Internally Calabi–Yau algebras and cluster-tilting objects

We describe what it means for an algebra to be internally d-Calabi–Yau with respect to an idempotent. This definition abstracts properties of endomorphism algebras of (d-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(d-1)$$\end{document}-cluster-tilting objects in certain stably (d-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(d-1)$$\end{document}-Calabi–Yau Frobenius categories, as observed by Keller–Reiten. We show that an internally d-Calabi–Yau algebra satisfying mild additional assumptions can be realised as the endomorphism algebra of a (d-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(d-1)$$\end{document}-cluster-tilting object in a Frobenius category. Moreover, if the algebra satisfies a stronger ‘bimodule’ internally d-Calabi–Yau condition, this Frobenius category is stably (d-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(d-1)$$\end{document}-Calabi–Yau. We pay special attention to frozen Jacobian algebras; in particular, we define a candidate bimodule resolution for such an algebra, and show that if this complex is indeed a resolution, then the frozen Jacobian algebra is bimodule internally 3-Calabi–Yau with respect to its frozen idempotent. These results suggest a new method for constructing Frobenius categories modelling cluster algebras with frozen variables, by first constructing a suitable candidate for the endomorphism algebra of a cluster-tilting object in such a category, analogous to Amiot’s construction in the coefficient-free case.


Introduction
Cluster categories, first introduced in special cases by Buan-Marsh-Reineke-Reiten-Todorov [BMR + 06] and later generalised by Amiot [Ami09], are certain Hom-finite 2-Calabi-Yau triangulated categories that model the combinatorics of cluster algebras without frozen variables. In particular, a cluster category C contains cluster-tilting objects, which are objects T satisfying add T = {X ∈ C : Ext 1 C (T, X) = 0} = {X ∈ C : Ext 1 C (X, T ) = 0}. Basic cluster-tilting objects, whose summands in any direct sum decomposition are pairwise non-isomorphic, model the clusters of the cluster algebra; from now on, any time we refer to a cluster-tilting object, we assume it to be basic.
The main reason that the cluster-tilting objects of C can be said to model the clusters of a cluster algebra is that, as with a cluster, it is possible to pass from one cluster-tilting object to another by a process of mutation. For any indecomposable summand T i of a cluster-tilting object T , there exists a unique indecomposable T ′ i ∈ C, not isomorphic to T i , such that T /T i ⊕ T ′ i is a cluster-tilting object. Moreover, T ′ i can be computed by either of the exchange triangles f g in which f is a minimal left add T /T i -approximation of T i , and g is a minimal right add T /T i -approximation of T i . Choosing an initial cluster-tilting object T 0 = n i=1 T 0 i of C yields a cluster character [CC06], [Pal08] ϕ : C → A (Q, x) from the objects of C to the cluster algebra A (Q, x) with initial seed given by the quiver Q of End C (T ) op with cluster variables x i = ϕ T 0 i . The exchange triangles correspond to the exchange relations ϕ T i ϕ T ′ i = ϕ X i + ϕ Y i in this cluster algebra. The cluster variables of A (Q, x) are precisely the elements of the form ϕ M for M an indecomposable reachable rigid object of C, where rigid means that Ext 1 C (M, M) = 0, and reachable means that M is a summand of a reachable clustertilting object, i.e. one obtained from T 0 by a finite sequence of mutations. The clusters are the sets of the form {ϕ T 1 , . . . , ϕ Tn } for T = n i=1 T i reachable cluster-tilting in C. For a more thorough introduction to the theory of cluster algebras and their categorification, we recommend Keller's survey [Kel10].
The categorification of cluster algebras by cluster categories has proved to be very useful in studying their combinatorics, since the cluster category C can be considered more globally than the associated cluster algebra A (Q, x). For example, to identify clusters or cluster variables of A (Q, x), one usually has to find a sequence of mutations from a known cluster, which is a highly computationally intensive procedure. By contrast, cluster-tilting or rigid objects of C are characterised intrinsically.
For this reason, it would be extremely useful to be able to more readily categorify cluster algebras that do have frozen variables, particularly as most of the examples occurring in nature, such as those on the coordinate rings of partial flag varieties and their unipotent cells, as studied by Geiß-Leclerc-Schröer [GLS08], are of this type. The natural candidate for such a categorification is a stably 2-Calabi-Yau Frobenius category, as we now describe.
A Frobenius category is an exact category with enough projective and injective objects, such that these two classes of objects coincide. If E is a Frobenius category, then the stable category E = E/ proj E is triangulated by a famous result of Happel [Hap88,§I.2]. It is immediate from the definition that if T ∈ E is cluster-tilting, then proj E ⊂ add T . In this case, we must have proj E = add P for some object P = n i=r+1 T i ; the intention is that the objects T i for r < i ≤ n, which occur as summands of every cluster-tilting object of E, will correspond to the frozen variables of a cluster algebra. Factoring out proj E corresponds to setting these frozen variables to 1 in the cluster algebra, to recover a cluster algebra without frozen variables. Thus the stable category E should be a cluster category-in particular it should be 2-Calabi-Yau. Such categorifications have been described for certain cluster algebras relating to partial flag varieties, for example by Geiss-Leclerc-Schröer [GLS08], Jensen-King-Su [JKS16] and Demonet-Iyama [DI16], and we will recall some of these constructions in Section 3. Nájera Chávez [NC16] has also categorified finite type cluster algebras with 'universal' coefficients. However, in all cases, the construction of the category depends on having at least a partial understanding of the overall structure of the cluster algebra. Thus the methods of these papers cannot be easily abstracted to produce categorifications for more general cluster algebras with frozen variables.
The main aim of this paper is to consider how one might be able to produce a categorification of a cluster algebra with frozen variables without understanding this global structure, instead starting only from the data of a single seed, which is how a cluster algebra is usually specified. This is analogous to Amiot's construction of cluster categories in the case that there are no frozen variables; given a seed, one has to find a rigid potential on the quiver of the seed such that the resulting Jacobian algebra (see Section 5) is finite dimensional, and then Amiot provides a general recipe for constructing a categorification of the cluster algebra. Our construction in the case that there are frozen variables is similar, but requires more data satisfying more conditions. Given the seed of a cluster algebra with frozen variables, we take its quiver, and aim to add arrows between frozen variables and choose a potential such that the resulting frozen Jacobian algebra satisfies a number of conditions, most importantly that of being bimodule internally 3-Calabi-Yau. If this can be achieved, the general machinery developed in this paper can take over to produce the desired Frobenius category, from which the frozen Jacobian algebra can be recovered as the endomorphism algebra of a cluster-tilting object.
For most of the paper, we will in fact work in a higher level of generality, and construct stably d-Calabi-Yau Frobenius categories admitting d-cluster-tilting objects (Definition 3.2); setting d = 2 recovers the definition of cluster-tilting given above.
The structure of the paper is as follows. In Section 2 we say what it means for an algebra to be internally d-Calabi-Yau with respect to an idempotent e (Definition 2.1), and also make a stronger, more symmetric, definition of bimodule internally d-Calabi-Yau with respect to e (Definition 2.4). An algebra A satisfying either of these definitions has, in particular, finite global dimension, and a duality for any pair M and N of A-modules such that M is finite dimensional, eM = 0 and both M and N are perfect when considered as stalk complexes in the bounded derived category of A. In much of the paper we will also assume that A is Noetherian, in which case this last condition reduces to M and N being finitely generated.
In Section 3, we introduce the class of Frobenius m-cluster categories for m ≥ 1, and exploit a result of Keller-Reiten [KR07] to show that the endomorphism algebra of an m-cluster-tilting object in a Frobenius m-cluster category is internally (m + 1)-Calabi-Yau. A Morita-type theorem of Iyama-Kalck-Wemyss-Yang [KIWY15] implies that any Frobenius m-cluster category admitting an m-cluster-tilting object with Noetherian endomorphism algebra is equivalent to the category of Gorenstein projective modules over some Iwanaga-Gorenstein ring (Definition 3.8), and so we pay special attention to categories of this form. We also give brief descriptions of some important families of Frobenius m-cluster categories already appearing in the literature.
In Section 4, we show that an algebra A that is internally d-Calabi-Yau with respect to an idempotent e (on both sides) for some d ≥ 2, and satisfies mild additional assumptions, necessarily arises as the endomorphism algebra of a (d − 1)-cluster-tilting object in some Frobenius category determined by A and e. Precisely, we prove the following theorem.
Under the stronger assumption that A is bimodule internally d-Calabi-Yau with respect to e, we can show more.
Theorem 2 (Theorem 4.10). Let A be a Noetherian algebra and let e ∈ A be an idempotent such that A/ e is finite dimensional, and A is bimodule internally d-Calabi-Yau with respect to e. Write B = eAe. Then all of the conclusions of Theorem 1 hold, and moreover GP(B) is (d − 1)-Calabi-Yau.
While, in general, checking that an algebra is internally d-Calabi-Yau with respect to an idempotent can be very difficult, there is more hope in the case that A is a frozen Jacobian algebra (Definition 5.1). Such an algebra is presented via a quiver with relations, in which the relations are dual to some of the arrows; the arrows which do not have any corresponding relations are called frozen, and their end-points are frozen vertices. In Section 5, we show that the required Calabi-Yau symmetry can be deduced from the exactness of a combinatorially defined complex P(A) → A (Definition 5.4), generalising work of Ginzburg [Gin07] for Jacobian algebras. More precisely, we prove the following.
Theorem 3 (Theorem 5.6). If A is a frozen Jacobian algebra such that P(A) is quasiisomorphic to A, then A is bimodule internally 3-Calabi-Yau with respect to the idempotent e = v∈F 0 e v , where F 0 denotes the set of frozen vertices.
The results of Section 5 are inspired by work of Broomhead [Bro12] on consistent dimer models (also known as brane tilings or bipartite field theories) on closed surfaces, which has applications to theoretical physics. We expect our results to have consequences for the more recent theory of dimer models on surfaces with boundary, studied for example by Franco [Fra12], and which has already appeared in the context of cluster categorification for the Grassmannian in work of Baur-King-Marsh [BKM16]. Dimer models on surfaces with boundary have also appeared, under the name 'plabic graphs', in work of Postnikov [Pos06] on the positive Grassmannian, and in recent work of Goncharov [Gon16].

Notation and conventions
Throughout, we fix a field K, and assume all categories are K-linear and all algebras are associative K-algebras with unit. If V is a K-vector space, we write DV = Hom K (V, K) for the dual space. All modules are unital, and are left modules unless otherwise indicated. Given an algebra A, we denote by Mod A the category of all Amodules, and by mod A the category of finitely generated A-modules. Given an object X of an exact category E, we denote by add X the full subcategory of E with objects isomorphic to finite direct sums of direct summands of X. We say an algebra A is Noetherian if it is Noetherian as both a left and right module over itself; this is stronger than requiring it to be Noetherian as an A-bimodule. We denote by DA and D b A the derived and bounded derived categories of A, and by D fd (A) the full subcategory of D b A consisting of objects with finite dimensional total cohomology. We denote by per A the thick triangulated subcategory of D b A generated by A (or equivalently by the finitely generated projective A-modules). Objects of per A are called perfect, and we will say that an A-module M is perfect if the stalk complex with M in degree 0 is perfect.
(ii) for each i ∈ Z, there is a functorial duality where M and N are perfect A-modules such that M is also a finite dimensional A/ e -module, where e = AeA is the two-sided ideal of A generated by e.
Recall that a module M is perfect if the stalk complex with M in degree 0 is perfect, i.e. quasi-isomorphic to a bounded complex of finitely generated projective A-modules. Thus a perfect module is always both finitely generated and finitely presented, but the converse statements need not hold. If A is Noetherian with finite global dimension (which will be the case for much of the paper, including the main theorems in Section 4), then an A-module is perfect if and only it is finitely generated. We will also consider other situations in which perfectness is equivalent to a more familiar notion, and will try to indicate these as they arise.
Note the lack of symmetry in the finiteness conditions on M and N; cf. [Kel08, Lem. 4.1]. These conditions are imposed in order to force the space Ext d−i A (N, M) on the right-hand side of the duality formula to be finite dimensional over K. Definition 2.1 is not necessarily left-right symmetric in this way if A is infinite dimensional, so we will also make a stronger definition that does have this property. Denote by A ε = A ⊗ K A op the enveloping algebra of A, so that an A-bimodule is the same as an A ε -module. Write Ω A = RHom A ε (A, A ε ). We view Ω A as a complex in DA ε via the 'inner' multiplication on A ε ; for any homomorphism f :

Remark 2.2. If
A is quasi-isomorphic to a bounded complex of finitely generated projective Abimodules) and there is an isomorphism in DA ε . This definition is slightly weaker than that of Ginzburg [Gin07, 3.2.5], as we will not need to impose any 'selfduality' condition on the isomorphism.
If A is an algebra with quotient A, write D A (A) for the full subcategory of DA consisting of complexes with homology groups in Mod A, and D fd,A (A) for the full subcategory of D A (A) consisting of objects with finite dimensional total cohomology. . When e = 0, this implication does not hold, and so we must make the stronger condition part of the definition. Note also that (i) does not imply (ii), since it asserts only that A has a finite resolution by projective A ε -modules, whereas (ii) requires additionally that these modules are finitely generated. An algebra A is bimodule internally d-Calabi-Yau with respect to 1 if and only if p. dim A ε A ≤ d and A ∈ per A ε ; in this case A = 0, so condition (iii) is satisfied for any ψ.

Definition 2.4. An algebra
Remark 2.6. There is an isomorphism A ε ∼ → (A op ) ε given by reversing the order of the tensor product. The resulting equivalence mod A ε ∼ → mod (A op ) ε takes A to A op (and A to A op ). As a result, Definition 2.4 is left-right symmetric, meaning that A is bimodule internally d-Calabi-Yau with respect to e if and only if the same is true of A op .
The following lemma, due to Keller, allows us to recover dualities of extension groups between A-modules from bimodule properties of A.
in DA. Now apply RHom A (−, M), to get a triangle Since M ∈ D fd,A (A), we have RHom A (C, M) = 0 by definition, and so and so applying − ⊗ A X to the above sequence gives a projective resolution . In particular, this duality applies when M and N are perfect A-modules, so A is internally d-Calabi-Yau with respect to e.
While Theorem 2.8 and 2.9 show that bimodule internally Calabi-Yau algebras have stronger properties than those required by the definition of internally Calabi-Yau, with the duality formula applying to a wider class of objects, we do not currently know of any examples of internally Calabi-Yau algebras which are not bimodule internally Calabi-Yau. It seems unlikely that the two classes of algebras coincide, but it is tempting to speculate based on work of Bocklandt [Boc08] that if A is an internally Calabi-Yau algebra admitting a suitable grading, then it is also bimodule internally Calabi-Yau.
Example 2.10. Consider the algebra A = KQ/I given by the quiver with ideal of relations I = α 2 α 1 , α 1 α 3 . This is an example of a frozen Jacobian algebra; see Definition 5.1 and Example 5.3. One can check (for example, by Theorem 5.6 below) that A is bimodule internally 3-Calabi-Yau with respect to e 1 + e 2 , and so both A and A op are internally 3-Calabi-Yau with respect to this idempotent.
Similarly, the algebra A ′ = KQ ′ /I ′ given by the quiver

Frobenius m-cluster categories
This section is devoted to describing a class of categories, which we term Frobenius m-cluster categories, providing us with a rich source of internally (m + 1)-Calabi-Yau algebras. Indeed, certain categories in this class motivated Definition 2.1.
Recall that an exact category E is Frobenius if it has enough projective objects and enough injective objects, and proj E = inj E. By a famous result of Happel [Hap88, §I.2], the stable category E = E/ proj E is triangulated, with suspension given by the inverse syzygy Ω −1 , taking an object to the cokernel of an injective hull.
Definition 3.1. Let E be a Frobenius category, and let m ≥ 1. We say that E is stably m-Calabi-Yau if its stable category E is m-Calabi-Yau, meaning that E is Hom-finite, and there is a functorial duality Definition 3.2. Let E be an exact category, and let T ⊂ E be a full and functorially finite subcategory closed under direct sums and direct summands. We say T is an m-cluster-tilting subcategory if We say an object T ∈ E is an m-cluster-tilting object if add T is an m-cluster-tilting subcategory.
If E is a Frobenius category, then for any X, Y ∈ E and i > 0, we have Thus if E is stably m-Calabi-Yau, the two equalities appearing in Definition 3.2 are equivalent to one another. Definition 3.3. Let E be a Frobenius category, and m ≥ 1. Then E is called a Frobenius m-cluster category if it is idempotent complete, stably m-Calabi-Yau, and gl. dim End E (T ) op ≤ m + 1 for any m-cluster-tilting object T , of which there is at least one. If these properties hold for m = 2, then E will be called simply a Frobenius cluster category.
Note that while E is Hom-finite for any Frobenius m-cluster category E, we do not assume that E itself is Hom-finite (cf. Example 3.12). By the following result, which follows immediately from a theorem of Keller-Reiten [KR07,5.4], Frobenius m-cluster categories provide a rich source of examples of internally Calabi-Yau algebras.
Theorem 3.4. Let E be a Frobenius m-cluster category, let T ∈ E be a basic mcluster-tilting object, and write A = End E (T ) op . Let P be a maximal projective-injective summand of T , and let e ∈ A be the idempotent given by projection onto P . Then A is internally (m + 1)-Calabi-Yau with respect to e.
Proof. Let M and N be perfect A-modules, and assume that M is a finite dimensional A/ e -module. Then by [KR07,5.4], there is a functorial duality for all i. Since gl. dim A ≤ m + 1 by assumption, we have the desired result.
Remark 3.5. If E is a Frobenius m-cluster category such that A = End E (T ) op is Noetherian for any m-cluster tilting object T , then E op is a Frobenius m-cluster category with the same m-cluster-tilting objects as E; the extra assumption is used to ensure that gl. dim A op = gl. dim A ≤ m + 1. Thus, under these circumstances, A op is also internally (m+1)-Calabi-Yau with respect to e.
In the context of Theorem 3.4, we have a more straightforward characterisation of the A/ e -modules that are perfect as A-modules, again coming from Keller-Reiten's work.
Proposition 3.6. Let E be a Frobenius m-cluster category, let T ∈ E be a basic mcluster-tilting object, and write A = End E (T ) op . Let P be a maximal projective-injective summand of T , let e ∈ A be the idempotent given by projection onto P and write

Then an A-module M is perfect as an A-module if and only if it is finitely generated.
Proof. Let M ∈ mod A. If M is perfect over A, then it is in particular finitely generated over A, and thus (by the same generating set) over A. Since A = End E (T ) op and E is Hom-finite, A is finite dimensional. It follows that any finitely generated A-module is finitely presented over A, and thus perfect over A by [KR07,5.4(c)].
The following proposition, which is based on work of Iyama [Iya07a,§2], [Iya07b, Thm. 3.6.2], gives sufficient conditions on a Frobenius category E for us to conclude that gl. dim End E (T ) op ≤ m + 1 for any m-cluster-tilting object T ∈ E, as well as providing a more straightforward characterisation of the perfect modules over this endomorphism algebra.
Proposition 3.7. Let E ⊆ A be a full, extension closed, Frobenius subcategory of an abelian category A such that (i) A has enough projectives, (ii) E contains proj A, (iii) E is closed under kernels of epimorphisms, and (iv) for any X ∈ A, and any exact sequence

Then an A-module M is perfect if and only if it is finitely presented, and for such
Proof. As already noted, any perfect A-module is finitely presented. For the converse, let M be a finitely presented A-module with presentation for some T 0 , T 1 ∈ add T , and form the exact sequence , the subcategory E is contravariantly finite in A, and hence so is add T . Working inductively for 1 ≤ j ≤ m − 1, let r j : T j+1 → K j be a right add T -approximation of K j , and let K j+1 be its kernel (in A). By additionally defining K 0 = im f and K −1 = coker f , we obtain exact sequences Combining these to form the exact sequence Next we check that K m ∈ add T , which we do by checking that Ext i E (T, K m ) = 0 for 0 < i < m. For any j, we may apply Hom E (T, −) to the short exact sequence It follows from the above calculations that the sequence For the final statement, if A is Noetherian, then every finitely generated A-module is also finitely presented, and thus has projective dimension at most m+1, as required.
We now recall some work of Iyama-Kalck-Wemyss-Yang [KIWY15], which gives a normal form for a Frobenius m-cluster category that is 'small enough', in the sense it admits an m-cluster-tilting object with Noetherian endomorphism algebra. It will follow from this description that any such Frobenius m-cluster category may be embedded into an abelian category in such a way that all of the assumptions of Proposition 3.7 hold. We begin with the following definitions.
Definition 3.8. An algebra B is Iwanaga-Gorenstein if it is Noetherian and has finite injective dimension as both a left and right module over itself. In this case, the left and right injective dimensions coincide, and are called the Gorenstein dimension of B. For brevity, an Iwanaga-Gorenstein algebra with Gorenstein dimension d will be called d-Iwanaga-Gorenstein. If B is such an algebra, we write If B is an (m + 1)-Iwanaga-Gorenstein algebra, then the assumptions of Proposition 3.7 hold for the exact subcategory E = GP(B) of the abelian category A = mod B. Thus the endomorphism algebra of an m-cluster-tilting object of GP(B) has global dimension at most m + 1 whenever it is Noetherian.
. Let E be an idempotent complete Frobenius category such that proj E = add P for some P ∈ E. Assume there exists M ∈ E such that P ∈ add M, the endomorphism algebra A = End E (M) op is Noetherian, and gl. dim A = d < ∞. Then B = End E (P ) op is Iwanaga-Gorenstein of Gorenstein dimension at most d, and there is an equivalence Corollary 3.10. Let E be a Frobenius m-cluster category and let T ∈ E be an mcluster-tilting object such that End E (T ) op is Noetherian. Let P be a maximal projective summand of T , and write B = End E (P ) op . Then B is Iwanaga-Gorenstein of Gorenstein dimension at most m + 1, and E ≃ GP(B).
Proof. By definition, E is idempotent complete and gl. dim End E (T ) op ≤ m + 1. Since T is m-cluster-tilting, proj E ⊆ add T , and so we have proj E = add P . Now the result follows from Theorem 3.9.
Under the notation and assumptions of 3.10, let e be the idempotent of A = End E (T ) op given by projection onto P , with respect to which A is internally (m + 1)-Calabi-Yau by Theorem 3.4. Then the algebra B = End E (P ) op is the idempotent subalgebra eAe. We will see below that the Gorenstein dimension of B may be strictly less than m + 1. An interesting question suggested by 3.10, to which we have no good answer at this stage, is the following: can one find reasonable conditions on an Iwanaga-Gorenstein algebra B under which GP(B) is a Frobenius m-cluster category?
The remainder of the section is devoted to examples. We describe two families of examples of Frobenius cluster categories, one Hom-finite and the other Hom-infinite, to which Theorem 3.4 applies to show that the endomorphism algebras of clustertilting objects are internally 3-Calabi-Yau with respect to projection onto a maximal projective-injective summand. We also give a family of examples of Frobenius 1-cluster categories arising as part of the algebraic McKay correspondence.
Example 3.11. Buan-Iyama-Reiten-Scott [BIRS09] construct a family of Hom-finite stably 2-Calabi-Yau Frobenius categories Sub Π ω . Here Π = Π(∆) is the preprojective algebra associated to a graph ∆, and ω is a finite product of simple reflections in the Weyl group of ∆. The algebra Π ω is a (finite dimensional) quotient of Π, and Sub Π ω is the full subcategory of mod Π ω given by objects isomorphic to submodules of direct sums of copies of Π ω . Then Sub Π ω is closed under extensions and subobjects (in particular under kernels of epimorphisms), and contains proj Π ω and Ω(mod Π ω ). Since Sub Π ω is Hom-finite, it follows from Proposition 3.7 that it is a Frobenius cluster category.
We note that the categories Sub Π ω constructed by Buan-Iyama-Reiten-Scott contain an important class of categories considered by Geiß-Leclerc-Schröer [GLS08], which we will also describe. For ∆ a Dynkin diagram, let Π be the preprojective algebra of type ∆. If j is a node of ∆, write Q j for the injective Π-module with socle at j. Then for any subset J of the nodes of ∆, we may write Q J = j∈J Q j , and consider the category Sub Q J of Π-modules isomorphic to a submodule of a direct sum of copies of Q J , or equivalently of those Π-modules with socle supported on J. The category Sub Q J models a cluster algebra structure on the coordinate ring of a dense open subset of the partial flag variety attached to the data of ∆ and J. For example, when ∆ is of type A n and J consists of a single node, this partial flag variety is a Grassmannian. If ω 0 is the longest word in the Weyl group of type ∆, and ω K 0 is the longest word in the subgroup generated by simple reflections at nodes not in J, then [GLS11, Lem. 17.2] we have Sub Q J = Sub Π ω K 0 ω 0 . In particular, the categories Sub Q J are Frobenius cluster categories.
If Π is the preprojective algebra of Dynkin type ∆, then we have mod Π = Sub Π = Sub Π ω 0 , where ω 0 is the longest word in the Weyl group of type ∆, so mod Π is a Frobenius cluster category. The algebra A appearing in Example 2.10 is isomorphic to the endomorphism algebra of a cluster-tilting object in mod Π for Π the preprojective algebra of type A 2 , and is thus internally 3-Calabi-Yau by Theorem 3.4. Similarly, the algebra A ′ from Example 2.10 is isomorphic to the endomorphism algebra of a clustertilting object in Sub Π s 2 s 1 s 3 s 2 = Sub Q 2 , where Π is the preprojective algebra of type A 3 and Q 2 is the indecomposable injective module with socle at the bivalent vertex 2, and so A ′ is also internally 3-Calabi-Yau.
The category of projective objects of Sub Π ω is given by add Π ω . Since Sub Π ω is a Hom-finite Frobenius cluster category, it follows from 3.10 that Sub Π ω ≃ GP(Π ω ). Since Π w has Gorenstein dimension at most 1 [BIRS09, Prop. III.2.2], we even have Sub Π ω = GP(Π ω ) as full subcategories of mod Π w . Note that the Gorenstein dimension of Π w is strictly smaller than the bound provided by 3.10.
Example 3.12. Our second family of examples was introduced by Jensen-King-Su [JKS16] to categorify the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian G n k of k-planes in C n . Each category in this family is of the form CM(B) for a Gorenstein order B (depending on positive integers 1 < k < n) over Z = C[[t]]. One description of B is as follows. Let ∆ be the graph (of affine type A n−1 ) with vertex set given by the cyclic group Z n , and edges between vertices i and i + 1 for all i. Let Π be the completion of the preprojective algebra on ∆ with respect to the arrow ideal. Write x for the sum of 'clockwise' arrows i → i + 1, and y for the sum of 'anti-clockwise' arrows i → i − 1. Then we have In this description, Z may be identified with the centre C[[xy]] of B.
Objects of CM(B) are B-modules that are free and finitely generated over Z. Since Z is a principal ideal domain, and hence Noetherian, any submodule of a free and finitely generated Z-module is also free and finitely generated, and so CM(B) is closed under subobjects. In particular, CM(B) is closed under kernels of epimorphisms. Moreover Baur-King-Marsh [BKM16, Thm. 10.3] show that for certain cluster-tilting objects T ∈ CM(B), the endomorphism algebra End B (T ) op is isomorphic to a frozen Jacobian algebra (Definition 5.1) associated to a dimer model on a disk, with the projection onto a maximal projective-injective summand corresponding to the sum of idempotents at the frozen vertices. By Theorem 3.4, these dimer algebras, which satisfy a natural consistency condition [BKM16,§5], are internally 3-Calabi-Yau with respect to their boundary idempotent; cf. Broomhead [Bro12,§7], who shows that consistent dimer models on closed surfaces give rise to 3-Calabi-Yau Jacobian algebras.
where ∆ is the extended Dynkin diagram given by the McKay graph of G, and ∆ is its unextended counterpart. It is well-known that Π(∆) is finite dimensional, so CM(R) is Hom-finite, and that gl. dim Π( ∆) ≤ 2. Thus CM(R) is a Frobenius 1-cluster category. The algebra End R (T ) op ∼ = Π( ∆) is bimodule 2-Calabi-Yau, which is consistent with (but stronger than) the conclusion of Theorem 3.4.

From internally d-Calabi-Yau algebras to d-cluster-tilting objects
Theorem 3.4 shows how internally (m + 1)-Calabi-Yau algebras arise as endomorphism algebras of cluster-tilting objects in Frobenius m-cluster categories. In this section we work in the opposite direction, and show how to construct a Frobenius category admitting a (d − 1)-cluster-tilting object from the data of an internally d-Calabi-Yau algebra, thus generalising a result of Amiot-Iyama-Reiten [AIR15, Thm. 2.3] on bimodule d-Calabi-Yau algebras. Since we will work only with Noetherian algebras, by 3.10 we should expect to produce categories of the form GP(B) for some Iwanaga-Gorenstein algebra B, and indeed this is what we shall do. Our main result is the following. The rest of the section is largely devoted to proving Theorem 4.1, so we let A, e, A and B be as in the assumptions of this theorem. We begin with the following straightforward observation.  Proof. Both A and Ae are finitely generated projective A-modules, and so are in particular perfect. Since A, and therefore X, is finite dimensional, X is a finitely generated A-module, and thus perfect since A is Noetherian of finite global dimension. Now we can use the internal Calabi-Yau duality of A to deduce that The assumption of part (i) of Proposition 4.4 is slightly more restrictive than that of [AIR15, Lem. 2.6(a)]. This is necessary for the result to hold in our setting, since our A is only internally d-Calabi-Yau. However, this stronger assumption is satisfied whenever [AIR15, Lem. 2.6(a)] is used in the proof of [AIR15, Thm. 2.3].

Theorem 4.1 (cf. [AIR15, Thm. 2.3]). Let A be a Noetherian algebra and let e ∈ A be an idempotent such that A/ e is finite dimensional, and both A and
The following results (Proposition 4.5 and Lemmas 4.6, 4.8 and 4.9) are now close analogues of [AIR15, Prop. 2.7, Lem. 2.9-2.11], with very similar proofs. For the convenience of the reader, and to make the paper more self-contained, we give a complete argument using our notation and conventions.  (1) Ae

modules, and isomorphisms
It follows that eH i (X) = 0, and hence H i (X) ∈ mod A for all i ∈ Z. Thus, by Proposition 4.4, Ext j A (H i (X), Ae) = 0 for all i, j ∈ Z. We can compute H k (RHom A (X, Ae)) via a hypercohomology spectral sequence II E ij r [Wei94, §5.7.9, see also Defn. 5.6.2], in which , Ae) = 0 as above. It follows that H k (RHom A (X, Ae)) = 0 for all k, and so RHom A (X, Ae) = 0. Now applying RHom A (−, Ae) to the triangle (1) yields the required isomorphism in DA ⊗ K B op , from which the first assertion follows by our initial calculations. Similarly, we have isomorphisms and so to obtain the second assertion we wish to show that Both Ae and eA are concentrated in degree 0, so by triangle (1)  From (1), we obtain the long exact sequence As Hom DA (X, A[i]) = 0 for i < d as above, it follows from our initial calculations that as required.
Proof. Since B is Noetherian by Proposition 4.3, X has a projective presentation P 1 → P 0 → X → 0 with P 0 and P 1 finitely generated. Applying Hom B (−, eA) gives the exact sequence 0 Hom B (X, eA) . If X ∈ GP(B) and Ext i B (X, eA) = 0 for all 0 < i < d − 1, then X ∈ add B (eA).
Proof. Pick an exact sequence  We are now ready to prove Theorem 4.1.
Proof of Theorem 4.1.
(i) We have already shown in Proposition 4.3 that B is Noetherian, so it remains to show that B has injective dimension at most d on each side. First we show that Ext d+1 B (X, B) = 0 for all X ∈ mod B. Given such an X, let Y = Ae ⊗ B X, and let P be a projective resolution of Y . Then eP is a bounded complex in the full subcategory add(eA) of mod B, quasi-isomorphic to eY = X. By Proposition 4.5, Ext i B (eA, B) = 0 for i > 0, so another spectral sequence argument (now using I E r pq from [Wei94, Defn. 5.6.1]) shows that Ext d+1 B (X, B) ∼ = H d+1 (Hom B (eP, B) where C denotes the ideal of maps factoring through the subcategory C.
In the setting of Theorem 4.1, we would also like to conclude that GP(B) is stably (d − 1)-Calabi-Yau. We now show, using descriptions by Kalck-Yang [KY16] of GP(B) in terms of (complexes of) A-modules, that this category is (d − 1)-Calabi-Yau when we strengthen the assumptions of Theorem 4.1 to require that A is bimodule internally d-Calabi-Yau. , and that the codomain of this equivalence coincides with thick(mod A). Since i * i * ≃ 1 DC , we see that i * restricts to an equivalence thick(mod A) ∼ → D fd (C), and so induces an equivalence where q denotes the projection D b A → D b A/ thick Ae, which restricts to an equivalence on thick(mod A) by the above observations. We also have equivalences It was necessary in the proof of Theorem 4.10 to use the bimodule internal Calabi-Yau symmetry of A to obtain a duality between spaces of maps of complexes of A-modules, so it is unclear whether the conclusion that GP(B) is (d − 1)-Calabi-Yau might hold only under the weaker assumptions of Theorem 4.1. As already stated, we do not currently have any examples of internally Calabi-Yau algebras that are not bimodule internally Calabi-Yau, but it seems unlikely that the two classes coincide.
Under the assumptions of Theorem 4.10, we would like to be able to conclude that the Frobenius category GP(B) is in fact a Frobenius (d − 1)-cluster category in the sense of Definition 3.3. It remains to check that gl. dim End B (T ) op ≤ d for any (d − 1)-clustertilting object T ∈ GP(B); a priori, we only know this for the (d − 1)-cluster-tilting object eA. Whenever End B (T ) op is Noetherian, we can apply Proposition 3.7 to get the desired conclusion. While this Noetherianity is automatic in some situations, such as if B is finite dimensional over K, in general it appears to be a more subtle issue. Remark 4.11. As indicated in the introduction, Theorems 4.1 and 4.10 are motivated by the problem of constructing Frobenius categories modelling cluster algebras. Given the seed of a cluster algebra with frozen variables, we can look for an algebra A with the same quiver (up to the addition of arrows between frozen vertices), satisfying the conditions of Theorem 4.10 for d = 3, and then apply this theorem to obtain the Frobenius category GP(B). Since constructing such an A can be very difficult, we wish to comment on the degree to which the conditions we are imposing are necessary.
Firstly, we consider it likely that the condition that A is Noetherian can be dropped, up to finding an appropriate replacement for the category GP(B). While B may not be Noetherian if A fails to be, it will still have injective dimension at most d on each side, so there should be a 'good' theory of Gorenstein projective modules over B. For our methods to work, we would need the analogues of [AIR15, Prop. 1.3] to hold in this setting. We would also hope for a more general version of the Iyama-Kalck-Wemyss-Yang equivalence stated here as Theorem 3.9, without the Noetherianity assumption, which would then apply to arbitrary Frobenius m-cluster categories, and Buchweitz's description of the stable category [Buc87,Thm. 4.4.1].
The other conditions are more essential; if A = End E (T ) op for T a cluster-tilting object in a Frobenius cluster category E, and e is the idempotent given by projecting onto a maximal projective summand of T , then A/ e must be finite dimensional since E is Hom-finite, and A is internally 3-Calabi-Yau with respect to e by Theorem 3.4. On the other hand, it may not be necessary for A to be bimodule internally 3-Calabi-Yau.

A bimodule complex for frozen Jacobian algebras
Given a Frobenius cluster category E and a cluster-tilting object T ∈ E, it is often the case that A = End E (T ) op takes the form of a frozen Jacobian algebra (see Definition 5.1 below). Indeed, this is the case for at least some cluster-tilting objects in the families of Frobenius cluster categories we described in Examples 3.11 and 3.12; see [BIRS11,Thm. 6.6], [BKM16,Thm. 10.3]. Thus these algebras, which also come with a preferred 'frozen' idempotent, are ideal candidates for constructing stably 2-Calabi-Yau Frobenius categories via the methods of Theorem 4.1 and Theorem 4.10. Moreover, 3-Calabi-Yau properties of ordinary Jacobian algebras have been widely studied, for example by Bocklandt [Boc08] and, in the context of dimer models on closed surfaces, by Broomhead [Bro12].
With this in mind, the main result of this section, Theorem 5.6, shows that a frozen Jacobian algebra admitting a particular bimodule resolution (analogous to one defined by Ginzburg [Gin07,5.1.5]) is bimodule internally 3-Calabi-Yau with respect to its frozen idempotent. ). An ice quiver (Q, F ) consists of a finite quiver Q without loops and a (not necessarily full) subquiver F of Q. Denote by KQ the completion of the path algebra of Q over K with respect to the arrow ideal. A potential on Q is a linear combination W of cycles of Q. 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 m 0 = Q 0 \ F 0 and Q m 1 = Q 1 \ F 1 for the sets of mutable vertices and unfrozen arrows respectively. For α ∈ Q 1 and α n · · · α 1 a cycle in Q, write and extend linearly. The ideal ∂ α W : α ∈ Q m 1 of KQ is called the Jacobian ideal, and we may take its closure ∂ α W : α ∈ Q m 1 since KQ is a topological algebra. We define the frozen Jacobian algebra associated to (Q, F, W ) by F, W ). The above presentation of A suggests a preferred idempotent e = v∈F 0 e v , which we call the frozen idempotent. We will call B = eAe the boundary algebra of A. the frozen subquiver F is the full subquiver on vertices 1 and 2, indicated by boxed vertices and a dashed arrow, and W = α 3 α 2 α 1 . Then the frozen Jacobian algebra J (Q, F, W ) is the quotient of KQ by the relations ∂ α 2 W = α 1 α 3 and ∂ α 3 W = α 2 α 1 , so J (Q, F, W ) is the algebra A from Example 2.10. (In this case A is finite dimensional, so it agrees with the quotient of the ordinary path algebra KQ by the same relations.) It will follow from Theorem 5.6 below that J (Q, F, W ) is bimodule internally 3-Calabi-Yau with respect to the idempotent e 1 + e 2 given by summing the idempotents corresponding to frozen vertices. The usual Jacobian algebra J (Q, W ) has the additional relation α 3 α 2 = 0 and is not bimodule 3-Calabi-Yau; indeed, it has infinite global dimension.
Given a quiver with potential (Q, W ), Ginzburg [Gin07, 5.1.5] (see also [Bro12,§7]) defines a complex of projective bimodules over the associated Jacobian algebra. For (Q, W ) a quiver with potential determined by a dimer model on a torus, Broomhead shows in [Bro12,Thm. 7.7] that if the dimer model is consistent, then this complex is isomorphic to A = J (Q, W ) in D b A ε , and thus provides a projective bimodule resolution of A. It follows in this case that A is 3-Calabi-Yau, with this property arising from a natural symmetry in the bimodule resolution.
We will now define an analogous complex P(A) for a frozen Jacobian algebra A = J (Q, F, W ). Our main result (Theorem 5.6) will be that if P(A) is isomorphic to A in D b A ε , then A is bimodule internally 3-Calabi-Yau with respect to the frozen idempotent e = v∈F 0 e v , in the sense of Definition 2.4. While we will write P(A) for this complex in order to save space, the definition depends not only on A but on the ice quiver with potential (Q, F, W ) giving the presentation of A as J (Q, F, W ).
Recall that we write Q m 0 = Q 0 \ F 0 for the set of mutable vertices and Q m 1 = Q 1 \ F 1 for the set of unfrozen arrows. We also write v + for the set of arrows with tail at v, and v − for the set of arrows with head at v. Denote the arrow ideal of A by m(A), and let S = A/m(A). For the remainder of this section, we write ⊗ = ⊗ S .
Introduce formal symbols ρ α for each α ∈ Q 1 and ω v for each v ∈ Q 0 , and define S-bimodule structures on the vector spaces Ke v , Kα, where hα and tα denote the head and tail of the arrow α. For each i, the S-bimodule KQ i splits as the direct sum is canonically isomorphic to A ⊗ A, and we will use the two descriptions interchangeably.
We define mapsμ i : For any path p = α m · · · α 1 of Q, we may define and extend by linearity to obtain a map ∆ α : KQ → A ⊗ KQ 1 ⊗ A. We then definē

is a complex of finitely generated projective A-bimodules, and there is a morphism
Proof. Each term of P(A) is a projective A-bimodule since A is a projective A-module on each side, and is finitely generated by finiteness of Q. By standard results on presentations of algebras, see for example Butler-King [BK99, 1.2], µ 0 is surjective, im µ 1 = ker µ 0 and im µ 2 = ker µ 1 . Thus the only statement left to check is that µ 2 • µ 3 = 0.
Let v ∈ Q 0 , and write We can calculate γ∈Q 1 ∆ γ (W v ) using each of the two expressions, to get If v ∈ Q m 0 , then all arrows incident with v are unfrozen, and so ∂ α W = 0 = ∂ β W in A for any α ∈ v + and β ∈ v − . Thus in this case we have It follows that This completes the proof.

If the map
from Lemma 5.5 is a quasi-isomorphism, then P(A) is a projective bimodule resolution of A. This means that, for the presentation of A as a frozen Jacobian algebra, with relations given by certain derivatives of the superpotential, the first syzygies are dual to the mutable vertices, and there are no higher syzygies. In particular, gl. dim A ≤ 3. By Lemma 5.5, this map is a quasi-isomorphism if and only if the cohomology of P(A) vanishes at A ⊗ KQ m 2 ⊗ A and A ⊗ KQ m 3 ⊗ A (cf. [Bro12,Rem. 7.4]). We will usually abuse notation and denote the map P(A) → A from Lemma 5.5 by µ 0 .
If F = ∅, the map µ 0 : P(A) → A being a quasi-isomorphism implies that A is 3-Calabi-Yau [Gin07, Cor. 5.3.3; Bro12, Thm. 7.7]. We now show that, in the general case, µ 0 being a quasi-isomorphism implies that A is bimodule internally 3-Calabi-Yau with respect to e. Theorem 5.6. If A is a frozen Jacobian algebra such that µ 0 : P(A) → A is a quasiisomorphism, then A is bimodule internally 3-Calabi-Yau with respect to the frozen idempotent e = v∈F 0 e v .
Define µ ∨ 2 to be the composition ofμ 2 with the projection A⊗KQ 1 ⊗A → A⊗KQ m 1 ⊗A; explicitly Finally, let µ ∨ 3 =μ 3 . Then one can check that the isomorphisms of A-bimodules defined above induce an isomorphism of Ω A with the complex As an example to illustrate the necessary calculations, we show that we get an isomorphism of µ * 2 : We then calculate for each β ∈ Q m 1 that We must then have and so the isomorphism as required.

Now consider the commutative diagram
(2) in which the columns are split exact, the second row is P(A), the third row is isomorphic to Ω A [3] by the preceding calculations, and the signs on the vertical arrows indicate whether the corresponding map is the inclusion or its negative. The diagram (2) provides us with a map of complexes A ∼ = P(A) → Ω A [3] in D b A ε , and shows that the cone of this map has the form Let M ∈ D A (A) have finite dimensional total cohomology. Since the cohomology of M is concentrated in some interval, M ∈ D b A. We pick a bounded representative M • of the quasi-isomorphism class of M, allowing us to compute the complex RHom A (C, M) as the total complex of the double complex with terms where V 1 = KF 3 , V 2 = KF 2 ⊕ KF 1 , V 3 = KF 0 and V i = 0 for all other i. Since each S-bimodule V i has the property that eV i e = V i , we have so the terms of the relevant double complex are isomorphic to Hom S (V i ⊗ eA, Hom A (Ae, M j )) = Hom S (V i ⊗ eA, eM j ).
Since M ∈ D A (A), the complex eM • is acyclic. Moreover, since S is semi-simple, Hom S (V i ⊗ eA, −) preserves acyclicity, so the vertical cohomology of the double complex vanishes. It follows that RHom A (C, M) = 0. A similar argument shows that RHom A op (C, N) = 0 for all N ∈ D A op (A op ), so we conclude that A is bimodule internally 3-Calabi-Yau with respect to e.
Example 5.7. Since the frozen Jacobian algebra A in Examples 2.10 and 5.3 is finite dimensional (of small dimension!) it is possible to check that P(A) ∼ → A directly by choosing a basis of A, whence this algebra is bimodule internally 3-Calabi-Yau with respect to this frozen idempotent. The algebra A ′ in Example 2.10 may also be presented as a frozen Jacobian algebra, and again the fact that A ′ is finite dimensional allows us to check directly that P(A ′ ) ∼ → A ′ , giving the bimodule internally 3-Calabi-Yau property. However, we also have many more examples, some of which are infinite dimensional. where the frozen subquiver is indicated by boxed vertices and dashed arrows as before, and let W be the potential given by the sum of the 3-cycles minus the sum of the 4cycles. WritingÂ = J (Q, F, W ), we have P(Â) ∼ →Â. We omit the calculation here, but observe that it there is a grading ofÂ in which all arrows have positive degree, meaning that by [Bro12,Prop. 7.5] it is sufficient to check that µ 0 ⊗Â S i : P(Â) ⊗Â S i → S i is a quasi-isomorphism for each simpleÂ-module S i , which is more straightforward. One can also check thatÂ is the endomorphism algebra of a cluster-tilting object in Jensen-King-Su's categorification of the cluster structure on the Grassmannian G 6 2 , described in Example 3.12.
The existence of a quasi-isomorphism P(A) ∼ → A allows us to deduce many homological properties of A and of the boundary algebra B. For example, any A-module M has a (usually non-minimal) projective resolution P(A) ⊗ A M. Using this, we see immediately that if M is any A-module such that M = eM, such as a simple module at a frozen vertex, then A ⊗ KQ m 3 ⊗ M = 0, and p. dim A M ≤ 2. It follows from Theorem 5.6 that if A is a frozen Jacobian algebra with the property that µ 0 : P(A) → A is a quasi-isomorphism, then both A and A op are internally 3-Calabi-Yau with respect to the frozen idempotent e. If additionally A is Noetherian and A/ e is finite dimensional, then we may apply Theorem 4.1. We may also make some observations about the homological algebra of the boundary algebra B in this situation. By Theorem 4.1(iii), the functor eA ⊗ A − ⊗ A Ae : proj A ε → mod B ε is fully faithful, and so Hom B ε (eP i e, B ε ) ∼ = Hom A ε (P i , Ae ⊗ K eA) = e Hom A ε (P i , A ε )e. Proof. The proof of Theorem 5.6 constructs a map A → Ω A [3] with mapping cone

It follows that
Since each S-bimodule KF i has the property that e(KF i )e = KF i , we can instead write C as