to apply this methodology in practice. In this work, we demonstrate that thisapproach is in fact practical, by using it to construct a Frobenius model for the cluster algebra with‘polarised principal coefficients’ associated to any acyclic quiver. Fomin–Zelevinsky have shownthat cluster algebras with principal coefficients play a ‘universal’ role in the theory, since theircombinatorics can be used to control that of any other cluster algebra with the same principalpart, meaning the cluster algebra obtained upon specialising all frozen variables to 1 [18]. Clusteralgebras with polarised principal coefficients, which we define in Section 2, differ from those withprincipal coefficients by adding more frozen variables, and so also have this universality property.This coefficient system, for Dynkin-type cluster algebras, also appeared in recent work of Borgesand Pierin [5], who define a modified cluster character on the ordinary, triangulated cluster category,with values in the corresponding cluster algebra with polarised principal coefficients. Our maintheorem is the following. CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS 3
Theorem 1.
Let Q be an acyclic quiver, and let A Q be the corresponding cluster algebra withoutfrozen variables. Then there exists a Frobenius cluster category E Q (constructed explicitly) such that(i) the stable category E Q is equivalent to the cluster category C Q , and(ii) there is a bijection between cluster-tilting objects of E Q and seeds of the polarised principalcoefficient cluster algebra with principal part A Q , commuting with mutation, such that theice quiver of the endomorphism algebra of each cluster-tilting object agrees, up to arrowsbetween frozen vertices, with the ice quiver of the corresponding seed. We interpret the quivers of endomorphism algebras in (ii), and indeed throughout the paper, asice quivers by declaring the frozen vertices to be those corresponding to projective indecomposablesummands. The definition of a Frobenius cluster category is stated below (Definition 2.8); suchcategories always admit a weak cluster structure in the sense of Buan–Iyama–Reiten–Scott [7, §II.1],by [7, Thm. II.1.1]. By Theorem 1(ii), this weak cluster structure on E Q is even a cluster structure,also defined in [7, §II.1].As we will indicate, while the acyclicity assumption is needed for all of the ingredients in theproof of Theorem 1 to be available simultaneously, many of these intermediate results hold in muchwider generality. In particular, we will see that the conclusions of the theorem remain valid when Q is a 3-cycle (replacing C Q in (i) by Amiot’s cluster category C Q,W , where W is the potential on Q given by the 3-cycle).It is necessary to pass from principal coefficients to some larger collection of frozen variables inorder for a result like Theorem 1 to be possible, since a Frobenius categorification of the principalcoefficient cluster algebra with principal part A Q cannot exist; see Proposition 2.3. However, it isstill possible to use the category E Q from Theorem 1 to study this cluster algebra, essentially by‘ignoring’ the extra projective-injective objects in the category, as we will illustrate in Section 9.On the other hand, one can also construct from E Q an extriangulated category, in the sense ofNakaoka and Palu [35], which is combinatorially closer to the principal coefficient cluster algebra, forexample by having indecomposable projective-injective objects in bijection with the frozen variablesof this cluster algebra. Many properties of the resulting category are then easy consequences of itsconstruction, via [35, Prop. 3.30], and Theorem 1. Corollary 2 (of Theorem 1) . Let Q be an acyclic quiver, and let A be the corresponding clus-ter algebra without frozen variables. Then there exists a Frobenius extriangulated category E + Q (constructed explicitly) such that(i) the stable category E + Q is equivalent to the cluster category C Q , and(ii) there is a bijection between cluster-tilting objects of E Q and seeds of the principal coefficientcluster algebra with principal part A Q , commuting with mutation, such that the ice quiver ofthe endomorphism algebra of each cluster-tilting object agrees, up to arrows between frozenvertices, with the ice quiver of the corresponding seed. We discuss this construction briefly in Section 7. Since cluster theory is better developed forFrobenius categories than for extriangulated categories, we focus more on E Q in the sequel, but itseems likely that much of the machinery used to study this category (for example, to obtain theresults of Section 9) could be extended in the future to cover suitably well-behaved extriangulatedcategories such as E + Q .On the way to proving Theorem 1, we will obtain other results that are of wider interest, suchas the following; here a positive grading of a quiver with potential ( Q, W ) is a Z -grading of theJacobian algebra J ( Q, W ) such that all arrows of Q have positive degree. (For the definitions ofordinary and frozen Jacobian algebras, see Definition 2.1.) Theorem 3 (Corollary 4.12) . Let ( Q, W ) be a quiver with potential admitting a positive grading,and let A = J ( Q, W ) . Then there is a frozen Jacobian algebra A = A Q,W (constructed explicitlyin Section 3), internally bimodule -Calabi–Yau with respect to its frozen idempotent e , such that A = A/ h e i . We see this statement as analogous to a result of Keller [30, Thm. 6.3, Thm. A.12], implyingthat any finite-dimensional Jacobian algebra may be realised as the 0-th homology of a bimodule
MATTHEW PRESSLAND A = A Q,W needed as input for this constructionis defined in Section 3, from the data of a quiver with potential (
Q, W ). In Section 4 we explainfurther results of [37] which allow one to check the bimodule internally 3-Calabi–Yau property for afrozen Jacobian algebra, and apply these to A under the assumption that ( Q, W ) admits a positivegrading. This establishes Theorem 3. In Section 5, we show that A is finite-dimensional, and henceNoetherian, when Q is acyclic. This allows us to conclude most of the statements of Theorem 1,and is the main reason for the acyclicity assumption in this theorem. The results on mutations arefound in Section 6, where we use them to complete the proof of Theorem 1. The construction ofthe extriangulated categorification of Corollary 2 then follows in Section 7.The Frobenius cluster category in Theorem 1 is a category of Gorenstein projective modulesover an Iwanaga–Gorenstein algebra B Q , which we describe explicitly via a quiver with relationsin Section 8. In Section 9 we show that this Frobenius cluster category may be graded, in thesense of [25], in a way that captures the grading of the principal coefficient cluster algebra byFomin–Zelevinsky [18]. This allows us to recover an identity of Fomin–Zelevinsky [18, Eq. 6.14],relating g-vectors and c-vectors. We close in Section 10 with some examples, in particular observingthat Theorem 1 remains true when Q is a 3-cycle.Throughout, algebras are assumed to be associative and unital. All modules are left modules,the composition of maps f : X → Y and g : Y → Z is denoted by gf , and we use the analogousconvention for compositions of arrows in quivers. The Jacobsen radical of a module X is denotedby m ( X ). If p is a path in a quiver, we denote its head by hp and its tail by tp .2. Polarised principal coefficients
Let A be a cluster algebra of geometric type without frozen variables, and let s be a seed of A , with quiver Q and cluster variables ( x , . . . , x n ). By definition, the quiver Q has no loops or2-cycles. The principal coefficient cluster algebra A • Q corresponding to this data is defined by thefollowing initial seed. The mutable cluster variables are again ( x , . . . , x n ), and the frozen variablesare ( y , . . . , y n ), where the indexing reveals a preferred bijection between the mutable and frozenvariables. The ice quiver Q • of this seed contains Q as a full subquiver, with mutable vertices, andfor each vertex i ∈ Q (corresponding to the variable x i ), Q • has a frozen vertex i + (correspondingto y i ) and an arrow i → i + . While A is isomorphic to the cluster algebra determined by any quivermutation equivalent to Q , this is not true of A • Q .Setting all frozen variables of A • Q to 1 to obtain a cluster algebra without frozen variables (aprocess known as taking the principal part ), recovers A , giving a bijection between the seeds of A • Q and those of A [12]; we write s • for the seed of A • Q corresponding to a seed s of A . Principalcoefficients are important, since knowledge of the cluster algebra A • Q can be used to study anycluster algebra A with principal part A , via the theory of g-vectors and F-polynomials [18].By choosing some extra data on Q , Amiot [1] is able to construct a categorical model of A . Theconstruction uses Jacobian algebras, so we recall some relevant definitions, which we will also needlater in the paper. Definition 2.1. An ice quiver ( Q, F ) consists of a finite quiver Q without loops and a (notnecessarily full) subquiver F of Q . Denote by K Q the completion of the path algebra of Q over afield K with respect to the arrow ideal. A potential on Q is W ∈ K Q/ [ K Q, K Q ], an element of thequotient of K Q by the closed ideal generated by commutators. One can think of W as a (possibly CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS 5 infinite) linear combination of cycles of Q , such that no two cycles with non-zero coefficient arecyclically equivalent, meaning they differ only by choosing a different starting vertex.A vertex or arrow of Q is called frozen if it is a vertex or arrow of F , and mutable or unfrozen otherwise. For brevity, we write Q m0 = Q \ F and Q m1 = Q \ F for the sets of mutable verticesand unfrozen arrows respectively. For α ∈ Q and α k · · · α a cycle in Q , write ∂ α α k · · · α = X α i = α α i − · · · α α k · · · α i +1 and extend linearly and continuously to define ∂ α W . The ideal h ∂ α W : α ∈ Q m1 i of K Q is calledthe Jacobian ideal , and we may take its closure h ∂ α W : α ∈ Q m1 i since K Q is a topological algebra.We define the frozen Jacobian algebra associated to ( Q, F, W ) by J ( Q, F, W ) = K Q/ h ∂ α W : α ∈ Q m1 i . Write A = J ( Q, F, W ). The above presentation of A suggests a preferred idempotent e = X v ∈ F e v , which we call the frozen idempotent . We will call B = eAe the boundary algebra of A . If F = ∅ ,then we refer to the pair ( Q, W ) as a quiver with potential, and call J ( Q, W ) = J ( Q, ∅ , W ) simplythe Jacobian algebra of ( Q, W ). Remark 2.2.
While K Q usually denotes the ordinary (uncompleted) path algebra of Q , we useit here as clean notation for the completed version since this is the version we will always take.Moreover, in much of the paper, we are interested in algebras K Q/ h R i which turn out to beisomorphic to the quotient of the ordinary path algebra of Q by the ideal h R i , the latter beingfinite-dimensional, and so the distinction was irrelevant. We take complete path algebras in thegeneral theory since we want the categories we construct to be Krull–Schmidt (cf. [28, Rem. 3.3]).Returning to the cluster algebra A Q , choose a potential W on Q such that J ( Q, W ) is finite-dimensional. Then by work of Amiot [1], there is a 2-Calabi–Yau triangulated category C Q,W categorifying A Q . If W is non-degenerate [16, Def. 7.2], then the seeds of A correspond bijectivelyto additive equivalence classes of cluster-tilting objects of C Q,W related by a finite sequence ofmutations from an initial cluster-tilting object T with End C Q,W ( T ) op = J ( Q, W ); we call suchcluster-tilting objects reachable , and denote the seed corresponding to such an object T by s T .(Non-degeneracy is needed here to see, using results of Buan–Iyama–Reiten–Smith [8], that mutationof cluster-tilting objects in the mutation class of T corresponds to Fomin–Zelevinsky mutation atthe level of quivers of endomorphism algebras.)A priori, to categorify the principal coefficient cluster algebra A • Q , we would like to to find aFrobenius category E such that(i) the stable category of E is triangle equivalent to C Q,W (so cluster-tilting objects of E maybe identified with those of C Q,W ), and(ii) for any reachable cluster-tilting object T , the quiver of End E ( T ) op is, up to arrows betweenfrozen vertices, the quiver of the seed s • T of A • Q .Unfortunately, this is not possible. Proposition 2.3.
There does not exist a Frobenius category E satisfying conditions (i)–(ii) above.Proof. Assume E were such a Frobenius category, and identify cluster-tilting objects of E with thoseof C Q,W by (i). We claim that the Gabriel quiver of End E ( T ) op , where T is the initial cluster-tiltingobject, has no paths from a frozen vertex, corresponding to a projective-injective indecomposablesummand of T , to a mutable vertex, corresponding to a non-projective indecomposable summand.Indeed, this is true of Q • , and by (ii) the relevant Gabriel quiver differs only by the possibleaddition of new arrows between frozen vertices, a process which does not affect the property underconsideration. It follows that there are no non-zero maps in E from a non-projective summandof T to any projective-injective object, all of which lie in add T , and so this summand has noinjective envelope, contradicting the requirement that E is a Frobenius category. (cid:3) MATTHEW PRESSLAND
To avoid this problem, Fu–Keller [20, §6.7] have modelled A • Q via an additive category, in the casethat Q is acyclic. Taking the extended quiver Q • , which is also acyclic, they consider a distinguishedsubcategory U Q ⊆ C Q • determined by the frozen vertices; precisely, it is the full subcategory onobjects Ext-orthogonal to the summands of the initial cluster-tilting object of C Q • corresponding tothese vertices. We note that the objects corresponding to frozen vertices of Q • are not characterisedintrinsically within C Q • , although it seems possible based on computations in small examples that U Q has a natural extriangulated structure (making it equivalent to the extriangulated category ofCorollary 2) in which these objects are the indecomposable projective-injectives; we do not pursuethis comparison in this paper, however.Here, we take a different approach—we simply add more frozen variables to A • Q and thentry to categorify the result using a Frobenius category E Q,W . While E Q,W will have too manyindecomposable objects to be a ‘strict’ model for A • Q , the objects corresponding to frozen variables(of our extended cluster algebra) will be intrinsically characterised by the property of beingindecomposable projective-injective. This approach is also more efficient, in the sense that E Q,W willonly have n = | Q | too many indecomposable objects, whereas the cluster category C Q • consideredby Fu–Keller can be of wild type even when C Q , and hence U Q , is of finite type.Since A • Q can be obtained from the extended cluster algebra by specialising some frozen variablesto 1, our category will still encode all of the combinatorial information about A • Q . By composingthe usual cluster character on E [20] with the projection to A • Q , one can even write down a functionwhich is morally a cluster character E → A • Q , but with the unusual feature that it ‘has kernel’ i.e.some non-zero objects of E have character 1. In particular, it will factor over the extriangulatedcategory of Corollary 2.Our chosen extension of A • Q is the ‘polarised principal coefficient’ cluster algebra f A Q , whichwe now define. Starting from our seed s of A , with quiver Q and cluster variables ( x , . . . , x n ),we construct an initial seed e s of f A Q as follows. The mutable variables are ( x , . . . , x n ), and thefrozen variables are ( y +1 , . . . , y + n , y − , . . . , y − n ). The ice quiver e Q contains Q as a full subquiver, withmutable vertices, and has two frozen vertices i + (corresponding to y + i ) and i − (corresponding to y − i ) for each mutable vertex i ∈ Q , with arrows i → i + and i − → i for each i . In Section 3 we willalso describe arrows between the frozen vertices of e Q , but since these play no role in the definitionof the cluster algebra f A Q we ignore them for now. This enlarged quiver (without the frozen arrows)is called the biframed quiver of Q in [5].We adopt the word ‘polarised’, referring to the partitioning of the frozen variables into two‘flavours’, to differentiate this coefficient system from the ‘double principal coefficients’ studied byRupel–Stella–Williams [39]. Since one encounters the same issues categorifying cluster algebraswith double principal coefficients as in the case of ordinary principal coefficients, namely that nocategorification may have enough injective objects, our preference here is for the polarised version. Remark 2.4.
Like the double principal coefficient cluster algebras of [39], the cluster algebra f A Q associated to a Dynkin quiver Q may (after inverting frozen variables) be realised as the coordinatering of a double Bruhat cell, as we now briefly explain.Let G be a simple connected complex Lie group of type ∆. After choosing a Borel subgroupand maximal torus, one may associate the double Bruhat cell G u,v to any pair u, v ∈ W , as in [19].Analogous to the classical Bruhat decomposition, G is then expressible as the disjoint union G = G ( u,v ) ∈ W G u,v . Berenstein–Fomin–Zelevinsky have shown that each coordinate ring C [ G u,v ] has the structure of anupper cluster algebra [4, Thm. 2.10] with invertible coefficients. In fact, by work of Goodearl andYakimov [23], the word ‘upper’ can be dropped.Assume Q is an orientation of a simply-laced Dynkin diagram ∆ with vertex set { , . . . , n } .Quivers with underlying graph ∆ are in bijection with Coxeter elements of the Weyl group W of ∆as follows. Let s , . . . , s n be the simple reflections generating W , and let i , . . . , i n be an orderingof Q = { , . . . , n } such that i j < i k whenever there is an arrow from j to k . Such an ordering is CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS 7 not unique, but any two determine the same Coxeter element c = s i · · · s i n of the Weyl group, andevery Coxeter element arises in this way from a unique orientation of Q .In the case of the double Bruhat cell G c,c associated to a Coxeter element c , the algebra C [ G c,c ]is isomorphic to the cluster algebra with (inverted) polarised principal coefficients associated to theorientation Q of ∆ determined by c , cf. [39, Thm. 2.13]. (In this case we do not need the results of[23] to know that the cluster algebra coincides with the upper cluster algebra, since this followsinstead from acyclicity of Q .) Our general results can thus be exploited to construct a Frobeniuscategorification of this cluster algebra.As discussed in the introduction, we will construct a categorification of f A Q using methodologyintroduced by the author in [37]. We now recall the key definitions and results needed to explainthis construction. Write D A for the derived category of an algebra A , and A ε = A ⊗ K A op for its enveloping algebra, modules of which are precisely A -bimodules. We denote by per A the perfect derived category of A , i.e. the thick subcategory of D A generated by A , and writeΩ A = R Hom A ε ( A, A ε ). Definition 2.5 ([37, Def. 2.4]) . Given an algebra A and idempotent e ∈ A , we say A is bimoduleinternally -Calabi–Yau with respect to e if(1) p . dim A ε A ≤ A ∈ per A ε , and(3) there exists a triangle A ψ −→ Ω A [3] −→ C −→ A [1] in D A ε , such that R Hom A ( C, M ) = 0 = R Hom A op ( C, N )whenever M ∈ D A is a complex whose total cohomology is a finite-dimensional A/ h e i -module, and analogously for N ∈ D A op . Remark 2.6.
Assume A is bimodule internally 3-Calabi–Yau with respect to e , and write A = A/ h e i .Then gl . dim A ≤
3, and there is a functorial dualityD Ext iA ( M, N ) = Ext − iA ( N, M )for finite-dimensional M ∈ mod A/ h e i and any N ∈ Mod A [37, Cor. 2.9]. Moreover, the same istrue of A op [37, Rem. 2.6]. In the language of [37, Def. 2.1], we say that A and A op are internally3-Calabi–Yau with respect to e (without the word ‘bimodule’).To construct our Frobenius categories, we will use the following theorem. Theorem 2.7 ([37, Thm. 4.1, Thm. 4.10]) . Let A be an algebra, and e ∈ A an idempotent. If A is Noetherian, A = A/ h e i is finite-dimensional, and A is bimodule internally -Calabi–Yau withrespect to e , then(i) B = eAe is Iwanaga–Gorenstein of injective dimension at most , so GP( B ) = { X ∈ mod B : Ext iB ( X, B ) = 0 , i > } is a Frobenius category,(ii) eA ∈ GP( B ) is cluster-tilting,(iii) there are natural isomorphisms End B ( eA ) op ∼ → A and End B ( eA ) op ∼ → A , and(iv) the stable category GP( B ) is -Calabi–Yau. Ideally, we would like GP( B ) as constructed above to be a Frobenius cluster category, in thesense of the following definition. Definition 2.8 (cf. [37, Def. 3.3]) . An exact category E is called a Frobenius cluster category ifit is Krull–Schmidt, stably 2-Calabi–Yau, and gl . dim End E ( T ) op ≤ T ∈ E , of which there is at least one.The category GP( B ) constructed in Theorem 2.7 is Frobenius by (i), idempotent completefor arbitrary B , and stably 2-Calabi–Yau by (iv). Since End B ( eA ) op ∼ = A is bimodule internally3-Calabi–Yau, it has global dimension at most 3, but we do not know a priori that this is true forendomorphism algebras of other cluster-tilting objects. However, this does hold whenever such an MATTHEW PRESSLAND algebra is Noetherian, by [37, Prop. 3.7]. In particular, if A is finite-dimensional, then so is B , andthen GP( B ) is a Frobenius cluster category.Returning to the problem of categorifying the cluster algebra f A Q , our aim now is to constructan algebra A satisfying the conditions of Theorem 2.7, such that the Gabriel quiver of A agreeswith the quiver e Q up to arrows between frozen vertices.3. An ice quiver with potential
Consider again our initial seed s for A , with quiver Q , and choose a potential W on Q . Inthis section, we will construct from ( Q, W ) an ice quiver with potential ( e Q, e F , f W ), and thus afrozen Jacobian algebra J ( e Q, e F , f W ). It is this algebra that we intend to use as the input for theconstruction of a Frobenius category by Theorem 2.7; at that point we will require that J ( Q, W )is finite-dimensional, but we do not assume this yet.
Definition 3.1.
Let (
Q, W ) be a quiver with potential. We define e Q to be the quiver with vertexset given by e Q = Q t Q +0 t Q − where Q +0 = { i + : i ∈ Q } is a set of formal symbols in bijection with Q , and similarly for Q − = { i − : i ∈ Q } . The set of arrows is given by e Q = Q t { α i : i ∈ Q } t { β i : i ∈ Q } t { δ i : i ∈ Q } t { δ a : a ∈ Q } . The head and tail functions h and t on e Q are extended from those on Q by defining hα i = i + , tα i = i, hβ i = i, tβ i = i − ,hδ i = i − , tδ i = i + , hδ a = ( ta ) − , tδ a = ( ha ) + . The frozen subquiver e F is defined by e F = Q +0 t Q − , e F = { δ i : i ∈ Q } t { δ a : a ∈ Q } . Note that the head and tail of any arrow in e F lies in e F , so these subsets describe a valid subquiverof e Q , that is in fact full. The quiver e F is also bipartite, meaning that every vertex is either a sourceor a sink, and so it has no paths of length greater than 1; precisely, vertices of the form i + aresources, and those of the form i − are sinks. When viewed in e Q , each vertex i + has unique incomingarrow α i and each i − has unique outgoing arrow β i .Finally, we define a potential f W on e Q by f W = W + X i ∈ Q β i δ i α i − X a ∈ Q aβ ta δ a α ha , and let A Q,W = J ( e Q, e F , f W )be the frozen Jacobian algebra determined by ( e Q, e F , f W ). We denote the boundary algebra of A Q,W by B Q,W = eA Q,W e , where e = P i ∈ Q ( e + i + e − i ) is the frozen idempotent of A Q,W . Note.
To aid legibility, if the vertices i + or i − appear as subscripts, we will usually move the signinto a superscript, as in the above expression for the frozen idempotent. When W = 0, we willtypically drop it from the notation; for example, we write A Q = A Q, . The reader is warned that e e Q .When Q is the quiver of the initial seed s of A , the quiver e Q ‘is’ by construction the ice quiverof the same name forming part of the data of our initial seed e s of f A Q , but with the additionaldata of arrows between frozen vertices.Since f W has a straightforward description in terms of W , so do the defining relations of A Q,W ;these form the set R consisting of(3.1) ∂ a f W = ∂ a W − β ta δ a α ha , ∂ α i f W = β i δ i − X γ ∈ Q hγ = i γβ tγ δ γ , ∂ β i f W = δ i α i − X γ ∈ Q tγ = i δ γ α hγ γ, CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS 9 for a ∈ Q and i ∈ Q . Having such an explicit generating set of relations will prove to be extremelyuseful later in the paper.To be able to apply Theorem 2.7, we wish to show that A Q,W is bimodule internally 3-Calabi–Yauwith respect to its frozen idempotent e , in the sense of Definition 2.5. We will do this, under a mildassumption on ( Q, W ), in Section 4, but first give some examples.
Example 3.2.
The quiver with potential ( Q, Q an A quiver, provides the most basicexample revealing all of the combinatorial features of the construction. In this case, we have e Q = 1 21 + − − + aα β α β δ δ δ a with e F indicated by the boxed vertices and dashed arrows. The potential on this ice quiver is f W = β δ α + β δ α − aβ δ a α . One can check that the frozen Jacobian algebra A Q attached to this data is isomorphic to theendomorphism algebra of a cluster-tilting object in the Frobenius cluster category of those modulesfor the preprojective algebra of type A with socle supported at a fixed bivalent vertex [21]. For anexplanation of why the categories arising in [21] are Frobenius cluster categories, see [37, Eg. 3.11].Now let ( Q, W ) be the quiver with potential in which Q = 31 2 ca b and W = cba . The Jacobian algebra is a cluster-tilted algebra of type A , and has infinite globaldimension (as indeed does any non-hereditary cluster-tilted algebra [32, Cor. 2.1]). In this case e Q = 31 2 2 − + − + − + ca bβ α β α β α δ c δ δ b δ δ a δ with e F again indicated by boxed vertices and dashed arrows. The potential is f W = cba + β δ α + β δ α + β δ α − aβ δ a α − bβ δ b α − cβ δ c α . The associated frozen Jacobian algebra A Q,W also arises from a dimer model on a disk with sixmarked points on its boundary [3], and is isomorphic to the endomorphism algebra of a cluster-tiltingobject in Jensen–King–Su’s categorification of the cluster algebra structure on the Grassmannian G [28]. This category is again a Frobenius cluster category [37, Eg. 3.12]. Unlike the first example,this algebra is infinite-dimensional. However, it is Noetherian, so Theorem 2.7 still applies. Since in both of these cases the algebra A Q,W is the endomorphism algebra of a cluster-tiltingobject in a Frobenius cluster category, it is internally 3-Calabi–Yau with respect to its frozenidempotent by a result of Keller–Reiten [32, §5.4] (see also [37, Cor. 3.10]). This foreshadowsTheorem 4.11 below, which states that the stronger bimodule internal Calabi–Yau property holds.
Remark 3.3.
Combinatorially, the algebra A Q,W seems to have a lot to do with the dg-algebraΓ
Q,W associated to (
Q, W ) by Ginzburg [22, §4.2]; the loops in cohomological degree − Q,W are replaced by the cycles i → i + → i − → i , and the degree − ha → ha + → ta − → ta . Here we use Amiot’s sign conventions [1, Def. 3.1], which are opposite toGinzburg’s.By a result of Keller [30, Thm. 6.3], the dg-algebra Γ Q,W is always bimodule 3-Calabi–Yau, andwe expect this to be related to the fact that (at least under a mild assumption on (
Q, W ); seeTheorem 4.11 below) A Q,W is bimodule internally 3-Calabi–Yau with respect to the idempotentdetermined by the vertices not appearing in Ginzburg’s construction. We will show in provingTheorem 1(i) that when Q is acyclic, the two constructions are also related via a triangle equivalenceGP( B Q ) ’ C Q = per Γ Q D b Γ Q . Calabi–Yau properties for frozen Jacobian algebras
We now recall a sufficient condition on an ice quiver with potential (
Q, F, W ) for the associatedfrozen Jacobian algebra to be bimodule internally 3-Calabi–Yau with respect to the frozen idempotent e = P v ∈ F e v [37, §5]. We will show, under mild assumptions on ( Q, W ), that the ice quiver withpotential ( e Q, e F , f W ) from Definition 3.1 satisfies this condition, and so A Q,W has the necessaryCalabi–Yau symmetry for us to be able to apply Theorem 2.7. Thus this section constitutes themain step in the proof of Theorem 1.In [37, §5], it is explained how an ice quiver with potential (
Q, F, W ) determines a complex ofprojective bimodules for the associated frozen Jacobian algebra A = J ( Q, F, W ). We denote thiscomplex by P ( A ), although strictly it depends on the presentation of A determined by ( Q, F, W ),and will now give its definition.Recall that Q m0 = Q \ F and Q m1 = Q \ F denote the sets of mutable vertices and unfrozenarrows of Q respectively. For v ∈ Q , we write out( v ) for the set of arrows of Q with tail v , and in( v )for the set of arrows of Q with head v . Denote the arrow ideal of A by m ( A ), and let S = A/ m ( A ).As a left A -module, S is the direct sum of the vertex simple left A -modules, and has a basis givenby the vertex idempotents e v . For the remainder of this section, we write ⊗ = ⊗ S .Introduce formal symbols ρ α for each α ∈ Q and ω v for each v ∈ Q , and define S -bimodulestructures on the vector spaces K Q = M v ∈ Q K e v , K Q m0 = M v ∈ Q m0 K e v , K F = M v ∈ F K e v , K Q = M α ∈ Q K α, K Q m1 = M α ∈ Q m1 K α, K F = M α ∈ F K α, K Q = M α ∈ Q K ρ α , K Q m2 = M α ∈ Q m1 K ρ α , K F = M α ∈ F K ρ α , K Q = M v ∈ Q K ω v , K Q m3 = M v ∈ Q m0 K ω v , K F = M v ∈ F K ω v , via the formulae e v · e v · e v = e v , e hα · α · e tα = α, e tα · ρ α · e hα = ρ α , e v · ω v · e v = ω v . For each i , the S -bimodule K Q i splits as the direct sum K Q i = K Q m i ⊕ K F i . Since K Q ∼ = S , the A -bimodule A ⊗ K Q ⊗ A is canonically isomorphic to A ⊗ A , and we will usethe two descriptions interchangeably. CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS 11
We define maps ¯ µ i : A ⊗ K Q i ⊗ A → A ⊗ K Q i − ⊗ A for 1 ≤ i ≤
3, starting with¯ µ ( x ⊗ α ⊗ y ) = x ⊗ e hα ⊗ αy − xα ⊗ e tα ⊗ y. Composing with the natural isomorphism A ⊗ K Q ⊗ A ∼ → A ⊗ A , we may instead write¯ µ ( x ⊗ α ⊗ y ) = x ⊗ αy − xα ⊗ y. For any path p = α m · · · α of K Q , we may define∆ α ( p ) = X α i = α α m · · · α i +1 ⊗ α i ⊗ α i − · · · α , and extend by linearity and continuity to obtain a map ∆ α : K Q → A ⊗ K Q ⊗ A . We then define¯ µ ( x ⊗ ρ α ⊗ y ) = X β ∈ Q x ∆ β ( ∂ α W ) y. Finally, let ¯ µ ( x ⊗ ω v ⊗ y ) = X α ∈ out( v ) x ⊗ ρ α ⊗ αy − X β ∈ in( v ) xβ ⊗ ρ β ⊗ y. Definition 4.1.
For an ice quiver with potential (
Q, F, W ), with frozen Jacobian algebra A = J ( Q, F, W ), let P ( A ) be the complex of A -bimodules with non-zero terms A ⊗ K Q m3 ⊗ A A ⊗ K Q m2 ⊗ A A ⊗ K Q ⊗ A A ⊗ K Q ⊗ A µ µ µ and A ⊗ K Q ⊗ A in degree 0, where µ = ¯ µ , and the maps µ and µ are obtained by restricting¯ µ and ¯ µ to A ⊗ K Q m2 ⊗ A and A ⊗ K Q m3 ⊗ A respectively. As out( v ) ∪ in( v ) ⊆ Q m1 for any v ∈ Q m0 ,the map µ takes values in A ⊗ K Q m2 ⊗ A as claimed. Theorem 4.2 ([37, Thm. 5.7]) . Let µ : A ⊗ A → A denote the multiplication map. If A is a frozenJacobian algebra such that → P ( A ) µ −→ A → is exact, then A is bimodule internally -Calabi–Yau with respect to the idempotent e = P v ∈ F e v . Remark 4.3.
By standard results on presentations of algebras, as in Butler–King [11], we needonly check exactness in degrees − −
3, exactness elsewhere holding for any A .Returning to our main goal, let ( Q, W ) be a quiver with potential, let ( e Q, e F , f W ) be the ice quiverwith potential associated to ( Q, W ) in Definition 3.1, and let A = A Q,W be its frozen Jacobianalgebra. In order to apply Theorem 4.2, we assume a little more about (
Q, W ). Definition 4.4.
Let (
Q, W ) be a quiver with potential. A positive grading of (
Q, W ) is a a functiondeg : Q → Z > , such that W is homogeneous in the induced Z -grading on K Q .Any Z -grading of K Q in which W is homogeneous descends to a Z -grading of the Jacobianalgebra J ( Q, W ), or of the frozen Jacobian algebra A = J ( Q, F, W ) for any subquiver F of Q . Ifthis grading is induced from a positive grading as in Definition 4.4, then the degree 0 part of A isisomorphic to S = A/ m ( A ).If W = 0, then any assignment of a positive integer to each arrow defines a positive grading.Given a positive grading in which W has degree d , multiplying the degree of every arrow appearingin W by k gives a new positive grading in which W has degree kd . This procedure will be usefulshortly, when we show that a positive grading of ( Q, W ) extends, up to such rescaling, to a positivegrading of ( e Q, f W ).A positive grading need not exist for an arbitrary quiver with potential ( Q, W ), such as if W = c + c for some cycle c . However, we are really interested in the structure of the algebra J ( Q, W ), which can be presented my many different quivers with potential. For example, any tworight equivalent quivers with potential [16, Def. 4.2] define isomorphic Jacobian algebras, and soit is enough that (
Q, W ) is right equivalent to some quiver with potential admitting a positivegrading. This can happen even if (
Q, W ) does not itself admit such a grading—for or example, thequiver with potential (
Q, W ) from Example 3.2 in which Q is a 3-cycle admits a positive grading in which every arrow has degree 1, but it is right equivalent to ( Q, cba + cbacba ), which does notadmit a positive grading. Lemma 4.5. If ( Q, W ) admits a positive grading, then so does ( e Q, f W ) .Proof. If W = 0, then ( e Q, f W ) has the positive grading deg withdeg a = 1 , deg α i = deg β i = deg δ a = 1 , deg δ i = 2for all i ∈ Q and a ∈ Q . From now on, we assume W = 0, and let deg be a positive grading of( Q, W ); since W = 0, we have deg ( W ) >
0. We begin by constructing a positive grading deg of(
Q, W ) with the property that deg( W ) − deg( a ) ≥ a ∈ Q .First, note that if deg ( W ) − deg ( a ) ≤ a ∈ Q , then a does not appear in W , since Q has no loops. Pick K ∈ Z such that K deg ( W ) − deg ( a ) ≥ a ∈ Q not appearing in W ; then defining deg ( a ) = ( K deg ( a ) , a appears in W , deg ( a ) , otherwise , we see that deg ( W ) = K deg ( W ), and so deg ( W ) − deg ( a ) ≥ a ∈ Q . Now definingdeg( a ) = 3 deg ( a ) for all a ∈ Q gives the required potential.Write d = deg( W ), and extend deg : Q → Z to the arrows of e Q , by definingdeg( α v ) = deg( β v ) = 1 , deg( δ v ) = d − , deg( δ a ) = d − − deg( a ) , for each v ∈ Q and a ∈ Q . Since d − deg( a ) ≥ a ∈ Q , all of these values are positiveintegers. It follows immediately from the definition of f W that this potential is homogeneous ofdegree d with respect to deg, and so deg is a positive grading for ( e Q, f W ). (cid:3) When A = A Q,W is graded in such a way that all arrows have positive degree, an argument byBroomhead [6, Prop. 7.5] can be used to show that exactness of the complex 0 −→ P ( A ) µ −→ A −→ P ( A ) ⊗ A S S . µ Indeed, the forward implication holds in general, since P ( A ) µ −→ A is perfect as a complex of right A -modules, and so remains exact under − ⊗ A M for any M ∈ Mod A . Broomhead’s argument, forthe case that F = ∅ , hinges on the property that, for any u ∈ A ⊗ K Q (m) i , 1 ≤ i ≤
3, we have µ i ( u ⊗
1) = v ⊗ X w ⊗ x for some v, w ∈ A ⊗ K Q (m) i − and x ∈ m ( A ); this is also the case in our setting, and so the rest ofBroomhead’s argument goes through without change (modulo switching from right to left modules,i.e. from S ⊗ A P ( A ) to P ( A ) ⊗ A S ).Now the complex (4.1) decomposes along with S , so that its exactness is equivalent to theexactness of 0 P ( A ) ⊗ A S v S v µ for each v ∈ e Q , where S v denotes the vertex simple left A -module at v . Thus when ( Q, W ), andhence ( e Q, f W ) by Lemma 4.5, admits a positive grading, we are able to reduce the problem ofcomputing a bimodule resolution of A to the simpler problem of computing a projective resolutionof each vertex simple left A -module. CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS 13
It will be useful to introduce some notation for elements of terms of P ( A ) ⊗ A S v . Note first thatwe have isomorphisms A ⊗ K e Q ⊗ A ⊗ A S v ∼ = M b ∈ out( v ) Ae hb ,A ⊗ K e Q m2 ⊗ A ⊗ A S v ∼ = M a ∈ in( v ) ∩ e Q m1 Ae ta ,A ⊗ K e Q m3 ⊗ A ⊗ A S v ∼ = ( Ae v , v ∈ Q m0 , , v ∈ F . (4.2)In the first two cases, the right-hand sides are of the form L α ∈ X Ae v ( a ) , where X is a set ofarrows, and v : X → Q . The map Ae v ( a ) → L a ∈ X Ae v ( a ) including the domain as the summandindexed by a will be denoted by x x ⊗ [ a ]; this helps us to distinguish these various inclusionswhen v is not injective. As a consequence, a general element of the direct sum is x = X a ∈ X x a ⊗ [ a ]for x a ∈ Ae v ( a ) .The most complicated map in the complex P ( A ) is µ , so we wish to spell out µ ⊗ A S v explicitly.Using the isomorphisms from (4.2), and our notation for elements of the direct sums, we have( µ ⊗ A S v )( x ) = X b ∈ out( v ) (cid:18) X a ∈ in( v ) ∩ e Q m1 x a ∂ rb ∂ a W (cid:19) ⊗ [ b ] , where ∂ rb , called the right derivative with respect to b , is defined on paths by(4.3) ∂ rb ( α k · · · α ) = ( α k · · · α , α = b, , α = b and extended linearly and continuously.We now prove the necessary exactness for the complex P ( A ) ⊗ A S v . To do this, we break intotwo cases depending on whether v is mutable or frozen, and use heavily the explicit set R of definingrelations for A given above in (3.1). Lemma 4.6.
Let i ∈ Q be a mutable vertex of e Q . If y ∈ Ae i satisfies yβ i = 0 , then y = 0 .Proof. Let e y be an arbitrary lift of y to K e Qe i . Now assume yβ i = 0, so e yβ i ∈ h R i . Since everyterm of e yβ i , when written in the basis of paths of e Q , ends with the arrow β i , but no term of anyelement of R has a term ending with β i , we must be able to write e yβ i = X j z j β i for z j ∈ h R i e i . Comparing terms, we see that e y = P j z j ∈ h R i , and so y = 0 in A . (cid:3) Proposition 4.7.
For i ∈ Q , the map µ ⊗ A S v : Ae v → M a ∈ in( v ) ∩ e Q m1 Ae ta is injective.Proof. We have ( µ ⊗ A S i )( y ) = X a ∈ in( i ) ∩ e Q m1 − ya ⊗ [ a ] . Since i is mutable, β i ∈ in( i ) ∩ e Q m1 by the construction of ( e Q, e F ). Thus if ( µ ⊗ S i )( x ) = 0, it followsfrom the above formula that − yβ i ⊗ [ β i ] = 0, hence yβ i = 0, and so y = 0 by Lemma 4.6. (cid:3) Lemma 4.8.
Let i ∈ Q be a mutable vertex of e Q . For each a ∈ in( i ) ∩ e Q m1 , pick x a ∈ Ae ta . If x β i δ i = X a ∈ in( i ) ∩ Q x a β ta δ a , then there exists y ∈ Ae i such that x a = ya for each a ∈ in( v ) ∩ e Q m1 .Proof. Pick a lift e x a ∈ K e Q of each x a . Writing p = e x β i δ i − X a ∈ in( v ) ∩ Q e x a β ta δ a , our assumption on the x a is equivalent to p ∈ h R i . Since every term of p ends with either δ i or β ta δ a for some a ∈ in( i ) ∩ Q , and the only element of R including terms ending with these arrowsis β i δ i − P a ∈ in( i ) ∩ Q aβ ta δ a , we can write p = z i δ i + X a ∈ in( i ) ∩ Q z a β ta δ a + y (cid:16) β i δ i − X a ∈ in( i ) ∩ Q aβ ta δ a (cid:17) , where z i ∈ h R i e − i , z a ∈ h R i e ta and y ∈ K e Qe i . Comparing terms in our two expressions for p , wesee that e x β i = z i + yβ i , e x a = z a + ya. Since z i , z a ∈ h R i , when we pass to the quotient algebra A = K e Q/ h R i we see that x β i = yβ i and x a = ya , as required. (cid:3) Proposition 4.9.
For i ∈ Q , we have ker( µ ⊗ A S i ) = im( µ ⊗ A S i ) .Proof. Since we already know that P ( A ) is a complex, it is enough to show thatker( µ ⊗ A S i ) ⊆ im( µ ⊗ A S i ) . Let x = P a ∈ in( i ) ∩ e Q m1 x a ⊗ [ a ] ∈ L a ∈ in( i ) ∩ e Q m1 Ae ta . Then( µ ⊗ A S i )( x ) = X b ∈ out( i ) (cid:18) X a ∈ in( i ) ∩ e Q m1 x a ∂ rb ∂ a W (cid:19) ⊗ [ b ] ∈ M b ∈ out( i ) Ae hb . In particular, α i ∈ out( i ), so if x ∈ ker( µ ⊗ A S i ), we have (cid:18) X a ∈ in( i ) ∩ e Q m1 x a ∂ rα i ∂ a W (cid:19) ⊗ [ α i ] = 0 . Using the explicit expressions for the relations ∂ a W computed in (3.1), we see that X a ∈ in( i ) ∩ e Q m x a ∂ rα i ∂ a W = x β i δ i − X a ∈ in( i ) ∩ Q x a β ta δ a = 0 , and so by Lemma 4.8 there exists y ∈ Ae i such that x a = ya for each a . It follows that x = ( µ ⊗ A S i )( y ), as required. (cid:3) Proposition 4.10. If v ∈ e F then P ( A ) ⊗ A S v is a projective resolution of S v .Proof. Since v ∈ e F , the complex P ( A ) ⊗ A S v is zero in degree −
3, so in view of Remark 4.3, it isonly necessary for us to check that µ ⊗ S v is injective. In fact, if v = i − for some i ∈ Q , then P ( A ) ⊗ A S v is also zero in degree − i − ), so we need onlyconsider v = i + for some i ∈ Q .Since in( i + ) ∩ e Q m1 = { α i } , we have µ ⊗ A S + i : Ae i → L b ∈ out( i ) Ae hb . Let x ∈ Ae i . Then,computing as above, we have( µ ⊗ A S + i )( x ) = X b ∈ out( i + ) x∂ rb ∂ α i W ⊗ [ b ] . Assume this is 0. Considering δ i ∈ out( i + ), we see that we must have 0 = x∂ rδ i ∂ α i W = xβ i . ByLemma 4.6, it follows that x = 0, and µ ⊗ A S + i is injective as required. (cid:3) CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS 15
Combining these results, we are able to establish the desired internal Calabi–Yau property for A Q,W whenever (
Q, W ) admits a positive grading.
Theorem 4.11. If ( Q, W ) admits a positive grading, then A = J ( e Q, e F , f W ) is bimodule internally -Calabi–Yau with respect to the frozen idempotent e = P i ∈ Q ( e + i + e − i ) .Proof. The combination of Propositions 4.7 and 4.9 shows that P ( A ) ⊗ A S i is exact for any i ∈ Q = e Q m0 , and Proposition 4.10 shows that the same is true of P ( A ) ⊗ A S v for v ∈ e F , so thedirect sum P ( A ) ⊗ A S µ −→ S of all of these complexes is exact. By Lemma 4.5, we can grade A sothat all arrows have positive degree. As already remarked, this allows us to apply Broomhead’sresult [6, Prop. 7.5] to deduce exactness of P ( A ) µ −→ A from that of P ( A ) ⊗ A S µ −→ S . Now theresult follows from Theorem 4.2. (cid:3) As a corollary, we obtain the promised result (Theorem 3) that any quiver with potentialadmitting a positive grading may be realised as the ‘interior’ of a bimodule internally 3-Calabi–Yaufrozen Jacobian algebra.
Corollary 4.12.
Let ( Q, W ) be a quiver with potential admitting a positive grading. Then J ( Q, W ) = A Q,W / h e i , where e is the frozen idempotent of A Q,W , and A Q,W is bimodule internally -Calabi–Yau with respect to e . (cid:3) As a second corollary, a Jacobi-finite quiver with potential (
Q, W ) admitting a positive gradingdetermines a stably 2-Calabi–Yau Frobenius category admitting a cluster-tilting object with stableendomorphism algebra J ( Q, W ), whenever A Q,W turns out to be Noetherian. This is how we willdefine the Frobenius cluster category E Q from Theorem 1 when Q is acyclic. Corollary 4.13.
Let ( Q, W ) be a Jacobi-finite quiver with potential admitting a positive grading.Assume that A = A Q,W is Noetherian, let e = P i ∈ Q ( e + i + e − i ) be the frozen idempotent of A , let B = eAe be its boundary algebra, and let T = eA . Then T is a cluster-tilting object of the stably -Calabi–Yau Frobenius category GP( B ) , with End B ( T ) op ∼ = A and End B ( T ) op ∼ = J ( Q, W ) .Proof. The existence of a positive grading implies that A is bimodule internally 3-Calabi–Yau withrespect to e by Theorem 4.11. Since A/ h e i = J ( Q, W ) by construction, the statement followsdirectly from Theorem 2.7. (cid:3)
Unfortunately, at the moment we have no general methods for determining when the algebra A Q,W is Noetherian unless Q is acyclic, and this is the main obstruction to extending Theorem 1to the case of quivers with cycles. When Q is acyclic, A Q is even finite-dimensional, as we will seein Section 5. On the other hand, it is certainly possible that A Q,W can still be Noetherian when Q has cycles; we already observed this for the 3-cycle with its natural potential in Example 3.2. Thisquiver with potential is also Jacobi-finite, and admits the positive grading in which all arrows havedegree 1, so Corollary 4.13 also applies in this case.We also have no evidence that the existence of a positive grading is necessary to obtain theconclusion of Theorem 4.11; rather, this condition was imposed in order to make the necessarycalculations more manageable. By analogy with Keller’s result [30, Thm. 6.3, Thm. A.12] that theJacobian algebra of any quiver with potential is the 0-th homology of a bimodule 3-Calabi–Yaudg-algebra, we conjecture that this assumption is not in fact needed. Conjecture 4.14.
The conclusion of Theorem 4.11, and hence those of Corollary 4.12 andCorollary 4.13, remains valid without the assumption that ( Q, W ) admits a positive grading. In support of this conjecture, we prove injectivity of µ directly, without any assumption on theexistence of a positive grading. Proposition 4.15.
The map µ : A ⊗ e Q m3 ⊗ A → A ⊗ e Q m2 ⊗ A is injective.Proof. We use the natural isomorphism A ⊗ K e Q m3 ⊗ A ∼ → M i ∈ Q Ae i ⊗ K e i A given by x ⊗ ω i ⊗ y x ⊗ y . For each i ∈ Q , let x i = ( x ji ) j ∈ J i and y i = ( y ji ) j ∈ J i be finite sets ofelements of Ae i and e i A respectively. These define an element x i ⊗ y i := X j ∈ J i x ji ⊗ y ji ∈ Ae i ⊗ K e i A, and all elements of Ae i ⊗ K e i A are of this form. Without loss of generality, i.e. without changingthe value of x i ⊗ y i , we may assume that { y ji : j ∈ J i } is a linearly independent set.Now assume P i ∈ Q x i ⊗ y i is in the kernel of µ . We aim to show that, in this case, x i ⊗ y i = 0for all i ∈ Q , and so in particular their sum is zero. Projecting onto the component A ⊗ K ρ β i ⊗ A ∼ = Ae i ⊗ K e + i A of A ⊗ e Q m2 ⊗ A , we see that x i β i ⊗ y i = X j ∈ J i x ji β i ⊗ y ji = 0 . Since the y ji are linearly independent, it follows that x ji β i = 0 for all j . By Lemma 4.6, it followsthat x ji = 0 for all j , and so x i ⊗ y i = 0. (cid:3) The acyclic case
Let (
Q, W ) be a quiver with potential such that J ( Q, W ) = A Q,W / h e i is finite-dimensional. Bythis assumption and Theorem 4.11, all of the assumptions of Theorem 2.7 are satisfied for A Q,W ,except possibly Noetherianity. Our goal in this section is to show that if Q is an acyclic quiver,then the algebra A = A Q = J ( e Q, e F , e
0) is even finite-dimensional, so we may apply Theorem 2.7 toobtain a categorification of the polarised principal coefficient cluster algebra f A Q . Lemma 5.1.
Let Q be any quiver, and let p be a path in e Q . Assume that p contains at least one arrownot in Q , and that hp, tp ∈ Q . Then p maps to zero under the projection K e Q → A = J ( e Q, e F , e .Moreover, if q is a path of length at least containing no arrows of Q , then q maps to zero underthis projection.Proof. Let γ be the first arrow of p in e Q \ Q ; there is such an arrow by assumption. If γ hasa predecessor in p , then this arrow is in Q , and so tγ ∈ Q . On the other hand, if γ is the firstarrow of p , then tγ = tp ∈ Q by assumption. By the construction of e Q , it follows that γ = α i forsome i ∈ Q . Since hp ∈ Q , the path p cannot terminate with α i , so this arrow has a successor.Looking again at the definition of e Q , the only options are δ a for some a ∈ Q with ha = i , or δ i .We break into two cases.First assume α i is followed in p by δ a for some a ∈ Q . This again cannot be the final arrow of p , since hp ∈ Q . The only arrow leaving hδ a = i − ta is β ta , so this must be the next arrow of p . Butafter projection to A , we have β ta δ a α ha = ∂ a W = 0 since W = 0, so p projects to zero.In the second case, α i is followed by δ i . As above, δ i must be followed in p by β i , and projectingto A yields β i δ i α i = (cid:18) X a ∈ Q ha = i aβ ta δ a (cid:19) α i = X a ∈ Q ha = i aβ ta δ a α ha = 0 , so that p again projects to zero.For the second statement, we have already shown that the paths β ta δ a α ha for some a ∈ Q or β i δ i α i for some i ∈ Q project to zero. Thus if q as in the statement does not project to zero, itmust not contain either of these subpaths. However, the bipartite property of e F means that anypath of arrows not in Q without either of these subpaths is itself a subpath of δ x α i β i δ y for some x, y ∈ Q ∪ Q and i ∈ Q with hx = i = ty , and so has length at most 4. (cid:3) Theorem 5.2.
Let Q be an acyclic quiver. Then A = J ( e Q, e F , e is finite-dimensional.Proof. We show that there are finitely many paths of e Q determining non-zero elements of A . ByLemma 5.1, any path p of e Q determining a non-zero element of A may not have any subpath withendpoints in Q and containing an arrow outside Q . Thus we must have p = q p q , where q and q are (possibly empty) paths not involving arrows of Q , and p is a path in Q . Since Q is acyclic, CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS 17 there are only finitely many possibilities for p . By Lemma 5.1 again, q and q have length at most4 if p is to be non-zero in A , and so there are again only finitely many possibilities. (cid:3) We have now established everything we need in order to construct the Frobenius cluster categoryrequired by Theorem 1, and to prove part (i) of this theorem.
Proof of Theorem 1(i).
Since Q is an acyclic quiver, J ( Q,
0) = K Q is finite-dimensional, and anyassignment of positive integers to the arrows of Q is a positive grading of ( Q, A Q = J ( e Q, e F , e
0) is finite-dimensional, and hence Noetherian.Now let e ∈ A Q be the frozen idempotent, and write B Q = eA Q e . Then Corollary 4.13 appliesto show that E Q := GP( B Q ) is a stably 2-Calabi–Yau Frobenius category. It is Krull–Schmidt since B is a finite-dimensional algebra.To see that GP( B ) is a Frobenius cluster category, it only remains to show that the endomorphismalgebra of any cluster-tilting object has global dimension 3, which follows from [37, Prop. 3.7], usingthat B is a finite-dimensional algebra to see that these endomorphism algebras are Noetherian.Now to see that Theorem 1(i) holds for this choice of E Q , we use Keller–Reiten’s ‘recognitiontheorem’ [31, Thm. 2.1]; by Corollary 4.13, E Q is a 2-Calabi–Yau triangulated category admitting acluster-tilting object T = eA Q with endomorphism algebra K Q , so E Q is triangle equivalent to thecluster category C Q as required. (cid:3) We already observed after stating Corollary 4.13 that our construction applies to (
Q, W ) givenby the 3-cycle with natural potential to produce a stably 2-Calabi–Yau Frobenius cluster category E Q,W = GP( B Q,W ). We claim that the analogous statement to Theorem 1(i) also holds here,namely that E Q,W is triangle equivalent to Amiot’s cluster category C Q,W . While Keller–Reiten’srecognition theorem no longer applies to the cluster-tilting object T provided by Corollary 4.13, sinceits stable endomorphism algebra is not hereditary, one can in this case find another cluster-tiltingobject T ∈ E Q,W with End E Q,W ( T ) op ∼ = K Q , for Q an orientation of the Dynkin diagram A .Thus the recognition theorem shows that E Q,W ’ C Q , and C Q ’ C Q,W either by the recognitiontheorem again or by [1, Cor. 3.11].We note that while Amiot–Reiten–Todorov have a recognition theorem [2, Thm. 3.1], whichidentifies certain stable categories of Frobenius categories admitting a cluster-tilting object withfrozen Jacobian endomorphism algebra as generalised cluster categories, this theorem does notapply to our object T when W = 0, because then ( e Q, e F , f W ) cannot satisfy their assumptions (H3)and (H4). Assumption (H3) requires a non-negative degree function on the arrows of e Q giving f W degree 1, and (H4) requires that the arrows α i , for i ∈ Q , all have degree 1. But if W = 0, then(H3) forces some arrow a ∈ Q to have degree 1, and then (H4) implies that the potential term aβ ta δ a α ha has degree at least 2, so (H3) does not hold.6. Mutation
To complete the proof of Theorem 1, we need to understand the relationship between mutation ofcluster-tilting objects in GP( B Q ) and Fomin–Zelevinsky mutation of their quivers. These operationsturn out to be compatible, in a much larger class of Frobenius cluster categories, in the sense of thefollowing theorem. Theorem 6.1 ([38, Thm. 5.14, Prop. 5.15]) . Let E be a Hom-finite Frobenius cluster category,and assume there is a cluster-tilting object T ∈ E such that End E ( T ) op ∼ = J ( Q, F, W ) , where Q theGabriel quiver of this algebra. Then(i) Q has no loops or -cycles incident with any mutable vertex, so the Iyama–Yoshino mutationof T at any indecomposable non-projective summand is well-defined,(ii) if T is obtained from T by such a mutation, then End E ( T ) op ∼ = J ( Q , F , W ) , where ( Q , F , W ) is obtained from ( Q, F, W ) by mutation at the vertex corresponding to themutated summand, and(iii) the quiver Q is the Gabriel quiver of End E ( T ) op and, up to arrows between frozen vertices,the Fomin–Zelevinsky mutation of Q at the appropriate vertex.These results then extend inductively to any cluster-tilting object in the mutation class of T . This statement can be proved in a similar way to an analogous result of Buan–Iyama–Reiten–Smith for triangulated categories [8]. The definition of mutation for an ice quiver with potential isgiven in [38, Def. 4.1], and is similar to that for ordinary quivers with potential [8, §1.2]. We alsoexplain in [38] how to extend the definition of Fomin–Zelevinsky mutation to ice quivers which mayhave arrows between their frozen vertices, allowing for a version of statement (iii) in which thesearrows are still considered. For now, we use this result to complete the proof of Theorem 1.
Proof of Theorem 1(ii).
As in the proof of part (i) in the previous section, we take E Q = GP( B Q ) tobe the category of Gorenstein projective modules over the boundary algebra of A Q , which we showedto be a Frobenius cluster category. It is Hom-finite, since A Q , and hence B Q , is a finite-dimensionalalgebra by Theorem 5.2. Corollary 4.13 provides a cluster-tilting object T = eA Q ∈ E Q withEnd B Q ( T ) op ∼ = A Q having Gabriel quiver e Q , which up to arrows between frozen vertices is the icequiver of the initial seed e s of f A Q .Now for any sequence µ k m · · · µ k of mutations, we associate the cluster-tilting object µ k m · · · µ k T ,which is well-defined by an induction using Theorem 6.1(i)–(ii), to the seed µ k m · · · µ k e s of f A Q .This map commutes with mutation by construction, and respects quivers by Theorem 6.1(iii). Sincecluster-tilting objects of GP( B Q ) are in bijection with those of the stable category GP( B Q ), andthe latter is triangle equivalent to the cluster category C Q by Theorem 1(i), the fact that thisassignment is a bijection follows from [10, Thm. A.1] and [12], the latter showing that the exchangegraph of A Q coincides with that of f A Q . (cid:3) Just as for Theorem 1(i), the conclusions of Theorem 1(ii) still hold when Q is a 3-cycle, replacing B Q by B Q,W for W the potential given by the 3-cycle. In this case f A Q is the cluster algebra of theGrassmannian G , and GP( B Q,W ) the Grassmannian cluster category constructed by Jensen–King–Su [28] (see Example 3.2), and Theorem 6.1 can be adapted to these categories, despite them beingHom-infinite—see [38, Prop. 5.16] and its preceding discussion.7.
An extriangulated categorification
Let (
Q, W ) be a Jacobi-finite quiver with potential admitting a positive grading, and assumethat A Q,W is Noetherian, so that we may apply Corollary 4.13 to obtain a stably 2-Calabi–YauFrobenius category E Q,W = GP( B Q,W ). Write e − = P i ∈ Q e − i , and P − = B Q,W e − . In this section,we consider the quotient category E + Q,W = E Q,W / [ P − ], obtained from E Q,W by taking the quotientby the the ideal of morphisms factoring through an object of add P − . The resulting category has anatural extriangulated structure—we do not define this notion for the brief discussion here, insteadreferring the reader to Nakaoka–Palu [35]. Proposition 7.1.
The category E + Q,W carries the structure of an extriangulated category. Moreover,it is a Frobenius extriangulated category [35, Def. 7.1], and the stable category E + Q,W coincides with E Q,W , meaning in particular that it is triangulated and -Calabi–Yau.Proof. Since P − is a projective-injective object of GP( B Q,W ), the extriangulated structure on E + Q,W is induced from the exact structure on E Q,W as in [35, Prop. 3.30]. Since the ‘extension groups’ E E + Q,W ( X, Y ) in this extriangulated category coincide with the extension groups Ext E Q,W ( X, Y ) inthe exact category E Q,W , we see that an object is projective in E + Q,W if and only if it is projectivein E Q,W , if and only if it is injective in E Q,W , if and only if it is injective in E + Q,W . Then thisagreement of extension groups also allows us to further deduce the existence of enough projectivesand injectives in E + Q,W [35, Def. 3.25] from the corresponding property of GP( B Q,W ). It also followsfrom the agreement of projective-injective objects in E + Q,W and E Q,W that E + Q,W = E Q,W , the latterbeing a 2-Calabi–Yau triangulated category by Corollary 4.13. (cid:3)
Now assume that Q is acyclic, so that E Q = GP( B Q ) is the Frobenius cluster category fromTheorem 1. In this case, E + Q = E Q / [ P − ] is the extriangulated category required by Corollary 2. CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS 19
Proof of Corollary 2.
The fact that E + Q is extriangulated with projective and injective objectscoinciding is already proved in Proposition 7.1. The same proposition states that E + Q = E Q , thelatter being equivalent to the cluster category C Q by Theorem 1(i), so statement (i) is proved.To see (ii), first note that E Q , E + Q and E Q all have the same cluster-tilting objects sinceExt E Q ( X, Y ) = E E + Q ( X, Y ) = Hom E Q ( X, Y [1]) . for any pair of objects X and Y of these categories. Moreover, since E + Q and E Q have the sameprojective-injective objects, the summands of a cluster-tilting object which are indecomposablenon-projective are also the same in all three categories. Thus two cluster-tilting objects fromthe common set of objects are related by mutation at an indecomposable summand in one ofthese categories if and only if they are related by mutation at the same summand in all three. Inparticular, the identity map is a bijection from the set of cluster-tilting objects of E + Q to the set ofcluster-tilting objects of E Q , commuting with mutation. A more detailed discussion of mutation ofcluster-tilting objects in extriangulated categories can be found in [13] (see also [41]).Now let T be a cluster-tilting object of E + Q . Since E + Q = E Q / [ P − ], the Gabriel quiver ofEnd E + Q ( T ) op is obtained from that of End E Q ( T ) op by deleting the vertices corresponding to thedirect summands of P − , and all adjacent arrows. On the cluster algebra side, let e s T be the seedof the polarised principal coefficient cluster algebra f A Q corresponding to T , now thought of asan object of E Q , under the bijection of Theorem 1(ii), so that the ice quiver of e s T is the Gabrielquiver of End E Q ( T ) op up to arrows between frozen vertices. We obtain a seed s T of the principalcoefficient cluster algebra A • Q by setting each frozen variable x − i to 1 and removing them from thecluster, while also removing the vertices i − from the ice quiver, so the ice quiver of s T coincides upto arrows between frozen vertices with the Gabriel quiver of End E + Q ( T ) op . Noting that all of thevariables x − i and vertices i − are frozen, this operation on seeds gives a bijection between seeds of f A Q and those of A • Q , commuting with mutation.Thus the the map T s T , from cluster-tilting objects of E + Q to seeds of A • Q , is the compositionof three bijections commuting with mutation, and so is itself such a bijection. (cid:3) Boundary algebras
Since the objects of the Frobenius categories constructed in Corollary 4.13 are modules for theidempotent subalgebra B Q,W of A Q,W determined by the frozen vertices, we wish to describe thissubalgebra more explicitly. In this section we will present B Q via a quiver with relations, in thecase that Q is acyclic, so that GP( B Q ) is the Frobenius category E Q from Theorem 1.Recall that the double quiver Q of a quiver Q has vertex set Q and arrows Q ∪ Q ∨ , where Q ∨ = { α ∨ : α ∈ Q } . The head and tail maps agree with those of Q on Q , and are defined by hα ∨ = tα and tα ∨ = hα on Q ∨ . The preprojective algebra of Q isΠ( Q ) = K Q/ D X α ∈ Q [ α, α ∨ ] E and, up to isomorphism, depends only on the underlying graph of Q . We begin with the followinggeneral statement for frozen Jacobian algebras, which reveals some of the relations of B Q,W for anarbitrary quiver with potential (
Q, W ). Proposition 8.1.
Let ( Q, F, W ) be an ice quiver with potential, let A = J ( Q, F, W ) and let B be the boundary algebra of A . Then there is an algebra homomorphism π : Π( F ) → B given by π ( e i ) = e i for all i ∈ F , and π ( α ) = α , π ( α ∨ ) = ∂ α W for all α ∈ F .Proof. It suffices to check that π ( P α ∈ F [ α, α ∨ ]) = 0, i.e. that P α ∈ F [ α, ∂ α W ] = 0. By construction,for any v ∈ Q we have X α ∈ in( v ) α∂ α W = X β ∈ out( v ) ∂ β W β in K Q . Projecting to A and summing over vertices, we see that0 = X α ∈ Q [ α, ∂ α W ] = X α ∈ Q m1 [ α, ∂ α W ] + X α ∈ F [ α, ∂ α W ] = X α ∈ F [ α, ∂ α W ] , where the final equality holds since ∂ α W = 0 in A whenever α ∈ Q m1 . (cid:3) Remark 8.2.
Familiarity with the constructions of [7, 21, 28] may make it tempting to conjecturethat the map π in Proposition 8.1 is surjective, at least when J ( Q, F, W ) is bimodule internally3-Calabi–Yau, but this is not the case. A small explicit counterexample is the frozen Jacobianalgebra A of 2 314where the frozen subquiver is indicated by boxed vertices and dashed arrows as usual, and thepotential is given by the difference of the two 3-cycles, so that the relations are generated by settingequal the two length two paths from 1 to 4. In this case the boundary algebra B is equal to A , andthe arrow from 4 to 1 is not in the image of π : Π( F ) → B . Since this algebra can be graded withevery arrow in degree 1, one can again check that the bimodule complex P ( A ) from Section 4 givesa resolution by computing a projective resolution of each simple module, so this frozen Jacobianalgebra is bimodule internally 3-Calabi–Yau. (Since every vertex is frozen, this reduces to thestatement that p . dim A ε A ≤
3, and in fact this projective dimension is 2.) We will see in thissection that π fails to be surjective for our frozen Jacobian algebras J ( e Q, e F , f W ) whenever Q isacyclic with a path of length at least 2; see Example 8.4 below.We now turn to our description of B = B Q , beginning with what will turn out to be its Gabrielquiver. Definition 8.3.
Let Q be an acyclic quiver, and consider the frozen subquiver e F of e Q , which hasvertex set e F = { i + , i − : i ∈ Q } and arrows δ i : i + → i − , δ a : ha + → ta − for each i ∈ Q and a ∈ Q . We define a quiver Γ Q by adjoining to e F an arrow δ ∨ p : tp − → hp + for each path p of Q . Since Q is acyclic, it has finitely many paths, and so Γ Q is a finite quiver.If p = e i is the trivial path at i ∈ Q , we write δ ∨ p = δ ∨ i to avoid a double subscript. The doublequiver of e F appears as the subquiver of Γ Q obtained by excluding the arrows δ ∨ p for p of length atleast two; the notation for the arrows of Γ Q is chosen to be consistent with that used earlier for thearrows of this double quiver. Example 8.4.
Let Q = 1 2 3 a b be a linearly oriented quiver of type A . Then Γ Q is the following quiver.Γ Q = 1 + − + − + − δ δ ∨ δ ∨ a δ a δ δ ∨ δ ∨ b δ b δ δ ∨ δ ∨ ba CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS 21
Definition 8.5.
Let Q be a quiver. A zig-zag in Q is a triple ( q, a, p ), where p and q are paths in Q , and a ∈ Q is an arrow such that hp = ha and tq = ta . Thus if a : v → w is an arrow of Q , azig-zag involving a is some configuration w v apq where the dotted arrows denote paths. We call the zig-zag strict if p = ap for any path p and q = q a for any path q , but do not exclude these possibilities in general. If z = ( q, a, p ) is a zig-zag,then we define hz = hq and tz = tp .We now write down elements of K Γ Q that will turn out to be a set of generating relations for B ,having three flavours, as follows.(R1) For each path p of Q , let r ( p ) = δ ∨ p δ tp − X a ∈ Q ha = tp δ ∨ pa δ a . (R2) For each path p of Q , let r ( p ) = δ hp δ ∨ p − X a ∈ Q ta = hp δ a δ ∨ ap . (R3) For each zig-zag ( q, a, p ) of Q , let r ( q, a, p ) = δ ∨ q δ a δ ∨ p . We write I for the closure of the ideal of K Γ Q generated by the union of these three sets of relations.This generating set is usually not minimal, as for certain zig-zags ( q, a, p ), the relation r ( q, a, p )may already lie in the ideal generated by relations of the form r and r . For example, if a ∈ Q isan arrow such that i = ta is incident with no other arrows of Q , then r ( a ) = δ ∨ a δ i , r ( e i ) = δ i δ ∨ i − δ a δ ∨ a , so in K Γ Q / h r ( a ) , r ( e i ) i we already have r ( a, a, a ) = δ ∨ a δ a δ ∨ a = δ ∨ a δ i δ ∨ i = 0 . One can check that if ( q, a, p ) is a strict zig-zag, then r ( q, a, p ) is not redundant, but this conditionis not necessary; if Q is a linearly oriented quiver of type A and a is the middle arrow, then thenon-strict zig-zag ( a, a, a ) yields an irredundant relation. Remark 8.6.
When p = e i is a vertex idempotent, the relations r ( e i ) and r ( e i ) reduce to‘preprojective’ relations on the double quiver of e F , of the form predicted by Proposition 8.1. Eacharrow a ∈ Q is part of the trivial strict zig-zag ( e ta , a, e ha ), and so contributes an irredundantrelation r ( e ta , a, e ha ) = δ ∨ ta δ a δ ∨ ha .Let Φ : K Γ Q → A Q be the map given by the identity on the vertices of Γ Q and the arrows δ i and δ a for i ∈ Q and a ∈ Q ; this makes sense as these are subsets of the vertices and arrows of e Q .On the remaining arrows δ ∨ p of Γ Q , we define Φ( δ ∨ p ) = α hp pβ tp . Proposition 8.7.
The map Φ above induces a well-defined map Φ : Γ Q /I → B .Proof. Since Φ sends every vertex or arrow of Γ Q to the image in A Q of a path in e Q with frozenhead and tail, it takes values in B . It remains to check that it is zero on each of the generating relations of I , which we do by explicit calculation. Let p be a path in Q . ThenΦ( r ( p )) = α hp pβ tp δ tp − X a ∈ Q ha = tp α hp paβ ta δ a = α hp p ( ∂ α tp W ) = 0 , Φ( r ( p )) = δ hp α hp pβ tp − X a ∈ Q ta = hp δ a α ha apβ tp = ( ∂ β hp W ) pβ tp = 0 . If ( q, a, p ) is a zig-zag, then Φ( r ( q, a, p )) = α hr qβ ta δ a α ha pβ tp = 0 , since 0 = ∂ a f W = ∂ a W − β ta δ a α ha = − β ta δ a α ha by acyclicity of Q . (cid:3) Theorem 8.8.
Let Q be an acyclic quiver. Then the map Φ : Γ Q /I → B Q , where Φ , Γ Q and I areall defined as above, is an isomorphism.Proof. We begin by showing surjectivity. As in the proof of Theorem 5.2, we may use Lemma 5.1to see that any path in e Q determining a non-zero element of A has the form p = q p q where q and q contain no arrows of Q , and p is a path of Q . If p has frozen head and tail, then q and q must be non-zero, so we even have p = q α hp p β tp q = q Φ( δ ∨ p ) q Now q and q are, like p , paths of e Q with frozen head or tail, but with the additional propertythat they include no arrows of Q . Let q be such a path. If q contains an arrow β i for some i ∈ Q ,then this arrow cannot be the final arrow of q , since its head is unfrozen, so it must be followed bythe arrow α i , as this is the only arrow outside of Q that composes with β i . It follows that q iseither a vertex idempotent e ± i = Φ( e ± i ), or is formed by composing paths of the form δ i = Φ( δ i )for i ∈ Q , δ a = Φ( δ a ) for a ∈ Q , or α i β i = Φ( δ ∨ i ) for i ∈ Q , and so is in the image of Φ. Weconclude that image in A of any path of e Q with frozen head or tail lies in the image of Φ. Sincesuch classes span B = B Q , we see that Φ is surjective.To complete the proof, we will use [8, Prop. 3.3]. In this context, this proposition states that Φis an isomorphism if and only if(8.1) M p path in Qtp = i Be + hp Be − i ⊕ (cid:16) M a ∈ Q ha = i Be − ta (cid:17) m ( Be + i ) 0 f ( · δ i , · δ a ) and(8.2) (cid:16) M p path in Qtp = i Be − hp (cid:17) ⊕ (cid:16) M z zig-zag tz = i Be + hz (cid:17) M p path in Qtp = i Be + hp m ( Be − i ) 0 g · Φ( δ ∨ p ) are exact sequences for all i ∈ Q . Here the left-most maps in each sequence are obtained from ourgenerators of I by right-differentiation (4.3) and the application of Φ, as proscribed in [8], so theyact on components by(8.3) f ( ye + hp ) = y Φ( δ ∨ p ) ⊗ [ i ] − X a ∈ Q ha = i y Φ( δ ∨ pa ) ⊗ [ a ] ,g ( ye − hp ) = yδ hp ⊗ [ p ] − X a ∈ Q ta = hp yδ a ⊗ [ ap ] g ( ye + hz ) = y Φ( δ ∨ q ) δ a ⊗ [ p ] , where z = ( q, a, p ).Here we have dealt with the ambiguity about which summand contains each term as in Section 4,denoting elements of L a ∈ X Be v ( a ) by P a ∈ X x a ⊗ [ a ], with x a ∈ Be v ( x ) and x a ⊗ [ a ] denoting itsinclusion into the summand of L a ∈ X Be v ( a ) indexed by a . We take the convention here that thesummand Be − i appearing in the middle term of (8.1) is indexed by the vertex i . CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS 23
Since Φ is well-defined and surjective, sequences (8.1) and (8.2) are complexes and exact at m ( Be + i ) and m ( Be − i ) respectively, so we need only check exactness at the middle term in each case.We proceed as in Section 4, using the explicit set R of relations for A Q from (3.1), and begin with(8.1). Let x i ∈ e K e Qe − i and x a ∈ e K e Qe − ta for each a ∈ Q with ha = i , determining the element x = x i ⊗ [ i ] + X a ∈ Q ha = i x a ⊗ [ a ]of the middle term of (8.1). Assume that x is a cycle, i.e. that x i δ i + X a ∈ Q ha = i x a δ a ∈ h R i . Since the only generating relation for A Q with terms ending in δ i or δ a for a ∈ Q with ha = i is ∂ α i W , it follows that in K e Q we have x i δ i + X a ∈ Q ha = i x a δ a = z i δ i + X a ∈ Q ha = i z a δ a + y∂ α i W = z i δ i + X a ∈ Q ha = i z a δ a + y (cid:18) β i δ i − X a ∈ Q ha = i aβ ta δ a (cid:19) for z i , z a ∈ h R i and y ∈ K e Qe i . Comparing terms, we see that x i = z i + yβ i and x a = z a − yaβ ta .Since hx i and hx a are frozen, but hβ v is unfrozen for all v ∈ Q , we must have y = X p path in Qtp = i y p α hp p. for some y p ∈ Be + hp . Projecting to B , we have x i = X p path in Qtp = i y p Φ( δ ∨ p ) , x a = X p path in Qtp = i − y p Φ( δ ∨ pa ) , Thus y = X p path in Qtp = i y p ⊗ p satisfies f ( y ) = x , and so sequence (8.1) is exact.Now we turn to (8.2). For each path p with tp = i , pick x p ∈ e K e Qe + hp , and assume that X p path in Qtq = i x p α hp pβ i ∈ h R i , so that x = X p path in Qtp = i x p ⊗ [ p ]is a cycle in the middle term of (8.2). By comparison with the generating relations, we see that wemay write X p path in Qtp = i x p α hp pβ i = X p path in Qtp = i z p α hp pβ i + y p (cid:18) δ hp α hp − X b ∈ Q tb = hp δ b α hb b (cid:19) pβ i − X a ∈ Q ha = hp y a,p ( β ta δ a α ha ) pβ i ! for some z p ∈ h R i , y p ∈ K e Qe − hp and y a,p ∈ e K e Qe ta . Note that either p = e i or we may write p = br for some arrow b and path r . By comparing terms, we deduce that after projection to B we have x i = y i δ i − X a ∈ Q ha = i y a,e i β ta δ a , x br = y br δ hb − y r δ b − X a ∈ Q ha = hb y a,br β ta δ a , Since y a,p ∈ e K e Qe ta , we must have y a,p = X q path in Qtq = ta y q,a,p α hq q for some y q,a,p ∈ e K e Qe + hq . The triple z = ( q, a, p ) occurring in a subscript here satisfies ha = hq and ta = tp , so it is a zig-zag. One may then calculate explicitly using (8.3) that the y p and y z = y q,a,p give a preimage y = X p path in Qtp = i y p ⊗ [ p ] + X z zig-zag tz = i y z ⊗ [ z ]of x under g . (cid:3) We close this section with the following curious property of the category E Q = GP( B Q ). Proposition 8.9.
Write P + = L k ∈ Q B Q e + k . If Q is acyclic and has no isolated vertices, then GP( B Q ) ⊆ Sub( P + ) .Proof. Write B = B Q . Since GP( B ) is a Frobenius category with injective objects those in add B ,it suffices to show that B ∈ Sub( P + ), or that Be ± k ∈ Sub( P + ) for each k . Since this is true of Be + k by definition of P + , it remains to show Be − k ∈ Sub( P + ) for each k . Since Q has no isolatedvertices, k cannot be both a source and a sink.First assume k is not a source in Q . Then the map Be − k → Be + k given by right multiplication by δ k is injective as follows. If x ∈ Be − k satisfies xδ k = 0 in Be + k , then lifting to K Γ Q and using theexplicit generating set of I , we have xδ k = zδ k + X p path in Qtp = k y p (cid:18) δ ∨ p δ k − X a ∈ Q ha = k δ ∨ pa δ a (cid:19) for some z ∈ I and y p ∈ e K e Qe + hp . Since k is not a source, the sum over arrows on the right-handside is non-empty. By comparing coefficients we see that y p = 0 for all p , and hence x = z = 0 in A .Now assume k is not a sink. Pick a ∈ Q with ta = k . Then the map Be − k → Be + ha given byright multiplication by δ a is injective as follows. If x ∈ Be − k satisfies xδ a = 0 in Be + ha , then liftingto K Γ Q and using our explicit relations, we have xδ a = z + X p path in Qtp = ha y p (cid:18) δ ∨ p δ ha − X b ∈ Q hb = ha δ ∨ pa δ a (cid:19) , for z ∈ I , and it follows by comparing coefficients that y p = 0 for all p , so x = 0. (cid:3) Example 10.2 below shows that we may have E Q (cid:40) Sub( P + ). The assumption on isolatedvertices is necessary, since such a vertex k of Q yields a direct summand C of E Q equivalent tomod Π for Π the preprojective algebra of type A , with indecomposable objects S ± k and P ± k , and S − k , P − k / ∈ Sub( P + ). 9. Gradings, indices and c-vectors
Let Q be an acyclic quiver. In this section, we show that E Q = GP( B Q ) may be equipped withthe structure of a Z n -graded Frobenius cluster category [25, Def. 3.7], corresponding to the gradingof the principal coefficient cluster algebra A • Q described by Fomin–Zelevinsky [18, §6] (or, moreprecisely, to the lift of this grading to f A Q by putting the extra frozen variables x − i in degree 0). By CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS 25 [25, Thm. 3.12], the data of this grading, when restricted to an initial seed, is equivalent to thedata of a group homomorphism K ( E Q ) → Z n , where K ( E Q ) denotes the Grothendieck group ofthe exact category E Q . We will be able to describe this group homomorphism in (almost) purelyhomological terms. The material in this section serves to illustrate how one can pass informationback and forth between a cluster algebra and its categorification, and how E Q may serve as a modelfor the principal coefficient cluster algebra A • Q . We use the Frobenius category here, rather thanthe extriangulated category E + Q of Section 7, for compatibility with the results of [25].First we briefly recall how to specify a grading on a cluster algebra, and how to interpret itcategorically. Here we will only consider Z n -gradings, where n is the rank of the cluster algebra; amore detailed explanation for gradings by arbitrary abelian groups has been given by Grabowskiand the author [25].To give a Z n -grading of a rank n cluster algebra (i.e. a grading of the underlying algebra inwhich all cluster variables are homogeneous elements), it suffices to pick a seed s with (extended) m × n integer exchange matrix e b , and an m × n integer matrix g such that e b t g = 0 [24, Def. 3.1].The i -th row of g is the degree of the cluster variable x i (which is frozen if i > n ), and the productcondition ensures that all of the exchange relations are homogeneous. If b is skew-symmetric, so wemay think of it as a quiver, this condition is equivalent to requiring, for each 1 ≤ k ≤ n , that X i → k deg x i = X k → j deg x j . Now assume that the upper n × n submatrix (principal part) of e b is skew-symmetric, and let Q bethe ice quiver corresponding to e b . If E is a Frobenius cluster category, and T ∈ E is a cluster-tiltingobject such that the quiver of A = End E ( T ) op agrees with Q up to arrows with frozen vertices,then one can interpret a grading g as the element G = m X i =1 g i ⊗ [ S i ] ∈ Z n ⊗ Z K (fd A ) = K (fd A ) n , where K (fd A ) is the Grothendieck group of finite-dimensional A -modules, and g i denotes the i -throw of g . The matrix identity e b t g = 0 implies, on the level of Grothendieck groups, that h M, G i = 0for all M ∈ mod A , where A = End E ( T ) op and h− , −i : K (mod A ) × K (fd A ) n → Z n is inducedfrom the Euler form of A by tensor product with Z n . This form is well-defined since E is a Frobeniuscluster category, so gl . dim A ≤
3. Moreover, any G ∈ K (fd A ) n with the above property arisesfrom a unique grading of the cluster algebra in this way.Now by [25, Thm. 3.12], given G ∈ K (fd A ) n as above, the mapdeg G : [ X ]
7→ h
Hom E ( T, X ) , G i is a group homomorphism K ( E ) → Z n , and, having fixed T , all such homomorphisms arise in thisway for a unique G . Thus the Z n -gradings of the cluster algebra generated by the seed s are inbijection with Hom Z (K ( E ) , Z n ).We now return to the case of polarised principal coefficient cluster algebras. Let ( Q, W ) be aquiver with potential such that A = A Q,W is Noetherian and the category E = GP( B Q,W ) is aFrobenius cluster category, such as if Q is acyclic. We also abbreviate B = B Q,W . To write downmatrices unambiguously, we pick a labelling of the vertices of Q by 1 , . . . , n , so that the vertices of e Q , the ice quiver of our preferred initial seed of f A Q , are labelled by 1 , . . . , n, + , . . . , n + , − , . . . , n − .The 3 n × n matrices we are about to write have rows indexed by these labels, in this order, andcolumns indexed by 1 , . . . , n .Let b be the skew-symmetric matrix associated to Q . Then the extended exchange matrix of ourinitial seed e s for f A Q is e b = b n − n . The principal coefficient grading of f A Q is defined via the matrix e g = n b n . It is straightforward to check that this is indeed a grading; e b t e g = b t + b = 0, since b is skew-symmetric.If one sets the cluster variables x − i of f A Q to 1, obtaining the principal coefficient cluster algebra A • Q , then this quotient map is homogeneous for the grading of the target cluster algebra defined byFomin–Zelevinsky [18, §6], given by the first 2 n rows of e g . The grading also extends to the Laurentpolynomial ring R generated by the x i and y ± i , since f A Q is a subring containing these elements.The degree of a homogeneous element of R is called its g-vector [18, §6].Let T = eA ∈ E be the cluster-tilting object of E from Corollary 4.13. In E , we have T ∼ = L ni =1 T i ,where T i = eAe i . Let G ∈ K (fd A ) n be the element corresponding to the grading e g , anddeg G : K ( E ) → Z n the associated group homomorphism.For any X ∈ E , its cluster character C TX with respect to T (defined by Fu–Keller [20, Thm. 3.3],see also [25, §3]) is a homogeneous element of R , with degree equal to deg G ( X ) [25, Prop. 3.11].If X has no non-zero projective summands, Fu–Keller [20, Prop. 6.2] (see also Plamondon [36,Prop. 3.6]) show how to compute the g-vector of C TX homologically in the stable category E , as wenow recall.Let X ∈ E have no non-zero projective summands. Using the monomial notation T x = L ni =1 T x i i ,we can find a triangle(9.1) T m X T p X X Ω − T m j f in E by choosing f to be a right (add T )-approximation, this property being equivalent to f havingdomain and mapping cylinder in add T . Then the index of X with respect to T is the vectorind T ( X ) = p X − m X ∈ Z n . This quantity is independent of the choice of f , although p X and m X individually are not. By [20, Prop. 6.2], we then havedeg G ( X ) = ind T ( X ) . Since deg G is additive on direct sums, this formula can be used to compute deg G ( X ) for any X ∈ E ,since the degrees of the projectives eAe ± i are given directly by the lower 2 n rows of e g . Remark 9.1.
By choosing a suitable map P → X , where P is projective, the triangle (9.1) mayalways be lifted to a short exact sequence0 T m X T p X ⊕ P X E , from which it follows that in K ( A ) we have[Hom B ( T, X )] = [Hom B ( T, T p X )] + [Hom B ( T, P )] − [Hom B ( T, T m X )] , and all of the A -modules on the right-hand side are projective. We haveind T ( X ) i = ( p X − m X ) i = h Hom B ( T, T p X ) , S i i − h Hom B ( T, T m X ) , S i i for all 1 ≤ i ≤ n , anddeg G ( X ) i = ( h Hom B ( T, T p X ) , G i − h Hom B ( T, T m X ) , G i + h Hom B ( T, P ) , G i ) i = h Hom B ( T, T p X ) , S i i − h Hom B ( T, T m X ) , S i i + ( h Hom B ( T, P ) , G i ) i , from the definition of G and the fact that T p X and T m X have no projective summands in E . Wetherefore deduce from the identity ind T ( X ) = deg G ( X ) that h Hom B ( T, P ) , G i = 0 , giving a linear relation between the rows of the (unextended) exchange matrix b , with coefficientsdetermined by the multiplicity of Be + i as a summand of P . When b has full rank, it follows thatit is always possible to choose P ∈ add P − , where P − = L i ∈ Q Be − i . We conjecture that thisis in fact true in general, and can observe it directly in Examples 10.2 and 10.3 below, in which CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS 27 the exchange matrices do not have full rank. Conversely, establishing that one can always choose P ∈ add P − would provide a new proof of the identity deg( C TX ) = ind T ( X ).Having understood the grading e g globally, as a function on K ( E ), we may now recover localinformation for any cluster-tilting object. Let T = L ni =1 T i ⊕ B , with T i indecomposable andnon-projective, be a cluster-tilting object of E , and write A = End E ( T ) op . Let e b be the extendedexchange matrix associated to the ice quiver Q of A , and let e g = g b n be the 3 n × n matrix whose first n rows are given bydeg G ( T i ) = h Hom B ( T, T i ) , G i = ind T ( T i ) . One can check [25, §3] that the corresponding element G ∈ K (fd A ) n satisfies h M, G i = 0 for all M ∈ mod End E ( T ) op , and the induced function K ( E ) → Z n coincides with deg G . In particular,this means that e g is a grading of the cluster algebra associated to Q , so ( e b ) t e g = 0. Writing e b = b c d , where b , c and d are n × n submatrices, and using skew-symmetry of b , we conclude that b g = ( c ) t b. For acyclic Q , the matrix e b is the extended exchange matrix of a seed of f A Q by Theorem 1(ii), sothe submatrix c is by definition the matrix of c-vectors of this seed. Thus in this case we recover(the transpose of) an identity of Fomin–Zelevinsky [18, Eq. 6.14] (see also [34, Rem. 2.1]), notingfor comparison that it is the rows, not columns, of our matrix g that are g-vectors.10. Examples
Example 10.1.
Let Q be an A quiver, so, as computed in Example 3.2,( e Q, e F ) = 1 21 + − − + aα β α β δ δ δ a and f W = β δ α + β δ α − aβ δ a α . Then A Q = J ( e Q, e F , f W ), and its boundary algebra is B Q ∼ = K Γ Q /I for Γ Q = 1 2 3 4 αα ∨ ββ ∨ γγ ∨ and I = h α ∨ α, αα ∨ − β ∨ β, ββ ∨ − γ ∨ γ, γγ ∨ , βα, γβ, α ∨ β ∨ γ ∨ i . Here we have relabelled thevertices by identifying the ordered sets (1 + , − , + , − ) and (1 , , ,
4) in the unique order preservingway. The Auslander–Reiten quiver of GP( B ) is shown in Figure 1, where we identify the leftand right sides of the picture so that the quiver is drawn on a Möbius band. To calculate theobjects of GP( B Q ) it is useful to observe that, in this example, B Q is 1-Iwanaga–Gorenstein, and soGP( B Q ) = Sub( B Q ). The stable category GP( B Q ) is the cluster category of type A , as expected.The cluster tilting object T = ⊕ ⊕ B Q Figure 1.
The Auslander–Reiten quiver of GP( B Q ) for Q of type A .of GP( B Q ) has endomorphism algebra A Q , and corresponds to the initial seed of the cluster algebrawith polarised principal coefficients associated to our initial A quiver Q . Example 10.2.
Let Q be a linearly oriented quiver of type A . We may then compute( e Q, e F ) = 1 2 31 + − + − − + a bα β α β α β δ δ δ δ a δ b Relabelling vertices similarly to Example 10.1, the boundary algebra B Q has quiverΓ Q = 1 2 3 4 5 6as computed before in Example 8.4. Explicit relations can be written down as in Section 8, but herewe will simply give radical filtrations for the projective modules. Note that despite the ‘geographical’separation of 2 and 5 in these filtrations, the arrow 2 → e B Q indicated by a 2 in the filtration to the 1-dimensionalsubspace of e B Q indicated by a 5 in the row below, when this configuration occurs. P = P =
21 3 52 4 P =
32 41 3 52 4 P =
43 52 4 P =
54 63 52 4 P = In this case the Gorenstein dimension of B Q is 2; the indecomposable projective P has injectivedimension 2, while all others have injective dimension 1. (This is in fact the first example knownto the author of a Frobenius cluster category GP( B ), with GP( B ) = 0, for which the Gorensteindimension of B is greater than 1.) The Auslander–Reiten quiver of GP( B Q ) is shown, again on aMöbius band, in Figure 2. The initial cluster tilting object from Corollary 4.13 is T = ⊕ ⊕ ⊕ B Q . Example 10.3.
Applying our construction to the quiver with potential (
Q, W ) from Example 3.2with Q a 3-cycle (which we may do, since while A is not finite-dimensional in this case, it is stillNoetherian) yields, as observed in Example 3.2, the Grassmannian cluster category GP( B Q,W ) =CM( B , ) [28]. This is a Hom-infinite category, and the Gorenstein projective B -modules are all CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS 2941 3 52 42 4 61 33 5522 44 2 41 3 52 454 63 52 4 232 41 3 52 454 123 52 4654 1 3 52 443 52 4 63 52 4 21 3 52 44 61 33 5522 44 2 4 61 33 5522 4441 3 52 4
Figure 2.
The Auslander–Reiten quiver of GP( B Q ) for Q linearly oriented of type A . In addition to the the usual mesh relations coming from Auslander–Reitensequences, the length two path from P to P represents the zero map.1346 231456 2415 342516 3526 453612 1346 Figure 3.
The Auslander–Reiten quiver of GP( B Q,W ), where (
Q, W ) is a 3-cycleand its usual potential, shown as the Grassmannian cluster category CM( B , ).infinite-dimensional. Representing these modules by Plücker labels as in [28], the Auslander–Reitenquiver of GP( B Q,W ) is shown, on the now familiar Möbius band, in Figure 3. In this case, thequiver of the endomorphism algebra of the object13 ⊕ ⊕ ⊕ B Q,W is e Q , as is the quiver of the (isomorphic) endomorphism algebra of 24 ⊕ ⊕ ⊕ B Q,W . We notethat the stable category GP( B Q,W ) is equivalent to the cluster category C Q,W ’ C Q where Q isany orientation of the Dynkin diagram A . Thus all of the conclusions of Theorem 1 (replacing K Q by J ( Q, W ) and C Q by C Q,W ) still hold for this example, despite the failure of acyclicity.
Acknowledgements
We thank Mikhail Gorsky, Alfredo Nájera Chávez, Hiroyuki Nakaoka, Yann Palu and SalvatoreStella for useful conversations and for providing references. These thanks are extended to theanonymous referee for suggesting various improvements, and the Max-Planck-Gesellschaft forfinancial support.
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