2 Coefficient systems and cluster categories

In this section, we introduce our main definitions. We will not give the full definition of a cluster algebra, since this is somewhat lengthy and can be found easily in many other sources; we recommend Keller’s survey [Reference Keller33]. For concreteness, our cluster algebras will be defined over $\mathbb {Q}$ , and we do not assume frozen variables to be invertible; that is, the cluster algebra $\mathscr {A}_{}$ generated by an initial seed with cluster $(x_1,\dotsc ,x_n,x_{n+1},\dotsc ,x_m)$ , in which the mutable variables are $x_1,\dotsc ,x_n$ , is defined to be a subalgebra of $\mathbb {Q}(x_1,\dotsc ,x_n)[x_{n+1},\dotsc ,x_m]$ . However, since our results are concerned with the set of cluster variables of $\mathscr {A}_{}$ , their grouping into clusters, and the exchange graph on these clusters, they are insensitive to changing these conventions by replacing $\mathbb {Q}$ by a field extension or inverting the frozen variables. As such, we de-emphasize these conventions below, and simply refer to the cluster algebra generated by a seed.

Let $\mathscr {A}_{}$ be a cluster algebra of geometric type without frozen variables, and let $s_0$ be a seed of $\mathscr {A}_{}$ , with quiver Q and cluster variables $(x_1,\dotsc ,x_n)$ . By definition, the quiver Q has no loops or $2$ -cycles. The principal coefficient cluster algebra $\mathscr {A}_{Q}^{\bullet }$ corresponding to these data is defined by the following initial seed. The mutable cluster variables are again $(x_1,\dotsc ,x_n)$ , and the frozen variables are $(x_1^+,\dotsc ,x_n^+)$ , where the indexing reveals a preferred bijection between the mutable and frozen variables. The ice quiver $Q^{\bullet }$ of this seed contains Q as a full subquiver, with mutable vertices, and for each vertex $i\in Q_0$ (corresponding to the variable  $x_i$ ), $Q^{\bullet }$ has a frozen vertex $i^+$ (corresponding to $x_i^+$ ) and an arrow $i\to i^+$ . While $\mathscr {A}_{}$ is isomorphic (as a cluster algebra) to the cluster algebra determined by any quiver mutation equivalent to Q, this is not true of $\mathscr {A}_{Q}^{\bullet }$ .

Specializing all frozen variables of $\mathscr {A}_{Q}^{\bullet }$ to $1$ to obtain a cluster algebra without frozen variables (which we refer to as taking the principal part) recovers $\mathscr {A}_{}$ , and gives a bijection between the seeds of $\mathscr {A}_{Q}^{\bullet }$ and those of $\mathscr {A}_{}$ [Reference Cerulli Irelli, Keller, Labardini-Fragoso and Plamondon14]; we write $s^{\bullet }$ for the seed of $\mathscr {A}_{Q}^{\bullet }$ corresponding to a seed s of $\mathscr {A}_{}$ . Precisely, if $s^{\bullet }$ has cluster $(v_1,\dotsc ,v_n,x_1^+,\dotsc ,x_n^+)$ , then s has cluster $(v_1,\dotsc ,v_n)|_{x_i^+=1,1\leq i\leq n}$ , and the quiver of s is the full subquiver of that of $s^{\bullet }$ on the mutable vertices. Principal coefficients are important since knowledge of the cluster algebra $\mathscr {A}_{Q}^{\bullet }$ can be used to study any cluster algebra $\mathscr {A}'$ with principal part $\mathscr {A}$ , via the theory of g-vectors and F-polynomials [Reference Fomin and Zelevinsky20].

By choosing some extra data on Q, Amiot [Reference Amiot1] constructs a categorification of $\mathscr {A}_{}$ . The construction uses Jacobian algebras, so we recall some relevant definitions, at the level of generality needed later in the paper.

Definition 2.1. An ice quiver $(Q,F)$ consists of a finite quiver Q without loops and a subquiver F of Q. Denote by $\mathbb {K}\langle \hspace {-0.1em}\langle Q\rangle \hspace {-0.1em}\rangle $ the completion of the path algebra of Q over our fixed algebraically closed field $\mathbb {K}$ with respect to the ideal $J_Q$ generated by arrows, treated as a topological algebra with the $J_Q$ -adic topology. A potential on Q is an element W of the vector space quotient $\mathbb {K}\langle \hspace {-0.1em}\langle Q\rangle \hspace {-0.1em}\rangle /\overline {\{\mathbb {K}\langle \hspace {-0.1em}\langle Q\rangle \hspace {-0.1em}\rangle ,\mathbb {K}\langle \hspace {-0.1em}\langle Q\rangle \hspace {-0.1em}\rangle \}}$ , of $\mathbb {K}\langle \hspace {-0.1em}\langle Q\rangle \hspace {-0.1em}\rangle $ by the closure of the subspace spanned by commutators, such that W lies in the image of the natural map from $J_Q$ to this vector space. (More informally, W is a possibly infinite linear combination of cyclic equivalence classes of cycles of positive length in 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 $Q_0^\circ =Q_0\setminus F_0$ and $Q_1^\circ =Q_1\setminus F_1$ for the sets of mutable vertices and unfrozen arrows, respectively. For $\alpha \in Q_1$ and $\alpha _k\dotsm \alpha _1$ a cycle in Q, write

$$\begin{align*}\partial_{\alpha}{\alpha_k\dotsm\alpha_1}=\sum_{\alpha_i=\alpha}\alpha_{i-1}\dotsm\alpha_1\alpha_k\dotsm\alpha_{i+1}.\end{align*}$$

Extending the map $\partial _{\alpha }$ linearly and continuously, and noting that it vanishes on commutators, allows us to define $\partial _{\alpha }{W}$ . The ideal $\langle \partial _{\alpha }{W}:\alpha \in Q_1^\circ \rangle $ of $\mathbb {K}\langle \hspace {-0.1em}\langle Q\rangle \hspace {-0.1em}\rangle $ is called the Jacobian ideal, and we define the frozen Jacobian algebra associated with $(Q,F,W)$ by

$$\begin{align*}J(Q,F,W)=\mathbb{K}\langle\hspace{-0.1em}\langle Q\rangle\hspace{-0.1em}\rangle/\overline{\langle\partial_{\alpha}{W}:\alpha\in Q_1^\circ\rangle}.\end{align*}$$

Write $A=J(Q,F,W)$ . The above presentation of A suggests a preferred idempotent

$$\begin{align*}e=\sum_{v\in F_0}e_{v},\end{align*}$$

which we call the frozen idempotent. We will call the subalgebra $B=eAe$ the boundary algebra of A. If $F=\varnothing $ , then we refer to the pair $(Q,W)$ as a quiver with potential, and call $J(Q,W)=J(Q,\varnothing ,W)$ simply the Jacobian algebra of $(Q,W)$ . If $J(Q,W)$ is finite-dimensional, we call both W and the pair $(Q,W)$ Jacobi-finite.

Assume that Q is a quiver admitting a Jacobi-finite potential W. Then, by work of Amiot [Reference Amiot1], there is a $2$ -Calabi–Yau triangulated category $\mathcal {C}_{Q,W}$ categorifying $\mathscr {A}_{Q}$ . If W is additionally nondegenerate [Reference Derksen, Weyman and Zelevinsky18, Def. 7.2], then the seeds of $\mathscr {A}_{}$ correspond bijectively to additive equivalence classes of cluster-tilting objects of $\mathcal {C}_{Q,W}$ related by a finite sequence of mutations to an initial cluster-tilting object $T_0$ with $\operatorname {End}_{\mathcal {C}_{Q,W}}(T_0)^{\mathrm {op}}=J(Q,W)$ ; such cluster-tilting objects are called reachable, and we denote the seed corresponding to such an object T by $s_T$ . Nondegeneracy is needed here to see, using results of Buan–Iyama–Reiten–Smith [Reference Buan, Iyama, Reiten and Smith9], that mutation of cluster-tilting objects in the mutation class of $T_0$ corresponds to Fomin–Zelevinsky mutation at the level of quivers of endomorphism algebras, and so in particular mutation is well defined at any indecomposable summand of any reachable cluster-tilting object.

A priori, to categorify the principal coefficient cluster algebra $\mathscr {A}_{Q}^{\bullet }$ , we would like to find a Frobenius category $\mathcal {E}$ such that:

1. (a) The stable category $\underline {\mathcal {E}}$ is triangle equivalent to $\mathcal {C}_{Q,W}$ (so that cluster-tilting objects of $\underline {\mathcal {E}}$ may be identified with those of $\mathcal {C}_{Q,W}$ , and hence cluster-tilting objects in $\mathcal {E}$ can be mutated at their nonprojective indecomposable summands).

2. (b) For any reachable cluster-tilting object T, there is an isomorphism $\mathbb {K}\langle \hspace {-0.1em}\langle Q'\rangle \hspace {-0.1em}\rangle /\overline {I}\stackrel {\sim }{\to }\operatorname {End}_{\mathcal {E}}(T)^{\mathrm {op}}$ , for some ideal $I\subseteq J_{Q'}^2$ , where $Q'$ is, up to arrows between frozen vertices, the quiver of the seed $s_T^{\bullet }$ of $\mathscr {A}_{Q}^{\bullet }$ . (As always, we define the frozen vertices of $Q'$ to be those whose image under the given isomorphism to $\operatorname {End}_{\mathcal {E}}(T)^{\mathrm {op}}$ is a projection onto an indecomposable projective summand.)

Unfortunately, this is not possible.

For this reason, we will first consider, and categorify, a different cluster algebra $\widetilde {\mathscr {A}}_{Q}$ with initial seed obtained from that of $\mathscr {A}_{Q}^{\bullet }$ by adding more frozen variables. Starting from our seed $s_0$ of $\mathscr {A}_{}$ , with quiver Q and cluster variables $(x_1,\dotsc ,x_n)$ , we construct an initial seed $\widetilde {s}_0$ of $\widetilde {\mathscr {A}}_{Q}$ as follows: the seed $\widetilde {s}_0$ has mutable variables $(x_1,\dotsc ,x_n)$ , and frozen variables $(x_1^+,\dotsc ,x_n^+,x_1^-,\dotsc ,x_n^-)$ . Its ice quiver $\widetilde {Q}$ is described fully in Definition 3.1; the relevant features for defining the cluster algebra are that $\widetilde {Q}$ contains Q as a full subquiver, with mutable vertices, and has two frozen vertices $i^+$ (corresponding to $x_i^+$ ) and $i^-$ (corresponding to $x_i^-)$ for each mutable vertex $i\in Q_0$ , with arrows $i\to i^+$ and $i^-\to i$ . (In Definition 3.1, we also describe arrows between the frozen vertices that will play a role in our categorification, but are not important for defining $\widetilde {\mathscr {A}}_{Q}$ .) We call $\widetilde {\mathscr {A}}_{Q}$ the polarized principal coefficient cluster algebra associated with Q.

We adopt the word polarised, referring to the partitioning of the frozen variables into two sets, to differentiate this coefficient system from the double principal coefficients studied by Rupel, Stella, and Williams [Reference Rupel, Stella and Williams46]. Since one encounters the same issues categorifying cluster algebras with double principal coefficients as in the case of ordinary principal coefficients, namely that no categorification may have enough injective objects, our preference here is for the polarized version.

As discussed in the introduction, we will construct a categorification of $\widetilde {\mathscr {A}}_{Q}$ using methodology introduced by the author in [Reference Pressland42]. We now recall the key definitions and results needed to explain this construction. Given a $\mathbb {K}$ -algebra A, we write ${\mathcal {D}}(A)$ for its derived category and ${A}^{\varepsilon }=A\mathbin {\otimes }_{\mathbb {K}} A^{\mathrm {op}}$ for its enveloping algebra, modules of which are precisely A-bimodules. We denote by $\operatorname {per}{A}$ the perfect derived category of A, that is, the thick subcategory of ${\mathcal {D}}(A)$ generated by A, and write $\Omega _{A}=\operatorname {\mathbf {R}Hom}_{{A}^{\varepsilon }}(A,{A}^{\varepsilon })$ .

Remark 2.5. Assume that A is bimodule internally $3$ -Calabi–Yau with respect to e. Then $\operatorname {gl.dim}{A}\leq 3$ , and there is a functorial duality

$$\begin{align*}\mathrm{D}\operatorname{Ext}^i_A(M,N)=\operatorname{Ext}^{3-i}_A(N,M) \end{align*}$$

for any finite-dimensional $A/AeA$ -module M and any A-module N (see [Reference Pressland42, Cor. 2.9]). Moreover, $A^{\mathrm {op}}$ has the same properties (see [Reference Pressland42, Rem. 2.6]).

To construct our Frobenius categories, we will use the following theorem.

In many cases—in particular in our application in the proof of Theorem 1—the Frobenius category $\operatorname {GP}(B)$ appearing in Theorem 2.6 is a Frobenius cluster category, as studied in [Reference Pressland42], and defined as follows.

Returning to the problem of categorifying the cluster algebra $\widetilde {\mathscr {A}}_{Q}$ , our aim now is to construct an algebra A satisfying the conditions of Theorem 2.6, defined as a quotient of the complete path algebra of the quiver $\widetilde {Q}$ used to define $\widetilde {\mathscr {A}}_{Q}$ .

3 An ice quiver with potential

Consider again our initial seed $s_0$ for $\mathscr {A}_{}$ , with quiver Q, and choose a potential W on Q. In this section, we will construct from $(Q,W)$ an ice quiver with potential $(\widetilde {Q},\widetilde {F},\widetilde {W})$ , and thus a frozen Jacobian algebra $J(\widetilde {Q},\widetilde {F},\widetilde {W})$ . It is this algebra that we intend to use as the input for the construction of a Frobenius category by Theorem 2.6; to apply this theorem, 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 $\widetilde {Q}$ to be the quiver with vertex set given by

$$\begin{align*}\widetilde{Q}_0=Q_0\sqcup Q_0^+\sqcup Q_0^-,\end{align*}$$

where $Q_0^+=\{i^+:i\in Q_0\}$ is a set of formal symbols in bijection with $Q_0$ , and similarly for $Q_0^-=\{i^-:i\in Q_0\}$ . The set of arrows is given by

$$\begin{align*}\widetilde{Q}_1=Q_1\sqcup\{\alpha_i:i\in Q_0\}\sqcup\{\beta_i:i\in Q_0\}\sqcup\{\delta_i:i\in Q_0\}\sqcup\{\delta_a:a\in Q_1\},\end{align*}$$

and the head and tail functions $h,t\colon \widetilde {Q}_1\to \widetilde {Q}_0$ are extended from those of Q by defining

$$ \begin{align*} h{\alpha_i}&=i^+,&h{\beta_i}&=i,&h{\delta_i}&=i^-,&h{\delta_a}&=(t{a})^-,\\ t{\alpha_i}&=i,&t{\beta_i}&=i^-,&t{\delta_i}&=i^+,&t{\delta_a}&=(h{a})^+. \end{align*} $$

The frozen subquiver $\widetilde {F}$ is defined by

$$\begin{align*}\widetilde{F}_0=Q_0^+\sqcup Q_0^-,\qquad \widetilde{F}_1=\{\delta_i:i\in Q_0\}\sqcup\{\delta_a:a\in Q_1\}. \end{align*}$$

Note that the head and tail of any arrow in $\widetilde {F}_1$ lie in $\widetilde {F}_0$ , so these subsets describe a valid subquiver of $\widetilde {Q}$ , which is even full. The quiver $\widetilde {F}$ is also bipartite, meaning that every vertex is either a source or a sink, and so it has no paths of length greater than $1$ ; precisely, the vertices in $Q_0^+$ are sources in $\widetilde {F}$ , and those in $Q_0^-$ are sinks. As a vertex of $\widetilde {Q}$ , each $i^+$ has unique incoming arrow $\alpha _i$ and each $i^-$ has unique outgoing arrow $\beta _i$ .

Finally, we define a potential $\widetilde {W}$ on $\widetilde {Q}$ by

$$\begin{align*}\widetilde{W}=W+\sum_{i\in Q_0}\beta_i\delta_i\alpha_i-\sum_{a\in Q_1}a\beta_{t{a}}\delta_a\alpha_{h{a}},\end{align*}$$

and let

$$\begin{align*}A_{Q,W}=J(\widetilde{Q},\widetilde{F},\widetilde{W})\end{align*}$$

be the frozen Jacobian algebra determined by $(\widetilde {Q},\widetilde {F},\widetilde {W})$ . We denote the boundary algebra of $A_{Q,W}$ by $B_{Q,W}=eA_{Q,W}e$ , where $e=\sum _{i\in Q_0}(e_i^++e_i^-)$ is the frozen idempotent of $A_{Q,W}$ .

Since $\widetilde {W}$ 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) $$ \begin{align} \partial_{a}{\widetilde{W}}=\partial_{a}{W}-\beta_{t{a}}\delta_a\alpha_{h{a}},\quad \partial_{\alpha_i}{\widetilde{W}}=\beta_i\delta_i-\sum_{\substack{\gamma\in Q_1\\h{\gamma}=i}}\gamma\beta_{t{\gamma}}\delta_{\gamma},\quad \partial_{\beta_i}{\widetilde{W}}=\delta_i\alpha_i-\sum_{\substack{\gamma\in Q_1\\t{\gamma}=i}}\delta_{\gamma}\alpha_{h{\gamma}}\gamma, \end{align} $$

for $a\in Q_1$ and $i\in Q_0$ . Having such an explicit generating set of relations will prove to be extremely useful later in the paper.

To be able to apply Theorem 2.6, we wish to show that $A_{Q,W}$ is bimodule internally $3$ -Calabi–Yau with respect to its frozen idempotent e, in the sense of Definition 2.4. We will do this, under a mild assumption on $(Q,W)$ , in §4, but first we give some examples.

Example 3.2. The quiver with potential $(Q,0)$ , for Q an $\mathsf {A}_2$ quiver, provides the most basic example revealing all of the combinatorial features of the construction. In this case, we have

with $\widetilde {F}$ indicated by the boxed vertices and dashed arrows. The potential on this ice quiver is

$$\begin{align*}\widetilde{W}=\beta_1\delta_1\alpha_1+\beta_2\delta_2\alpha_2-a\beta_1\delta_a\alpha_2.\end{align*}$$

One can check that the frozen Jacobian algebra $A_Q$ attached to these data is isomorphic to the endomorphism algebra of a cluster-tilting object in the Frobenius cluster category consisting of those modules for the preprojective algebra of type $\mathsf {A}_4$ with socle supported at a fixed bivalent vertex [Reference Geiß, Leclerc and Schröer24]. For an explanation of why the categories arising in [Reference Geiß, Leclerc and Schröer24] are Frobenius cluster categories, see [Reference Pressland42, Exam. 3.11].

Example 3.3. Now, let $(Q,W)$ be the quiver with potential in which

and $W=cba$ . The Jacobian algebra is a cluster-tilted algebra of type $\mathsf {A}_3$ , and has infinite global dimension (like any nonhereditary cluster-tilted algebra [Reference Keller and Reiten35, Cor. 2.1]). In this case,

with $\widetilde {F}$ again indicated by boxed vertices and dashed arrows. The potential is

$$\begin{align*}\widetilde{W}=cba+\beta_1\delta_1\alpha_1+\beta_2\delta_2\alpha_2+\beta_3\delta_3\alpha_3-a\beta_1\delta_a\alpha_2-b\beta_2\delta_b\alpha_3-c\beta_3\delta_c\alpha_1.\end{align*}$$

The associated frozen Jacobian algebra $A_{Q,W}$ also arises from a dimer model on a disk with six marked points on its boundary [Reference Baur, King and Marsh3], and is isomorphic to the endomorphism algebra of a cluster-tilting object in Jensen–King–Su’s categorification of the cluster algebra structure on the Grassmannian $\mathrm {Gr}_{2}^{6}$ [Reference Jensen, King and Su32]. This category is again a Frobenius cluster category (see [Reference Pressland42, Exam. 3.12]). Unlike the first example, this algebra is infinite-dimensional. However, it is Noetherian, so Theorem 2.6 still applies.

Since, in Examples 3.2 and 3.3, the algebra $A_{Q,W}$ is the endomorphism algebra of a cluster-tilting object in a Frobenius cluster category, it is internally $3$ -Calabi–Yau with respect to its frozen idempotent—that is, it has the properties in Remark 2.5—by a result of Keller–Reiten [Reference Keller and Reiten35, §5.4] (see also [Reference Pressland42, Cor. 3.10]). This foreshadows Theorem 4.14, which states that under mild assumptions on $(Q,W)$ , the algebra $A_{Q,W}$ has the (a priori stronger) bimodule internal Calabi–Yau property from Definition 2.4.

4 Calabi–Yau properties for frozen Jacobian algebras

In this section, we will recall from [Reference Pressland42, §5] a bimodule complex $0\to \mathbf {P}(A)\to A\to 0$ for a frozen Jacobian algebra A, exactness of which implies that A is bimodule internally $3$ -Calabi–Yau with respect to its frozen idempotent. We show that this complex is exact when $A=A_{Q,W}$ is as in Definition 3.1. Thus, these frozen Jacobian algebras satisfy the most complicated condition in Theorem 2.6. This constitutes the main step in the proof of Theorem 1.

Our arguments exploit the fact that the algebras we consider are pseudocompact, and so we begin with some useful generalities on such algebras. Further background can be found in [Reference Brumer7], [Reference Gabriel22], [Reference Iusenko and MacQuarrie30], [Reference Simson47], [Reference Van den Bergh51], for example.

Definition 4.1. Let A be a topological $\mathbb {K}$ -algebra. We say that A is pseudocompact if there is a system $\mathcal {I}=\{I\}$ of neighborhoods of $0\in A$ consisting of (open) ideals such that $A/I$ is finite-dimensional for all $I\in \mathcal {I}$ and the natural map

$$\begin{align*}A\to{\varprojlim}_IA/I\end{align*}$$

is an isomorphism. The Jacobson radical of A, denoted by J, is the intersection of maximal closed left ideals of A, and we say that A is J-adically complete if the natural map $A\to \varprojlim _nA/J^n$ is an isomorphism.

Proof. This statement (in a more general form) can be found in The Stacks Project §00M9Footnote ² , but we give a direct argument. For each $n\in \mathbb {Z}$ , there is a short exact sequence

(4.1)

where $\pi \colon \mathbb {K}\langle \hspace {-0.1em}\langle Q\rangle \hspace {-0.1em}\rangle \to A=\mathbb {K}\langle \hspace {-0.1em}\langle Q\rangle \hspace {-0.1em}\rangle /I$ is the projection. In particular, it follows that $A/\pi J^n$ is finite-dimensional. Note that the ideals $\pi J^n$ form a system of open neighborhoods of $0$ in A, since $\pi ^{-1}(\pi J^n)=I+J^n$ is open in $\mathbb {K}\langle \hspace {-0.1em}\langle Q\rangle \hspace {-0.1em}\rangle $ .

The sequence (4.1) is the nth component of a short exact sequence of inverse systems, each of which has only surjective morphisms and so satisfies (in a trivial way) the Mittag-Leffler condition. Thus, we obtain a short exact sequence of inverse limits

The left-hand term of this sequence may be naturally identified with $\bigcap _{n\geq 1}(I+J^n)$ , which is the closure of I in the J-adic topology, and is thus equal to I since I is closed. Moreover, the middle term is $\mathbb {K}\langle \hspace {-0.1em}\langle Q\rangle \hspace {-0.1em}\rangle $ since this algebra is J-adically complete by Proposition 4.2. Hence, the right-hand term is A, which is therefore pseudocompact.

We now return to the setting of ice quivers with potential. Let $(Q,F,W)$ be an ice quiver with potential and write $A=J(Q,F,W)$ . Recall that $Q_0^\circ =Q_0\setminus F_0$ and $Q_1^\circ =Q_1\setminus F_1$ denote the sets of mutable vertices and unfrozen arrows of Q, respectively. For $i\in Q_0$ , we write ${\mathrm {H}_{i}}$ for the set of arrows of Q with head i, and ${\mathrm {T}_{i}}$ for the set of arrows of Q with tail i. We denote by $\mathrm {H}_{i}^{\circ }$ and $\mathrm {T}_{i}^{\circ }$ the intersection of $Q_1^\circ $ with ${\mathrm {H}_{i}}$ and ${\mathrm {T}_{i}}$ , respectively. Recall from Remark 2.2 that the quotient $S:=A/J(A)$ of A by its Jacobson radical is a semisimple algebra, isomorphic as a left A-module to the direct sum of the vertex simple A-modules. (This property is sometimes [Reference Iusenko and MacQuarrie30, §3.2] called pointedness.) Thus, S has a basis given by the vertex idempotents $e_i$ for $i\in Q_0$ . For the remainder of this section, we write $\mathbin {\otimes }=\mathbin {\otimes }_S$ .

Introduce formal symbols $\rho _{\alpha }$ for each $\alpha \in Q_1$ and $\omega _v$ for each $v\in Q_0$ , and define S-bimodule structures on the vector spaces

$$ \begin{align*} V_0&=\bigoplus_{i\in Q_0}\mathbb{K}e_{i},& V_1&=\bigoplus_{\alpha\in Q_1}\mathbb{K}\alpha,& V_2&=\bigoplus_{\alpha\in Q_1^\circ}\mathbb{K}\rho_{\alpha},& V_3&=\bigoplus_{i\in Q_0^\circ}\mathbb{K}\omega_i, \end{align*} $$

via the formulae

$$\begin{align*}e_{i}\cdot e_{i}\cdot e_{i}=e_{i},\qquad e_{h{\alpha}}\cdot\alpha\cdot e_{t{\alpha}}=\alpha,\qquad e_{t{\alpha}}\cdot\rho_{\alpha}\cdot e_{h{\alpha}}=\rho_{\alpha},\qquad e_{i}\cdot\omega_i\cdot e_{i}=\omega_i. \end{align*}$$

Next, we describe various morphisms $\mu _k$ of A-bimodules. Since $V_0\cong S$ , the A-bimodule $A\mathbin {\otimes } V_0\mathbin {\otimes } A$ is canonically isomorphic to $A\mathbin {\otimes } A$ . Composing this isomorphism with the multiplication map for A, we obtain a map $\mu _0\colon A\mathbin {\otimes } V_0\mathbin {\otimes } A\to A$ . Define $\mu _1\colon A\mathbin {\otimes } V_1\mathbin {\otimes } A\to A\mathbin {\otimes } V_0\mathbin {\otimes } A$ by

$$\begin{align*}\mu_1(x\mathbin{\otimes}\alpha\mathbin{\otimes} y)=x\mathbin{\otimes}e_{h{\alpha}}\mathbin{\otimes}\alpha y-x\alpha\mathbin{\otimes}e_{t{\alpha}}\mathbin{\otimes} y.\end{align*}$$

For any path $p=\alpha _m\dotsm \alpha _1$ of $\mathbb {K} Q$ , we may define

$$\begin{align*}\Delta_{\alpha}(p)=\sum_{\alpha_i=\alpha}\alpha_m\dotsm\alpha_{i+1}\mathbin{\otimes}\alpha_i\mathbin{\otimes}\alpha_{i-1}\dotsm\alpha_1,\end{align*}$$

and extend by linearity and continuity to obtain a map $\Delta _{\alpha }\colon \mathbb {K}\langle \hspace {-0.1em}\langle Q\rangle \hspace {-0.1em}\rangle \to A\mathbin {\otimes } V_1\mathbin {\otimes } A$ . We then define $\mu _2\colon A\mathbin {\otimes } V_2\mathbin {\otimes } A\to A\mathbin {\otimes } V_1\mathbin {\otimes } A$ by

$$\begin{align*}\mu_2(x\mathbin{\otimes}\rho_{\alpha}\mathbin{\otimes} y)=\sum_{\beta\in Q_1}x\Delta_{\beta}(\partial_{\alpha}{W})y.\end{align*}$$

Finally, define $\mu _3\colon A\mathbin {\otimes } V_3\mathbin {\otimes } A\to A\mathbin {\otimes } V_2\mathbin {\otimes } A$ by

$$\begin{align*}\mu_3(x\mathbin{\otimes}\omega_v\mathbin{\otimes} y)=\sum_{\alpha\in{\mathrm{T}_{v}}}x\mathbin{\otimes}\rho_{\alpha}\mathbin{\otimes}\alpha y-\sum_{\beta\in{\mathrm{H}_{v}}}x\beta\mathbin{\otimes}\rho_{\beta}\mathbin{\otimes} y.\end{align*}$$

Note that we have $A\otimes V_k\otimes A\otimes _AS=A\otimes V_k\otimes S=A\otimes V_k$ for $0\leq k\leq 3$ . Thus, for each $1\leq k\leq 3$ , we obtain a map $\bar {\mu }_k:=(\mu _k\otimes _AS)\colon A\otimes V_k\to A\otimes V_{k-1}$ .

Lemma 4.5. Let $1\leq k\leq 3$ and consider $x\otimes v\otimes y\in A\otimes V_k\otimes A$ , where $x,y\in A$ and $v\in V_k$ . Then

$$\begin{align*}\mu_k(x\otimes v\otimes y)=\bar{\mu}_k(x\otimes v)\otimes y + r,\end{align*}$$

where $r\in A\otimes V_{k-1}\otimes Jy$ .

Definition 4.6. For an ice quiver with potential $(Q,F,W)$ with frozen Jacobian algebra A, denote by $\mathbf {P}(A)$ the complex of A-bimodules with nonzero terms

and $A\mathbin {\otimes }\mathbb {K} Q_0\mathbin {\otimes } A$ in degree $0$ . That this sequence is indeed a complex is [Reference Pressland42, Lem. 5.5].

By [Reference Pressland42, Th. 5.7], if A is a frozen Jacobian algebra such that

(4.2)

is exact, then A is bimodule internally $3$ -Calabi–Yau with respect to the idempotent $e=\sum _{i\in F_0}e_{i}$ . Our goal for the remainder of the section is to check exactness of (4.2) when $A=A_{Q,W}$ is the algebra from Definition 3.1. The following lemma allows us to simplify this calculation; its proof is based on [Reference Broomhead6, Prop. 7.5], but adapted to exploit pseudocompactness of A in place of a suitable grading.

Lemma 4.7. The complex (4.2) is exact if and only if

(4.3)

is exact.

Proof. The forward implication holds since $\mathbf {P}(A)\stackrel {\mu _0}{\longrightarrow }A$ is perfect as a complex of right A-modules, and so remains exact under $-\mathbin {\otimes }_AM$ for any $M\in \operatorname{Mod}{A}$ .

For the converse implication, we use pseudocompactness. Abbreviate $\bar {\mu }_{k,n}=\mu _k\otimes _AA/J^n$ (so that $\bar {\mu }_{k,1}=\bar {\mu }_k$ as defined above). We will show inductively that

(4.4)

is exact for all n; the assumed exactness of (4.3) gives the base case $n=1$ of this induction. Then the result will follow as in the proof of Proposition 4.3 from the fact that (4.4) remains exact under taking inverse limits, noting again that the relevant inverse systems satisfy the Mittag-Leffler condition since all their morphisms are surjective.

That (4.4) is exact at $A/J^n$ (and indeed, that (4.2) is exact at A) follows since $\mu _0$ is multiplication in the unital algebra A, so we concentrate on the other terms, each of which has the form $A\otimes V_k\otimes A/J^n$ for some S-bimodule $V_k$ , adopting the convention that $V_k=0$ for $k>3$ .

Choose $k\geq 0$ , let $\phi _0\in A\otimes V_k\otimes A/J^n$ for some $n\geq 2$ , and write $\phi _0^{\prime }$ for the projection of $\phi _0$ to $A\otimes V_k\otimes A/J^{n-1}$ . If $\phi _0$ is closed, that is, if $\bar {\mu }_{k,n}(\phi _0)=0$ , then $\phi ^{\prime }_0$ is also closed, and so by induction there is $\psi _0^{\prime }\in A\otimes V_{k+1}\otimes A/J^{n-1}$ with $\bar {\mu }_{k+1,n-1}(\psi ')=\phi '$ . Choose any $\psi _0\in A\otimes V_{k+1}\otimes A/J^n$ projecting to $\psi ^{\prime }_0$ . Then $\phi _1:=\phi _0-\bar {\mu }_{k+1,n}(\psi _0)$ projects to $\phi _0^{\prime }-\bar {\mu }_{k,n-1}(\psi _0^{\prime })=0$ , and hence $\phi _1\in A\otimes V_k\otimes J^{n-1}/J^n$ . Moreover, $\phi _1$ is closed.

Now, let Y be a basis of the finite-dimensional vector space $J^{n-1}/J^n$ , so that we may write

$$\begin{align*}\phi_1=\sum_{y\in Y}u_y\otimes y\end{align*}$$

for some $u_y\in A\otimes V_k$ . Since each $y\in J^{n-1}/J^n$ and $\phi _1$ is closed, we have

$$\begin{align*}0=\sum_{y\in Y}\bar{\mu}_{k,n}(u_y\otimes y)=\sum_{y\in Y}\bar{\mu}_k(u_y)\otimes y\end{align*}$$

by Lemma 4.5. It then follows by linear independence of the set Y that $\bar {\mu }_k(u_y)=0$ for all $y\in Y$ . By the assumed exactness of (4.3), there are therefore $v_y\in A\otimes V_{k+1}$ with $\bar {\mu }_{k+1}(v_y)=u_y$ .

Now, let

$$\begin{align*}\psi_1=\sum_{y\in Y}v_y\otimes y\in A\otimes V_{k+1}\otimes A/J^n.\end{align*}$$

Using Lemma 4.5 again, we may calculate

$$\begin{align*}\bar{\mu}_{k+1,n}(\psi_1)=\sum_{y\in Y}\bar{\mu}_{k+1}(v_y)\otimes y=\sum_{y\in Y}u_y\otimes y=\phi_1,\end{align*}$$

and so $\phi _0=\bar {\mu }_{k+1,n}(\psi _0+\psi _1)$ is exact. Thus, each sequence (4.4), and hence the limit (4.2), is exact as required.

Thus, it suffices for us to show that (4.3) is exact. This complex decomposes along with S, so that its exactness is equivalent to the exactness of

(4.5)

for each $i\in \widetilde {Q}_0$ , where $S_{i}$ denotes the vertex simple left A-module at i. We may calculate the terms in this complex as

(4.6) $$ \begin{align} \begin{aligned} A\mathbin{\otimes}\mathbb{K} Q_1\mathbin{\otimes} A\mathbin{\otimes}_AS_{i}&\cong\bigoplus_{b\in{\mathrm{T}_{i}}}Ae_{h{b}}, \\ A\mathbin{\otimes}\mathbb{K} Q_2^\circ\mathbin{\otimes} A\mathbin{\otimes}_AS_{i}&\cong\bigoplus_{a\in\mathrm{H}_{i}^{\circ}}Ae_{t{a}}, \\ A\mathbin{\otimes}\mathbb{K} Q_3^\circ\mathbin{\otimes} A\mathbin{\otimes}_AS_{i}&\cong{\begin{cases}Ae_{i},&i\in Q_0^\circ,\\ 0,&i\in F_0.\end{cases}} \end{aligned} \end{align} $$

In the first two cases, the right-hand sides are of the form $\bigoplus _{a\in X}Ae_{v(a)}$ , where X is a set of arrows, and $v\colon X\to Q_0$ . We will denote by $x\mapsto x\otimes [a]$ the inclusion $Ae_{v(a)}\to \bigoplus _{a\in X}Ae_{v(a)}$ mapping the domain to the summand indexed by a; this helps us to distinguish these various inclusions when v is not injective. As a consequence, a general element of the direct sum is $x=\sum _{a\in X}x_a\otimes [a]$ for $x_a\in Ae_{v(a)}$ .

Each arrow $b\in Q_1$ has an associated right derivative $\partial ^r_{b}\colon \mathbb {K}\langle \hspace {-0.1em}\langle Q\rangle \hspace {-0.1em}\rangle \to \mathbb {K}\langle \hspace {-0.1em}\langle Q\rangle \hspace {-0.1em}\rangle $ , which is linear and continuous and given by

(4.7) $$ \begin{align} \partial^r_{b}(\alpha_{\ell}\dotsm\alpha_1)=\begin{cases}\alpha_{\ell}\dotsm\alpha_2,&\alpha_1=b,\\ 0,&\alpha_1\ne b\end{cases} \end{align} $$

on paths. Using this map, we may write

$$\begin{align*}(\mu_2\mathbin{\otimes}_AS_{i})(x)=\sum_{b\in{\mathrm{T}_{i}}}\bigg(\sum_{a\in\mathrm{H}_{i}^{\circ}}x_a\partial^r_{b}{\partial_{a}{W}}\bigg)\mathbin{\otimes}[b].\end{align*}$$

Calculating $\mu _k\otimes _A S_i$ for the other values of k is similar, and more straightforward.

Now, let $(Q,W)$ be a quiver with potential, and let $A=A_{Q,W}=J(\widetilde {Q},\widetilde {F},\widetilde {W})$ . The following series of results will establish exactness of the complex (4.3) for A for each vertex $i\in \widetilde {Q}_0$ , using heavily the explicit set R of defining relations for A given in (3.1).

Proof. Pick a lift $\widetilde {x}_a\in \mathbb {K}\langle \hspace {-0.1em}\langle \widetilde {Q}\rangle \hspace {-0.1em}\rangle $ of each $x_a$ . Writing $p=\widetilde {x}_{\beta _i}\delta _i-\sum _{a\in \mathrm {H}_{i}^{\circ }\setminus \{\beta _i\}}\widetilde {x}_a\beta _{t{a}}\delta _a$ , our assumption on the $x_a$ is equivalent to $p\in \overline {\langle R\rangle }$ . Since every term of p ends with either $\delta _i$ or $\beta _{t{a}}\delta _a$ for some $a\in \mathrm {H}_{i}^{\circ }\setminus \{\beta _i\}$ , and the only element of R including terms ending with these arrows is $\beta _i\delta _i-\sum _{a\in \mathrm {H}_{i}^{\circ }\setminus \{\beta _i\}}a\beta _{t{a}}\delta _a$ , we can write

$$\begin{align*}p=z_i\delta_i+\sum_{a\in\mathrm{H}_{i}^{\circ}\setminus\{\beta_i\}}z_a\beta_{t{a}}\delta_a+y\Big(\beta_i\delta_i-\sum_{a\in\mathrm{H}_{i}^{\circ}\setminus\{\beta_i\}}a\beta_{t{a}}\delta_a\Big),\end{align*}$$

where $z_i\in \overline {\langle R\rangle }e_i^-$ , $z_a\in \overline {\langle R\rangle }e_{t{a}}$ , and $y\in \mathbb {K}\langle \hspace {-0.1em}\langle \widetilde {Q}\rangle \hspace {-0.1em}\rangle e_i$ . Comparing terms in our two expressions for p, we see that $\widetilde {x}_{\beta _i}=y\beta _i+z_i$ and $\widetilde {x}_{a}=ya-z_a$ . Since $z_i,z_a\in \overline {\langle R\rangle }$ , when we pass to the quotient algebra A, we see that $x_{\beta _i}=y\beta _i$ and $x_a=ya$ , as required.

Proof. Since $\mathbf {P}(A)$ is a complex, it is enough to show that $\ker (\mu _2\mathbin {\otimes }_AS_{i})\subseteq \operatorname {im}(\mu _3\mathbin {\otimes }_AS_{i})$ . Let $x=\sum _{a\in \mathrm {H}_{i}^{\circ }}x_a\mathbin {\otimes }[a]\in \bigoplus _{a\in \mathrm {H}_{i}^{\circ }}Ae_{t{a}}$ . Then

$$\begin{align*}(\mu_2\mathbin{\otimes}_AS_{i})(x)=\sum_{b\in{\mathrm{T}_{i}}}\bigg(\sum_{a\in\mathrm{H}_{i}^{\circ}}x_a\partial^r_{b}{\partial_{a}{\widetilde{W}}}\bigg)\mathbin{\otimes}[b]\in\bigoplus_{b\in{\mathrm{T}_{i}}}Ae_{h{b}}.\end{align*}$$

In particular, $\alpha _i\in {\mathrm {T}_{i}}$ , so if $x\in \ker (\mu _2\otimes _AS_{i})$ , we have $\left (\sum _{a\in \mathrm {H}_{i}^{\circ }}x_a\partial ^r_{\alpha _i}{\partial _{a}{\widetilde {W}}}\right )\otimes [\alpha _i]=0$ . Using the explicit expressions for the relations $\partial _{a}{\widetilde {W}}$ computed in (3.1), we see that

$$\begin{align*}\sum_{a\in\mathrm{H}_{i}^{\circ}}x_a\partial^r_{\alpha_i}\partial_{a}{\widetilde{W}}=x_{\beta_i}\delta_i-\sum_{a\in\mathrm{H}_{i}^{\circ}\setminus\{\beta_i\}}x_a\beta_{t{a}}\delta_a=0,\end{align*}$$

and so by Lemma 4.11 there exists $y\in Ae_{i}$ such that $x_a=ya$ for each a. It follows that $x=(\mu _3\mathbin {\otimes }_AS_{i})(y)$ , as required.

Proposition 4.13. If $i^{\pm }\in \widetilde {F}_0$ , then the complex

is exact.

Proof. Since $i^{\pm }\in \widetilde {F}_0$ , our complex is zero in degree $-3$ . By Remark 4.8, we need only check that $\mu _2\mathbin {\otimes }_AS_{i}^{\pm }$ is injective. Since $\mathrm {H}_{i^-}^{\circ }=\varnothing $ , the complex $\mathbf {P}(A)\mathbin {\otimes }_AS_{i}^-$ is also zero in degree $-2$ , so we are left to consider $\mu _2\mathbin {\otimes }_AS_{i}^+$ .

Since $\mathrm {H}_{i^+}^{\circ }=\{\alpha _i\}$ , we have $\mu _2\mathbin {\otimes }_AS_{i}^+\colon Ae_i\to \bigoplus _{b\in {\mathrm {T}_{i}}^+}Ae_{h{b}}$ . Let $x\in Ae_i$ , for which we calculate

$$\begin{align*}(\mu_2\mathbin{\otimes}_AS_{i}^+)(x)=\sum_{b\in{\mathrm{T}_{i}}^+}x\partial^r_{b}{\partial_{\alpha_i}{\widetilde{W}}}\otimes[b].\end{align*}$$

Assume that this is $0$ . Considering $\delta _i\in {\mathrm {T}_{i}}^+$ , we see that we must have $0=x\partial ^r_{\delta _i}{\partial _{\alpha _i}{\widetilde {W}}}=x\beta _i$ . By Lemma 4.9, it follows that $x=0$ , and $\mu _2\mathbin {\otimes }_AS_{i}^+$ is injective as required.

Summarizing the discussion above, we are able to establish the desired internal Calabi–Yau property for $A_{Q,W}$ .

Since for $A=A_{Q,W}$ we have $A/AeA=J(Q,W)$ by construction, Theorem 2 from the introduction is an immediate consequence. We record a further corollary, obtained by combining Theorem 4.14 with Theorem 2.6.

It is this corollary that we will use to define the Frobenius category $\mathcal {E}_Q$ appearing in Theorem 1 for an acyclic quiver Q; to be able to do so, it only remains to check that $A_Q$ is Noetherian in this case, and we will show in §5 that it is even finite-dimensional. Unfortunately, at the moment, we lack methods for showing that $A_{Q,W}$ is Noetherian in more cases (and expect that $A_{Q,W}$ is not Noetherian general), preventing us from extending Theorem 1 to the case of quivers with cycles. We already observed in Example 3.3 that $A_{Q,W}$ is Noetherian (but not finite-dimensional) when $(Q,W)$ is the $3$ -cycle with its natural potential. This quiver with potential is Jacobi-finite, so Corollary 4.15 also applies in this case.

Remark 4.16. Combinatorially, the algebra $A_{Q,W}$ seems to be related the dg-algebra $\Gamma _{Q,W}$ associated with $(Q,W)$ by Ginzburg [Reference Ginzburg25, §4.2]; the loops in cohomological degree $-2$ in $\Gamma _{Q,W}$ are replaced by the cycles $i\to i^+\to i^-\to i$ , and the degree $-1$ arrows are replaced by the paths $h{a}\to h{a}^+\to t{a}^-\to t{a}$ . Here, we use Amiot’s sign conventions [Reference Amiot1, Def. 3.1], which are opposite to Ginzburg’s. By a result of Van den Bergh [Reference Keller34, Th. A.12], the dg-algebra $\Gamma _{Q,W}$ is bimodule $3$ -Calabi–Yau, and we expect this to be related to the fact that $A_{Q,W}$ is bimodule internally $3$ -Calabi–Yau with respect to the idempotent determined by the vertices not appearing in Ginzburg’s construction. We will show in proving Theorem 1 (b) that, when Q is acyclic, the two constructions are also related via a triangle equivalence

(4.8) $$ \begin{align} \underline{\operatorname{GP}}(B_Q)\simeq\mathcal{C}_{Q}=\operatorname{per}{\Gamma_Q}/\mathcal{D}^{\mathrm{b}}(\Gamma_Q). \end{align} $$

By a result of Wu [Reference Wu52, Lem. 5.10], it follows from exactness of (4.2) that the relative Ginzburg algebra $\Gamma _{\widetilde {Q},\widetilde {F},\widetilde {W}}$ [Reference Wu52, Def. 4.20] is concentrated in degree $0$ , its zeroth cohomology being the algebra $A_{Q,W}$ . There is thus [Reference Wu53, Prop. 3.19] a homotopy pushout diagram

taken in the category of k-linear dg-categories with Tabuada’s model structure [Reference Tabuada50], where the functor G is the deformed relative $3$ -Calabi–Yau completion, with respect to $\widetilde {W}$ , of the natural map $\mathbb {K}\widetilde {F}\to \mathbb {K}\widetilde {Q}$ . In particular, this makes $\mathbf {\Pi }_{\widetilde {F}}$ the $2$ -Calabi–Yau completion of $\mathbb {K}\widetilde {F}$ (a dg version of the preprojective algebra of $\widetilde {F}$ ). Thus, we can reconstruct the Ginzburg dg-algebra $\Gamma _{Q,W}$ from $A_{Q,W}$ (plus enough additional data to recover the dg-functor G). It is also possible to obtain the triangle equivalence (4.8) using this style of argument (see Remark 5.4).

However, the only properties of the ice quiver with potential $(\widetilde {Q},\widetilde {F},\widetilde {W})$ that we use here are that $\Gamma _{\widetilde {Q},\widetilde {F},\widetilde {W}}$ is concentrated in degree $0$ , and that removing the frozen vertices recovers $(Q,W)$ ; typically, there are many ice quivers with potential with these two properties. At present, we do not know a natural construction of the specific frozen Jacobian algebra $A_{Q,W}$ from $\Gamma _{Q,W}$ , despite the combinatorial similarity of their definitions.

5 The acyclic case

Let $(Q,W)$ be a Jacobi-finite quiver with potential, so that the algebra $A_{Q,W}$ from Definition 3.1 satisfies all of the assumptions of Corollary 4.15 except possibly Noetherianity. Our goal in this section is to show that if Q is an acyclic quiver, then the algebra $A=A_Q=J(\widetilde {Q},\widetilde {F},\widetilde {0})$ is even finite-dimensional, so that this result does indeed apply.

Proof. Let $\gamma $ be the first arrow of p in $\widetilde {Q}_1\setminus Q_1$ . If $\gamma $ has a predecessor in p, then this arrow is in $Q_1$ , and so $t{\gamma }\in Q_0$ . On the other hand, if $\gamma $ is the first arrow of p, then $t{\gamma }=t{p}\in Q_0$ by assumption. By the construction of $\widetilde {Q}$ , it follows that $\gamma =\alpha _i$ for some $i\in Q_0$ . Since $h{p}\in Q_0$ , the path p cannot terminate with $\alpha _i$ , so this arrow has a successor. Looking again at the definition of $\widetilde {Q}$ , the only options are $\delta _a$ for some $a\in Q_1$ with $h{a}=i$ , or $\delta _i$ . We break into two cases.

First, assume $\alpha _i$ is followed in p by $\delta _a$ for some $a\in Q_0$ . This again cannot be the final arrow of p, since $h{p}\in Q_0$ . The only arrow leaving $h{\delta _a}=i_{t{a}}^-$ is $\beta _{t{a}}$ , so this must be the next arrow of p. But after projection to A by $\pi $ , we have $\beta _{t{a}}\delta _a\alpha _{h{a}}=\partial _{a}{W}=0$ since $W=0$ , so $p\in \ker {\pi }$ .

In the second case, $\alpha _i$ is followed by $\delta _i$ . As above, $\delta _i$ must be followed in p by $\beta _i$ , and projecting to A yields

$$\begin{align*}\beta_i\delta_i\alpha_i=\bigg(\sum_{\substack{a\in Q_1\\ h{a}=i}}a\beta_{t{a}}\delta_a\bigg)\alpha_i=\sum_{\substack{a\in Q_1\\ h{a}=i}}a\beta_{t{a}}\delta_a\alpha_{h{a}}=0,\end{align*}$$

so that again $p\in \ker {\pi }$ .

For the second statement, we have already shown that $\beta _{t{a}}\delta _a\alpha _{h{a}}\in \ker {\pi }$ for any $a\in Q_1$ and $\beta _i\delta _i\alpha _i\in \ker {\pi }$ for any $i\in Q_0$ . It then follows from the bipartite property of $\widetilde {F}$ that any path which is not in $\ker {\pi }$ and involves only arrows not in $Q_1$ must be a subpath of $\delta _x\alpha _i\beta _i\delta _y$ for some $x,y\in Q_0\cup Q_1$ and $i\in Q_0$ with $h{x}=i=t{y}$ . In particular, such a path has length at most $4$ .

We have now established everything we need in order to construct the Frobenius cluster category required by Theorem 1, and to prove the first two parts of this theorem.

We already observed after stating Corollary 4.15 that our construction applies to $(Q,W)$ given by the $3$ -cycle with natural potential to produce a stably $2$ -Calabi–Yau Frobenius cluster category $\mathcal {E}_{Q,W}=\operatorname {GP}(B_{Q,W})$ . We claim that the analogous statement to Theorem 1 (b) also holds here, namely that $\underline {\mathcal {E}}_{Q,W}$ is triangle equivalent to Amiot’s cluster category $\mathcal {C}_{Q,W}$ . While Keller–Reiten’s recognition theorem no longer applies to the cluster-tilting object T provided by Corollary 4.15, since its stable endomorphism algebra is not hereditary, one can in this case find another cluster-tilting object $T'\in \mathcal {E}_{Q,W}$ with $\underline {\operatorname {End}}_{\mathcal {E}_{Q,W}}(T')^{\mathrm {op}}\cong \mathbb {K} Q'$ , for $Q'$ an orientation of the Dynkin diagram $\mathsf {A}_{3}$ . Thus, the recognition theorem shows that $\underline {\mathcal {E}}_{Q,W}\simeq \mathcal {C}_{Q'}$ , and $\mathcal {C}_{Q'}\simeq \mathcal {C}_{Q,W}$ either by the recognition theorem again or by [Reference Amiot1, Cor. 3.11].

We note that while Amiot, Reiten, and Todorov have a recognition theorem [Reference Amiot, Reiten and Todorov2, Th. 3.1], which identifies certain stable categories of Frobenius categories admitting a cluster-tilting object with frozen Jacobian endomorphism algebra as generalized cluster categories, this theorem does not apply to our object T when $W\ne 0$ , because then $(\widetilde {Q},\widetilde {F},\widetilde {W})$ cannot satisfy their assumptions (H3) and (H4). Assumption (H3) requires a nonnegative degree function on the arrows of $\widetilde {Q}$ giving $\widetilde {W}$ degree $1$ , and (H4) requires that the arrows $\alpha _i$ , for $i\in Q_0$ , all have degree $1$ . But if $W\ne 0$ , then (H3) forces some arrow $a\in Q_1$ to have degree $1$ , and then (H4) implies that the potential term $a\beta _{t{a}}\delta _a\alpha _{h{a}}$ has degree at least $2$ , so (H3) does not hold.

6 Mutation and the cluster character

To complete the proof of Theorem 1, we need to understand the relationship between mutation of cluster-tilting objects in $\operatorname {GP}(B_Q)$ and Fomin–Zelevinsky mutation of their quivers. These operations turn out to be compatible, in a much larger class of Frobenius cluster categories, in the sense of the following theorem.

The proof of [Reference Pressland43, Th. 5.15] referred to in that of Theorem 6.1 is similar to an analogous result of Buan–Iyama–Reiten–Smith for triangulated categories [Reference Buan, Iyama, Reiten and Smith9]. The definition of mutation for an ice quiver with potential is given in [Reference Pressland43, Def. 4.1], and is similar to that for ordinary quivers with potential [Reference Buan, Iyama, Reiten and Smith9, §1.2]. We also explain in [Reference Pressland43] how to extend the definition of Fomin–Zelevinsky mutation to ice quivers which may have arrows between their frozen vertices. Thus, the same argument proves a stronger version of Theorem 6.1 (c) in which these arrows are still considered, but we will not use this here.

A consequence of Theorem 6.1 is that the Frobenius category $\mathcal {E}_Q$ we attach to an acyclic quiver has a cluster structure in the sense of Buan–Iyama–Reiten–Scott [Reference Buan, Iyama, Reiten and Scott8, §II.1], which will allow us to use results of Fu and Keller [Reference Fu and Keller21] to complete the proof of Theorem 1.

Just as for Theorem 1 (b), the conclusions of Theorem 1 (c) and (d) still hold when Q is a $3$ -cycle, replacing $B_Q$ by $B_{Q,W}$ for W the natural potential on Q. In this case, $\widetilde {\mathscr {A}}_{Q}$ is the cluster algebra of the Grassmannian $G_2^6$ , and $\operatorname {GP}(B_{Q,W})$ is the Grassmannian cluster category constructed by Jensen–King–Su [Reference Jensen, King and Su32] (see Example 3.3). The analogue of Theorem 6.1 holds for Grassmannian cluster categories, despite them being Hom-infinite (see [Reference Pressland43, Prop. 5.17] and its preceding discussion).

For a general quiver with potential $(Q,W)$ , even assuming that $A_{Q,W}$ is Noetherian so that the category $\mathcal {E}_{Q,W}$ is well defined, it may still be the case that the cluster-tilting objects in this category form several different mutation classes. However, this would not significantly affect our arguments here: while we can only rule out loops and $2$ -cycles in the mutation class of $T=eA_{Q,W}$ via our methods, and so we cannot show that $\mathcal {E}_{Q,W}$ has a cluster structure in the strict sense, Fu–Keller’s bijections (with the word reachable reinstated) are still valid in this situation. A more detailed discussion of this issue can be found in [Reference Pressland45, §6].

7 An extriangulated categorification

In this section, we will prove Corollary 1, but first we need to define the extriangulated category appearing in the statement. To keep our discussion brief, and since we only deal with very particular examples, we will not define extriangulated categories fully here, and instead refer the reader to [Reference Nakaoka and Palu40]. The most important piece of data for us will be that of the extension group $\mathbb {E}(X,Y)$ for any two objects X and Y in the extriangulated category; in our context, the objects of the category will be the same as those of the exact category $\mathcal {E}_Q$ , and we will have $\mathbb {E}(X,Y)=\operatorname {Ext}^1_{\mathcal {E}_Q}(X,Y)$ .

Note that if $(Q,W)$ is a Jacobi-finite quiver with potential such that $A_{Q,W}$ is Noetherian and internally Calabi–Yau with respect to e (such as the $3$ -cycle with its usual potential), so that we have a stably $2$ -Calabi–Yau Frobenius category $\mathcal {E}_{Q,W}$ , we can define $\mathcal {E}_{Q,W}^+=\mathcal {E}_{Q,W}/[B_{Q,W}e^-]$ . The statement of Proposition 7.2 also holds for this category, with the same proof.

Proof of Corollary 1

Statement (a) follows from Proposition 7.2, using Theorem 1 (b) to see that $\underline {\mathcal {E}}_Q^+=\underline {\mathcal {E}}_Q$ is equivalent to the cluster category $\mathcal {C}_{Q}$ .

To deduce the remaining statements from Theorem 1, first note that

$$\begin{align*}\operatorname{Ext}^1_{\mathcal{E}_Q}(X,Y)=\mathbb{E}_{\mathcal{E}^+_Q}(X,Y)=\underline{\operatorname{Hom}}_{\mathcal{E}_Q}(X,Y[1])\end{align*}$$

for any X and Y in the common set of objects of the three categories $\mathcal {E}_Q$ , $\mathcal {E}^+_Q$ , and $\underline {\mathcal {E}}_Q$ . This means that a set of representatives of isomorphism classes of cluster-tilting objects in $\mathcal {E}_Q$ is also such a set for $\mathcal {E}_Q^+$ and for $\underline {\mathcal {E}}_Q$ . Moreover, since $\mathcal {E}^+_Q$ and $\mathcal {E}_Q$ have the same projective–injective objects, a direct summand of one of these cluster-tilting objects is indecomposable nonprojective in one of these subcategories if and only if this is the case in all three. Thus, two of these cluster-tilting objects are related by mutation at an indecomposable summand in one of the categories if and only if they are related by mutation at the same summand in all three. Consequently, the identity map is a bijection from the set of cluster-tilting objects of $\mathcal {E}_Q$ to the set of cluster-tilting objects of $\mathcal {E}^+_Q$ , commuting with mutation. A more detailed discussion of mutation of cluster-tilting objects in extriangulated categories can be found in [Reference Chang, Zhou and Zhu15] (see also [Reference Zhou and Zhu54]).

Finally, the indecomposable rigid objects of $\mathcal {E}_Q^+$ are, up to isomorphism, precisely the indecomposable rigid objects of $\mathcal {E}_Q$ which are not summands of $P^-$ .

It now follows from Theorem 1 (c) and (d) that composing Fu–Keller’s cluster character $\operatorname {Ob}(\mathcal {E}_Q)\to \widetilde {\mathscr {A}}_{Q}$ with the map $\widetilde {\mathscr {A}}_{Q}\to \mathscr {A}_{Q}^{\bullet }$ evaluating the frozen variables $x_i^-$ at $1$ provides the required bijections. To see Corollary 1 (c), note additionally that each seed of $\mathscr {A}_{Q}^{\bullet }$ is obtained from one of $\widetilde {\mathscr {A}}_{Q}$ by removing the frozen variables $x_i^-$ and deleting the corresponding vertices (together with any incident arrows) from the quiver. Similarly, one obtains the quiver of $\operatorname {End}_{\mathcal {E}_Q^+}(T)^{\mathrm {op}}$ from that of $\operatorname {End}_{\mathcal {E}_Q}(T)^{\mathrm {op}}$ by removing the vertices corresponding to summands of $P^-$ .

8 Boundary algebras

Since the objects of the Frobenius category $\mathcal {E}_Q$ in Theorem 1 are modules for the idempotent subalgebra $B_Q=eA_Qe$ determined by the frozen vertices of $\widetilde {Q}$ , we wish to describe this subalgebra more explicitly. In this section, we will give a presentation of $B_Q$ via a quiver with relations.

Recall that the double quiver $\overline {Q}$ of a quiver Q has vertex set $Q_0$ and arrows $Q_1\cup Q^*_1$ , where $Q^*_1=\{\alpha ^*:\alpha \in Q_1\}$ . The head and tail maps agree with those of Q on $Q_1$ , and are defined by $h{\alpha ^*}=t{\alpha }$ and $t{\alpha ^*}=h{\alpha }$ on $Q^*_1$ . The complete preprojective algebra of Q is

$$\begin{align*}\Pi_{Q}=\mathbb{K}\langle\hspace{-0.1em}\langle \overline{Q}\rangle\hspace{-0.1em}\rangle/\overline{\langle\textstyle\sum_{\alpha\in Q_1}[\alpha,\alpha^*]\rangle}\end{align*}$$

and, up to isomorphism, depends only on the underlying graph of Q. We begin with the following general statement for frozen Jacobian algebras, which reveals some of the relations of $B_{Q,W}$ for an arbitrary quiver with potential $(Q,W)$ .

Proof. It suffices to check that $\pi (\sum _{\alpha \in F_1}[\alpha ,\alpha ^*])=0$ , that is, $\sum _{\alpha \in F_1}[\alpha ,\partial _{\alpha }{W}]=0$ . By construction, for any $i\in Q_0$ , we have

$$\begin{align*}\sum_{\alpha\in\mathrm{H}_v}\alpha\partial_{\alpha}{W}-\sum_{\beta\in\mathrm{T}_v}\partial_{\beta}{W}\beta=0\end{align*}$$

in $\mathbb {K}\langle \hspace {-0.1em}\langle Q\rangle \hspace {-0.1em}\rangle $ . Projecting to A and summing over vertices, we see that

$$\begin{align*}0=\sum_{\alpha\in Q_1}[\alpha,\partial_{\alpha}{W}]=\sum_{\alpha\in Q_1^\circ}[\alpha,\partial_{\alpha}{W}]+\sum_{\alpha\in F_1}[\alpha,\partial_{\alpha}{W}]=\sum_{\alpha\in F_1}[\alpha,\partial_{\alpha}{W}],\end{align*}$$

where the final equality holds since $\partial _{\alpha }{W}=0$ in A whenever $\alpha \in Q_1^\circ $ .

We now turn to our description of $B_Q$ , beginning with what will turn out to be its Gabriel quiver.

Definition 8.3. Let Q be an acyclic quiver, and consider the frozen subquiver $\widetilde {F}$ of $\widetilde {Q}$ , which has vertex set $\widetilde {F}_0=\{i^+,i^-:i\in Q_0\}$ and arrows

$$\begin{align*}\delta_i\colon i^+\to i^-,\qquad \delta_a\colon h{a}^+\to t{a}^- \end{align*}$$

for each $i\in Q_0$ and $a\in Q_1$ . We define a quiver $\Lambda _Q$ by adjoining to $\widetilde {F}$ an arrow

$$\begin{align*}\delta_p^*\colon t{p}^-\to h{p}^+\end{align*}$$

for each path p of Q. Since Q is acyclic, it has finitely many paths, and so $\Lambda _Q$ is a finite quiver.

If $p=e_{i}$ is the trivial path at $i\in Q_0$ , we write $\delta ^*_p=\delta ^*_i$ to avoid a double subscript. The double quiver of $\widetilde {F}$ appears as the subquiver of $\Lambda _Q$ obtained by excluding the arrows $\delta ^*_p$ for p of length 2 or more; the notation for the arrows of $\Lambda _Q$ is chosen to be consistent with that used earlier for the arrows of this double quiver.

Example 8.4. Let

be a linearly oriented quiver of type $\mathsf {A}_3$ . Then $\Lambda _Q$ is the following quiver.

Definition 8.5. Let Q be a quiver. A zigzag in Q is a triple $(q,a,p)$ , where p and q are paths in Q, and $a\in Q_1$ is an arrow such that $h{p}=h{a}$ and $t{q}=t{a}$ . Thus, if $a\colon v\to w$ is an arrow of Q, a zigzag involving a is some configuration

where the dotted arrows denote paths. We call the zigzag strict if $p\ne ap'$ for any path $p'$ and $q\ne q'a$ for any path $q'$ , but do not exclude these possibilities in general. If $z=(q,a,p)$ is a zigzag, then we define $h{z}=h{q}$ and $t{z}=t{p}$ .

We now write down elements of $\mathbb {K}\langle \hspace {-0.1em}\langle \Lambda _Q\rangle \hspace {-0.1em}\rangle $ that will turn out to be a set of generating relations for $B_Q$ , having three flavors, as follows:

1. (R1) For each path p of Q, let

   $$\begin{align*}r_1(p)=\delta^*_p\delta_{t{p}}-\sum_{\substack{a\in Q_1\\h{a}=t{p}}}\delta^*_{pa}\delta_a.\end{align*}$$

2. (R2) For each path p of Q, let

   $$\begin{align*}r_2(p)=\delta_{h{p}}\delta^*_p-\sum_{\substack{a\in Q_1\\t{a}=h{p}}}\delta_a\delta_{ap}^*.\end{align*}$$

3. (R3) For each zigzag $(q,a,p)$ of Q, let

   $$\begin{align*}r_3(q,a,p)=\delta^*_{q}\delta_a\delta^*_{p}.\end{align*}$$

We write I for the ideal of $\mathbb {K}\langle \hspace {-0.1em}\langle \Lambda _Q\rangle \hspace {-0.1em}\rangle $ generated by the union of these three sets of relations. This generating set is usually not minimal, as for certain zigzags $(q,a,p)$ , the relation $r_3(q,a,p)$ may already lie in the ideal generated by relations of the forms $r_1$ and $r_2$ . For example, if $a\in Q_1$ is an arrow such that $i=t{a}$ is not incident with any other arrows of Q, then

$$\begin{align*}r_1(a)=\delta^*_a\delta_i,\qquad r_2(e_i)=\delta_i\delta_i^*-\delta_a\delta^*_a, \end{align*}$$

so in $\mathbb {K}\langle \hspace {-0.1em}\langle \Lambda _Q\rangle \hspace {-0.1em}\rangle /\overline {\langle r_1(a),r_2(e_i)\rangle }$ we already have

$$\begin{align*}r_3(a,a,a)=\delta_a^*\delta_a\delta^*_a=\delta^*_a\delta_{i}\delta^*_{i}=0.\end{align*}$$

One can check that if $(q,a,p)$ is a strict zigzag, then $r_3(q,a,p)$ is not redundant, but this condition is not necessary; if Q is a linearly oriented quiver of type $\mathsf {A}_4$ and a is the middle arrow, then the nonstrict zigzag $(a,a,a)$ yields an irredundant relation.

Let $\Phi \colon \mathbb {K}\langle \hspace {-0.1em}\langle \Lambda _Q\rangle \hspace {-0.1em}\rangle \to A_Q$ be the map given by the identity on the vertices of $\Lambda _Q$ and the arrows $\delta _i$ and $\delta _a$ for $i\in Q_0$ and $a\in Q_1$ ; this makes sense as these are subsets of the vertices and arrows of $\widetilde {Q}$ . On the remaining arrows $\delta ^*_p$ of $\Lambda _Q$ , we define $\Phi (\delta ^*_p)=\alpha _{h{p}}p\beta _{t{p}}$ .

Proof. Since $\Phi $ sends every vertex or arrow of $\Lambda _Q$ to the image in $A_Q$ of a path in $\widetilde {Q}$ with frozen head and tail, it takes values in $B_Q$ . 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. Then

$$ \begin{align*} \Phi(r_1(p))&=\alpha_{h p}p\beta_{t{p}}\delta_{t{p}}-\sum_{\substack{a\in Q_1\\h{a}=t{p}}}\alpha_{h p}pa\beta_{t{a}}\delta_a=\alpha_{h{p}}p(\partial_{\alpha_{t{p}}}{\widetilde{W}})=0,\\ \Phi(r_2(p))&=\delta_{h{p}}\alpha_{h p}p\beta_{t{p}}-\sum_{\substack{a\in Q_1\\t{a}=h{p}}}\delta_a\alpha_{h{a}}ap\beta_{t{p}}=(\partial_{\beta_{h{p}}}{\widetilde{W}})p\beta_{t{p}}=0. \end{align*} $$

If $(q,a,p)$ is a zigzag, then

$$\begin{align*}\Phi(r_3(q,a,p))=\alpha_{h{r}}q\beta_{t{a}}\delta_a\alpha_{h{a}}p\beta_{t{p}}=0,\end{align*}$$

since $0=\partial _{a}{\widetilde {W}}=\partial _{a}{W}-\beta _{t{a}}\delta _a\alpha _{h{a}}=-\beta _{t{a}}\delta _a\alpha _{h{a}}$ by acyclicity of Q.

Proof. We begin by showing surjectivity. As in the proof of Theorem 5.2, we may use Lemma 5.1 to see that any path in $\widetilde {Q}$ determining a nonzero element of A has the form $p=q_1p'q_2$ where $q_1$ and $q_2$ contain no arrows of $Q_1$ , and $p'$ is a path of Q. If p has frozen head and tail, then $q_1$ and $q_2$ must be nonzero, so we even have

$$\begin{align*}p=q_1^{\prime}\alpha_{h{p'}} p'\beta_{t{p'}} q_2^{\prime} = q_1^{\prime}\Phi(\delta_{p'}^*)q_2^{\prime}.\end{align*}$$

Now, $q_1^{\prime }$ and $q_2^{\prime }$ are, like p, paths of $\widetilde {Q}$ with frozen head and tail, but with the additional property that they include no arrows of $Q_1$ . Let q be such a path. If q contains the arrow $\beta _i$ for some $i\in Q_0$ , then this arrow must have a successor in q, since its head is unfrozen. Moreover, this successor must be $\alpha _i$ , since this is the only arrow outside of $Q_1$ that may follow $\beta _i$ . Similarly, any occurrence of $\alpha _i$ in q must be preceded by $\beta _i$ . It follows that q is either a vertex idempotent $e_i^\pm =\Phi (e_i^\pm )$ , or is formed by composing paths of the form $\delta _i=\Phi (\delta _i)$ for $i\in Q_0$ , $\delta _a=\Phi (\delta _a)$ for $a\in Q_1$ , or $\alpha _i\beta _i=\Phi (\delta ^*_i)$ for $i\in Q_0$ , and so is in the image of $\Phi $ . We conclude that the projection to $A_Q$ of any path in $\widetilde {Q}$ with frozen head and tail lies in the image of $\Phi $ . Since these projections span $B_Q$ , we see that $\Phi $ is surjective.

From now on, we abbreviate $B=B_Q$ . To complete the proof, we will use [Reference Buan, Iyama, Reiten and Smith9, Prop. 3.3], which in this context states that $\Phi $ is an isomorphism if and only if

(8.1)

and

(8.2)

are exact sequences for all $i\in Q_0$ . Here, the left-most maps in each sequence are obtained from our generators for the ideal I by right-differentiation (4.7) and the application of $\Phi $ , as prescribed in [Reference Buan, Iyama, Reiten and Smith9], so they act on components by

(8.3) $$ \begin{align} \begin{aligned} f(ye_{h{p}}^+)&=y\Phi(\delta^*_p)\mathbin{\otimes}[i]-\sum_{\substack{a\in Q_1\\h{a}=i}}y\Phi(\delta^*_{pa})\mathbin{\otimes}[a],\\ g(ye_{h{p}}^-)&=y\delta_{h{p}}\mathbin{\otimes}[p]-\sum_{\substack{a\in Q_1\\t{a}=h{p}}}y\delta_a\mathbin{\otimes}[ap],\\ g(ye_{h{z}}^+)&=y\Phi(\delta^*_{q})\delta_a\otimes[p],\ \text{where } z=(q,a,p). \end{aligned} \end{align} $$

Here, we have dealt with the ambiguity about which summand contains each term as in §4, denoting elements of $\bigoplus _{a\in X}Be_{v(a)}$ by $\sum _{a\in X}x_a\otimes [a]$ , with $x_a\in Be_{v(x)}$ and $x_a\otimes [a]$ denoting its inclusion into the summand of $\bigoplus _{a\in X}Be_{v(a)}$ indexed by a. We adopt the convention here that the summand $Be_{i}^-$ appearing in the middle term of (8.1) is indexed by the vertex i.

Since $\Phi $ is well defined and surjective, sequences (8.1) and (8.2) are complexes and exact at $\operatorname {rad}{Be_i^+}$ and $\operatorname {rad}{Be_i^-}$ , respectively, so we need only check exactness at the middle term in each case. We proceed as in §4, using the explicit set R of relations for $A_Q$ from (3.1), and begin with (8.1). Let $x_i\in e\mathbb {K}\langle \hspace {-0.1em}\langle \widetilde {Q}\rangle \hspace {-0.1em}\rangle e_i^-$ and $x_a\in e\mathbb {K}\langle \hspace {-0.1em}\langle \widetilde {Q}\rangle \hspace {-0.1em}\rangle e_{t{a}}^-$ for each $a\in Q_1$ with $h{a}=i$ , determining the element

$$\begin{align*}x=x_i\otimes[i]+\sum_{\substack{a\in Q_1\\h{a}=i}}x_a\otimes[a]\end{align*}$$

of the middle term of (8.1). Assume that x is a cycle for this complex, that is,

$$\begin{align*}x_i\delta_i+\sum_{\substack{a\in Q_1\\h{a}=i}}x_a\delta_a\in\overline{\langle R\rangle}.\end{align*}$$

Since the only generating relation for $A_Q$ with terms ending in $\delta _i$ or $\delta _a$ for $a\in Q_1$ with $h{a}=i$ is $\partial _{\alpha _i}{W}$ , it follows that in $\mathbb {K}\langle \hspace {-0.1em}\langle \widetilde {Q}\rangle \hspace {-0.1em}\rangle $ we have

$$ \begin{align*} x_i\delta_i+\sum_{\substack{a\in Q_1\\h{a}=i}}x_a\delta_a&=z_i\delta_i+\sum_{\substack{a\in Q_1\\h{a}=i}}z_a\delta_a+y\partial_{\alpha_i}{W}\\ &=z_i\delta_i+\sum_{\substack{a\in Q_1\\h{a}=i}}z_a\delta_a+y\bigg(\beta_i\delta_i-\sum_{\substack{a\in Q_1\\h{a}=i}}a\beta_{t{a}}\delta_a\bigg)\end{align*} $$

for $z_i,z_a\in \overline {\langle R\rangle }$ and $y\in \mathbb {K}\langle \hspace {-0.1em}\langle \widetilde {Q}\rangle \hspace {-0.1em}\rangle e_{i}$ . Comparing terms, we see that $x_i=z_i+y\beta _i$ and $x_a=z_a-ya\beta _{t{a}}$ . Since $h{x_i}$ and $h{x_a}$ are frozen, but $h{\beta _j}$ is unfrozen for all $j\in Q_0$ , we must have

$$\begin{align*}y=\sum_{\substack{p \text{ path in } Q\\t{p}=i}}y_p\alpha_{h p}p\end{align*}$$

for some $y_p\in Be_{h{p}}^+$ . Projecting to B, we have

$$\begin{align*}x_i=\sum_{\substack{p \text{ path in } Q\\t{p}=i}}y_p\Phi(\delta^*_p),\qquad x_a=\sum_{\substack{p \text{ path in} Q\\t{p}=i}}-y_p\Phi(\delta^*_{pa}). \end{align*}$$

Thus,

$$\begin{align*}y=\sum_{\substack{p \text{ path in } Q\\t{p}=i}}y_p\otimes p\end{align*}$$

satisfies $f(y)=x$ , and so sequence (8.1) is exact.

Now, we turn to (8.2). For each path p in Q with $t{p}=i$ , pick $x_p\in e\mathbb {K}\langle \hspace {-0.1em}\langle \widetilde {Q}\rangle \hspace {-0.1em}\rangle e_{h{p}}^+$ , and assume that

$$\begin{align*}\sum_{\substack{p \text{ path in } Q\\t{q}=i}}x_p\alpha_{h{p}}p\beta_i\in\overline{\langle R\rangle},\end{align*}$$

so that

$$\begin{align*}x=\sum_{\substack{p \text{ path in } Q\\t{p}=i}}x_p\otimes[p]\end{align*}$$

is a cycle in the middle term of (8.2). By comparison with the generating relations, we see that we may write

$$\begin{align*}&\sum_{\substack{p \text{ path in } Q\\t{p}=i}}x_p\alpha_{h{p}}p\beta_i\\&\quad=\sum_{\substack{p \text{ path in } Q\\t{p}=i}}\Bigg(z_p\alpha_{h{p}}p\beta_i+y_p\bigg(\delta_{h{p}}\alpha_{h{p}}-\sum_{\substack{b\in Q_1\\t{b}=h{p}}}\delta_b\alpha_{h{b}}b\bigg)p\beta_i-\sum_{\substack{a\in Q_1\\h{a}=h{p}}}y_{a,p}(\beta_{t{a}}\delta_a\alpha_{h{a}})p\beta_i\Bigg)\end{align*}$$

for some $z_p\in \overline {\langle R\rangle }$ , $y_p\in \mathbb {K}\langle \hspace {-0.1em}\langle \widetilde {Q}\rangle \hspace {-0.1em}\rangle e_{h{p}}^-$ and $y_{a,p}\in e\mathbb {K}\langle \hspace {-0.1em}\langle \widetilde {Q}\rangle \hspace {-0.1em}\rangle e_{t{a}}$ . Each path p in the sum is either the trivial path $e_i$ or of the form $p=br$ for some arrow b and path r. By comparing terms, we deduce that after projection to B we have

$$\begin{align*}x_i=y_i\delta_i-\sum_{\substack{a\in Q_1\\h{a}=i}}y_{a,e_i}\beta_{t{a}}\delta_a,\qquad x_{br}=y_{br}\delta_{h{b}}-y_{r}\delta_b-\sum_{\substack{a\in Q_1\\h{a}=h{b}}}y_{a,br}\beta_{t{a}}\delta_a. \end{align*}$$

Since $y_{a,p}\in e\mathbb {K}\langle \hspace {-0.1em}\langle \widetilde {Q}\rangle \hspace {-0.1em}\rangle e_{t{a}}$ , we must have

$$\begin{align*}y_{a,p}=\sum_{\substack{q \text{ path in } Q\\t{q}=t{a}}}y_{q,a,p}\alpha_{h{q}}q\end{align*}$$

for some $y_{q,a,p}\in e\mathbb {K} \widetilde {Q}e_{h{q}}^+$ . The triple $z=(q,a,p)$ occurring in a subscript here satisfies $h{a}=h{q}$ and $t{a}=t{p}$ , so it is a zigzag. One may then calculate explicitly using (8.3) that the $y_p$ and $y_z=y_{q,a,p}$ give a preimage

$$\begin{align*}y=\sum_{\substack{p \text{ path in } Q\\t{p}=i}}y_p\otimes[p]+\sum_{\substack{z \text{ zig-zag}\\t{z}=i}}y_z\otimes[z]\end{align*}$$

of x under g.

We close this section with the following curious property of the category $\mathcal {E}_Q=\operatorname {GP}(B_Q)$ . Recall that $\operatorname {Sub}(X)$ , for $X\in \operatorname {mod}{B_Q}$ , denotes the full subcategory of $\operatorname {mod}{B_Q}$ on objects M admitting a monomorphism $M\to X^n$ for some n.

Proof. Write $B=B_Q$ . Since $\operatorname {GP}(B)$ is a Frobenius category with injective objects those in $\operatorname {add}{B}$ , it suffices to show that $B\in \operatorname {Sub}(P^+)$ , or that $Be_i^\pm \in \operatorname {Sub}(P^+)$ for each i. Since this is true of $Be_i^+$ by definition of $P^+$ , it remains to show that $Be_i^-\in \operatorname {Sub}(P^+)$ for each $i\in Q_0$ . Since Q has no isolated vertices, i cannot be both a source and a sink.

First, assume that i is not a source in Q. Then the map $Be_i^-\to Be_i^+$ given by right multiplication by $\delta _i$ is injective, as follows. If $x\in Be_i^-$ satisfies $x\delta _i=0$ in $Be_i^+$ , then lifting to $\mathbb {K}\langle \hspace {-0.1em}\langle \Lambda _Q\rangle \hspace {-0.1em}\rangle $ and using the explicit generating set of I, we have

$$\begin{align*}x\delta_i=z\delta_i+\sum_{\substack{p \text{ path in } Q\\t{p}=i}}y_p\bigg(\delta^*_p\delta_i-\sum_{\substack{a\in Q_1\\h{a}=i}}\delta_{pa}^*\delta_a\bigg)\end{align*}$$

for some $z\in I$ and $y_p\in e\mathbb {K}\langle \hspace {-0.1em}\langle \Lambda _Q\rangle \hspace {-0.1em}\rangle e_{h{p}}^+$ . Since i is not a source, the sum over arrows on the right-hand side is nonempty. By comparing coefficients, we see that $y_p=0$ for all p, and hence $x=z=0$ in A.

Now, assume that i is not a sink. Pick $a\in Q_1$ with $t{a}=i$ . Then the map $Be_i^-\to Be_{h{a}}^+$ given by right multiplication by $\delta _a$ is injective as follows. If $x\in Be_i^-$ satisfies $x\delta _a=0$ in $Be_{h{a}}^+$ , then lifting to $\mathbb {K}\langle \hspace {-0.1em}\langle \Lambda _Q\rangle \hspace {-0.1em}\rangle $ and using our explicit relations, we have

$$\begin{align*}x\delta_a=z+\sum_{\substack{p \text{ path in } Q\\ t{p}=h{a}}}y_p\bigg(\delta_p^*\delta_{h{a}}-\sum_{\substack{b\in Q_1\\h{b}= h{a}}}\delta^*_{pa}\delta_a\bigg),\end{align*}$$

for $z\in I$ , and it follows by comparing coefficients that $y_p=0$ for all p, so $x=0$ .

Example 10.2 shows that we may have $\mathcal {E}_Q\subsetneq \operatorname {Sub}(P^+)$ . The assumption on isolated vertices is necessary, since such a vertex i of Q results in a direct summand of $\mathcal {E}_Q$ equivalent to $\operatorname {mod}\ {\Pi }$ for $\Pi $ the preprojective algebra of type $\mathsf {A}_2$ , with indecomposable objects $S_i^\pm $ and $P_i^\pm $ , and $S_i^-,P_i^-\notin \operatorname {Sub}(P^+)=\operatorname {Sub}(P_i^+)$ .

9 g-vectors, indices, and dimension vectors

In this section, we show how representation-theoretic information about the category $\mathcal {E}_Q$ from Theorem 1 can be used to recover information about the principal coefficient cluster algebra $\mathscr {A}_{Q}^{\bullet }$ . These are not new results (unsurprisingly, since this cluster algebra is very well-studied), but the style of proofs is different and demonstrates how the categorification given here can be used. Our results are also not restricted to the acyclic case, but hold whenever an appropriate Frobenius categorification can be constructed, for example, in the context of Corollary 4.15.

Let $(Q,W)$ be a Jacobi-finite quiver with potential such that the frozen Jacobian algebra $A=A_{Q,W}$ is Noetherian; for example, one can take Q to be an acyclic quiver. Then, by Corollary 4.15, the category $\mathcal {E}=\operatorname {GP}(eAe)$ is a stably $2$ -Calabi–Yau category with a cluster-tilting object $T=eA$ , and there are compatible isomorphisms $\operatorname {End}_{\mathcal {E}}(T)^{\mathrm {op}}\cong A$ and $\underline {\operatorname {End}}_{\mathcal {E}}(T)^{\mathrm {op}}\cong J(Q,W)$ . We keep the abbreviated notation A, T, and $\mathcal {E}$ for the rest of the section.

The isomorphism $\operatorname {End}_{\mathcal {E}}(T)^{\mathrm {op}}\cong A$ allows us to write the indecomposable summands of T as $T_i$ , for $i\in Q_0$ , where $T_i$ corresponds to $Ae_i$ under the isomorphism. We write $P_i^\pm $ instead of $T_i^\pm $ , noting that these are the indecomposable projective–injective objects of $\mathcal {E}$ , and write $P^\pm =\bigoplus _{i\in Q_0}P_i^\pm $ , which is compatible with the notation in §8. We abbreviate and , these being functors $\mathcal {E}\to \operatorname {mod}\ {A}$ via the isomorphism $\operatorname {End}_{\mathcal {E}}(T)^{\mathrm {op}}\cong A$ . Finally, we write $[M:N]$ for the multiplicity of N as an indecomposable summand of M (for M and N objects in any category in which this makes sense).

Lemma 9.1. Let $X\in \mathcal {E}$ . Then the minimal projective resolution

(9.1)

of $GX$ has the property that, for each $j\in Q_0$ ,

$$ \begin{align*} [P^k:Ae_j^+]&={\begin{cases}\dim_{\mathbb{K}}(e_jGX),&k=1,\\0,&\text{otherwise},\end{cases}}\\ [P^k:Ae_j^-]&={\begin{cases}\dim_{\mathbb{K}}(e_jGX),&k=2,\\0,&\text{otherwise}.\end{cases}} \end{align*} $$

Proof. Note that $eGX=\operatorname {Ext}^1_{\mathcal {E}}(P^+\oplus P^-,X)=0$ , so the composition series of $GX$ includes only the simple modules $S_i$ for $i\in Q_0$ a mutable vertex. In this case, the sequence (4.5), which is exact by Propositions 4.10 and 4.12, is a projective resolution of the simple A-module $S_{i}$ . We write this resolution as

(9.2)

for projective A-modules $P_i^k$ which can be described explicitly using (4.6). In particular, one sees using this description that the only appearances of the projective A-modules $Ae_j^\pm $ in (9.2) are one copy of $Ae_i^+$ in $P_i^1$ , and one copy of $Ae_i^-$ in $P_i^2$ .

By the horseshoe lemma, we can thus construct a (possibly nonminimal) projective resolution of $GX$ in which the degree k term $\hat {P}^k$ is a direct sum of the $P_i^k$ over composition factors $S_i$ of $GX$ . As a consequence, this resolution includes $\dim _{\mathbb {K}}(e_jGX)$ copies of $Ae_j^+$ , all summands of $\hat {P}^1$ , and $\dim _{\mathbb {K}}(e_jGX)$ copies of $Ae_j^-$ , all summands of $\hat {P}^2$ . Since each $Ae_j^\pm $ appears in only one homological degree in this resolution, it cannot be part of an acyclic summand, and so it appears with the same multiplicity and degree in the minimal projective resolution (9.1) of $GX$ , as required.

Recall that any $X\in \mathcal {E}$ fits into an exact sequence

(9.3)

with $K_X,R_X\in \operatorname {add}{T}$ . Indeed, this holds for any Frobenius cluster category and cluster-tilting object, and the choice of sequence (9.3) is equivalent to the choice of the map $R_X\to X$ , subject to the condition that this map is a right $\operatorname {add}{T}$ -approximation. Dually, X fits into an exact sequence

(9.4)

with $L_X,C_X\in \operatorname {add}{T}$ , with the choice of sequence being equivalent to the choice of left $\operatorname {add}{T}$ -approximation $X\to L_X$ . We say that the sequence (9.3) is minimal if the map ${R_X\to X}$ is a minimal right $\operatorname {add}{T}$ -approximation of X, or equivalently if no nonzero component of the map $K_X\to R_X$ is an isomorphism, and define minimality for (9.4) dually.

Lemma 9.2. For any $X\in \mathcal {E}$ , choose minimal sequences of the forms (9.3) and (9.4), and let $i\in Q_0$ . Then

$$ \begin{align*} [K_X:P_i^\pm]=[C_X:P_i^\pm]&=0,\\ [R_X:P_i^+]=[L_X:P_i^-]&=0,\\ [R_X:P_i^-]=[L_X:P_i^+]&=\dim_{\mathbb{K}}(e_iGX). \end{align*} $$

Proof. The first line of equalities follows immediately from minimality, using that $P_i^\pm $ is projective–injective. For the remaining equalities, recall that F restricts to an equivalence $\operatorname {add}{T}\stackrel {\sim }{\to }\operatorname {proj}{A}$ by the Yoneda lemma. Applying F to (9.3) yields the exact sequence

(9.5)

which is a projective resolution of $FX$ . Applying F to (9.3) yields

and taking the cup product with (9.5) yields

(9.6)

which is a projective resolution of $GX$ . Furthermore, since the $\operatorname {add}{T}$ -approximations $R_X\to X$ and $X\to L_X$ were chosen to be minimal, the projective resolutions (9.5) and (9.6) are also minimal. Noting that $[R_X:P_i^\pm ]=[FR_X:FP_i^\pm ]$ and $[L_X:P_i^\pm ]=[FL_X:FP_i^\pm ]$ , and that $FP_i^\pm =Ae_i^\pm $ by Corollary 4.15, the result follows from Lemma 9.1.

These observations will allow us to compute the g-vectors of any cluster algebra arising from a seed with principal part Q in terms of in terms of the category $\mathcal {E}=\mathcal {E}_Q$ . We define g-vectors via a grading on the principal coefficient cluster algebra $\mathscr {A}_{Q}^{\bullet }$ , as in [Reference Fomin and Zelevinsky20, §6]. Write the initial cluster variables of this cluster algebra as $x_i$ and $x_i^+$ for $i\in Q_0$ , where the $x_i^+$ are frozen. We then give these variables $\mathbb {Z}^{Q_0}$ -degrees

$$\begin{align*}\deg{x_i}=\varepsilon_i,\qquad \deg{x_i^+}=b_i,\end{align*}$$

where $\varepsilon _i$ is the ith standard basis vector (named to avoid confusion with idempotents) and $b_i$ is the ith row of the exchange matrix b of Q. We extend this to the initial cluster variables of $\widetilde {\mathscr {A}}_{Q}$ by setting $\deg {x_i^-}=0$ . This assignment puts a grading on the Laurent polynomial ring in variables $x_i$ , $x_i^+$ , and $x_i^-$ , and thus on its subalgebra $\widetilde {\mathscr {A}}_{Q}$ ; note that the specialization $x_i^-=1$ , which restricts to a map $\widetilde {\mathscr {A}}_{Q}\to \mathscr {A}_{Q}^{\bullet }$ , preserves degrees. We refer to the degree of any homogeneous element of the Laurent polynomial ring in $x_i$ and $x_i^+$ as the g-vector of the element.

All cluster variables of $\widetilde {\mathscr {A}}_{Q}$ are homogeneous in the above grading; by work of Grabowski [Reference Grabowski26, Prop. 3.2], this follows from the matrix identity

(9.7) $$ \begin{align} \begin{pmatrix}-b&1_n&-1_n\end{pmatrix}\begin{pmatrix}1_n\\b\\0_n\end{pmatrix}, \end{align} $$

where $1_n$ is the $n\times n$ identity matrix and $0_n$ is the $n\times n$ zero matrix, by observing that the first matrix in the product is the transposed exchange matrix of the initial seed of $\widetilde {\mathscr {A}}_{Q}$ , and the second has rows given by the degrees of the cluster variables in this seed. Thus, the cluster variables of $\mathscr {A}_{Q}^{\bullet }$ are also homogeneous and hence have well-defined g-vectors.

Given any seed s with principal part Q, the mutable cluster variables of $\mathscr {A}_{Q}^{\bullet }$ are in natural bijection with those of the cluster algebra $\mathscr {A}_{s}$ with initial seed s, and so we define the g-vector of a cluster variable in $\mathscr {A}_{s}$ to be the g-vector of the corresponding cluster variable in $\mathscr {A}_{Q}^{\bullet }$ . This applies in particular to the coefficient-free cluster algebra $\mathscr {A}_{Q}$ .

The main results of this section involve the Fu–Keller cluster character $\varphi ^{T}$ as in [Reference Fu and Keller21] and §6. While we are only assuming that $A=\operatorname {End}_{\mathcal {E}}(T)^{\mathrm {op}}$ is Noetherian, and not necessarily finite-dimensional, this is sufficient for the cluster character to be well defined (cf. [Reference Grabowski and Pressland27, §3] and [Reference Pressland45, §6]).

Theorem 9.3. Assume that $A=A_{Q,W}$ is Noetherian, and consider the corresponding Frobenius category $\mathcal {E}=\mathcal {E}_{Q,W}$ and cluster-tilting object $T=eA$ . Let $X\in \mathcal {E}$ , and choose a minimal short exact sequence of the form (9.3). Write

$$\begin{align*}R_X=P\mathbin{\oplus}\bigoplus_{i\in Q_0}T_i^{r_i},\qquad K_X=\bigoplus_{i\in Q_0}T_i^{k_i}, \end{align*}$$

where $P\in \mathcal {E}$ is projective and $T_i=eAe_i$ . Then $\varphi ^{T}(X)$ is homogeneous, and its g-vector is $\sum _{i\in Q_0}(r_i-k_i)\varepsilon _i$ .

Proof. We first observe that $K_X$ has no projective summands, since this would violate minimality, and so $R_X$ and $K_X$ have expressions of the required form. The element

$$\begin{align*}G=\sum_{i\in Q_0}[S_i]\otimes \varepsilon_i+\sum_{i\in Q_0}[S_i^+]\otimes b_i\in\mathrm{K}_0(\operatorname{fd}{A})\otimes\mathbb{Z}^{Q_0},\end{align*}$$

where $S_j$ denotes the simple A-module at vertex $j\in \widetilde {Q}_0$ , is a grading of $\mathcal {E}$ in the sense of [Reference Grabowski and Pressland27, Def. 3.7]; this follows again from the matrix identity (9.7), as in [Reference Grabowski and Pressland27, Rem. 3.9(ii)]. We write this grading of $\mathcal {E}$ as $\deg _G$ . It then follows from [Reference Grabowski and Pressland27, Prop. 3.11(i)] that $\varphi ^{T}(X)$ is homogeneous and its g-vector is equal to $\deg _G(X)$ .

Now, by [Reference Grabowski and Pressland27, Prop. 3.11(ii)], we have

$$\begin{align*}\deg_G(X)=\deg_G(R_X)-\deg_G(K_X).\end{align*}$$

By construction, we have

$$ \begin{align*} \deg_G(R_X)&=\deg_G(P)+\sum_{i\in Q_0}r_i\deg_G(T_i)=\deg_G(P)+\sum_{i\in Q_0}r_i\varepsilon_i,\\ \deg_G(K_X)&=\sum_{i\in Q_0}k_i\deg_G(T_i)=\sum_{i\in Q_0}k_i\varepsilon_i, \end{align*} $$

so that

$$\begin{align*}\deg_G(X)=\deg_G(P)+\sum_{i\in Q_0}(r_i-k_i)\varepsilon_i.\end{align*}$$

But, by Lemma 9.2, we have $P\in \operatorname {add}{P^-}$ , so $\deg _G(P)=0$ and the result follows.

We may use similar methods to show how the cluster character $\varphi ^{T}$ on $\mathcal {E}$ may be computed solely in terms of the representation theory of the stable category $\underline {\mathcal {E}}$ , and of the finite-dimensional algebra $J(Q,W)$ . This is the first point at which we use an explicit formula for $\varphi ^{T}$ , and so we recall this from [Reference Fu and Keller21].