A categorification of acyclic principal coefficient cluster algebras
Matthew Pressland
In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables, requiring as input a suitable candidate for the endomorphism algebra of a cluster-tilting object in such a category. In this paper, we construct such candidates in the case of cluster algebras with `polarised' principal coefficients, and obtain Frobenius categorifications in the acyclic case. Since cluster algebras with principal coefficients are obtained from those with polarised principal coefficients by setting half of the frozen variables to $1$, our categories also indirectly model principal coefficient cluster algebras, for which no Frobenius categorification exists in general. Moreover, partial stabilisation provides an extriangulated category, in the sense of Nakaoka and Palu, that may more directly model the principal coefficient cluster algebra. Many of the intermediate results remain valid without the acyclicity assumption, as we will indicate. Along the way, we establish a Frobenius version of Keller's result that the Ginzburg dg-algebra of a quiver with potential is bimodule $3$-Calabi--Yau.
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\title{A categorification of acyclic principal coefficient cluster algebras}
\author{Matthew Pressland}
\address{Matthew Pressland\\Institut für Algebra und Zahlentheorie\\Universität Stuttgart\\Pfaffenwaldring 57\\70569 Stuttgart\\Germany}
\email{presslmw@mathematik.uni-stuttgart.de}
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\keywords{cluster algebra, cluster-tilting object, principal coefficients, Calabi--Yau algebra, Frobenius category, Jacobian algebra, quiver with potential}
\date{October 1, 2018} %fix for arXiv
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\begin{abstract}
In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables, requiring as input a suitable candidate for the endomorphism algebra of a cluster-tilting object in such a category. In this paper, we construct such candidates in the case of cluster algebras with `polarised' principal coefficients, and obtain Frobenius categorifications in the acyclic case. Since cluster algebras with principal coefficients are obtained from those with polarised principal coefficients by setting half of the frozen variables to $1$, our categories also indirectly model principal coefficient cluster algebras, for which no Frobenius categorification exists in general. Moreover, partial stabilisation provides an extriangulated category, in the sense of Nakaoka and Palu, that may more directly model the principal coefficient cluster algebra. Many of the intermediate results remain valid without the acyclicity assumption, as we will indicate. Along the way, we establish a Frobenius version of Keller's result that the Ginzburg dg-algebra of a quiver with potential is bimodule $3$-Calabi--Yau.
\end{abstract}
\maketitle

\section{Introduction}

Cluster algebras, introduced by Fomin--Zelevinsky \cite{fomincluster1}, are combinatorially defined algebras with applications to many areas of mathematics, and have been the subject of intense study; see Keller \cite{kellercluster} for a survey of connections between cluster algebras and the representation theory of associative algebras, and the references therein for applications to other fields.

A key obstruction to studying cluster algebras is their recursive definition---one is given some initial data (a seed), and constructs the cluster algebra inductively via sequences of mutations of this seed. Typically one may obtain an infinite number of seeds in this way, and so the output is not easily controllable. To gain a better understanding of the cluster algebra, it has been fruitful to construct categorical models, which allow the combinatorics to be understood in a more global way. In such a model, given by an exact or triangulated category $\clustcat{}$, the clusters are replaced by cluster-tilting objects $T$, defined by the property that
\[\add{T}=\set{X\in\cat:\Ext^1_\cat(T,X)=0}=\set{X\in\cat:\Ext^1_\cat(X,T)=0}.\]
When $\clustcat{}$ is $2$-Calabi--Yau in a suitable sense, these objects may be mutated via a process analogous to that of the mutation of seeds \cite{iyamamutation}. Categorification allows one to give clean, conceptual proofs of many key statements for any cluster algebra admitting such a categorical model, such as `cluster determines seed' \cite{buanclusters}, linear independence of cluster monomials \cite{cerullilinear}, sign coherence of c-vectors \cite{speyeracyclic}, and so on.

We consider skew-symmetric cluster algebras of geometric type, so that a seed is given by the data of a collection of cluster variables forming the vertices of a quiver. Mutation of seeds replaces a single cluster variable by a different one, and alters the quiver by Fomin--Zelevinsky mutation (see for example \cite[\S3.2]{kellercluster}). A subset of the variables may be frozen, indicating that they may not be mutated, and thus occur in every seed. We use the terminology `ice quiver' to refer to a quiver with a specified `frozen' subquiver; from the point of view of cluster algebras, the only relevant additional data is the set of frozen vertices, since arrows between these play no role in the cluster structure, but we will want to consider such arrows later.

For cluster algebras without frozen variables, categorical models, known as cluster categories, have been constructed in great generality, beginning with Buan--Marsh--Reineke--Reiten--Todorov's construction \cite{buantilting} for the case of acyclic quivers, later generalised by Amiot \cite{amiotcluster} to allow for the existence of cycles. Unfortunately, most cluster algebras appearing naturally in other contexts, such as the cluster structures on the coordinate rings of partial flag varieties \cite{geisspartial} and their double Bruhat cells \cite{berensteincluster3}, do have frozen variables, which we would like to capture in a categorical model.

This can be achieved by replacing cluster categories, which are $2$-Calabi--Yau triangulated categories, by stably $2$-Calabi--Yau Frobenius categories. A Frobenius category is, by definition, an exact category with enough projective objects and enough injective objects, such that these two classes of objects coincide. It is the indecomposable projective-injective objects, which necessarily appear as summands of any cluster-tilting object, that will model the frozen variables. The fact that setting all frozen variables to $1$ in a cluster algebra produces a new cluster algebra without frozen variables corresponds to the fact that the stable category of a Frobenius category, given by taking the quotient by the ideal of morphisms factoring through a projective object (`setting the projective objects to $0$') produces a triangulated category \cite[\S I.2]{happeltriangulated}, which we require to be $2$-Calabi--Yau.

Such Frobenius models for cluster algebras with frozen variables have been constructed sporadically for families of examples, often geometric in nature. For example, Geiß--Leclerc--Schröer \cite{geisspartial} construct cluster structures on coordinate rings of open cells in partial flag varieties using a Frobenius model. They then combinatorially lift such structures to the homogeneous coordinate ring of the whole flag variety. Frobenius categorifications of the resulting structures have been obtained in the case of Grassmannians of planes by Demonet--Luo \cite{demoneticequivers1}, all Grassmannians by Jensen--King--Su \cite{jensencategorification}, and in general by Demonet--Iyama \cite{demonetlifting}. However, these constructions are all somewhat specialised to the case at hand, and depend to some extent on using the geometry of the partial flag varieties to gain some insight into the global structure of the cluster algebra, before constructing the categorification. Nájera Chávez \cite{najerachavez2calabiyau} has constructed Frobenius models for finite-type cluster algebras with universal coefficients, in this case using the finiteness to control the global structure.

By contrast, the constructions of cluster categories for cluster algebras without frozen variables by Buan--Marsh--Reineke--Reiten--Todorov and Amiot did not depend on such global information, but instead start from the data of a single seed, as in the original definition of a cluster algebra, possibly enhanced by some additional (but still local) data, such as a potential on the quiver in Amiot's case. In earlier work \cite{presslandinternally}, the author introduced a similar framework for constructing Frobenius models of a cluster algebra, starting again from the data of an initial seed. Starting from the quiver $Q$ of this seed (some of the vertices of which are frozen), one attempts to find a Noetherian algebra $A$ such that the Gabriel quiver of $A$ agrees with $Q$ up to the addition of arrows between frozen vertices, the quotient of $A$ by paths passing through these vertices is finite-dimensional, and, most importantly, $A$ is bimodule internally $3$-Calabi--Yau with respect to these vertices \cite[Def.~2.4]{presslandinternally}. Such an algebra $A$ then determines a candidate Frobenius model of the cluster algebra \cite[Thm.~4.1, Thm.~4.10]{presslandinternally}.

Passing from $Q$ to $A$ requires (mostly necessarily \cite[Rem.~4.11]{presslandinternally}) the choice of a great deal of extra data, satisfying restrictive conditions. Thus it was not clear from the results of \cite{presslandinternally} how realistic it would be to apply this methodology in practice. In this work, we demonstrate that this approach is in fact practical, by using it to construct a Frobenius model for the cluster algebra with `polarised principal coefficients' associated to any acyclic quiver. Fomin--Zelevinsky have shown that cluster algebras with principal coefficients play a `universal' role in the theory, since their combinatorics can be used to control that of any other cluster algebra with the same principal part, meaning the cluster algebra obtained upon specialising all frozen variables to $1$ \cite{fomincluster4}. Cluster algebras with polarised principal coefficients, which we define in \Cref{pp-coeffs}, differ from those with principal coefficients by adding more frozen variables, and so also have this universality property. This coefficient system, for Dynkin-type cluster algebras, also appeared in recent work of Borges and Pierin \cite{borgesnew}, who define a modified cluster character on the ordinary, triangulated cluster category, with values in the corresponding cluster algebra with polarised principal coefficients. Our main theorem is the following.

\begin{thm*}
\label{main-cat-thm}
Let $Q$ be an acyclic quiver, and let $\clust{Q}$ be the corresponding cluster algebra without frozen variables. Then there exists a Frobenius cluster category $\frobcat_Q$ (constructed explicitly) such that
\begin{enumerate}[label=(\roman*)]
\item\label{main-cat-thm-cc} the stable category $\stab{\frobcat}_Q$ is equivalent to the cluster category $\clustcat{Q}$, and
\item\label{main-cat-thm-mut} there is a bijection between cluster-tilting objects of $\frobcat_Q$ and seeds of the polarised principal coefficient cluster algebra with principal part $\clust{Q}$, commuting with mutation, such that the ice quiver of the endomorphism algebra of each cluster-tilting object agrees, up to arrows between frozen vertices, with the ice quiver of the corresponding seed.
\end{enumerate}
\end{thm*}

We interpret the quivers of endomorphism algebras in \ref{main-cat-thm-mut}, and indeed throughout the paper, as ice quivers by declaring the frozen vertices to be those corresponding to projective indecomposable summands. The definition of a Frobenius cluster category is stated below (\Cref{frobclustcat}); such categories always admit a weak cluster structure in the sense of Buan--Iyama--Reiten--Scott \cite[\S II.1]{buancluster}, by \cite[Thm.~II.1.1]{buancluster}. By \Cref{main-cat-thm}\ref{main-cat-thm-mut}, this weak cluster structure on $\frobcat_Q$ is even a cluster structure, also defined in  \cite[\S II.1]{buancluster}.

As we will indicate, while the acyclicity assumption is needed for all of the ingredients in the proof of \Cref{main-cat-thm} to be available simultaneously, many of these intermediate results hold in much wider generality. In particular, we will see that the conclusions of the theorem remain valid when $Q$ is a $3$-cycle (replacing $\clustcat{Q}$ in \ref{main-cat-thm-cc} by Amiot's cluster category $\clustcat{Q,W}$, where $W$ is the potential on $Q$ given by the $3$-cycle).

It is necessary to pass from principal coefficients to some larger collection of frozen variables in order for a result like \Cref{main-cat-thm} to be possible, since a Frobenius categorification of the principal coefficient cluster algebra with principal part $\clust{Q}$ cannot exist; see \Cref{p:no-prin-cat}. However, it is still possible to use the category $\frobcat_Q$ from \Cref{main-cat-thm} to study this cluster algebra, essentially by `ignoring' the extra projective-injective objects in the category, as we will illustrate in \Cref{gradings}. On the other hand, one can also construct from $\frobcat_Q$ an extriangulated category, in the sense of Nakaoka and Palu \cite{nakaokamutation}, which is combinatorially closer to the principal coefficient cluster algebra, for example by having indecomposable projective-injective objects in bijection with the frozen variables of this cluster algebra. Many properties of the resulting category are then easy consequences of its construction, via \cite[Prop.~3.30]{nakaokamutation}, and \Cref{main-cat-thm}.

\begin{cor*}[of \Cref{main-cat-thm}]
\label{c:extriangulated}
Let $Q$ be an acyclic quiver, and let $\clust{}$ be the corresponding cluster algebra without frozen variables. Then there exists a Frobenius extriangulated category $\frobcat^+_Q$ (constructed explicitly) such that
\begin{enumerate}[label=(\roman*)]
\item\label{c:extriangulated-cc} the stable category $\stab{\frobcat}^+_Q$ is equivalent to the cluster category $\clustcat{Q}$, and
\item\label{c:extriangulated-mut} there is a bijection between cluster-tilting objects of $\frobcat_Q$ and seeds of the principal coefficient cluster algebra with principal part $\clust{Q}$, commuting with mutation, such that the ice quiver of the endomorphism algebra of each cluster-tilting object agrees, up to arrows between frozen vertices, with the ice quiver of the corresponding seed.
\end{enumerate}
\end{cor*}

We discuss this construction briefly in \Cref{extriangulated}. Since cluster theory is better developed for Frobenius categories than for extriangulated categories, we focus more on $\frobcat_Q$ in the sequel, but it seems likely that much of the machinery used to study this category (for example, to obtain the results of \Cref{gradings}) could be extended in the future to cover suitably well-behaved extriangulated categories such as $\frobcat^+_Q$.

On the way to proving \Cref{main-cat-thm}, we will obtain other results that are of wider interest, such as the following; here a positive grading of a quiver with potential $(Q,W)$ is a $\ZZ$-grading of the Jacobian algebra $\jac{Q}{W}$ such that all arrows of $Q$ have positive degree. (For the definitions of ordinary and frozen Jacobian algebras, see \Cref{frjacalg}.)

\begin{thm*}[\Cref{QPinterior}]
\label{keller-analogue}
Let $(Q,W)$ be a quiver with potential admitting a positive grading, and let $\stab{A}=\jac{Q}{W}$. Then there is a frozen Jacobian algebra $A=A_{Q,W}$ (constructed explicitly in \Cref{bigQP}), internally bimodule $3$-Calabi--Yau with respect to its frozen idempotent $e$, such that $\stab{A}=A/\Span{e}$.
\end{thm*}

We see this statement as analogous to a result of Keller \cite[Thm.~6.3, Thm.~A.12]{kellerdeformed}, implying that any finite-dimensional Jacobian algebra may be realised as the $0$-th homology of a bimodule $3$-Calabi--Yau dg-algebra constructed by Ginzburg \cite{ginzburgcalabiyau}, and based on this analogy conjecture that it remains true without the assumption on the grading.

Proving \Cref{main-cat-thm}\ref{main-cat-thm-mut} involves extending results of Buan--Iyama--Reiten--Smith \cite{buanmutation}, on mutations of cluster-tilting objects and quivers with potential, to the Frobenius setting. The details of these arguments, which were included in the first arXiv version of the present paper, are now given in a separate article \cite{presslandmutation}, and also apply to other Frobenius cluster categories, such as the Grassmannian cluster categories of Jensen, King and Su \cite{jensencategorification}.

The main results of the paper are contained in Sections~\ref{pp-coeffs}--\ref{extriangulated}. In \Cref{pp-coeffs} we describe the cluster algebras with polarised principal coefficients that we will categorify, and recall the results of \cite{presslandinternally}, which we will use to produce the model. The algebra $A=A_{Q,W}$ needed as input for this construction is defined in \Cref{bigQP}, from the data of a quiver with potential $(Q,W)$. In \Cref{frjacCY} we explain further results of \cite{presslandinternally} which allow one to check the bimodule internally $3$-Calabi--Yau property for a frozen Jacobian algebra, and apply these to $A$ under the assumption that $(Q,W)$ admits a positive grading. This establishes \Cref{keller-analogue}. In \Cref{acyclic}, we show that $A$ is finite-dimensional, and hence Noetherian, when $Q$ is acyclic. This allows us to conclude most of the statements of \Cref{main-cat-thm}, and is the main reason for the acyclicity assumption in this theorem. The results on mutations are found in \Cref{mutation}, where we use them to complete the proof of \Cref{main-cat-thm}. The construction of the extriangulated categorification of \Cref{c:extriangulated} then follows in \Cref{extriangulated}.

The Frobenius cluster category in \Cref{main-cat-thm} is a category of Gorenstein projective modules over an Iwanaga--Gorenstein algebra $B_Q$, which we describe explicitly via a quiver with relations in \Cref{bdy-algs}. In \Cref{gradings} we show that this Frobenius cluster category may be graded, in the sense of \cite{grabowskigradedfrobenius}, in a way that captures the grading of the principal coefficient cluster algebra by Fomin--Zelevinsky \cite{fomincluster4}. This allows us to recover an identity of Fomin--Zelevinsky \cite[Eq.~6.14]{fomincluster4}, relating g-vectors and c-vectors. 
We close in \Cref{egs} with some examples, in particular observing that \Cref{main-cat-thm} remains true when $Q$ is a $3$-cycle.

Throughout, algebras are assumed to be associative and unital. All modules are left modules, the composition of maps $f\colon X\to Y$ and $g\colon Y\to Z$ is denoted by $gf$, and we use the analogous convention for compositions of arrows in quivers. The Jacobsen radical of a module $X$ is denoted by $\rad{X}$. If $p$ is a path in a quiver, we denote its head by $\head{p}$ and its tail by $\tail{p}$.

\section{Polarised principal coefficients}
\label{pp-coeffs}
Let $\clust{}$ be a cluster algebra of geometric type without frozen variables, and let $s_0$ be a seed of $\clust{}$, 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 $\princlust{Q}$ corresponding to this data is defined by the following initial seed. The mutable cluster variables are again $(x_1,\dotsc,x_n)$, and the frozen variables are $(y_1,\dotsc,y_n)$, 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 $y_i$) and an arrow $i\to i^+$. While $\clust{}$ is isomorphic to the cluster algebra determined by any quiver mutation equivalent to $Q$, this is not true of $\princlust{Q}$.

Setting all frozen variables of $\princlust{Q}$ to $1$ to obtain a cluster algebra without frozen variables (a process known as taking the \emph{principal part}), recovers $\clust{}$, giving a bijection between the seeds of $\princlust{Q}$ and those of $\clust{}$ \cite{cerullilinear}; we write $s^\bullet$ for the seed of $\princlust{Q}$ corresponding to a seed $s$ of $\clust{}$. Principal coefficients are important, since knowledge of the cluster algebra $\princlust{Q}$ can be used to study any cluster algebra $\clust{}'$ with principal part $\clust{}$, via the theory of g-vectors and F-polynomials \cite{fomincluster4}.

By choosing some extra data on $Q$, Amiot \cite{amiotcluster} is able to construct a categorical model of $\clust{}$. The construction uses Jacobian algebras, so we recall some relevant definitions, which we will also need later in the paper.

\begin{defn}
\label{frjacalg}
An \emph{ice quiver} $(Q,F)$ consists of a finite quiver $Q$ without loops and a (not necessarily full) subquiver $F$ of $Q$. Denote by $\KK Q$ the completion of the path algebra of $Q$ over a field $\KK$ with respect to the arrow ideal. A \emph{potential} on $Q$ is $W\in\KK Q/\close{[\KK Q,\KK Q]}$, an element of the quotient of $\KK Q$ by the closed ideal generated by commutators. One can think of $W$ as a (possibly infinite) linear combination of cycles of $Q$, such that no two cycles with non-zero coefficient are cyclically equivalent, meaning they differ only by choosing a different starting vertex.

A vertex or arrow of $Q$ is called \emph{frozen} if it is a vertex or arrow of $F$, and \emph{mutable} or \emph{unfrozen} otherwise. For brevity, we write $Q_0^\mut=Q_0\setminus F_0$ and $Q_1^\mut=Q_1\setminus F_1$ 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
\[\der{\alpha}{\alpha_k\dotsm\alpha_1}=\sum_{\alpha_i=\alpha}\alpha_{i-1}\dotsm\alpha_1\alpha_k\dotsm\alpha_{i+1}\]
and extend linearly and continuously to define $\der{\alpha}{W}$. The ideal $\Span{\der{\alpha}{W}:\alpha\in Q_1^\mut}$ of $\KK Q$ is called the \emph{Jacobian ideal}, and we may take its closure $\close{\Span{\der{\alpha}{W}:\alpha\in Q_1^\mut}}$ since $\KK Q$ is a topological algebra. We define the \emph{frozen Jacobian algebra} associated to $(Q,F,W)$ by
\[\frjac{Q}{F}{W}=\KK Q/\close{\Span{\der{\alpha}{W}:\alpha\in Q_1^\mut}}.\]
Write $A=\frjac{Q}{F}{W}$. The above presentation of $A$ suggests a preferred idempotent
\[e=\sum_{v\in F_0}\idemp{v},\]
which we call the \emph{frozen idempotent}. We will call $B=eAe$ the \emph{boundary algebra} of $A$. If $F=\varnothing$, then we refer to the pair $(Q,W)$ as a quiver with potential, and call $\jac{Q}{W}=\frjac{Q}{\varnothing}{W}$ simply the Jacobian algebra of $(Q,W)$.
\end{defn}

\begin{rem}
While $\KK Q$ usually denotes the ordinary (uncompleted) path algebra of $Q$, we use it here as clean notation for the completed version since this is the version we will always take. Moreover, in much of the paper, we are interested in algebras $\KK Q/\close{\Span{R}}$ which turn out to be isomorphic to the quotient of the ordinary path algebra of $Q$ by the ideal $\Span{R}$, the latter being finite-dimensional, and so the distinction was irrelevant. We take complete path algebras in the general theory since we want the categories we construct to be Krull--Schmidt (cf. \cite[Rem.~3.3]{jensencategorification}).
\end{rem}

Returning to the cluster algebra $\clust{Q}$, choose a potential $W$ on $Q$ such that $\jac{Q}{W}$ is finite-dimensional. Then by work of Amiot \cite{amiotcluster}, there is a $2$-Calabi--Yau triangulated category $\clustcat{Q,W}$ categorifying $\clust{Q}$. If $W$ is non-degenerate \cite[Def.~7.2]{derksenquivers1}, then the seeds of $\clust{}$ correspond bijectively to additive equivalence classes of cluster-tilting objects of $\clustcat{Q,W}$ related by a finite sequence of mutations from an initial cluster-tilting object $T_0$ with $\Endalg{\clustcat{Q,W}}{T_0}=\jac{Q}{W}$; we call such cluster-tilting objects \emph{reachable}, and denote the seed corresponding to such an object $T$ by $s_T$. (Non-degeneracy is needed here to see, using results of Buan--Iyama--Reiten--Smith \cite{buanmutation}, 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.)

A priori, to categorify the principal coefficient cluster algebra $\princlust{Q}$, we would like to to find a Frobenius category $\frobcat$ such that
\begin{enumerate}[label=(\roman*)]
\item\label{prin-cat-cc} the stable category of $\frobcat$ is triangle equivalent to $\clustcat{Q,W}$ (so cluster-tilting objects of $\frobcat$ may be identified with those of $\clustcat{Q,W}$), and
\item\label{prin-cat-mut} for any reachable cluster-tilting object $T$, the quiver of $\Endalg{\frobcat}{T}$ is, up to arrows between frozen vertices, the quiver of the seed $s_T^\bullet$ of $\princlust{Q}$.
\end{enumerate}
Unfortunately, this is not possible.

\begin{prop}
\label{p:no-prin-cat}
There does not exist a Frobenius category $\frobcat$ satisfying conditions \ref{prin-cat-cc}--\ref{prin-cat-mut} above.
\end{prop}
\begin{proof}
Assume $\frobcat$ were such a Frobenius category, and identify cluster-tilting objects of $\frobcat$ with those of $\clustcat{Q,W}$ by \ref{prin-cat-cc}. We claim that the Gabriel quiver of $\Endalg{\frobcat}{T_0}$, where $T_0$ is the initial cluster-tilting object, has no paths from a frozen vertex, corresponding to a projective-injective indecomposable summand of $T_0$, to a mutable vertex, corresponding to a non-projective indecomposable summand. Indeed, this is true of $Q^\bullet$, and by \ref{prin-cat-mut} the relevant Gabriel quiver differs only by the possible addition of new arrows between frozen vertices, a process which does not affect the property under consideration. It follows that there are no non-zero maps in $\frobcat$ from a non-projective summand of $T_0$ to any projective-injective object, all of which lie in $\add{T_0}$, and so this summand has no injective envelope, contradicting the requirement that $\frobcat$ is a Frobenius category.
\end{proof}

To avoid this problem, Fu--Keller \cite[\S6.7]{fucluster} have modelled $\princlust{Q}$ via an additive category, in the case that $Q$ is acyclic. Taking the extended quiver $Q^\bullet$, which is also acyclic, they consider a distinguished subcategory $\mathcal{U}_Q\subseteq\clustcat{Q^\bullet}$ determined by the frozen vertices; precisely, it is the full subcategory on objects Ext-orthogonal to the summands of the initial cluster-tilting object of $\clustcat{Q^{\bullet}}$ corresponding to these vertices. We note that the objects corresponding to frozen vertices of $Q^\bullet$ are not characterised intrinsically within $\clustcat{Q^{\bullet}}$, although it seems possible based on computations in small examples that $\mathcal{U}_Q$ has a natural extriangulated structure (making it equivalent to the extriangulated category of \Cref{c:extriangulated}) in which these objects are the indecomposable projective-injectives; we do not pursue this comparison in this paper, however.

Here, we take a different approach---we simply add more frozen variables to $\princlust{Q}$ and then try to categorify the result using a Frobenius category $\frobcat_{Q,W}$. While $\frobcat_{Q,W}$ will have too many indecomposable objects to be a `strict' model for $\princlust{Q}$, the objects corresponding to frozen variables (of our extended cluster algebra) will be intrinsically characterised by the property of being indecomposable projective-injective. This approach is also more efficient, in the sense that $\frobcat_{Q,W}$ will only have $n=|Q_0|$ too many indecomposable objects, whereas the cluster category $\clustcat{Q^\bullet}$ considered by Fu--Keller can be of wild type even when $\clustcat{Q}$, and hence $\mathcal{U}_Q$, is of finite type.

Since $\princlust{Q}$ can be obtained from the extended cluster algebra by specialising some frozen variables to $1$, our category will still encode all of the combinatorial information about $\princlust{Q}$. By composing the usual cluster character on $\frobcat$ \cite{fucluster} with the projection to $\princlust{Q}$, one can even write down a function which is morally a cluster character $\frobcat\to\princlust{Q}$, but with the unusual feature that it `has kernel' i.e.\ some non-zero objects of $\frobcat$ have character $1$. In particular, it will factor over the extriangulated category of \Cref{c:extriangulated}.

Our chosen extension of $\princlust{Q}$ is the `polarised principal coefficient' cluster algebra $\pprinclust{Q}$, which we now define. Starting from our seed $s_0$ of $\clust{}$, with quiver $Q$ and cluster variables $(x_1,\dotsc,x_n)$, we construct an initial seed $\lift{s}_0$ of $\pprinclust{Q}$ as follows. The mutable variables are $(x_1,\dotsc,x_n)$, and the frozen variables are $(y_1^+,\dotsc,y_n^+,y_1^-,\dotsc,y_n^-)$. The ice quiver $\lift{Q}$ contains $Q$ as a full subquiver, with mutable vertices, and has two frozen vertices $i^+$ (corresponding to $y_i^+$) and $i^-$ (corresponding to $y_i^-)$ for each mutable vertex $i\in Q_0$, with arrows $i\to i^+$ and $i^-\to i$ for each $i$. In \Cref{bigQP} we will also describe arrows between the frozen vertices of $\lift{Q}$, but since these play no role in the definition of the cluster algebra $\pprinclust{Q}$ we ignore them for now. This enlarged quiver (without the frozen arrows) is called the \emph{biframed quiver} of $Q$ in \cite{borgesnew}.

We adopt the word `polarised', referring to the partitioning of the frozen variables into two `flavours', to differentiate this coefficient system from the `double principal coefficients' studied by Rupel--Stella--Williams \cite{rupelgeneralized}. 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 polarised version.

\begin{rem}
\label{bruhatremark}
Like the double principal coefficient cluster algebras of \cite{rupelgeneralized}, the cluster algebra $\pprinclust{Q}$ associated to a Dynkin quiver $Q$ may (after inverting frozen variables) be realised as the coordinate ring of a double Bruhat cell, as we now briefly explain.

Let $G$ be a simple connected complex Lie group of type $\Delta$. After choosing a Borel subgroup and maximal torus, one may associate the \emph{double Bruhat cell} $G^{u,v}$ to any pair $u,v\in W$, as in \cite{fomindouble}. Analogous to the classical Bruhat decomposition, $G$ is then expressible as the disjoint union
\[G=\bigsqcup_{(u,v)\in W^2}G^{u,v}.\]
Berenstein--Fomin--Zelevinsky have shown that each coordinate ring $\CC[G^{u,v}]$ has the structure of an upper cluster algebra \cite[Thm.~2.10]{berensteincluster3} with invertible coefficients. In fact, by work of Goodearl and Yakimov \cite{goodearlberenstein}, the word `upper' can be dropped.

Assume $Q$ is an orientation of a simply-laced Dynkin diagram $\Delta$ with vertex set $\{1,\dotsc,n\}$. Quivers with underlying graph $\Delta$ are in bijection with Coxeter elements of the Weyl group $W$ of $\Delta$ as follows. Let $s_1,\dotsc,s_n$ be the simple reflections generating $W$, and let $i_1,\dotsc,i_n$ be an ordering of $Q_0=\{1,\dotsc,n\}$ such that $i_j<i_k$ whenever there is an arrow from $j$ to $k$. Such an ordering is not unique, but any two determine the same Coxeter element $c=s_{i_1}\dotsm s_{i_n}$ of the Weyl group, and every Coxeter element arises in this way from a unique orientation of $Q$.

In the case of the double Bruhat cell $G^{c,c}$ associated to a Coxeter element $c$, the algebra $\CC[G^{c,c}]$ is isomorphic to the cluster algebra with (inverted) polarised principal coefficients associated to the orientation $Q$ of $\Delta$ determined by $c$, cf.\ \cite[Thm.~2.13]{rupelgeneralized}. (In this case we do not need the results of \cite{goodearlberenstein} to know that the cluster algebra coincides with the upper cluster algebra, since this follows instead from acyclicity of $Q$.) Our general results can thus be exploited to construct a Frobenius categorification of this cluster algebra.
\end{rem}

As discussed in the introduction, we will construct a categorification of $\pprinclust{Q}$ using methodology introduced by the author in \cite{presslandinternally}. We now recall the key definitions and results needed to explain this construction. Write $\dcat{A}$ for the derived category of an algebra $A$, and $\env{A}=A\tens_\KK A^{\op}$ for its enveloping algebra, modules of which are precisely $A$-bimodules. We denote by $\per{A}$ the perfect derived category of $A$, i.e.\ the thick subcategory of $\dcat{A}$ generated by $A$, and write $\rhom{A}=\RHom_{\env{A}}(A,\env{A})$.

\begin{defn}[{\cite[Def.~2.4]{presslandinternally}}]
\label{bi3cy-def}
Given an algebra $A$ and idempotent $e\in A$, we say $A$ is \emph{bimodule internally $3$-Calabi--Yau} with respect to $e$ if
\begin{enumerate}
\item $\pdim_{\env{A}}{A}\leq 3$,
\item $A\in\per{\env{A}}$, and
\item there exists a triangle $A\map{\psi}\rhom{A}[3]\map{}C\map{}A[1]$ in $\dcat{\env{A}}$, such that
\[\RHom_A(C,M)=0=\RHom_{A^{\op}}(C,N)\]
whenever $M\in\dcat{A}$ is a complex whose total cohomology is a finite-dimensional $A/\Span{e}$-module, and analogously for $N\in\dcat{A^{\op}}$.
\end{enumerate}
\end{defn}

\begin{rem}
Assume $A$ is bimodule internally $3$-Calabi--Yau with respect to $e$, and write $\stab{A}=A/\Span{e}$. Then $\gldim{A}\leq 3$, and there is a functorial duality
\[\Kdual\Ext^i_A(M,N)=\Ext^{3-i}_A(N,M)\]
for finite-dimensional $M\in\fgmod{A/\Span{e}}$ and any $N\in\Mod{A}$ \cite[Cor.~2.9]{presslandinternally}. Moreover, the same is true of $A^{\op}$ \cite[Rem.~2.6]{presslandinternally}. In the language of \cite[Def.~2.1]{presslandinternally}, we say that $A$ and $A^{\op}$ are internally $3$-Calabi--Yau with respect to $e$ (without the word `bimodule').
\end{rem}

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

\begin{thm}[{\cite[Thm.~4.1, Thm.~4.10]{presslandinternally}}]
\label{icytofrobcat}
Let $A$ be an algebra, and $e\in A$ an idempotent. If $A$ is Noetherian, $\stab{A}=A/\Span{e}$ is finite-dimensional, and $A$ is bimodule internally $3$-Calabi--Yau with respect to $e$, then
\begin{enumerate}[label=(\roman*)]
\item\label{icytofrobcat-frob} $B=eAe$ is Iwanaga--Gorenstein of injective dimension at most $3$, so
\[\GP(B)=\set{X\in\fgmod{B}:\Ext^i_B(X,B)=0,\ i>0}\]
is a Frobenius category,
\item\label{icytofrobcat-ct} $eA\in\GP(B)$ is cluster-tilting,
\item\label{icytofrobcat-isos} there are natural isomorphisms $\Endalg{B}{eA}\isoto A$ and $\stabEndalg{B}{eA}\isoto\stab{A}$, and
\item\label{icytofrobcat-cy} the stable category $\stabGP(B)$ is $2$-Calabi--Yau.
\end{enumerate}
\end{thm}

Ideally, we would like $\GP(B)$ as constructed above to be a Frobenius cluster category, in the sense of the following definition.

\begin{defn}[cf.~{\cite[Def.~3.3]{presslandinternally}}]
\label{frobclustcat}
An exact category $\frobcat$ is called a Frobenius cluster category if it is Krull--Schmidt, stably $2$-Calabi--Yau, and $\gldim{\Endalg{\frobcat}{T}}\leq3$ for any cluster-tilting object $T\in\frobcat$, of which there is at least one.
\end{defn}

The category $\GP(B)$ constructed in \Cref{icytofrobcat} is Frobenius by \ref{icytofrobcat-frob}, idempotent complete for arbitrary $B$, and stably $2$-Calabi--Yau by \ref{icytofrobcat-cy}. Since $\Endalg{B}{eA}\iso A$ is bimodule internally $3$-Calabi--Yau, it has global dimension at most $3$, but we do not know a priori that this is true for endomorphism algebras of other cluster-tilting objects. However, this does hold whenever such an algebra is Noetherian, by \cite[Prop.~3.7]{presslandinternally}. In particular, if $A$ is finite-dimensional, then so is $B$, and then $\GP(B)$ is a Frobenius cluster category.

Returning to the problem of categorifying the cluster algebra $\pprinclust{Q}$, our aim now is to construct an algebra $A$ satisfying the conditions of \Cref{icytofrobcat}, such that the Gabriel quiver of $A$ agrees with the quiver $\lift{Q}$ up to arrows between frozen vertices.

\section{An ice quiver with potential}
\label{bigQP}

Consider again our initial seed $s_0$ for $\clust{}$, with quiver $Q$, and choose a potential $W$ on $Q$. In this section, we will construct from $(Q,W)$ an ice quiver with potential $(\lift{Q},\lift{F},\lift{W})$, and thus a frozen Jacobian algebra $\frjac{\lift{Q}}{\lift{F}}{\lift{W}}$. It is this algebra that we intend to use as the input for the construction of a Frobenius category by \Cref{icytofrobcat}; at that point we will require that $\jac{Q}{W}$ is finite-dimensional, but we do not assume this yet.

\begin{defn}
\label{bigQP-def}
Let $(Q,W)$ be a quiver with potential. We define $\lift{Q}$ to be the quiver with vertex set given by
\[\lift{Q}_0=Q_0\sqcup Q_0^+\sqcup Q_0^-\]
where $Q_0^+=\set{i^+:i\in Q_0}$ is a set of formal symbols in bijection with $Q_0$, and similarly for $Q_0^-=\set{i^-:i\in Q_0}$. The set of arrows is given by
\[\lift{Q}_1=Q_1\sqcup\set{\alpha_i:i\in Q_0}\sqcup\set{\beta_i:i\in Q_0}\sqcup\set{\delta_i:i\in Q_0}\sqcup\set{\delta_a:a\in Q_1}.\]
The head and tail functions $\head{}$ and $\tail{}$ on $\lift{Q}_1$ are extended from those on $Q_1$ by defining
\begin{align*}
\head{\alpha_i}&=i^+,&\tail{\alpha_i}&=i,&\head{\beta_i}&=i,&\tail{\beta_i}&=i^-,\\
\head{\delta_i}&=i^-,&\tail{\delta_i}&=i^+,&\head{\delta_a}&=(\tail{a})^-,&\tail{\delta_a}&=(\head{a})^+.
\end{align*}
The frozen subquiver $\lift{F}$ is defined by
\[
\lift{F}_0=Q_0^+\sqcup Q_0^-,\qquad
\lift{F}_1=\set{\delta_i:i\in Q_0}\sqcup\set{\delta_a:a\in Q_1}.
\]
Note that the head and tail of any arrow in $\lift{F}_1$ lies in $\lift{F}_0$, so these subsets describe a valid subquiver of $\lift{Q}$, that is in fact full. The quiver $\lift{F}$ 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, vertices of the form $i^+$ are sources, and those of the form $i^-$ are sinks. When viewed in $\lift{Q}$, each vertex $i^+$ has unique incoming arrow $\alpha_i$ and each $i^-$ has unique outgoing arrow $\beta_i$.

Finally, we define a potential $\lift{W}$ on $\lift{Q}$ by
\[\lift{W}=W+\sum_{i\in Q_0}\beta_i\delta_i\alpha_i-\sum_{a\in Q_1}a\beta_{\tail{a}}\delta_a\alpha_{\head{a}},\]
and let
\[A_{Q,W}=\frjac{\lift{Q}}{\lift{F}}{\lift{W}}\]
be the frozen Jacobian algebra determined by $(\lift{Q},\lift{F},\lift{W})$. 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 $A_{Q,W}$.
\end{defn}

\begin{nb}
To aid legibility, if the vertices $i^+$ or $i^-$ appear as subscripts, we will usually move the sign into a superscript, as in the above expression for the frozen idempotent. When $W=0$, we will typically drop it from the notation; for example, we write $A_Q=A_{Q,0}$. The reader is warned that $\lift{0}$ is not the zero potential on $\lift{Q}$.
\end{nb}

When $Q$ is the quiver of the initial seed $s_0$ of $\clust{}$, the quiver $\lift{Q}$ `is' by construction the ice quiver of the same name forming part of the data of our initial seed $\lift{s}_0$ of $\pprinclust{Q}$, but with the additional data of arrows between frozen vertices.

Since $\lift{W}$ has a straightforward description in terms of $W$, so do the defining relations of $A_{Q,W}$; these form the set $R$ consisting of
\begin{equation}
\label{relations}
\der{a}{\lift{W}}=\der{a}{W}-\beta_{\tail{a}}\delta_a\alpha_{\head{a}},\quad
\der{\alpha_i}{\lift{W}}=\beta_i\delta_i-\sum_{\substack{\gamma\in Q_1\\\head{\gamma}=i}}\gamma\beta_{\tail{\gamma}}\delta_\gamma,\quad
\der{\beta_i}{\lift{W}}=\delta_i\alpha_i-\sum_{\substack{\gamma\in Q_1\\\tail{\gamma}=i}}\delta_\gamma\alpha_{\head{\gamma}}\gamma,
\end{equation}
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 \Cref{icytofrobcat}, 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 \Cref{bi3cy-def}. We will do this, under a mild assumption on $(Q,W)$, in \Cref{frjacCY}, but first give some examples.

\begin{eg}
\label{construction-egs}
The quiver with potential $(Q,0)$, for $Q$ an $\typeA{2}$ quiver, provides the most basic example revealing all of the combinatorial features of the construction. In this case, we have
\[\lift{Q}=\mathord{\begin{tikzpicture}[baseline={(current bounding box.center)},yscale=-1]
\node at (-1,0) (1) {$1$};
\node at (1,0) (2) {$2$};
\node at (-2,1) (1+) {$\boxed{1^+}$};
\node at (-2,-1) (1-) {$\boxed{1^-}$};
\node at (2,1) (2-) {$\boxed{2^-}$};
\node at (2,-1) (2+) {$\boxed{2^+}$};
\path[-angle 90,font=\scriptsize]
	(1) edge node[below] {$a$} (2)
	(1) edge node[below right] {$\alpha_1$} (1+)
	(1-) edge node[above right] {$\beta_1$} (1)
	(2) edge node[above left] {$\alpha_2$} (2+)
	(2-) edge node[below left] {$\beta_2$} (2);
\path[\frozen,-angle 90,font=\scriptsize]
	(1+) edge[bend right] node[left] {$\delta_1$} (1-)
	(2+) edge[bend right] node[right] {$\delta_2$} (2-)
	(2+) edge[bend left] node[above] {$\delta_a$} (1-);
\end{tikzpicture}}\]
with $\lift{F}$ indicated by the boxed vertices and dashed arrows. The potential on this ice quiver is
\[\lift{W}=\beta_1\delta_1\alpha_1+\beta_2\delta_2\alpha_2-a\beta_1\delta_a\alpha_2.\]
One can check that the frozen Jacobian algebra $A_Q$ attached to this data is isomorphic to the endomorphism algebra of a cluster-tilting object in the Frobenius cluster category of those modules for the preprojective algebra of type $\typeA{4}$ with socle supported at a fixed bivalent vertex \cite{geisspartial}. For an explanation of why the categories arising in \cite{geisspartial} are Frobenius cluster categories, see \cite[Eg.~3.11]{presslandinternally}.

Now let $(Q,W)$ be the quiver with potential in which
\[Q=\mathord{\begin{tikzpicture}[baseline=0,xscale=1.5]
\node at (-30:0.75) (3) {$3$};
\node at (-150:0.75) (1) {$1$};
\node at (90:0.75) (2) {$2$};
\path[-angle 90,font=\scriptsize]
(3) edge node[below] {$c$} (1)
(1) edge node[left] {$a$} (2)
(2) edge node[right] {$b$} (3);
\end{tikzpicture}}\]
and $W=cba$. The Jacobian algebra is a cluster-tilted algebra of type $\typeA{3}$, and has infinite global dimension (as indeed does any non-hereditary cluster-tilted algebra \cite[Cor.~2.1]{kellerclustertilted}). In this case
\[\widetilde{Q}=\mathord{\begin{tikzpicture}[baseline={(current bounding box.center)},scale=1.2]
\node at (-30:0.75) (3) {$3$};
\node at (-150:0.75) (1) {$1$};
\node at (90:0.75) (2) {$2$};
\node at (60:2) (2-) {$\boxed{2^-}$};
\node at (0:2) (3+) {$\boxed{3^+}$};
\node at (-60:2) (3-) {$\boxed{3^-}$};
\node at (-120:2) (1+) {$\boxed{1^+}$};
\node at (180:2) (1-) {$\boxed{1^-}$};
\node at (120:2) (2+) {$\boxed{2^+}$};
\path[-angle 90,font=\scriptsize]
(3) edge node[below] {$c$} (1)
(1) edge node[left] {$a$} (2)
(2) edge node[right] {$b$} (3)
(2-) edge node[right] {$\beta_2$} (2)
(3) edge node[above] {$\alpha_3$} (3+)
(3-) edge node[right] {$\beta_3$} (3)
(1) edge node[left] {$\alpha_1$} (1+)
(1-) edge node[above] {$\beta_1$} (1)
(2) edge node [left] {$\alpha_2$} (2+);
\path[-angle 90,\frozen,font=\scriptsize]
(1+) edge [bend right=20pt] node[below] {$\delta_c$} (3-)
(3+) edge [bend left=20pt] node[right] {$\delta_3$} (3-)
(3+) edge [bend right=20pt] node[right] {$\delta_b$} (2-)
(2+) edge [bend left=20pt] node[above] {$\delta_2$} (2-)
(2+) edge [bend right=20pt] node[left] {$\delta_a$} (1-)
(1+) edge [bend left=20pt] node[left] {$\delta_1$} (1-);
\end{tikzpicture}}\]
with $\lift{F}$ again indicated by boxed vertices and dashed arrows. The potential is
\[\lift{W}=cba+\beta_1\delta_1\alpha_1+\beta_2\delta_2\alpha_2+\beta_3\delta_3\alpha_3-a\beta_1\delta_a\alpha_2-b\beta_2\delta_b\alpha_3-c\beta_3\delta_c\alpha_1.\]
The associated frozen Jacobian algebra $A_{Q,W}$ also arises from a dimer model on a disk with six marked points on its boundary \cite{baurdimer}, 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 $\Grass{2}{6}$ \cite{jensencategorification}. This category is again a Frobenius cluster category \cite[Eg.~3.12]{presslandinternally}. Unlike the first example, this algebra is infinite-dimensional. However, it is Noetherian, so \Cref{icytofrobcat} still applies.

Since in both of these cases 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 by a result of Keller--Reiten \cite[\S5.4]{kellerclustertilted} (see also \cite[Cor.~3.10]{presslandinternally}). This foreshadows \Cref{bi3cy} below, which states that the stronger bimodule internal Calabi--Yau property holds.
\end{eg}

\begin{rem}
Combinatorially, the algebra $A_{Q,W}$ seems to have a lot to do with the dg-algebra $\Gamma_{Q,W}$ associated to $(Q,W)$ by Ginzburg \cite[\S4.2]{ginzburgcalabiyau}; 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 $\head{a}\to\head{a}^+\to\tail{a}^-\to\tail{a}$. Here we use Amiot's sign conventions \cite[Def.~3.1]{amiotcluster}, which are opposite to Ginzburg's.

By a result of Keller \cite[Thm.~6.3]{kellerdeformed}, the dg-algebra $\Gamma_{Q,W}$ is always bimodule $3$-Calabi--Yau, and we expect this to be related to the fact that (at least under a mild assumption on $(Q,W)$; see \Cref{bi3cy} below) $A_{Q,W}$ 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 \Cref{main-cat-thm}\ref{main-cat-thm-cc} that when $Q$ is acyclic, the two constructions are also related via a triangle equivalence
\[\stabGP(B_Q)\simeq\clustcat{Q}=\frac{\per{\Gamma_Q}}{\bdcat{\Gamma_Q}}.\]
\end{rem}

\section{Calabi--Yau properties for frozen Jacobian algebras}
\label{frjacCY}

We now recall a sufficient condition on an ice quiver with potential $(Q,F,W)$ for the associated frozen Jacobian algebra to be bimodule internally $3$-Calabi--Yau with respect to the frozen idempotent $e=\sum_{v\in F_0}e_v$ \cite[\S5]{presslandinternally}. We will show, under mild assumptions on $(Q,W)$, that the ice quiver with potential $(\lift{Q},\lift{F},\lift{W})$ from \Cref{bigQP-def} satisfies this condition, and so $A_{Q,W}$ has the necessary Calabi--Yau symmetry for us to be able to apply \Cref{icytofrobcat}. Thus this section constitutes the main step in the proof of \Cref{main-cat-thm}.

In \cite[\S5]{presslandinternally}, it is explained how an ice quiver with potential $(Q,F,W)$ determines a complex of projective bimodules for the associated frozen Jacobian algebra $A=\frjac{Q}{F}{W}$. We denote this complex by $\res{A}$, although strictly it depends on the presentation of $A$ determined by $(Q,F,W)$, and will now give its definition.

Recall that $Q_0^\mut=Q_0\setminus F_0$ and $Q_1^\mut=Q_1\setminus F_1$ denote the sets of mutable vertices and unfrozen arrows of $Q$ respectively. For $v\in Q_0$, we write $\vout{v}$ for the set of arrows of $Q$ with tail $v$, and $\vin{v}$ for the set of arrows of $Q$ with head $v$. Denote the arrow ideal of $A$ by $\rad{A}$, and let $S=A/\rad{A}$. As a left $A$-module, $S$ is the direct sum of the vertex simple left $A$-modules, and has a basis given by the vertex idempotents $e_v$. For the remainder of this section, we write $\tens=\tens_S$.

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*}
\KK Q_0&=\bigdsum_{v\in Q_0}\KK\idemp{v},&\KK Q_0^\mut&=\bigdsum_{v\in Q_0^\mut}\KK\idemp{v},&\KK F_0&=\bigdsum_{v\in F_0}\KK\idemp{v},\\
\KK Q_1&=\bigdsum_{\alpha\in Q_1}\KK\alpha,&\KK Q_1^\mut&=\bigdsum_{\alpha\in Q_1^\mut}\KK\alpha,&\KK F_1&=\bigdsum_{\alpha\in F_1}\KK\alpha,\\
\KK Q_2&=\bigdsum_{\alpha\in Q_1}\KK\rel{\alpha},&\KK Q_2^\mut&=\bigdsum_{\alpha\in Q_1^\mut}\KK\rel{\alpha},&\KK F_2&=\bigdsum_{\alpha\in F_1}\KK\rel{\alpha},\\
\KK Q_3&=\bigdsum_{v\in Q_0}\KK\omega_v,&\KK Q_3^\mut&=\bigdsum_{v\in Q_0^\mut}\KK\omega_v,&\KK F_3&=\bigdsum_{v\in F_0}\KK\omega_v,
\end{align*}
via the formulae
\[
\idemp{v}\cdot\idemp{v}\cdot\idemp{v}=\idemp{v},\qquad
\idemp{\head{\alpha}}\cdot\alpha\cdot\idemp{\tail{\alpha}}=\alpha,\qquad
\idemp{\tail{\alpha}}\cdot\rel{\alpha}\cdot\idemp{\head{\alpha}}=\rel{\alpha},\qquad
\idemp{v}\cdot\omega_v\cdot\idemp{v}=\omega_v.
\]
For each $i$, the $S$-bimodule $\KK Q_i$ splits as the direct sum
\[\KK Q_i=\KK Q_i^\mut\dsum\KK F_i.\]
Since $\KK Q_0\cong S$, the $A$-bimodule $A\tens \KK Q_0\tens A$ is canonically isomorphic to $A\tens A$, and we will use the two descriptions interchangeably.

We define maps $\bar{\mu}_i\colon A\tens \KK Q_i\tens A\to A\tens \KK Q_{i-1}\tens A$ for $1\leq i\leq 3$, starting with
\[\bar{\mu}_1(x\tens\alpha\tens y)=x\tens\idemp{\head{\alpha}}\tens\alpha y-x\alpha\tens\idemp{\tail{\alpha}}\tens y.\]
Composing with the natural isomorphism $A\tens\KK Q_0\tens A\isoto A\tens A$, we may instead write
\[\bar{\mu}_1(x\tens\alpha\tens y)=x\tens\alpha y-x\alpha\tens y.\]
For any path $p=\alpha_m\dotsm\alpha_1$ of $\KK Q$, we may define
\[\Delta_\alpha(p)=\sum_{\alpha_i=\alpha}\alpha_m\dotsm\alpha_{i+1}\tens\alpha_i\tens\alpha_{i-1}\dotsm\alpha_1,\]
and extend by linearity and continuity to obtain a map $\Delta_\alpha\colon\KK Q\to A\tens\KK Q_1\tens A$. We then define
\[\bar{\mu}_2(x\tens\rel{\alpha}\tens y)=\sum_{\beta\in Q_1}x\Delta_\beta(\der{\alpha}{W})y.\]
Finally, let
\[\bar{\mu}_3(x\tens\omega_v\tens y)=\sum_{\alpha\in\vout{v}}x\tens\rel{\alpha}\tens\alpha y-\sum_{\beta\in\vin{v}}x\beta\tens\rel{\beta}\tens y.\]

\begin{defn}
\label{frjacres}
For an ice quiver with potential $(Q,F,W)$, with frozen Jacobian algebra $A=\frjac{Q}{F}{W}$, let $\res{A}$ be the complex of $A$-bimodules with non-zero terms
\[\begin{tikzcd}[column sep=20pt]
A\tens \KK Q_3^\mut\tens A\arrow{r}{\mu_3}&A\tens \KK Q_2^\mut\tens A\arrow{r}{\mu_2}&A\tens \KK Q_1\tens A\arrow{r}{\mu_1}&A\tens\KK Q_0\tens A
\end{tikzcd}\]
and $A\tens\KK Q_0\tens A$ in degree $0$, where $\mu_1=\bar{\mu}_1$, and the maps $\mu_2$ and $\mu_3$ are obtained by restricting $\bar{\mu}_2$ and $\bar{\mu}_3$ to $A\tens \KK Q_2^\mut\tens A$ and $A\tens \KK Q_3^\mut\tens A$ respectively. As $\vout{v}\union\vin{v}\subseteq Q_1^\mut$ for any $v\in Q_0^\mut$, the map $\mu_3$ takes values in $A\tens \KK Q_2^\mut\tens A$ as claimed.
\end{defn}

\begin{thm}[{\cite[Thm.~5.7]{presslandinternally}}]
\label{frjaci3cy}
Let $\mu_0\colon A\tens A\to A$ denote the multiplication map. If $A$ is a frozen Jacobian algebra such that
\[0\to\res{A}\map{\mu_0}A\to0\]
is exact, then $A$ is bimodule internally $3$-Calabi--Yau with respect to the idempotent $e=\sum_{v\in F_0}\idemp{v}$.
\end{thm}

\begin{rem}
\label{standard-res}
By standard results on presentations of algebras, as in Butler--King \cite{butlerminimal}, we need only check exactness in degrees $-2$ and $-3$, exactness elsewhere holding for any $A$.
\end{rem}

Returning to our main goal, let $(Q,W)$ be a quiver with potential, let $(\lift{Q},\lift{F},\lift{W})$ be the ice quiver with potential associated to $(Q,W)$ in \Cref{bigQP-def}, and let $A=A_{Q,W}$ be its frozen Jacobian algebra. In order to apply \Cref{frjaci3cy}, we assume a little more about $(Q,W)$.

\begin{defn}
\label{posgrad-def}
Let $(Q,W)$ be a quiver with potential. A \emph{positive grading} of $(Q,W)$ is a a function $\deg\colon Q_1\to\ZZ_{>0}$, such that $W$ is homogeneous in the induced $\ZZ$-grading on $\KK Q$.
\end{defn}

Any $\ZZ$-grading of $\KK Q$ in which $W$ is homogeneous descends to a $\ZZ$-grading of the Jacobian algebra $\jac{Q}{W}$, or of the frozen Jacobian algebra $A=\frjac{Q}{F}{W}$ for any subquiver $F$ of $Q$. If this grading is induced from a positive grading as in \Cref{posgrad-def}, then  the degree $0$ part of $A$ is isomorphic to $S=A/\rad{A}$.

If $W=0$, then any assignment of a positive integer to each arrow defines a positive grading. Given a positive grading in which $W$ has degree $d$, multiplying the degree of every arrow appearing in $W$ by $k$ gives a new positive grading in which $W$ has degree $kd$. This procedure will be useful shortly, when we show that a positive grading of $(Q,W)$ extends, up to such rescaling, to a positive grading of $(\lift{Q},\lift{W})$.

A positive grading need not exist for an arbitrary quiver with potential $(Q,W)$, such as if $W=c+c^2$ for some cycle $c$. However, we are really interested in the structure of the algebra $\jac{Q}{W}$, which can be presented my many different quivers with potential. For example, any two right equivalent quivers with potential \cite[Def.~4.2]{derksenquivers1} define isomorphic Jacobian algebras, and so it is enough that $(Q,W)$ is right equivalent to some quiver with potential admitting a positive grading. This can happen even if $(Q,W)$ does not itself admit such a grading---for or example, the quiver with potential $(Q,W)$ from \Cref{construction-egs} in which $Q$ is a $3$-cycle admits a positive grading in which every arrow has degree $1$, but it is right equivalent to $(Q,cba+cbacba)$, which does not admit a positive grading.

\begin{lem}
\label{posgrading}
If $(Q,W)$ admits a positive grading, then so does $(\lift{Q},\lift{W})$.
\end{lem}
\begin{proof}
If $W=0$, then $(\lift{Q},\lift{W})$ has the positive grading $\deg$ with
\[
\deg{a}=1,\qquad
\deg{\alpha_i}=\deg{\beta_i}=\deg{\delta_a}=1,\qquad
\deg{\delta_i}=2
\]
for all $i\in Q_0$ and $a\in Q_1$. From now on, we assume $W\ne 0$, and let $\deg_0$ be a positive grading of $(Q,W)$; since $W\ne0$, we have $\deg_0(W)>0$. We begin by constructing a positive grading $\deg$ of $(Q,W)$ with the property that $\deg(W)-\deg(a)\geq3$ for all $a\in Q_1$.

First, note that if $\deg_0(W)-\deg_0(a)\leq0$ for some $a\in Q_1$, then $a$ does not appear in $W$, since $Q$ has no loops. Pick $K\in\ZZ$ such that $K\deg_0(W)-\deg_0(a)\geq1$ for all $a\in Q_1$ not appearing in $W$; then defining
\[\deg_1(a)=\begin{cases}K\deg_0(a),&\text{$a$ appears in $W$},\\\deg_0(a),&\text{otherwise},\end{cases}\]
we see that $\deg_1(W)=K\deg_0(W)$, and so $\deg_1(W)-\deg_1(a)\geq1$ for all $a\in Q_1$. Now defining $\deg(a)=3\deg_1(a)$ for all $a\in Q_1$ gives the required potential.

Write $d=\deg(W)$, and extend $\deg\colon Q_1\to\ZZ$ to the arrows of $\lift{Q}$, by defining
\[\deg(\alpha_v)=\deg(\beta_v)=1,\qquad
\deg(\delta_v)=d-2,\qquad
\deg(\delta_a)=d-2-\deg(a),\]
for each $v\in Q_0$ and $a\in Q_1$. Since $d-\deg(a)\geq3$ for all $a\in Q_1$, all of these values are positive integers. It follows immediately from the definition of $\lift{W}$ that this potential is homogeneous of degree $d$ with respect to $\deg$, and so $\deg$ is a positive grading for $(\lift{Q},\lift{W})$.
\end{proof}

When $A=A_{Q,W}$ is graded in such a way that all arrows have positive degree, an argument by Broomhead \cite[Prop.~7.5]{broomheaddimer} can be used to show that exactness of the complex $0\map{}\res{A}\map{\mu_0}A\map{}0$
is equivalent to the exactness of
\begin{equation}
\label{eq:proj-res-simples}
\begin{tikzcd}[column sep=20pt]
0\arrow{r}&\res{A}\tens_AS\arrow{r}{\mu_0}&S\arrow{r}&0.
\end{tikzcd}
\end{equation}
Indeed, the forward implication holds in general, since $\res{A}\map{\mu_0}A$ is perfect as a complex of right $A$-modules, and so remains exact under $-\tens_AM$ for any $M\in\Mod{A}$. Broomhead's argument, for the case that $F=\varnothing$, hinges on the property that, for any $u\in A\tens \KK Q_i^{(\mut)}$, $1\leq i\leq 3$, we have
\[\mu_i(u\tens 1)=v\tens 1+\sum w\tens x\]
for some $v,w\in A\tens\KK Q_{i-1}^{(\mut)}$ and $x\in\rad{A}$; this is also the case in our setting, and so the rest of Broomhead's argument goes through without change (modulo switching from right to left modules, i.e.\ from $S\tens_A\res{A}$ to $\res{A}\tens_AS$).

Now the complex \eqref{eq:proj-res-simples} decomposes along with $S$, so that its exactness is equivalent to the exactness of
\[\begin{tikzcd}[column sep=20pt]
0\arrow{r}&\res{A}\tens_A\simp{v}\arrow{r}{\mu_0}&\simp{v}\arrow{r}&0
\end{tikzcd}\]
for each $v\in \lift{Q}_0$, where $\simp{v}$ denotes the vertex simple left $A$-module at $v$. Thus when $(Q,W)$, and hence $(\lift{Q},\lift{W})$ by \Cref{posgrading}, admits a positive grading, we are able to reduce the problem of computing a bimodule resolution of $A$ to the simpler problem of computing a projective resolution of each vertex simple left $A$-module.

It will be useful to introduce some notation for elements of terms of $\res{A}\tens_A\simp{v}$. Note first that we have isomorphisms
\begin{align}
\label{eq:resA-isos}
\begin{split}
A\tens\KK \lift{Q}_1\tens A\tens_A\simp{v}&\iso\bigdsum_{b\in\vout{v}}Ae_{\head{b}},\\
A\tens\KK \lift{Q}_2^\mut\tens A\tens_A\simp{v}&\iso\bigdsum_{a\in\vin{v}\cap\lift{Q}_1^\mut}Ae_{\tail{a}},\\
A\tens\KK\lift{Q}_3^\mut\tens A\tens_A\simp{v}&\iso{\begin{cases}A\idemp{v},&v\in Q_0^\mut,\\0,&v\in F_0.\end{cases}}
\end{split}
\end{align}

In the first two cases, the right-hand sides are of the form $\bigoplus_{\alpha\in X}A\idemp{v(a)}$, where $X$ is a set of arrows, and $v\colon X\to Q_0$. The map $A\idemp{v(a)}\to\bigoplus_{a\in X}A\idemp{v(a)}$ including the domain as the summand indexed by $a$ will be denoted by $x\mapsto x\otimes[a]$; 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 $x_a\in A\idemp{v(a)}$.

The most complicated map in the complex $\res{A}$ is $\mu_2$, so we wish to spell out $\mu_2\tens_A\simp{v}$ explicitly. Using the isomorphisms from \eqref{eq:resA-isos}, and our notation for elements of the direct sums, we have
\[(\mu_2\tens_A\simp{v})(x)=\sum_{b\in\vout{v}}\bigg(\sum_{a\in\vin{v}\cap\lift{Q}_1^\mut}x_a\rightder{b}{\der{a}{W}}\bigg)\tens[b],\]
where $\rightder{b}$, called the \emph{right derivative} with respect to $b$, is defined on paths by
\begin{equation}
\label{eq:rightder}
\rightder{b}(\alpha_k\dotsm\alpha_1)=\begin{cases}\alpha_k\dotsm\alpha_2,&\alpha_1=b,\\0,&\alpha_1\ne b\end{cases}
\end{equation}
and extended linearly and continuously.

We now prove the necessary exactness for the complex $\res{A}\tens_A\simp{v}$. To do this, we break into two cases depending on whether $v$ is mutable or frozen, and use heavily the explicit set $R$ of defining relations for $A$ given above in \eqref{relations}.

\begin{lem}
\label{vertex-test}
Let $i\in Q_0$ be a mutable vertex of $\lift{Q}$. If $y\in Ae_i$ satisfies $y\beta_i=0$, then $y=0$.
\end{lem}
\begin{proof}
Let $\lift{y}$ be an arbitrary lift of $y$ to $\KK\lift{Q}e_i$. Now assume $y\beta_i=0$, so $\lift{y}\beta_i\in\close{\Span{R}}$. Since every term of $\lift{y}\beta_i$, when written in the basis of paths of $\lift{Q}$, ends with the arrow $\beta_i$, but no term of any element of $R$ has a term ending with $\beta_i$, we must be able to write
\[\lift{y}\beta_i=\sum_jz_j\beta_i\]
for $z_j\in\close{\Span{R}}e_i$. Comparing terms, we see that $\lift{y}=\sum_jz_j\in\close{\Span{R}}$, and so $y=0$ in $A$.
\end{proof}

\begin{prop}
\label{mut-exact-3}
For $i\in Q_0$, the map $\mu_3\tens_A\simp{v}\colon Ae_v\to\displaystyle\bigdsum_{a\in\vin{v}\cap\lift{Q}_1^\mut} Ae_{\tail{a}}$ is injective.
\end{prop}
\begin{proof}
We have
\[(\mu_3\tens_A\simp{i})(y)=\sum_{a\in\vin{i}\cap\lift{Q}_1^\mut}-ya\otimes[a].\]
Since $i$ is mutable, $\beta_i\in\vin{i}\cap\lift{Q}_1^\mut$ by the construction of $(\lift{Q},\lift{F})$. Thus if $(\mu_3\tens\simp{i})(x)=0$, it follows from the above formula that $-y\beta_i\tens[\beta_i]=0$, hence $y\beta_i=0$, and so $y=0$ by \Cref{vertex-test}.
\end{proof}

\begin{lem}
\label{arrow-test}
Let $i\in Q_0$ be a mutable vertex of $\lift{Q}$. For each $a\in\vin{i}\cap\lift{Q}_1^\mut$, pick $x_a\in Ae_{\tail a}$. If
\[x_{\beta_i}\delta_i=\sum_{a\in\vin{i}\cap Q_1}x_a\beta_{\tail{a}}\delta_a,\]
then there exists $y\in Ae_i$ such that $x_a=ya$ for each $a\in\vin{v}\cap\lift{Q}_1^\mut$.
\end{lem}

\begin{proof}
Pick a lift $\lift{x}_a\in\KK\lift{Q}$ of each $x_a$. Writing
\[p=\lift{x}_{\beta_i}\delta_i-\sum_{a\in\vin{v}\cap Q_1}\lift{x}_a\beta_{\tail{a}}\delta_a,\]
our assumption on the $x_a$ is equivalent to $p\in\close{\Span{R}}$. Since every term of $p$ ends with either $\delta_i$ or $\beta_{\tail{a}}\delta_a$ for some $a\in\vin{i}\cap Q_1$, and the only element of $R$ including terms ending with these arrows is $\beta_i\delta_i-\sum_{a\in\vin{i}\cap Q_1}a\beta_{\tail{a}}\delta_a$, we can write
\[p=z_i\delta_i+\sum_{a\in\vin{i}\cap Q_1}z_a\beta_{\tail{a}}\delta_a+y\Big(\beta_i\delta_i-\sum_{a\in\vin{i}\cap Q_1}a\beta_{\tail{a}}\delta_a\Big),\]
where $z_i\in\close{\Span{R}}e_i^-$, $z_a\in\close{\Span{R}}e_{\tail{a}}$ and $y\in\KK\lift{Q}e_i$. Comparing terms in our two expressions for $p$, we see that
\[
\lift{x}_{\beta_i}=z_i+y\beta_i,\qquad
\lift{x}_{a}=z_a+ya.
\]
Since $z_i,z_a\in\close{\Span{R}}$, when we pass to the quotient algebra $A=\KK\lift{Q}/\close{\Span{R}}$ we see that $x_{\beta_i}=y\beta_i$ and $x_a=ya$, as required.
\end{proof}

\begin{prop}
\label{mut-exact-2}
For $i\in Q_0$, we have $\ker(\mu_2\tens_A\simp{i})=\im(\mu_3\tens_A\simp{i})$.
\end{prop}

\begin{proof}
Since we already know that $\res{A}$ is a complex, it is enough to show that
\[\ker(\mu_2\tens_A\simp{i})\subseteq\im(\mu_3\tens_A\simp{i}).\]
Let $x=\sum_{a\in\vin{i}\cap\lift{Q}_1^\mut}x_a\tens[a]\in\bigdsum_{a\in\vin{i}\cap\lift{Q}_1^\mut}Ae_{\tail{a}}$. Then
\[(\mu_2\tens_A\simp{i})(x)=\sum_{b\in\vout{i}}\bigg(\sum_{a\in\vin{i}\cap\lift{Q}_1^\mut}x_a\rightder{b}{\der{a}{W}}\bigg)\tens[b]\in\bigdsum_{b\in\vout{i}}Ae_{\head{b}}.\]
In particular, $\alpha_i\in\vout{i}$, so if $x\in\ker(\mu_2\otimes_A\simp{i})$, we have
\[\bigg(\sum_{a\in\vin{i}\cap\lift{Q}_1^\mut}x_a\rightder{\alpha_i}{\der{a}{W}}\bigg)\otimes[\alpha_i]=0.\]
Using the explicit expressions for the relations $\der{a}{W}$ computed in \eqref{relations}, we see that
\[\sum_{a\in\vin{i}\cap\lift{Q}_1^m}x_a\rightder{\alpha_i}\der{a}{W}=x_{\beta_i}\delta_i-\sum_{a\in\vin{i}\cap Q_1}x_a\beta_{\tail{a}}\delta_a=0,\]
and so by \Cref{arrow-test} there exists $y\in A\idemp{i}$ such that $x_a=ya$ for each $a$. It follows that $x=(\mu_3\tens_A\simp{i})(y)$, as required.
\end{proof}

\begin{prop}
\label{frozen-res}
If $v\in\lift{F}_0$ then $\res{A}\tens_A\simp{v}$ is a projective resolution of $\simp{v}$.
\end{prop}
\begin{proof}
Since $v\in\lift{F}_0$, the complex $\res{A}\tens_A\simp{v}$ is zero in degree $-3$, so in view of \Cref{standard-res}, it is only necessary for us to check that $\mu_2\tens\simp{v}$ is injective. In fact, if $v=i^-$ for some $i\in Q_0$, then $\res{A}\tens_A\simp{v}$ is also zero in degree $-2$ since there are no unfrozen arrows in $\vin{i^-}$, so we need only consider $v=i^+$ for some $i\in Q_0$.

Since $\vin{i^+}\cap\lift{Q}_1^\mut=\set{\alpha_i}$, we have $\mu_2\tens_A\simp{i}^+\colon Ae_i\to\bigdsum_{b\in\vout{i}}Ae_{\head{b}}$. Let $x\in Ae_i$. Then, computing as above, we have
\[(\mu_2\tens_A\simp{i}^+)(x)=\sum_{b\in\vout{i^+}}x\rightder{b}{\der{\alpha_i}{W}}\otimes[b].\]
Assume this is $0$. Considering $\delta_i\in\vout{i^+}$, we see that we must have $0=x\rightder{\delta_i}{\der{\alpha_i}{W}}=x\beta_i$. 
By \Cref{vertex-test}, it follows that $x=0$, and $\mu_2\tens_A\simp{i}^+$ is injective as required.
\end{proof}

Combining these results, we are able to establish the desired internal Calabi--Yau property for $A_{Q,W}$ whenever $(Q,W)$ admits a positive grading.

\begin{thm}
\label{bi3cy}
If $(Q,W)$ admits a positive grading, then $A=\frjac{\lift{Q}}{\lift{F}}{\lift{W}}$ is bimodule internally $3$-Calabi--Yau with respect to the frozen idempotent $e=\sum_{i\in Q_0}(e_i^++e_i^-)$.
\end{thm}
\begin{proof}
The combination of \Cref{mut-exact-3,mut-exact-2} shows that $\res{A}\tens_A\simp{i}$ is exact for any $i\in Q_0=\lift{Q}_0^\mut$, and \Cref{frozen-res} shows that the same is true of $\res{A}\tens_A\simp{v}$ for $v\in\lift{F}_0$, so the direct sum $\res{A}\tens_AS\map{\mu_0}S$ of all of these complexes is exact. By \Cref{posgrading}, we can grade $A$ so that all arrows have positive degree. As already remarked, this allows us to apply Broomhead's result \cite[Prop.~7.5]{broomheaddimer} to deduce exactness of $\res{A}\map{\mu_0}A$ from that of $\res{A}\tens_AS\map{\mu_0}S$. Now the result follows from \Cref{frjaci3cy}.
\end{proof}

As a corollary, we obtain the promised result (\Cref{keller-analogue}) that any quiver with potential admitting a positive grading may be realised as the `interior' of a bimodule internally $3$-Calabi--Yau frozen Jacobian algebra.

\begin{cor}
\label{QPinterior}
Let $(Q,W)$ be a quiver with potential admitting a positive grading. Then $\jac{Q}{W}=A_{Q,W}/\Span{e}$, where $e$ is the frozen idempotent of $A_{Q,W}$, and $A_{Q,W}$ is bimodule internally $3$-Calabi--Yau with respect to $e$.\qed
\end{cor}

As a second corollary, a Jacobi-finite quiver with potential $(Q,W)$ admitting a positive grading determines a stably $2$-Calabi--Yau Frobenius category admitting a cluster-tilting object with stable endomorphism algebra $\jac{Q}{W}$, whenever $A_{Q,W}$ turns out to be Noetherian. This is how we will define the Frobenius cluster category $\frobcat_Q$ from \Cref{main-cat-thm} when $Q$ is acyclic.

\begin{cor}
\label{frobcat-constr}
Let $(Q,W)$ be a Jacobi-finite quiver with potential admitting a positive grading. Assume that $A=A_{Q,W}$ is Noetherian, let $e=\sum_{i\in Q_0}(e_i^++e_i^-)$ be the frozen idempotent of $A$, let $B=eAe$ be its boundary algebra, and let $T=eA$. Then $T$ is a cluster-tilting object of the stably $2$-Calabi--Yau Frobenius category $\GP(B)$, with $\Endalg{B}{T}\cong A$ and $\stabEndalg{B}{T}\cong\jac{Q}{W}$.
\end{cor}
\begin{proof}
The existence of a positive grading implies that $A$ is bimodule internally $3$-Calabi--Yau with respect to $e$ by \Cref{bi3cy}. Since $A/\Span{e}=\jac{Q}{W}$ by construction, the statement follows directly from \Cref{icytofrobcat}.
\end{proof}

Unfortunately, at the moment we have no general methods for determining when the algebra $A_{Q,W}$ is Noetherian unless $Q$ is acyclic, and this is the main obstruction to extending \Cref{main-cat-thm} to the case of quivers with cycles. When $Q$ is acyclic, $A_Q$ is even finite-dimensional, as we will see in \Cref{acyclic}. On the other hand, it is certainly possible that $A_{Q,W}$ can still be Noetherian when $Q$ has cycles; we already observed this for the $3$-cycle with its natural potential in \Cref{construction-egs}. This quiver with potential is also Jacobi-finite, and admits the positive grading in which all arrows have degree $1$, so \Cref{frobcat-constr} also applies in this case.

We also have no evidence that the existence of a positive grading is necessary to obtain the conclusion of \Cref{bi3cy}; rather, this condition was imposed in order to make the necessary calculations more manageable. By analogy with Keller's result  \cite[Thm.~6.3, Thm.~A.12]{kellerdeformed} that the Jacobian algebra of any quiver with potential is the $0$-th homology of a bimodule $3$-Calabi--Yau dg-algebra, we conjecture that this assumption is not in fact needed.

\begin{conj}
The conclusion of \Cref{bi3cy}, and hence those of \Cref{QPinterior} and \Cref{frobcat-constr}, remains valid without the assumption that $(Q,W)$ admits a positive grading.
\end{conj}

In support of this conjecture, we prove injectivity of $\mu_3$ directly, without any assumption on the existence of a positive grading.

\begin{prop}
\label{bimod-exact-3}
The map $\mu_3\colon A\tens \lift{Q}_3^\mut\tens A\to A\tens \lift{Q}_2^\mut\tens A$ is injective.
\end{prop}
\begin{proof}
We use the natural isomorphism
\[A\tens \KK \lift{Q}_3^\mut\tens A\isoto\bigoplus_{i\in Q_0}A\idemp{i}\tens_\KK\idemp{i}A\]
given by $x\tens \omega_i\tens y\mapsto x\tens y$. For each $i\in Q_0$, let $x_i=(x_i^j)_{j\in J_i}$ and $y_i=(y_i^j)_{j\in J_i}$ be finite sets of elements of $A\idemp{i}$ and $\idemp{i}A$ respectively. These define an element
\[x_i\tens y_i:=\sum_{j\in J_i}x_i^j\tens y_i^j\in A\idemp{i}\otimes_\KK\idemp{i}A,\]
and all elements of $A\idemp{i}\tens_\KK\idemp{i}A$ are of this form. Without loss of generality, i.e.\ without changing the value of $x_i\tens y_i$, we may assume that $\set{y_i^j:j\in J_i}$ is a linearly independent set.

Now assume $\sum_{i\in Q_0}x_i\tens y_i$ is in the kernel of $\mu_3$. We aim to show that, in this case, $x_i\tens y_i=0$ for all $i\in Q_0$, and so in particular their sum is zero. Projecting onto the component $A\tens\KK\rho_{\beta_i}\tens A\iso A\idemp{i}\tens_\KK\idemp{i}^+A$ of $A\tens \lift{Q}_2^\mut\tens A$, we see that
\[x_i\beta_i\tens y_i=\sum_{j\in J_i}x_i^j\beta_i\tens y_i^j=0.\]
Since the $y_i^j$ are linearly independent, it follows that $x_i^j\beta_i=0$ for all $j$. By \Cref{vertex-test}, it follows that $x_i^j=0$ for all $j$, and so $x_i\tens y_i=0$.
\end{proof}

\section{The acyclic case}
\label{acyclic}

Let $(Q,W)$ be a quiver with potential such that $\jac{Q}{W}=A_{Q,W}/\Span{e}$ is finite-dimensional. By this assumption and \Cref{bi3cy}, all of the assumptions of \Cref{icytofrobcat} are satisfied for $A_{Q,W}$, except possibly Noetherianity. Our goal in this section is to show that if $Q$ is an acyclic quiver, then the algebra $A=A_Q=\frjac{\lift{Q}}{\lift{F}}{\lift{0}}$ is even finite-dimensional, so we may apply \Cref{icytofrobcat} to obtain a categorification of the polarised principal coefficient cluster algebra $\pprinclust{Q}$.

\begin{lem}
\label{Q0-to-Q0}
Let $Q$ be any quiver, and let $p$ be a path in $\lift{Q}$. Assume that $p$ contains at least one arrow not in $Q_1$, and that $\head{p},\tail{p}\in Q_0$. Then $p$ maps to zero under the projection $\KK\lift{Q}\to A=\frjac{\lift{Q}}{\lift{F}}{\lift{0}}$. Moreover, if $q$ is a path of length at least $5$ containing no arrows of $Q_1$, then $q$ maps to zero under this projection.
\end{lem}
\begin{proof}
Let $\gamma$ be the first arrow of $p$ in $\lift{Q}_1\setminus Q_1$; there is such an arrow by assumption. If $\gamma$ has a predecessor in $p$, then this arrow is in $Q_1$, and so $\tail{\gamma}\in Q_0$. On the other hand, if $\gamma$ is the first arrow of $p$, then $\tail{\gamma}=\tail{p}\in Q_0$ by assumption. By the construction of $\lift{Q}$, it follows that $\gamma=\alpha_i$ for some $i\in Q_0$. Since $\head{p}\in Q_0$, the path $p$ cannot terminate with $\alpha_i$, so this arrow has a successor. Looking again at the definition of $\lift{Q}$, the only options are $\delta_a$ for some $a\in Q_1$ with $\head{a}=i$, 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 $\head{p}\in Q_0$. The only arrow leaving $\head{\delta_a}=i_{\tail{a}}^-$ is $\beta_{\tail{a}}$, so this must be the next arrow of $p$. But after projection to $A$, we have $\beta_{\tail{a}}\delta_a\alpha_{\head{a}}=\der{a}{W}=0$ since $W=0$, so $p$ projects to zero.

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
\[\beta_i\delta_i\alpha_i=\bigg(\sum_{\substack{a\in Q_1\\\head{a}=i}}a\beta_{\tail{a}}\delta_a\bigg)\alpha_i=\sum_{\substack{a\in Q_1\\\head{a}=i}}a\beta_{\tail{a}}\delta_a\alpha_{\head{a}}=0,\]
so that $p$ again projects to zero.

For the second statement, we have already shown that the paths $\beta_{\tail{a}}\delta_a\alpha_{\head{a}}$ for some $a\in Q_1$ or $\beta_i\delta_i\alpha_i$ for some $i\in Q_0$ project to zero. Thus if $q$ as in the statement does not project to zero, it must not contain either of these subpaths. However, the bipartite property of $\lift{F}$ means that any path of arrows not in $Q_1$ without either of these subpaths is itself 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 $\head{x}=i=\tail{y}$, and so has length at most $4$.
\end{proof}

\begin{thm}
\label{acyclic-fd}
Let $Q$ be an acyclic quiver. Then $A=\frjac{\lift{Q}}{\lift{F}}{\lift{0}}$ is finite-dimensional.
\end{thm}
\begin{proof}
We show that there are finitely many paths of $\lift{Q}$ determining non-zero elements of $A$. By \Cref{Q0-to-Q0}, any path $p$ of $\lift{Q}$ determining a non-zero element of $A$ may not have any subpath with endpoints in $Q_0$ and containing an arrow outside $Q_1$. Thus we must have $p=q_2p'q_1$, where $q_1$ and $q_2$ are (possibly empty) paths not involving arrows of $Q_1$, and $p'$ is a path in $Q$. Since $Q$ is acyclic, there are only finitely many possibilities for $p'$. By \Cref{Q0-to-Q0} again, $q_1$ and $q_2$ have length at most $4$ if $p$ is to be non-zero in $A$, and so there are again only finitely many possibilities.
\end{proof}

We have now established everything we need in order to construct the Frobenius cluster category required by \Cref{main-cat-thm}, and to prove part \ref{main-cat-thm-cc} of this theorem.

\begin{proof}[Proof of \Cref{main-cat-thm}\ref{main-cat-thm-cc}]
Since $Q$ is an acyclic quiver, $\jac{Q}{0}=\KK Q$ is finite-dimensional, and any assignment of positive integers to the arrows of $Q$ is a positive grading of $(Q,0)$. By \Cref{acyclic-fd}, the algebra $A_Q=\frjac{\lift{Q}}{\lift{F}}{\lift{0}}$ is finite-dimensional, and hence Noetherian.

Now let $e\in A_Q$ be the frozen idempotent, and write $B_Q=eA_Qe$. Then \Cref{frobcat-constr} applies to show that $\frobcat_Q:=\GP(B_Q)$ is a stably $2$-Calabi--Yau Frobenius category. It is Krull--Schmidt since $B$ is a finite-dimensional algebra.

To see that $\GP(B)$ is a Frobenius cluster category, it only remains to show that the endomorphism algebra of any cluster-tilting object has global dimension $3$, which follows from \cite[Prop.~3.7]{presslandinternally}, using that $B$ is a finite-dimensional algebra to see that these endomorphism algebras are Noetherian.

Now to see that \Cref{main-cat-thm}\ref{main-cat-thm-cc} holds for this choice of $\frobcat_Q$, we use Keller--Reiten's `recognition theorem' \cite[Thm.~2.1]{kelleracyclic}; by \Cref{frobcat-constr}, $\stab{\frobcat}_Q$ is a $2$-Calabi--Yau triangulated category admitting a cluster-tilting object $T=eA_Q$ with endomorphism algebra $\KK Q$, so $\stab{\frobcat}_Q$ is triangle equivalent to the cluster category $\clustcat{Q}$ as required.
\end{proof}

We already observed after stating \Cref{frobcat-constr} 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 $\frobcat_{Q,W}=\GP(B_{Q,W})$. We claim that the analogous statement to \Cref{main-cat-thm}\ref{main-cat-thm-cc} also holds here, namely that $\stab{\frobcat}_{Q,W}$ is triangle equivalent to Amiot's cluster category $\clustcat{Q,W}$. While Keller--Reiten's recognition theorem no longer applies to the cluster-tilting object $T$ provided by \Cref{frobcat-constr}, since its stable endomorphism algebra is not hereditary, one can in this case find another cluster-tilting object $T'\in\frobcat_{Q,W}$ with $\stabEndalg{\frobcat_{Q,W}}{T'}\iso\KK Q'$, for $Q'$ an orientation of the Dynkin diagram $\typeA{3}$. Thus the recognition theorem shows that $\stab{\frobcat}_{Q,W}\simeq\clustcat{Q'}$, and $\clustcat{Q'}\simeq\clustcat{Q,W}$ either by the recognition theorem again or by \cite[Cor.~3.11]{amiotcluster}.

We note that while Amiot--Reiten--Todorov have a recognition theorem \cite[Thm.~3.1]{amiotubiquity}, which identifies certain stable categories of Frobenius categories admitting a cluster-tilting object with frozen Jacobian endomorphism algebra as generalised cluster categories, this theorem does not apply to our object $T$ when $W\ne0$, because then $(\lift{Q},\lift{F},\lift{W})$ cannot satisfy their assumptions (H3) and (H4). Assumption (H3) requires a non-negative degree function on the arrows of $\lift{Q}$ giving $\lift{W}$ degree $1$, and (H4) requires that the arrows $\alpha_i$, for $i\in Q_0$, all have degree $1$. But if $W\ne0$, then (H3) forces some arrow $a\in Q_1$ to have degree $1$, and then (H4) implies that the potential term $a\beta_{\tail{a}}\delta_a\alpha_{\head{a}}$ has degree at least $2$, so (H3) does not hold.

\section{Mutation}
\label{mutation}

To complete the proof of Theorem~\ref{main-cat-thm}, we need to understand the relationship between mutation of cluster-tilting objects in $\GP(B_Q)$ 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.

\begin{thm}[{\cite[Thm.~5.14, Prop.~5.15]{presslandmutation}}]
\label{mutation-thm}
Let $\frobcat$ be a Hom-finite Frobenius cluster category, and assume there is a cluster-tilting object $T\in\frobcat$ such that $\Endalg{\frobcat}{T}\cong\frjac{Q}{F}{W}$, where $Q$ the Gabriel quiver of this algebra. Then
\begin{enumerate}[label=(\roman*)]
\item\label{mutation-thm-iy} $Q$ has no loops or $2$-cycles incident with any mutable vertex, so the Iyama--Yoshino mutation of $T$ at any indecomposable non-projective summand is well-defined,
\item\label{mutation-thm-qpmut} if $T'$ is obtained from $T$ by such a mutation, then $\End_\frobcat(T')^{\op}\cong\frjac{Q'}{F'}{W'}$, where $(Q',F',W')$ is obtained from $(Q,F,W)$ by mutation at the vertex corresponding to the mutated summand, and
\item\label{mutation-thm-fzmut} the quiver $Q'$ is the Gabriel quiver of $\End_\frobcat(T')^{\op}$ and, up to arrows between frozen vertices, the Fomin--Zelevinsky mutation of $Q$ at the appropriate vertex.
\end{enumerate}
These results then extend inductively to any cluster-tilting object in the mutation class of $T$.
\end{thm}

This statement can be proved in a similar way to an analogous result of Buan--Iyama--Reiten--Smith for triangulated categories \cite{buanmutation}. The definition of mutation for an ice quiver with potential is given in \cite[Def.~4.1]{presslandmutation}, and is similar to that for ordinary quivers with potential \cite[\S1.2]{buanmutation}. We also explain in \cite{presslandmutation} how to extend the definition of Fomin--Zelevinsky mutation to ice quivers which may have arrows between their frozen vertices, allowing for a version of statement \ref{mutation-thm-fzmut} in which these arrows are still considered. For now, we use this result to complete the proof of Theorem~\ref{main-cat-thm}.

\begin{proof}[Proof of \Cref{main-cat-thm}\ref{main-cat-thm-mut}]
As in the proof of part \ref{main-cat-thm-cc} in the previous section, we take $\frobcat_Q=\GP(B_Q)$ to be the category of Gorenstein projective modules over the boundary algebra of $A_Q$, which we showed to be a Frobenius cluster category. It is Hom-finite, since $A_Q$, and hence $B_Q$, is a finite-dimensional algebra by \Cref{acyclic-fd}. \Cref{frobcat-constr} provides a cluster-tilting object $T=eA_Q\in\frobcat_Q$ with $\Endalg{B_Q}{T}\cong A_Q$ having Gabriel quiver $\lift{Q}$, which up to arrows between frozen vertices is the ice quiver of the initial seed $\lift{s}_0$ of $\pprinclust{Q}$.

Now for any sequence $\mu_{k_m}\dotsm\mu_{k_1}$ of mutations, we associate the cluster-tilting object $\mu_{k_m}\dotsm\mu_{k_1}T$, which is well-defined by an induction using \Cref{mutation-thm}\ref{mutation-thm-iy}--\ref{mutation-thm-qpmut}, to the seed $\mu_{k_m}\dotsm\mu_{k_1}\lift{s}_0$ of $\pprinclust{Q}$. This map commutes with mutation by construction, and respects quivers by \Cref{mutation-thm}\ref{mutation-thm-fzmut}. Since cluster-tilting objects of $\GP(B_Q)$ are in bijection with those of the stable category $\stabGP(B_Q)$, and the latter is triangle equivalent to the cluster category $\clustcat{Q}$ by \Cref{main-cat-thm}\ref{main-cat-thm-cc}, the fact that this assignment is a bijection follows from \cite[Thm.~A.1]{buanclusters} and \cite{cerullilinear}, the latter showing that the exchange graph of $\clust{Q}$ coincides with that of $\pprinclust{Q}$.
\end{proof}

Just as for \Cref{main-cat-thm}\ref{main-cat-thm-cc}, the conclusions of \Cref{main-cat-thm}\ref{main-cat-thm-mut} still hold when $Q$ is a $3$-cycle, replacing $B_Q$ by $B_{Q,W}$ for $W$ the potential given by the $3$-cycle. In this case $\pprinclust{Q}$ is the cluster algebra of the Grassmannian $G_2^6$, and $\GP(B_{Q,W})$ the Grassmannian cluster category constructed by Jensen--King--Su \cite{jensencategorification} (see Example~\ref{construction-egs}), and \Cref{mutation-thm} can be adapted to these categories, despite them being Hom-infinite---see \cite[Prop.~5.16]{presslandmutation} and its preceding discussion.

\section{An extriangulated categorification}
\label{extriangulated}

Let $(Q,W)$ be a Jacobi-finite quiver with potential admitting a positive grading, and assume that $A_{Q,W}$ is Noetherian, so that we may apply \Cref{frobcat-constr} to obtain a stably $2$-Calabi--Yau Frobenius category $\frobcat_{Q,W}=\GP(B_{Q,W})$. Write $e^-=\sum_{i\in Q_0}e_i^-$, and $P^-=B_{Q,W}e^-$. In this section, we consider the quotient category $\frobcat^+_{Q,W}=\frobcat_{Q,W}/[P^-]$, obtained from $\frobcat_{Q,W}$ by taking the quotient by the the ideal of morphisms factoring through an object of $\add{P^-}$. The resulting category has a natural extriangulated structure---we do not define this notion for the brief discussion here, instead referring the reader to Nakaoka--Palu \cite{nakaokamutation}.

\begin{prop}
\label{p:extriangulated}
The category $\frobcat^+_{Q,W}$ carries the structure of an extriangulated category. Moreover, it is a Frobenius extriangulated category \cite[Def.~7.1]{nakaokamutation}, and the stable category $\stab{\frobcat}^+_{Q,W}$ coincides with $\stab{\frobcat}_{Q,W}$, meaning in particular that it is triangulated and $2$-Calabi--Yau.
\end{prop}
\begin{proof}
Since $P^-$ is a projective-injective object of $\GP(B_{Q,W})$, the extriangulated structure on $\frobcat_{Q,W}^+$ is induced from the exact structure on $\frobcat_{Q,W}$ as in \cite[Prop.~3.30]{nakaokamutation}. Since the `extension groups' $\EE_{\frobcat^+_{Q,W}}(X,Y)$ in this extriangulated category coincide with the extension groups $\Ext^1_{\frobcat_{Q,W}}(X,Y)$ in the exact category $\frobcat_{Q,W}$, we see that an object is projective in $\frobcat^+_{Q,W}$ if and only if it is projective in $\frobcat_{Q,W}$, if and only if it is injective in $\frobcat_{Q,W}$, if and only if it is injective in $\frobcat^+_{Q,W}$. Then this agreement of extension groups also allows us to further deduce the existence of enough projectives and injectives in $\frobcat^+_{Q,W}$ \cite[Def.~3.25]{nakaokamutation} from the corresponding property of $\GP(B_{Q,W})$. It also follows from the agreement of projective-injective objects in $\frobcat^+_{Q,W}$ and $\frobcat_{Q,W}$ that $\stab{\frobcat}^+_{Q,W}=\stab{\frobcat}_{Q,W}$, the latter being a $2$-Calabi--Yau triangulated category by \Cref{frobcat-constr}.
\end{proof}

Now assume that $Q$ is acyclic, so that $\frobcat_Q=\GP(B_Q)$ is the Frobenius cluster category from \Cref{main-cat-thm}. In this case, $\frobcat^+_Q=\frobcat_Q/[P^-]$ is the extriangulated category required by \Cref{c:extriangulated}.

\begin{proof}[Proof of \Cref{c:extriangulated}]
The fact that $\frobcat^+_Q$ is extriangulated with projective and injective objects coinciding is already proved in \Cref{p:extriangulated}. The same proposition states that $\stab{\frobcat}^+_Q=\stab{\frobcat}_Q$, the latter being equivalent to the cluster category $\clustcat{Q}$ by \Cref{main-cat-thm}\ref{main-cat-thm-cc}, so statement \ref{c:extriangulated-cc} is proved.

To see \ref{c:extriangulated-mut}, first note that $\frobcat_Q$, $\frobcat^+_Q$ and $\stab{\frobcat}_Q$ all have the same cluster-tilting objects since
\[\Ext^1_{\frobcat_Q}(X,Y)=\EE_{\frobcat^+_Q}(X,Y)=\stabHom_{\frobcat_Q}(X,Y[1]).\]
for any pair of objects $X$ and $Y$ of these categories. Moreover, since $\frobcat^+_Q$ and $\frobcat_Q$ have the same projective-injective objects, the summands of a cluster-tilting object which are indecomposable non-projective are also the same in all three categories. Thus two cluster-tilting objects from the common set of objects are related by mutation at an indecomposable summand in one of these categories if and only if they are related by mutation at the same summand in all three. In particular, the identity map is a bijection from the set of cluster-tilting objects of $\frobcat^+_Q$ to the set of cluster-tilting objects of $\frobcat_Q$, commuting with mutation. A more detailed discussion of mutation of cluster-tilting objects in extriangulated categories can be found in \cite{changcluster} (see also \cite{zhoutriangulated}).

Now let $T$ be a cluster-tilting object of $\frobcat^+_Q$. Since $\frobcat^+_Q=\frobcat_Q/[P^-]$, the Gabriel quiver of $\Endalg{\frobcat^+_Q}{T}$ is obtained from that of $\Endalg{\frobcat_Q}{T}$ by deleting the vertices corresponding to the direct summands of $P^-$, and all adjacent arrows. On the cluster algebra side, let $\lift{s}_T$ be the seed of the polarised principal coefficient cluster algebra $\pprinclust{Q}$ corresponding to $T$, now thought of as an object of $\frobcat_Q$, under the bijection of \Cref{main-cat-thm}\ref{main-cat-thm-mut}, so that the ice quiver of $\lift{s}_T$ is the Gabriel quiver of $\Endalg{\frobcat_Q}{T}$ up to arrows between frozen vertices. We obtain a seed $s_T$ of the principal coefficient cluster algebra $\princlust{Q}$ by setting each frozen variable $x_i^-$ to $1$ and removing them from the cluster, while also removing the vertices $i^-$ from the ice quiver, so the ice quiver of $s_T$ coincides up to arrows between frozen vertices with the Gabriel quiver of $\Endalg{\frobcat^+_Q}{T}$. Noting that all of the variables $x_i^-$ and vertices $i^-$ are frozen, this operation on seeds gives a bijection between seeds of $\pprinclust{Q}$ and those of $\princlust{Q}$, commuting with mutation.

Thus the the map $T\mapsto s_T$, from cluster-tilting objects of $\frobcat^+_Q$ to seeds of $\princlust{Q}$, is the composition of three bijections commuting with mutation, and so is itself such a bijection.
\end{proof}

\section{Boundary algebras}
\label{bdy-algs}

Since the objects of the Frobenius categories constructed in \Cref{frobcat-constr} are modules for the idempotent subalgebra $B_{Q,W}$ of $A_{Q,W}$ determined by the frozen vertices, we wish to describe this subalgebra more explicitly. In this section we will present $B_Q$ via a quiver with relations, in the case that $Q$ is acyclic, so that $\GP(B_Q)$ is the Frobenius category $\frobcat_Q$ from \Cref{main-cat-thm}.

Recall that the double quiver $\overline{Q}$ of a quiver $Q$ has vertex set $Q_0$ and arrows $Q_1\cup\dual{Q}_1$, where $\dual{Q}_1=\set{\dual{\alpha}:\alpha\in Q_1}$. The head and tail maps agree with those of $Q$ on $Q_1$, and are defined by $\head{\dual{\alpha}}=\tail{\alpha}$ and $\tail{\dual{\alpha}}=\head{\alpha}$ on $\dual{Q}_1$. The preprojective algebra of $Q$ is
\[\preproj{Q}=\KK\overline{Q}/\close{\Big\langle\sum_{\alpha\in Q_1}[\alpha,\dual{\alpha}]\Big\rangle}\]
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 $(Q,W)$.

\begin{prop}
\label{universal-preproj}
Let $(Q,F,W)$ be an ice quiver with potential, let $A=\frjac{Q}{F}{W}$ and let $B$ be the boundary algebra of $A$. Then there is an algebra homomorphism $\pi\colon\preproj{F}\to B$ given by $\pi(\idemp{i})=\idemp{i}$ for all $i\in F_0$, and $\pi(\alpha)=\alpha$, $\pi(\dual{\alpha})=\der{\alpha}{W}$ for all $\alpha\in F_1$.
\end{prop}
\begin{proof}
It suffices to check that $\pi(\sum_{\alpha\in F_1}[\alpha,\dual{\alpha}])=0$, i.e.\ that $\sum_{\alpha\in F_1}[\alpha,\der{\alpha}{W}]=0$.
By construction, for any $v\in Q_0$ we have
\[\sum_{\alpha\in\vin{v}}\alpha\der{\alpha}{W}=\sum_{\beta\in\vout{v}}\der{\beta}{W}\beta\]
in $\KK Q$. Projecting to $A$ and summing over vertices, we see that
\[0=\sum_{\alpha\in Q_1}[\alpha,\der{\alpha}{W}]=\sum_{\alpha\in Q_1^\mut}[\alpha,\der{\alpha}{W}]+\sum_{\alpha\in F_1}[\alpha,\der{\alpha}{W}]=\sum_{\alpha\in F_1}[\alpha,\der{\alpha}{W}],\]
where the final equality holds since $\der{\alpha}{W}=0$ in $A$ whenever $\alpha\in Q_1^\mut$.
\end{proof}

\begin{rem}
Familiarity with the constructions of \cites{geisspartial,buancluster,jensencategorification} may make it tempting to conjecture that the map $\pi$ in \Cref{universal-preproj} is surjective, at least when $\frjac{Q}{F}{W}$ is bimodule internally $3$-Calabi--Yau, but this is not the case. A small explicit counterexample is the frozen Jacobian algebra $A$ of
\[\begin{tikzpicture}[scale=1.2]
\node at (180:1) (2) {$\boxed{2}$};
\node at (0:1) (3) {$\boxed{3}$};
\node at (90:1) (1) {$\boxed{1}$};
\node at (-90:1) (4) {$\boxed{4}$};
\path[-angle 90,]
(4) edge (1);
\path[-angle 90,\frozen]
(1) edge (3)
(3) edge (4)
(1) edge (2)
(2) edge (4);
\end{tikzpicture}\]
where the frozen subquiver is indicated by boxed vertices and dashed arrows as usual, and the potential is given by the difference of the two $3$-cycles, so that the relations are generated by setting equal the two length two paths from $1$ to $4$. In this case the boundary algebra $B$ is equal to $A$, and the arrow from $4$ to $1$ is not in the image of $\pi\colon\preproj{F}\to B$. Since this algebra can be graded with every arrow in degree $1$, one can again check that the bimodule complex $\res{A}$ from \Cref{frjacCY} gives a resolution by computing a projective resolution of each simple module, so this frozen Jacobian algebra is bimodule internally $3$-Calabi--Yau. (Since every vertex is frozen, this reduces to the statement that $\pdim_{\env{A}}{A}\leq 3$, and in fact this projective dimension is $2$.) We will see in this section that $\pi$ fails to be surjective for our frozen Jacobian algebras $\frjac{\lift{Q}}{\lift{F}}{\lift{W}}$ whenever $Q$ is acyclic with a path of length at least $2$; see \Cref{a3-bdy-eg} below.
\end{rem}

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

\begin{defn}
Let $Q$ be an acyclic quiver, and consider the frozen subquiver $\lift{F}$ of $\lift{Q}$, which has vertex set $\lift{F}_0=\set{i^+,i^-:i\in Q_0}$ 
and arrows
\[
\delta_i\colon i^+\to i^-,\qquad
\delta_a\colon \head{a}^+\to\tail{a}^-
\]
for each $i\in Q_0$ and $a\in Q_1$. We define a quiver $\Gamma_Q$ by adjoining to $\lift{F}$ an arrow
\[\dual{\delta_p}\colon\tail{p}^-\to\head{p}^+\]
for each path $p$ of $Q$. Since $Q$ is acyclic, it has finitely many paths, and so $\Gamma_Q$ is a finite quiver.
\end{defn}

If $p=\idemp{i}$ is the trivial path at $i\in Q_0$, we write $\dual{\delta}_p=\dual{\delta}_i$ to avoid a double subscript. The double quiver of $\lift{F}$ appears as the subquiver of $\Gamma_Q$ obtained by excluding the arrows $\dual{\delta}_p$ for $p$ of length at least two; the notation for the arrows of $\Gamma_Q$ is chosen to be consistent with that used earlier for the arrows of this double quiver.

\begin{eg}
\label{a3-bdy-eg}
Let
\[Q=\mathord{\begin{tikzpicture}[baseline={([yshift=-1.2ex]current bounding box.center)}]
\node at (-1,0) (1) {$1$};
\node at (1,0) (2) {$2$};
\node at (3,0) (3) {$3$};
\path[-angle 90,font=\scriptsize]
	(1) edge node[above] {$a$} (2)
	(2) edge node[above] {$b$} (3);
\end{tikzpicture}}\]
be a linearly oriented quiver of type $\typeA{3}$. Then $\Gamma_Q$ is the following quiver.
\[\Gamma_Q=\mathord{\begin{tikzpicture}[baseline={([yshift=3.2ex]current bounding box.center)},xscale=1.5]
\node at (0,0) (1) {$1^+$};
\node at (1,0) (2) {$1^-$};
\node at (2,0) (3) {$2^+$};
\node at (3,0) (4) {$2^-$};
\node at (4,0) (5) {$3^+$};
\node at (5,0) (6) {$3^-$};
\path[-angle 90,font=\scriptsize]
	(1) edge[bend left] node[above] {$\delta_1$} (2)
	(2) edge[bend left] node[below] {$\dual{\delta}_1$} (1)
	(2) edge[bend right] node[below] {$\dual{\delta}_a$} (3)
	(3) edge[bend right] node[above] {$\delta_a$} (2)
	(3) edge[bend left] node[above] {$\delta_2$} (4)
	(4) edge[bend left] node[below] {$\dual{\delta}_2$} (3)
	(4) edge[bend right] node[below] {$\dual{\delta}_b$} (5)
	(5) edge[bend right] node[above] {$\delta_b$}  (4)
	(5) edge[bend left] node[above] {$\delta_3$} (6)
	(6) edge[bend left] node[below] {$\dual{\delta}_3$} (5)
	(2) edge[bend right=70] node[below] {$\dual{\delta}_{ba}$} (5);
\end{tikzpicture}}\]
\end{eg}

\begin{defn}
Let $Q$ be a quiver. A \emph{zig-zag} 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 $\head{p}=\head{a}$ and $\tail{q}=\tail{a}$.
Thus if $a\colon v\to w$ is an arrow of $Q$, a zig-zag involving $a$ is some configuration
\[\begin{tikzpicture}[xscale=2,yscale=1.3]
\node at (0,0) (w) {$w$};
\node at (1,0) (v) {$v$};
\path[-angle 90,font=\scriptsize]
	(v) edge node[above] {$a$} (w);
\path[-angle 90,font=\scriptsize,dotted]
	(1,1) edge node [above left] {$p$} (w)
	(v) edge node [below right] {$q$} (0,-1);
\end{tikzpicture}\]
where the dotted arrows denote paths. We call the zig-zag \emph{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 zig-zag, then we define $\head{z}=\head{q}$ and $\tail{z}=\tail{p}$.
\end{defn}

We now write down elements of $\KK\Gamma_Q$ that will turn out to be a set of generating relations for $B$, having three flavours, as follows.
\begin{enumerate}[label=({R}\arabic*)]
\item\label{path-rel-t} For each path $p$ of $Q$, let
\[r_1(p)=\dual{\delta}_p\delta_{\tail{p}}-\sum_{\substack{a\in Q_1\\\head{a}=\tail{p}}}\dual{\delta}_{pa}\delta_a.\]
\item\label{path-rel-h} For each path $p$ of $Q$, let
\[r_2(p)=\delta_{\head{p}}\dual{\delta}_p-\sum_{\substack{a\in Q_1\\\tail{a}=\head{p}}}\delta_a\dual{\delta_{ap}}.\]
\item\label{zigzag-rel} For each zig-zag $(q,a,p)$ of $Q$, let
\[r_3(q,a,p)=\dual{\delta}_{q}\delta_a\dual{\delta}_{p}.\]
\end{enumerate}
We write $I$ for the closure of the ideal of $\KK \Gamma_Q$ generated by the union of these three sets of relations. This generating set is usually not minimal, as for certain zig-zags $(q,a,p)$, the relation $r_3(q,a,p)$ may already lie in the ideal generated by relations of the form $r_1$ and $r_2$. For example, if $a\in Q_1$ is an arrow such that $i=\tail{a}$ is incident with no other arrows of $Q$, then
\[
r_1(a)=\dual{\delta}_a\delta_i,\qquad
r_2(e_i)=\delta_i\dual{\delta_i}-\delta_a\dual{\delta}_a,
\]
so in $\KK \Gamma_Q/\Span{r_1(a),r_2(e_i)}$ we already have
\[r_3(a,a,a)=\dual{\delta_a}\delta_a\dual{\delta}_a=\dual{\delta}_a\delta_{i}\dual{\delta}_{i}=0.\]
One can check that if $(q,a,p)$ is a strict zig-zag, then $r_3(q,a,p)$ is not redundant, but this condition is not necessary; if $Q$ is a linearly oriented quiver of type $\typeA{4}$ and $a$ is the middle arrow, then the non-strict zig-zag $(a,a,a)$ yields an irredundant relation.

\begin{rem}
When $p=e_i$ is a vertex idempotent, the relations $r_1(e_i)$ and $r_2(e_i)$ reduce to `preprojective' relations on the double quiver of $\lift{F}$, of the form predicted by \Cref{universal-preproj}. Each arrow $a\in Q_1$ is part of the trivial strict zig-zag $(e_{\tail{a}},a,e_{\head{a}})$, and so contributes an irredundant relation $r_3(e_{\tail{a}},a,e_{\head{a}})=\dual{\delta}_{\tail{a}}\delta_a\dual{\delta}_{\head{a}}$.
\end{rem}

Let $\Phi\colon\KK \Gamma_Q\to A_Q$ be the map given by the identity on the vertices of $\Gamma_Q$ 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 $\lift{Q}$. On the remaining arrows $\dual{\delta}_p$ of $\Gamma_Q$, we define $\Phi(\dual{\delta}_p)=\alpha_{\head{p}}p\beta_{\tail{p}}$.

\begin{prop}
\label{well-def}
The map $\Phi$ above induces a well-defined map $\Phi\colon \Gamma_Q/I\to B$.
\end{prop}
\begin{proof}
Since $\Phi$ sends every vertex or arrow of $\Gamma_Q$ to the image in $A_Q$ of a path in $\lift{Q}$ with frozen head and tail, it takes values in $B$. It remains to check that it is zero on each of the generating relations of $I$, which we do by explicit calculation. Let $p$ be a path in $Q$. Then
\begin{align*}
\Phi(r_1(p))&=\alpha_{\head p}p\beta_{\tail{p}}\delta_{\tail{p}}-\sum_{\substack{a\in Q_1\\\head{a}=\tail{p}}}\alpha_{\head p}pa\beta_{\tail{a}}\delta_a=\alpha_{\head{p}}p(\der{\alpha_{\tail{p}}}{W})=0,\\
\Phi(r_2(p))&=\delta_{\head{p}}\alpha_{\head p}p\beta_{\tail{p}}-\sum_{\substack{a\in Q_1\\\tail{a}=\head{p}}}\delta_a\alpha_{\head{a}}ap\beta_{\tail{p}}=(\der{\beta_{\head{p}}}{W})p\beta_{\tail{p}}=0.
\end{align*}
If $(q,a,p)$ is a zig-zag, then
\[\Phi(r_3(q,a,p))=\alpha_{\head{r}}q\beta_{\tail{a}}\delta_a\alpha_{\head{a}}p\beta_{\tail{p}}=0,\]
since $0=\der{a}{\lift{W}}=\der{a}{W}-\beta_{\tail{a}}\delta_a\alpha_{\head{a}}=-\beta_{\tail{a}}\delta_a\alpha_{\head{a}}$ by acyclicity of $Q$.
\end{proof}

\begin{thm}
Let $Q$ be an acyclic quiver. Then the map $\Phi\colon \Gamma_Q/I\to B_Q$, where $\Phi$, $\Gamma_Q$ and $I$ are all defined as above, is an isomorphism.
\end{thm}
\begin{proof}
We begin by showing surjectivity. As in the proof of \Cref{acyclic-fd}, we may use \Cref{Q0-to-Q0} to see that any path in $\lift{Q}$ determining a non-zero 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 non-zero, so we even have
\[p=q_1'\alpha_{\head{p'}}p'\beta_{\tail{p'}}q_2'=q_1'\Phi(\dual{\delta_{p'}})q_2'\]
Now $q_1'$ and $q_2'$ are, like $p$, paths of $\lift{Q}$ with frozen head or tail, but with the additional property that they include no arrows of $Q_1$. Let $q$ be such a path. If $q$ contains an arrow $\beta_i$ for some $i\in Q_0$, then this arrow cannot be the final arrow of $q$, since its head is unfrozen, so it must be followed by the arrow $\alpha_i$, as this is the only arrow outside of $Q_1$ that composes with $\beta_i$. 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 $\alpha_i\beta_i=\Phi(\dual{\delta}_i)$ for $i\in Q_0$, and so is in the image of $\Phi$. We conclude that image in $A$ of any path of $\lift{Q}$ with frozen head or tail lies in the image of $\Phi$. Since such classes span $B=B_Q$, we see that $\Phi$ is surjective.

To complete the proof, we will use \cite[Prop.~3.3]{buanmutation}. In this context, this proposition states that $\Phi$ is an isomorphism if and only if
\begin{equation}
\label{B-complex+}
\begin{tikzcd}[column sep=40pt]
\displaystyle\bigdsum_{\substack{\text{$p$ path in $Q$}\\\tail{p}=i}}Be_{\head{p}}^+\arrow{r}{f}
&Be_i^-\oplus\Big(\displaystyle\bigdsum_{\substack{a\in Q_1\\\head{a}=i}}Be_{\tail{a}}^-\Big)\arrow{r}{(\cdot\delta_i,
\cdot\delta_a)}&\rad{Be_i^+}\arrow{r}&0
\end{tikzcd}
\end{equation}
and
\begin{equation}
\label{B-complex-}
\begin{tikzcd}[column sep=28pt]
\displaystyle\Big(\bigdsum_{\substack{\text{$p$ path in $Q$}\\\tail{p}=i}}Be_{\head{p}}^-\Big)\dsum\Big(\bigdsum_{\substack{\text{$z$ zig-zag}\\\tail{z}=i}}Be_{\head{z}}^+\Big)\arrow{r}{g}
&\displaystyle\bigdsum_{\substack{\text{$p$ path in $Q$}\\\tail{p}=i}}Be_{\head{p}}^+\arrow{r}{\cdot\Phi(\dual{\delta}_p)}&\rad{Be_i^-}\arrow{r}&0
\end{tikzcd}
\end{equation}
are exact sequences for all $i\in Q_0$. Here the left-most maps in each sequence are obtained from our generators of $I$ by right-differentiation \eqref{eq:rightder} and the application of $\Phi$, as proscribed in \cite{buanmutation}, so they act on components by
\begin{equation}
\label{rel-maps}
\begin{aligned}
f(ye_{\head{p}}^+)&=y\Phi(\dual{\delta}_p)\tens[i]-\sum_{\substack{a\in Q_1\\\head{a}=i}}y\Phi(\dual{\delta}_{pa})\tens[a],\\
g(ye_{\head{p}}^-)&=y\delta_{\head{p}}\tens[p]-\sum_{\substack{a\in Q_1\\\tail{a}=\head{p}}}y\delta_a\tens[ap]\\
g(ye_{\head{z}}^+)&=y\Phi(\dual{\delta}_{q})\delta_a\otimes[p],\ \text{where $z=(q,a,p)$.}
\end{aligned}
\end{equation}
Here we have dealt with the ambiguity about which summand contains each term as in \Cref{frjacCY}, denoting elements of $\bigdsum_{a\in X}Be_{v(a)}$ 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 $\bigdsum_{a\in X}Be_{v(a)}$ indexed by $a$. We take the convention here that the summand $B\idemp{i}^-$ appearing in the middle term of \eqref{B-complex+} is indexed by the vertex $i$.

Since $\Phi$ is well-defined and surjective, sequences \eqref{B-complex+} and \eqref{B-complex-} are complexes and exact at $\rad{Be_i^+}$ and $\rad{Be_i^-}$ respectively, so we need only check exactness at the middle term in each case.

We proceed as in \Cref{frjacCY}, using the explicit set $R$ of relations for $A_Q$ from \eqref{relations}, and begin with \eqref{B-complex+}. Let $x_i\in e\KK \lift{Q}e_i^-$ and $x_a\in e\KK \lift{Q}e_{\tail{a}}^-$ for each $a\in Q_1$ with $\head{a}=i$, determining the element
\[x=x_i\otimes[i]+\sum_{\substack{a\in Q_1\\\head{a}=i}}x_a\otimes[a]\]
of the middle term of \eqref{B-complex+}. Assume that $x$ is a cycle, i.e.\ that
\[x_i\delta_i+\sum_{\substack{a\in Q_1\\\head{a}=i}}x_a\delta_a\in\close{\Span{R}}.\]
Since the only generating relation for $A_Q$ with terms ending in $\delta_i$ or $\delta_a$ for $a\in Q_1$ with $\head{a}=i$ is $\der{\alpha_i}{W}$, it follows that in $\KK\lift{Q}$ we have
\begin{align*}
x_i\delta_i+\sum_{\substack{a\in Q_1\\\head{a}=i}}x_a\delta_a&=z_i\delta_i+\sum_{\substack{a\in Q_1\\\head{a}=i}}z_a\delta_a+y\der{\alpha_i}{W}\\
&=z_i\delta_i+\sum_{\substack{a\in Q_1\\\head{a}=i}}z_a\delta_a+y\bigg(\beta_i\delta_i-\sum_{\substack{a\in Q_1\\\head{a}=i}}a\beta_{\tail{a}}\delta_a\bigg)\end{align*}
for $z_i,z_a\in\close{\Span{R}}$ and $y\in\KK \lift{Q}\idemp{i}$. Comparing terms, we see that $x_i=z_i+y\beta_i$ and $x_a=z_a-ya\beta_{\tail{a}}$.
Since $\head{x_i}$ and $\head{x_a}$ are frozen, but $\head{\beta_v}$ is unfrozen for all $v\in Q_0$, we must have
\[y=\sum_{\substack{\text{$p$ path in $Q$}\\\tail{p}=i}}y_p\alpha_{\head p}p.\]
for some $y_p\in Be_{\head{p}}^+$. Projecting to $B$, we have
\[
x_i=\sum_{\substack{\text{$p$ path in $Q$}\\\tail{p}=i}}y_p\Phi(\dual{\delta}_p),\qquad
x_a=\sum_{\substack{\text{$p$ path in $Q$}\\\tail{p}=i}}-y_p\Phi(\dual{\delta}_{pa}),
\]
Thus 
\[y=\sum_{\substack{\text{$p$ path in $Q$}\\\tail{p}=i}}y_p\otimes p\]
satisfies $f(y)=x$, and so sequence \eqref{B-complex+} is exact.

Now we turn to \eqref{B-complex-}. For each path $p$ with $\tail{p}=i$, pick $x_p\in e\KK\lift{Q}e_{\head{p}}^+$, and assume that
\[\sum_{\substack{\text{$p$ path in $Q$}\\\tail{q}=i}}x_p\alpha_{\head{p}}p\beta_i\in\close{\Span{R}},\]
so that
\[x=\sum_{\substack{\text{$p$ path in $Q$}\\\tail{p}=i}}x_p\otimes[p]\]
is a cycle in the middle term of \eqref{B-complex-}.
By comparison with the generating relations, we see that we may write
\[\sum_{\substack{\text{$p$ path in $Q$}\\\tail{p}=i}}x_p\alpha_{\head{p}}p\beta_i=\sum_{\substack{\text{$p$ path in $Q$}\\\tail{p}=i}}\Bigg(z_p\alpha_{\head{p}}p\beta_i+y_p\bigg(\delta_{\head{p}}\alpha_{\head{p}}-\sum_{\substack{b\in Q_1\\\tail{b}=\head{p}}}\delta_b\alpha_{\head{b}}b\bigg)p\beta_i-\sum_{\substack{a\in Q_1\\\head{a}=\head{p}}}y_{a,p}(\beta_{\tail{a}}\delta_a\alpha_{\head{a}})p\beta_i\Bigg)\]
for some $z_p\in\close{\Span{R}}$, $y_p\in\KK\lift{Q}\idemp{\head{p}}^-$ and $y_{a,p}\in e\KK\lift{Q}\idemp{\tail{a}}$. Note that either $p=e_i$ or we may write $p=br$ for some arrow $b$ and path $r$. By comparing terms, we deduce that after projection to $B$ we have
\[
x_i=y_i\delta_i-\sum_{\substack{a\in Q_1\\\head{a}=i}}y_{a,e_i}\beta_{\tail{a}}\delta_a,\qquad
x_{br}=y_{br}\delta_{\head{b}}-y_{r}\delta_b-\sum_{\substack{a\in Q_1\\\head{a}=\head{b}}}y_{a,br}\beta_{\tail{a}}\delta_a,
\]
Since $y_{a,p}\in e\KK \lift{Q}\idemp{\tail{a}}$, we must have
\[y_{a,p}=\sum_{\substack{\text{$q$ path in $Q$}\\\tail{q}=\tail{a}}}y_{q,a,p}\alpha_{\head{q}}q\]
for some $y_{q,a,p}\in e\KK \lift{Q}e_{\head{q}}^+$. The triple $z=(q,a,p)$ occurring in a subscript here satisfies $\head{a}=\head{q}$ and $\tail{a}=\tail{p}$, so it is a zig-zag. One may then calculate explicitly using \eqref{rel-maps} that the $y_p$ and $y_z=y_{q,a,p}$ give a preimage
\[y=\sum_{\substack{\text{$p$ path in $Q$}\\\tail{p}=i}}y_p\otimes[p]+\sum_{\substack{\text{$z$ zig-zag}\\\tail{z}=i}}y_z\otimes[z]\]
of $x$ under $g$.
\end{proof}

We close this section with the following curious property of the category $\frobcat_Q=\GP(B_Q)$.

\begin{prop}
Write $P^+=\bigdsum_{k\in Q_0}B_Qe_k^+$. If $Q$ is acyclic and has no isolated vertices, then $\GP(B_Q)\subseteq\Sub(P^+)$.
\end{prop}
\begin{proof}
Write $B=B_Q$. Since $\GP(B)$ is a Frobenius category with injective objects those in $\add{B}$, it suffices to show that $B\in\Sub(P^+)$, or that $Be_k^\pm\in\Sub(P^+)$ for each $k$. Since this is true of $Be_k^+$ by definition of $P^+$, it remains to show $Be_k^-\in\Sub(P^+)$ for each $k$. Since $Q$ has no isolated vertices, $k$ cannot be both a source and a sink.

First assume $k$ is not a source in $Q$. Then the map $Be_k^-\to Be_k^+$ given by right multiplication by $\delta_k$ is injective as follows. If $x\in Be_k^-$ satisfies $x\delta_k=0$ in $Be_k^+$, then lifting to $\KK \Gamma_Q$ and using the explicit generating set of $I$, we have
\[x\delta_k=z\delta_k+\sum_{\substack{\text{$p$ path in $Q$}\\\tail{p}=k}}y_p\bigg(\dual{\delta}_p\delta_k-\sum_{\substack{a\in Q_1\\\head{a}=k}}\dual{\delta_{pa}}\delta_a\bigg)\]
for some $z\in I$ and $y_p\in e\KK\lift{Q}\idemp{\head{p}}^+$. Since $k$ is not a source, the sum over arrows on the right-hand side is non-empty. By comparing coefficients we see that $y_p=0$ for all $p$, and hence $x=z=0$ in $A$.

Now assume $k$ is not a sink. Pick $a\in Q_1$ with $\tail{a}=k$. Then the map $Be_k^-\to Be_{\head{a}}^+$ given by right multiplication by $\delta_a$ is injective as follows. If $x\in Be_k^-$ satisfies $x\delta_a=0$ in $Be_{\head{a}}^+$, then lifting to $\KK \Gamma_Q$ and using our explicit relations, we have
\[x\delta_a=z+\sum_{\substack{\text{$p$ path in $Q$}\\\tail{p}=\head{a}}}y_p\bigg(\dual{\delta_p}\delta_{\head{a}}-\sum_{\substack{b\in Q_1\\\head{b}=\head{a}}}\dual{\delta}_{pa}\delta_a\bigg),\]
for $z\in I$, and it follows by comparing coefficients that $y_p=0$ for all $p$, so $x=0$.
\end{proof}
\Cref{a3-eg} below shows that we may have $\frobcat_Q\subsetneq\Sub(P^+)$. The assumption on isolated vertices is necessary, since such a vertex $k$ of $Q$ yields a direct summand $\cat$ of $\frobcat_Q$ equivalent to $\fgmod{\Pi}$ for $\Pi$ the preprojective algebra of type $\typeA{2}$, with indecomposable objects $S_k^\pm$ and $P_k^\pm$, and $S_k^-,P_k^-\notin\Sub(P^+)$.

\section{Gradings, indices and c-vectors}
\label{gradings}

Let $Q$ be an acyclic quiver. In this section, we show that $\frobcat_Q=\GP(B_Q)$ may be equipped with the structure of a $\ZZ^n$-graded Frobenius cluster category \cite[Def.~3.7]{grabowskigradedfrobenius}, corresponding to the grading of the principal coefficient cluster algebra $\princlust{Q}$ described by Fomin--Zelevinsky \cite[\S6]{fomincluster4} (or, more precisely, to the lift of this grading to $\pprinclust{Q}$ by putting the extra frozen variables $x_i^-$ in degree $0$). By \cite[Thm.~3.12]{grabowskigradedfrobenius}, the data of this grading, when restricted to an initial seed, is equivalent to the data of a group homomorphism $\K_0(\frobcat_Q)\to\ZZ^n$, where $\K_0(\frobcat_Q)$ denotes the Grothendieck group of the exact category $\frobcat_Q$. We will be able to describe this group homomorphism in (almost) purely homological terms. The material in this section serves to illustrate how one can pass information back and forth between a cluster algebra and its categorification, and how $\frobcat_Q$ may serve as a model for the principal coefficient cluster algebra $\princlust{Q}$. We use the Frobenius category here, rather than the extriangulated category $\frobcat_Q^+$ of \Cref{extriangulated}, for compatibility with the results of \cite{grabowskigradedfrobenius}.

First we briefly recall how to specify a grading on a cluster algebra, and how to interpret it categorically. Here we will only consider $\ZZ^n$-gradings, where $n$ is the rank of the cluster algebra; a more detailed explanation for gradings by arbitrary abelian groups has been given by Grabowski and the author \cite{grabowskigradedfrobenius}.

To give a $\ZZ^n$-grading of a rank $n$ cluster algebra (i.e.\ a grading of the underlying algebra in which all cluster variables are homogeneous elements), it suffices to pick a seed $s$ with (extended) $m\times n$ integer exchange matrix $\lift{b}$, and an $m\times n$ integer matrix $g$ such that $\lift{b}^tg=0$ \cite[Def.~3.1]{grabowskigraded}. The $i$-th row of $g$ is the degree of the cluster variable $x_i$ (which is frozen if $i>n$), and the product condition ensures that all of the exchange relations are homogeneous. If $b$ is skew-symmetric, so we may think of it as a quiver, this condition is equivalent to requiring, for each $1\leq k\leq n$, that
\[\sum_{i\to k}\deg{x_i}=\sum_{k\to j}\deg{x_j}.\]

Now assume that the upper $n\times n$ submatrix (principal part) of $\lift{b}$ is skew-symmetric, and let $Q$ be the ice quiver corresponding to $\lift{b}$. If $\frobcat$ is a Frobenius cluster category, and $T\in\frobcat$ is a cluster-tilting object such that the quiver of $A=\Endalg{\frobcat}{T}$ agrees with $Q$ up to arrows with frozen vertices, then one can interpret a grading $g$ as the element
\[G=\sum_{i=1}^mg_i\tens[S_i]\in\ZZ^n\tens_\ZZ\K_0(\fd{A})=\K_0(\fd{A})^n,\]
where $\K_0(\fd{A})$ is the Grothendieck group of finite-dimensional $A$-modules, and $g_i$ denotes the $i$-th row of $g$. The matrix identity $\lift{b}^tg=0$ implies, on the level of Grothendieck groups, that $\bil{M}{G}=0$ for all $M\in\fgmod{\stab{A}}$, where $\stab{A}=\stabEndalg{\frobcat}{T}$ and $\bil{-}{-}\colon\K_0(\fgmod{A})\times\K_0(\fd{A})^n\to\ZZ^n$ is induced from the Euler form of $A$ by tensor product with $\ZZ^n$. This form is well-defined since $\frobcat$ is a Frobenius cluster category, so $\gldim{A}\leq 3$. Moreover, any $G\in\K_0(\fd{A})^n$ with the above property arises from a unique grading of the cluster algebra in this way.

Now by \cite[Thm.~3.12]{grabowskigradedfrobenius}, given $G\in\K_0(\fd{A})^n$ as above, the map
\[\deg_G\colon[X]\mapsto\bil{\Hom_\frobcat(T,X)}{G}\]
is a group homomorphism $\K_0(\frobcat)\to\ZZ^n$, and, having fixed $T$, all such homomorphisms arise in this way for a unique $G$. Thus the $\ZZ^n$-gradings of the cluster algebra generated by the seed $s$ are in bijection with $\Hom_\ZZ(\K_0(\frobcat),\ZZ^n)$.

We now return to the case of polarised principal coefficient cluster algebras. Let $(Q,W)$ be a quiver with potential such that $A=A_{Q,W}$ is Noetherian and the category $\frobcat=\GP(B_{Q,W})$ is a Frobenius cluster category, such as if $Q$ is acyclic. We also abbreviate $B=B_{Q,W}$. To write down matrices unambiguously, we pick a labelling of the vertices of $Q$ by $1,\dotsc,n$, so that the vertices of $\lift{Q}$, the ice quiver of our preferred initial seed of $\pprinclust{Q}$, are labelled by $1,\dotsc,n,1^+,\dotsc,n^+,1^-,\dotsc,n^-$. The $3n\times n$ matrices we are about to write have rows indexed by these labels, in this order, and columns indexed by $1,\dotsc,n$.

Let $b$ be the skew-symmetric matrix associated to $Q$. Then the extended exchange matrix of our initial seed $\lift{s}_0$ for $\pprinclust{Q}$ is
\[\lift{b}=\begin{pmatrix}b\\\id{n}\\-\id{n}\end{pmatrix}.\]
The principal coefficient grading of $\pprinclust{Q}$ is defined via the matrix
\[\lift{g}=\begin{pmatrix}\id{n}\\b\\0_n\end{pmatrix}.\]
It is straightforward to check that this is indeed a grading; $\lift{b}^t\lift{g}=b^t+b=0$, since $b$ is skew-symmetric. If one sets the cluster variables $x_i^-$ of $\pprinclust{Q}$ to $1$, obtaining the principal coefficient cluster algebra $\princlust{Q}$, then this quotient map is homogeneous for the grading of the target cluster algebra defined by Fomin--Zelevinsky \cite[\S6]{fomincluster4}, given by the first $2n$ rows of $\lift{g}$. The grading also extends to the Laurent polynomial ring $R$ generated by the $x_i$ and $y_i^\pm$, since $\pprinclust{Q}$ is a subring containing these elements. The degree of a homogeneous element of $R$ is called its g-vector \cite[\S6]{fomincluster4}.

Let $T=eA\in\frobcat$ be the cluster-tilting object of $\frobcat$ from \Cref{frobcat-constr}. In $\stab{\frobcat}$, we have $T\iso\bigdsum_{i=1}^nT_i$, where $T_i=eAe_i$. Let $G\in\K_0(\fd{A})^n$ be the element corresponding to the grading $\lift{g}$, and $\deg_G\colon\K_0(\frobcat)\to\ZZ^n$ the associated group homomorphism.

For any $X\in\frobcat$, its cluster character $C^T_X$ with respect to $T$ (defined by Fu--Keller \cite[Thm.~3.3]{fucluster}, see also \cite[\S3]{grabowskigradedfrobenius}) is a homogeneous element of $R$, with degree equal to $\deg_G(X)$ \cite[Prop.~3.11]{grabowskigradedfrobenius}. If $X$ has no non-zero projective summands, Fu--Keller \cite[Prop.~6.2]{fucluster} (see also Plamondon \cite[Prop.~3.6]{plamondonclusteralgebras}) show how to compute the g-vector of $C^T_X$ homologically in the stable category $\stab{\frobcat}$, as we now recall.

Let $X\in\frobcat$ have no non-zero projective summands. Using the monomial notation $T^x=\bigoplus_{i=1}^nT_i^{x_i}$, we can find a triangle
\begin{equation}
\label{index}
\begin{tikzcd}[column sep=20pt]
T^{m_X}\arrow{r}&T^{p_X}\arrow{r}{f}&X\arrow{r}&\Omega^{-1}T^{m_j}
\end{tikzcd}
\end{equation}
in $\stab{\frobcat}$ by choosing $f$ to be a right $(\add{T})$-approximation, this property being equivalent to $f$ having domain and mapping cylinder in $\add{T}$. Then the index of $X$ with respect to $T$ is the vector $\ind{T}{X}=p_X-m_X\in\ZZ^n$. This quantity is independent of the choice of $f$, although $p_X$ and $m_X$ individually are not. By \cite[Prop.~6.2]{fucluster}, we then have
\[\deg_G(X)=\ind{T}{X}.\]
Since $\deg_G$ is additive on direct sums, this formula can be used to compute $\deg_G(X)$ for any $X\in\frobcat$, since the degrees of the projectives $eAe_i^{\pm}$ are given directly by the lower $2n$ rows of $\lift{g}$.

\begin{rem}
By choosing a suitable map $P\to X$, where $P$ is projective, the triangle \eqref{index} may always be lifted to a short exact sequence
\[\begin{tikzcd}[column sep=20pt]
0\arrow{r}&T^{m_X}\arrow{r}&T^{p_X}\dsum P\arrow{r}&X\arrow{r}&0
\end{tikzcd}\]
in $\frobcat$, from which it follows that in $\K_0(A)$ we have
\[[\Hom_B(T,X)]=[\Hom_B(T,T^{p_X})]+[\Hom_B(T,P)]-[\Hom_B(T,T^{m_X})],\]
and all of the $A$-modules on the right-hand side are projective. We have
\[\ind{T}{X}_i=(p_X-m_X)_i=\bil{\Hom_B(T,T^{p_X})}{S_i}-\bil{\Hom_B(T,T^{m_X})}{S_i}\]
for all $1\leq i\leq n$, and
\begin{align*}
\deg_G(X)_i&=(\bil{\Hom_B(T,T^{p_X})}{G}-\bil{\Hom_B(T,T^{m_X})}{G}+\bil{\Hom_B(T,P)}{G})_i\\
&=\bil{\Hom_B(T,T^{p_X})}{S_i}-\bil{\Hom_B(T,T^{m_X})}{S_i}+(\bil{\Hom_B(T,P)}{G})_i,
\end{align*}
from the definition of $G$ and the fact that $T^{p_X}$ and $T^{m_X}$ have no projective summands in $\frobcat$. We therefore deduce from the identity $\ind{T}{X}=\deg_G(X)$ that
\[\bil{\Hom_B(T,P)}{G}=0,\]
giving a linear relation between the rows of the (unextended) exchange matrix $b$, with coefficients determined by the multiplicity of $Be_i^+$ as a summand of $P$. When $b$ has full rank, it follows that it is always possible to choose $P\in\add{P^-}$, where $P^-=\bigdsum_{i\in Q_0}Be_i^-$. We conjecture that this is in fact true in general, and can observe it directly in \Cref{a3-eg,3cycle-eg} below, in which the exchange matrices do not have full rank. Conversely, establishing that one can always choose $P\in\add{P^-}$ would provide a new proof of the identity $\deg(C^T_X)=\ind{T}{X}$.
\end{rem}

Having understood the grading $\lift{g}$ globally, as a function on $\K_0(\frobcat)$, we may now recover local information for any cluster-tilting object. 
Let $T'=\bigoplus_{i=1}^nT_i'\oplus B$, with $T_i'$ indecomposable and non-projective, be a cluster-tilting object of $\frobcat$, and write $A'=\Endalg{\frobcat}{T'}$. Let $\lift{b}'$ be the extended exchange matrix associated to the ice quiver $Q'$ of $A'$, and let
\[\lift{g}'=\begin{pmatrix}g'\\b\\0_n\end{pmatrix}\]
be the $3n\times n$ matrix whose first $n$ rows are given by
\[\deg_G(T_i')=\bil{\Hom_B(T,T_i')}{G}=\ind{T}{T_i'}.\]
One can check \cite[\S3]{grabowskigradedfrobenius} that the corresponding element $G'\in\K_0(\fd{A'})^n$ satisfies $\bil{M}{G'}=0$
for all $M\in\fgmod{\stabEndalg{\frobcat}{T'}}$, and the induced function $\K_0(\frobcat)\to\ZZ^n$ coincides with $\deg_G$. In particular, this means that $\lift{g}'$ is a grading of the cluster algebra associated to $Q'$, so $(\lift{b}')^t\lift{g}'=0$.
Writing
\[\lift{b}'=\begin{pmatrix}b'\\c'\\d'\end{pmatrix},\]
where $b'$, $c'$ and $d'$ are $n\times n$ submatrices, and using skew-symmetry of $b'$, we conclude that
\[b'g'=(c')^tb.\]
For acyclic $Q$, the matrix $\lift{b}'$ is the extended exchange matrix of a seed of $\pprinclust{Q}$ by \Cref{main-cat-thm}\ref{main-cat-thm-mut}, so the submatrix $c'$ is by definition the matrix of c-vectors of this seed. Thus in this case we recover (the transpose of) an identity of Fomin--Zelevinsky \cite[Eq.~6.14]{fomincluster4} (see also \cite[Rem.~2.1]{nakanishitropical}), noting for comparison that it is the rows, not columns, of our matrix $g'$ that are g-vectors.

\section{Examples}
\label{egs}

\begin{eg}
\label{a2eg}
Let $Q$ be an $\typeA{2}$ quiver, so, as computed in \Cref{construction-egs},
\[(\lift{Q},\lift{F})=\mathord{\begin{tikzpicture}[baseline={(current bounding box.center)},yscale=-1]
\node at (-1,0) (1) {$1$};
\node at (1,0) (2) {$2$};
\node at (-2,1) (1+) {$\boxed{1^+}$};
\node at (-2,-1) (1-) {$\boxed{1^-}$};
\node at (2,1) (2-) {$\boxed{2^-}$};
\node at (2,-1) (2+) {$\boxed{2^+}$};
\path[-angle 90,font=\scriptsize]
	(1) edge node[below] {$a$} (2)
	(1) edge node[below right] {$\alpha_1$} (1+)
	(1-) edge node[above right] {$\beta_1$} (1)
	(2) edge node[above left] {$\alpha_2$} (2+)
	(2-) edge node[below left] {$\beta_2$} (2);
\path[\frozen,-angle 90,font=\scriptsize]
	(1+) edge[bend right] node[left] {$\delta_1$} (1-)
	(2+) edge[bend right] node[right] {$\delta_2$} (2-)
	(2+) edge[bend left] node[above] {$\delta_a$} (1-);
\end{tikzpicture}}\]
and $\lift{W}=\beta_1\delta_1\alpha_1+\beta_2\delta_2\alpha_2-a\beta_1\delta_a\alpha_2$. Then $A_Q=\frjac{\lift{Q}}{\lift{F}}{\lift{W}}$, and its boundary algebra is $B_Q\iso\KK \Gamma_Q/I$ for
\[\Gamma_Q=\mathord{\begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)},scale=1.5]
\node at (0,0) (1) {$1$};
\node at (1,0) (2) {$2$};
\node at (2,0) (3) {$3$};
\node at (3,0) (4) {$4$};
\path[-angle 90,font=\scriptsize]
	(1) edge[bend left] node[above] {$\alpha$} (2)
	(2) edge[bend left] node[below] {$\dual{\alpha}$} (1)
	(2) edge[bend left] node[above] {$\beta$} (3)
	(3) edge[bend left] node[below] {$\dual{\beta}$} (2)
	(3) edge[bend left] node[above] {$\gamma$} (4)
	(4) edge[bend left] node[below] {$\dual{\gamma}$} (3);
\end{tikzpicture}}\]
and $I=\Span{\dual{\alpha}\alpha,\ \alpha\dual{\alpha}-\dual{\beta}\beta,\ \beta\dual{\beta}-\dual{\gamma}\gamma,\ \gamma\dual{\gamma},\ \beta\alpha,\ \gamma\beta,\ \dual{\alpha}\dual{\beta}\dual{\gamma}}$. Here we have relabelled the vertices by identifying the ordered sets $(1^+,1^-,2^+,2^-)$ and $(1,2,3,4)$ in the unique order preserving way.
The Auslander--Reiten quiver of $\GP(B)$ is shown in \Cref{a2-AR}, where we identify the left and right sides of the picture so that the quiver is drawn on a Möbius band.
\begin{figure}[h]
\begin{tikzpicture}[yscale=1.2,xscale=1.4]
\node at (0,0) (radP3) {$\begin{radfilt}&2&&4\\1&&3\\&2\end{radfilt}$};
\node at (1,1) (P3) {$\boxed{\begin{radfilt}&&3\\&2&&4\\1&&3\\&2\end{radfilt}}$};
\node at (1,-1) (S2) {$\begin{radfilt}2\end{radfilt}$};
\node at (2,0) (radP4) {$\begin{radfilt}&3\\2\end{radfilt}$};
\node at (2,-2) (P1) {$\boxed{\begin{radfilt}1\\&2\end{radfilt}}$};
\node at (3,1) (P4) {$\boxed{\begin{radfilt}&&4\\&3\\2\end{radfilt}}$};
\node at (3,-1) (radP2) {$\begin{radfilt}1&&3\\&2\end{radfilt}$};
\node at (4,0) (M) {$\begin{radfilt}&&&4\\1&&3\\&2\end{radfilt}$};
\node at (4,-2) (P2) {$\boxed{\begin{radfilt}&2\\1&&3\\&2\end{radfilt}}$};
\node at (5,-1) (radP3') {$\begin{radfilt}&2&&4\\1&&3\\&2\end{radfilt}$};
\path[-angle 90]
	(radP3) edge (P3)
	(radP3) edge (S2)
	(P3) edge (radP4)
	(S2) edge (radP4)
	(S2) edge (P1)
	(radP4) edge (P4)
	(radP4) edge (radP2)
	(P1) edge (radP2)
	(P4) edge (M)
	(radP2) edge (M)
	(radP2) edge (P2)
	(M) edge (radP3')
	(P2) edge (radP3');
\path[dashed]
	(0,-1) edge (S2)
	(radP3) edge (radP4)
	(S2) edge (radP2)
	(radP4) edge (M)
	(radP2) edge (radP3')
	(M) edge (5,0);
\path[dotted]
	(0,1) edge (radP3)
	(radP3) edge (0,-2)
	(5,1) edge (radP3')
	(radP3') edge (5,-2);
\end{tikzpicture}
\caption{\label{a2-AR}The Auslander--Reiten quiver of $\GP(B_Q)$ for $Q$ of type $\typeA{2}$.}
\end{figure}
To calculate the objects of $\GP(B_Q)$ it is useful to observe that, in this example, $B_Q$ is $1$-Iwanaga--Gorenstein, and so $\GP(B_Q)=\Sub(B_Q)$. The stable category $\stabGP(B_Q)$ is the cluster category of type $\typeA{2}$, as expected. The cluster tilting object
\[T=\begin{radfilt}&3\\2\end{radfilt}\oplus\begin{radfilt}1&&3\\&2\end{radfilt}\oplus B_Q\]
of $\GP(B_Q)$ has endomorphism algebra $A_Q$, and corresponds to the initial seed of the cluster algebra with polarised principal coefficients associated to our initial $\typeA{2}$ quiver $Q$.
\end{eg}

\begin{eg}
\label{a3-eg}
Let $Q$ be a linearly oriented quiver of type $\mathsf{A}_3$. We may then compute
\[(\lift{Q},\lift{F})=\mathord{\begin{tikzpicture}[baseline={(current bounding box.center)},xscale=1.2,yscale=-1.1]
\node at (-1,0) (1) {$1$};
\node at (1,0) (2) {$2$};
\node at (3,0) (3) {$3$};
\node at (-2,1) (1+) {$\boxed{1^+}$};
\node at (-2,-1) (1-) {$\boxed{1^-}$};
\node at (0,-1) (2+) {$\boxed{2^+}$};
\node at (2,-1) (2-) {$\boxed{2^-}$};
\node at (4,1) (3-) {$\boxed{3^-}$};
\node at (4,-1) (3+) {$\boxed{3^+}$};
\path[-angle 90,font=\scriptsize]
	(1) edge node[above] {$a$} (2)
	(2) edge node[above] {$b$} (3)
	(1) edge node[below right] {$\alpha_1$} (1+)
	(1-) edge node[above right] {$\beta_1$} (1)
	(2) edge node[left] {$\alpha_2$} (2+)
	(2-) edge node[right] {$\beta_2$} (2)
	(3) edge node[above left] {$\alpha_3$} (3+)
	(3-) edge node[below left] {$\beta_3$} (3);
\path[\frozen,-angle 90,font=\scriptsize]
	(1+) edge[bend right] node[left] {$\delta_1$} (1-)
	(2+) edge[bend right] node[above] {$\delta_2$} (2-)
	(3+) edge[bend right] node[right] {$\delta_3$} (3-)
	(2+) edge[bend left] node[above] {$\delta_a$} (1-)
	(3+) edge[bend left] node[above] {$\delta_b$} (2-);
\end{tikzpicture}}\]
Relabelling vertices similarly to \Cref{a2eg}, the boundary algebra $B_Q$ has quiver
\[\Gamma_Q=\mathord{\begin{tikzpicture}[baseline={([yshift=3.6ex]current bounding box.center)},xscale=1.5,yscale=0.8]
\node at (0,0) (1) {$1$};
\node at (1,0) (2) {$2$};
\node at (2,0) (3) {$3$};
\node at (3,0) (4) {$4$};
\node at (4,0) (5) {$5$};
\node at (5,0) (6) {$6$};
\path[-angle 90,font=\scriptsize]
	(1) edge[bend left] (2)
	(2) edge[bend left] (1)
	(2) edge[bend right] (3)
	(3) edge[bend right] (2)
	(3) edge[bend left] (4)
	(4) edge[bend left] (3)
	(4) edge[bend right] (5)
	(5) edge[bend right] (4)
	(5) edge[bend left] (6)
	(6) edge[bend left] (5)
	(2) edge[bend right=60] (5);
\end{tikzpicture}}\]
as computed before in \Cref{a3-bdy-eg}. Explicit relations can be written down as in \Cref{bdy-algs}, but here we will simply give radical filtrations for the projective modules. Note that despite the `geographical' separation of $2$ and $5$ in these filtrations, the arrow $2\to 5$ always acts as a vector space isomorphism from the $1$-dimensional subspace of $e_2B_Q$ indicated by a $2$ in the filtration to the $1$-dimensional subspace of $e_5B_Q$ indicated by a $5$ in the row below, when this configuration occurs.
\begin{align*}
P_1&=\begin{radfilt}1\\&2\end{radfilt}&P_2&=\begin{radfilt}&2\\1&&3&&5\\&2&&4\end{radfilt}&P_3&=\begin{radfilt}&&3\\&2&&4\\1&&3&&5\\&2&&4\end{radfilt}\\[1ex]
P_4&=\begin{radfilt}&&4\\&3&&5\\2&&4\end{radfilt}&P_5&=\begin{radfilt}&&&5\\&&4&&6\\&3&&5\\2&&4\end{radfilt}&P_6&=\begin{radfilt}&&6\\&5\\4\end{radfilt}
\end{align*}

In this case the Gorenstein dimension of $B_Q$ is $2$; the indecomposable projective $P_2$ has injective dimension $2$, while all others have injective dimension $1$. (This is in fact the first example known to the author of a Frobenius cluster category $\GP(B)$, with $\stabGP(B)\ne0$, for which the Gorenstein dimension of $B$ is greater than $1$.) The Auslander--Reiten quiver of $\GP(B_Q)$ is shown, again on a Möbius band, in \figref{a3-AR}.
\begin{figure}[t]
\begin{tikzpicture}[yscale=1.8,xscale=1.9]
\node at (0,0) (M4) {$\begin{radfilt}&&&4\\1&&3&&5\\&2&&4\end{radfilt}$};
\node at (0,-2) (M9) {$\begin{radfilt}&2&&4&&6\\1&&33&&55\\&22&&44\end{radfilt}$};
\node at (1,-1) (M8) {$\begin{radfilt}&2&&4\\1&&3&&5\\&2&&4\end{radfilt}$};
\node at (1,-3) (P5) {$\boxed{\begin{radfilt}&&&5\\&&4&&6\\&3&&5\\2&&4\end{radfilt}}$};
\node at (2,0) (M7) {$\begin{radfilt}2\end{radfilt}$};
\node at (2,-0.6) (P3) {$\boxed{\begin{radfilt}&&3\\&2&&4\\1&&3&&5\\&2&&4\end{radfilt}}$};
\node at (2,-2) (M3) {$\begin{radfilt}&5\\4\end{radfilt}$};
\node at (3,1) (P1) {$\boxed{\begin{radfilt}1\\&2\end{radfilt}}$};
\node at (3,-1) (M2) {$\begin{radfilt}&3&&5\\2&&4\end{radfilt}$};
\node at (3,-3) (P6) {$\boxed{\begin{radfilt}&&6\\&5\\4\end{radfilt}}$};
\node at (4,0) (M1) {$\begin{radfilt}1&&3&&5\\&2&&4\end{radfilt}$};
\node at (4,-0.6) (P4) {$\boxed{\begin{radfilt}&&4\\&3&&5\\2&&4\end{radfilt}}$};
\node at (4,-2) (M6) {$\begin{radfilt}&&&&6\\&3&&5\\2&&4\end{radfilt}$};
\node at (5,1) (P2) {$\boxed{\begin{radfilt}&2\\1&&3&&5\\&2&&4\end{radfilt}}$};
\node at (5,-1) (M5) {$\begin{radfilt}&&&4&&6\\1&&33&&55\\&22&&44\end{radfilt}$};
\node at (6,0) (M9') {$\begin{radfilt}&2&&4&&6\\1&&33&&55\\&22&&44\end{radfilt}$};
\node at (6,-2) (M4') {$\begin{radfilt}&&&4\\1&&3&&5\\&2&&4\end{radfilt}$};

\path[-angle 90]
	(M4) edge (M8)
	(M9) edge (M8)
	(M9) edge (P5)
	(M8) edge (M7)
	(M8) edge (P3)
	(M8) edge (M3)
	(P5) edge (M3)
	(M7) edge (P1)
	(M7) edge (M2)
	(P3) edge (M2)
	(M3) edge (M2)
	(M3) edge (P6)
	(P1) edge (M1)
	(M2) edge (M1)
	(M2) edge (P4)
	(M2) edge (M6)
	(P6) edge (M6)
	(M1) edge (P2)
	(M1) edge (M5)
	(P4) edge (M5)
	(M6) edge (M5)
	(P2) edge (M9')
	(M5) edge (M9')
	(M5) edge (M4');
\path[dashed]
	(0,-1) edge (M8)
	(M4) edge (M7)
	(M9) edge (M3)
	(M8) edge (M2)
	(M7) edge (M1)
	(M3) edge (M6)
	(M2) edge (M5)
	(M1) edge (M9')
	(M6) edge (M4')
	(M5) edge (6,-1);
\path[dotted]
	(0,1) edge (M4)
	(M4) edge (M9)
	(M9) edge (0,-3)
	(6,1) edge (M9')
	(M9') edge (M4')
	(M4') edge (6,-3);
\end{tikzpicture}
\caption{\label{a3-AR}The Auslander--Reiten quiver of $\GP(B_Q)$ for $Q$ linearly oriented of type $\typeA{3}$. In addition to the the usual mesh relations coming from Auslander--Reiten sequences, the length two path from $P_2$ to $P_5$ represents the zero map.}
\end{figure}
The initial cluster tilting object from \Cref{frobcat-constr} is
\[T=\begin{radfilt}1&&3&&5\\&2&&4\end{radfilt}\oplus\begin{radfilt}&3&&5\\2&&4\end{radfilt}\oplus\begin{radfilt}&5\\4\end{radfilt}\oplus B_Q.\]
\end{eg}

\begin{eg}
\label{3cycle-eg}
Applying our construction to the quiver with potential $(Q,W)$ from \Cref{construction-egs} with $Q$ a $3$-cycle (which we may do, since while $A$ is not finite-dimensional in this case, it is still Noetherian) yields, as observed in \Cref{construction-egs}, the Grassmannian cluster category $\GP(B_{Q,W})=\CM(B_{2,6})$ \cite{jensencategorification}. This is a Hom-infinite category, and the Gorenstein projective $B$-modules are all infinite-dimensional. Representing these modules by Plücker labels as in \cite{jensencategorification}, the Auslander--Reiten quiver of $\GP(B_{Q,W})$ is shown, on the now familiar Möbius band, in \Cref{3cycle-AR}.
\begin{figure}[t]
\begin{tikzpicture}[scale=1.1]
\node at (0,1) (13) {$13$};
\node at (0,-1) (46) {$46$};
\node at (1,2) (23) {$\boxed{23}$};
\node at (1,0) (14) {$14$};
\node at (1,-2) (56) {$\boxed{56}$};
\node at (2,1) (24) {$24$};
\node at (2,-1) (15) {$15$};
\node at (3,2) (34) {$\boxed{34}$};
\node at (3,0) (25) {$25$};
\node at (3,-2) (16) {$\boxed{16}$};
\node at (4,1) (35) {$35$};
\node at (4,-1) (26) {$26$};
\node at (5,2) (45) {$\boxed{45}$};
\node at (5,0) (36) {$36$};
\node at (5,-2) (12) {$\boxed{12}$};
\node at (6,1) (13') {$13$};
\node at (6,-1) (46') {$46$};

\path[-angle 90]
	(13) edge (23)
	(13) edge (14)
	(46) edge (14)
	(46) edge (56)
	(23) edge (24)
	(14) edge (24)
	(14) edge (15)
	(56) edge (15)
	(24) edge (34)
	(24) edge (25)
	(15) edge (25)
	(15) edge (16)
	(34) edge (35)
	(25) edge (35)
	(25) edge (26)
	(16) edge (26)
	(35) edge (45)
	(35) edge (36)
	(26) edge (36)
	(26) edge (12)
	(45) edge (13')
	(36) edge (13')
	(36) edge (46')
	(12) edge (46');
\path[dashed]
	(0,0) edge (14)
	(13) edge (24)
	(46) edge (15)
	(14) edge (25)
	(24) edge (35)
	(15) edge (26)
	(25) edge (36)
	(35) edge (13')
	(26) edge (46')
	(36) edge (6,0);
\path[dotted]
	(0,2.5) edge (13)
	(13) edge (46)
	(46) edge (0,-2.5)
	(6,2.5) edge (13')
	(13') edge (46')
	(46') edge (6,-2.5);
\end{tikzpicture}
\caption{\label{3cycle-AR}The Auslander--Reiten quiver of $\GP(B_{Q,W})$, where $(Q,W)$ is a $3$-cycle and its usual potential, shown as the Grassmannian cluster category $\CM(B_{2,6})$.}
\end{figure}
In this case, the quiver of the endomorphism algebra of the object
\[13\dsum 15\dsum 35\dsum B_{Q,W}\]
is $\lift{Q}$, as is the quiver of the (isomorphic) endomorphism algebra of $24\dsum 26\dsum 46\dsum B_{Q,W}$. We note that the stable category $\stabGP(B_{Q,W})$ is equivalent to the cluster category $\clustcat{Q,W}\simeq\clustcat{Q'}$ where $Q'$ is any orientation of the Dynkin diagram $\typeA{3}$. Thus all of the conclusions of \Cref{main-cat-thm} (replacing $\KK Q$ by $\jac{Q}{W}$ and $\clustcat{Q}$ by $\clustcat{Q,W}$) still hold for this example, despite the failure of acyclicity.
\end{eg}

\section*{Acknowledgements}
We thank Mikhail Gorsky, Alfredo Nájera Chávez, Hiroyuki Nakaoka, Yann Palu and Salvatore Stella for useful conversations and for providing references. These thanks are extended to the anonymous referee for suggesting various improvements, and the Max-Planck-Gesellschaft for financial support.

\defbibheading{bibliography}[\refname]{\section*{#1}}
\printbibliography
\end{document}

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