# Calabi–Yau properties of Postnikov diagrams

Matthew Pressland  Matthew Pressland
School of Mathematics
University of Leeds
Leeds LS2 9JT
United Kingdom<EMAIL_ADDRESS>
###### Abstract.

We show that the dimer algebra of a connected Postnikov diagram in the disc is
bimodule internally $3$-Calabi–Yau in the sense of the author’s earlier work
[42]. As a consequence, we obtain an additive categorification of the cluster
algebra associated to the diagram, which (after inverting frozen variables) is
isomorphic to the homogeneous coordinate ring of a positroid variety in the
Grassmannian by a recent result of Galashin and Lam [17]. We show that our
categorification can be realised as a full extension closed subcategory of
Jensen–King–Su’s Grassmannian cluster category [27], in a way compatible with
their bijection between rank $1$ modules and Plücker coordinates.

###### Key words and phrases:

Postnikov diagram, dimer model, Jacobian algebra, Calabi–Yau, positroid

###### 2010 Mathematics Subject Classification:

16G20 (Primary) 13F60, 14M15, 16S38, 18E10, 18E30 (Secondary)

##  1\. Introduction

###  1.1. Main results

A Postnikov diagram, or alternating strand diagram [40], is a combinatorial
object consisting of a collection of strands in the disc. It encodes a great
deal of further combinatorial, algebraic and geometric data, mostly connected
to questions regarding total positivity in the Grassmannian. From the point of
view of this paper, the key piece of data encoded by a Postnikov diagram is
that of an ice quiver with potential $(Q_{D},F_{D},W_{D})$. Here $Q_{D}$ is a
quiver with a distinguished subquiver $F_{D}$ which we call frozen, so that
the pair $(Q_{D},F_{D})$ is an ice quiver, and the potential $W_{D}$ is,
loosely speaking, a linear combination of cycles in $Q_{D}$ (see Definition
2.3 for a precise definition). The quiver $Q_{D}$ and the vertex set of
$F_{D}$ determine a cluster algebra $\mathscr{A}_{D}$ with frozen variables;
the reader unfamiliar with this construction can find details in, for example,
Keller’s survey [28]. In this paper, we will explain how to use the triple
$(Q_{D},F_{D},W_{D})$ to construct an additive categorification of this
cluster algebra, in the case that $D$ is connected.

The construction proceeds via the frozen Jacobian algebra $A_{D}$ associated
to $(Q_{D},F_{D},W_{D})$, also known as the dimer algebra of $D$. This algebra
is a quotient of the complete path algebra of $Q_{D}$ by the closure of an
ideal generated by relations determined by $W_{D}$ and the arrows of $F_{D}$;
again, precise definitions are found in Definition 2.3. This algebra has a
distinguished idempotent $e$, given by the sum of vertex idempotents over the
vertices of $F_{D}$, from which we obtain a _boundary algebra_
$B_{D}=eA_{D}e$. We will show that the category $\operatorname{GP}(B_{D})$ of
Gorenstein projective $B_{D}$-modules is our desired categorification.

######  Theorem 1 (cf. Theorem 4.3 and Theorem 6.11).

Let $D$ be a connected Postnikov diagram, and let $B_{D}=eA_{D}e$ be the
boundary algebra of its dimer algebra. Then the category
$\operatorname{GP}(B_{D})$ is an additive categorification of the cluster
algebra $\mathscr{A}_{D}$.

While we use the term ‘additive categorification’ for brevity in the
introduction, we acknowledge that it does not have a widely accepted formal
definition in this context, and refer the reader to the more precise
statements below, which we hope justify our use of this terminology.

The main step in proving Theorem 1 is to establish a certain Calabi–Yau
property for the algebra $A_{D}$, relative to the boundary idempotent $e$, as
referred to in the title. This result is likely to be of independent interest,
and is analogous to Broomhead’s theorem [3, Thm. 7.1] concerning Calabi–Yau
properties of algebraically consistent dimer models on the torus.

######  Theorem 2 (cf. Theorem 3.7).

Let $D$ be a connected Postnikov diagram, with dimer algebra $A_{D}$. Then
$A_{D}$ is bimodule internally $3$-Calabi–Yau with respect to the idempotent
$e$ given by summing the vertex idempotents for the vertices of $F_{D}$.

The general definition of a bimodule internally $3$-Calabi–Yau algebra, and
the role of this property in constructing categorifications, is explained in
[42]. We will not give this general definition here; since $A_{D}$ is
presented as a frozen Jacobian algebra, we can work instead with a sufficient
condition implying the Calabi–Yau property, which appears in Theorem 3.7.

###  1.2. Positroid varieties

Postnikov diagrams and their associated cluster algebras are important in
studying positroid varieties in the Grassmannian—while this connection has
only a minor logical role in the paper, we recall it here to help
contextualise and motivate our results. Positroid varieties were introduced by
Knutson, Lam and Speyer [31], and are closely related to various
stratifications of the Grassmannian [35, 45, 4, 20]; see [31, §5.3] for a
discussion of this relationship. Intersecting a positroid variety with the
totally non-negative Grassmannian produces a cell in Postnikov’s decomposition
of this space, the study of which is one of the main motivations of the work
in [40].

In this context, a positroid is a subset $\mathcal{P}$ of the set
$\binom{[n]}{k}$ of $k$-element subsets of $\left\\{1,\dotsc,n\right\\}$ (see
[37, §1.4] for the defining properties). Each positroid is uniquely determined
by a _necklace_ , a cyclically ordered set of $n$ elements of $\binom{[n]}{k}$
(see [38, Defn. 4.2]). Strictly speaking, it is more compatible with our
conventions to use the reverse necklace (see [37, §2.1] and [11, Rem. 8.6]),
but we will abuse terminology by referring to the set
$\mathcal{I}\subseteq\mathcal{P}$ of elements of this reverse necklace as
simply ‘the necklace $\mathcal{I}$ of $\mathcal{P}$’. Recall that the
Grassmannian $\mathrm{Gr}_{k}^{n}$ of $k$-dimensional subspaces of
$\mathbb{C}^{n}$ has as a standard generating set of homogenous coordinates
the Plücker coordinates $\Delta_{I}$ for $I\in\binom{[n]}{k}$.

######  Definition 1.1.

Given a positroid $\mathcal{P}\subseteq\binom{[n]}{k}$, the (closed)
_positroid variety_ $\Pi(\mathcal{P})$ is the subvariety of the Grassmannian
$\mathrm{Gr}_{k}^{n}$ on which $\Delta_{I}=0$ for $I\notin\mathcal{P}$. The
_open positroid variety_ $\Pi^{\circ}(\mathcal{P})$ is the subvariety of
$\Pi(\mathcal{P})$ on which additionally $\Delta_{I}\neq 0$ for
$I\in\mathcal{I}$. We denote the cones on these varieties, which are defined
by the same conditions inside the affine cone
$\widehat{\mathrm{Gr}}{\vphantom{\mathrm{Gr}}}_{k}^{n}$ on the Grassmannian,
by $\widehat{\Pi}(\mathcal{P})$ and $\widehat{\Pi}^{\circ}(\mathcal{P})$
respectively.

A Postnikov diagram $D$ determines a positroid $\mathcal{P}_{D}$ with necklace
$\mathcal{I}_{D}$, and a maximal weakly separated (or maximal non-crossing)
collection [38, Defn. 4.3] $\mathcal{C}_{D}$ with
$\mathcal{I}_{D}\subseteq\mathcal{C}_{D}\subseteq\mathcal{P}_{D}$. In
particular, $D$ determines the numbers $k$ and $n$ such that
$\mathcal{P}_{D}$, $\mathcal{I}_{D}$ and $\mathcal{C}_{D}$ consist of
$k$-element subsets of $\left\\{1,\dotsc,n\right\\}$; see Definition 2.1. The
elements of $\mathcal{C}_{D}$ are attached to the alternating regions of $D$,
or equivalently to the vertices of the associated ice quiver $Q_{D}$; the
precise construction is given in Remark 2.7. While many different Postnikov
diagrams can determine the same positroid and necklace, it is rarer for two
diagrams to determine the same weakly separated collection—indeed, given a
positroid $\mathcal{P}$ with necklace $\mathcal{I}$, Oh–Postnikov–Speyer [38,
Thm. 1.5] show that the maximal weakly separated collections $\mathcal{C}$
with $\mathcal{I}\subseteq\mathcal{C}\subseteq\mathcal{P}$ are in bijection
with Postnikov diagrams $D$ such that $\mathcal{P}_{D}=\mathcal{P}$
(equivalently, $\mathcal{I}_{D}=\mathcal{I}$), up to isotopy and certain moves
(see Figure 4).

We may interpret the collection $\mathcal{C}_{D}$ as a set of functions on
$\widehat{\Pi}^{\circ}(\mathcal{P}_{D})$ by restricting the Plücker
coordinates $\Delta_{I}$ for $I\in\mathcal{C}_{D}$ to this subvariety of
$\widehat{\mathrm{Gr}}{\vphantom{\mathrm{Gr}}}_{k}^{n}$. Since these functions
are attached to the vertices of $Q_{D}$, we thus obtain a map
$\mathscr{A}_{D}\to\mathbb{C}(\widehat{\Pi}^{\circ}(\mathcal{P}_{D}))$, from
the abstract cluster algebra $\mathscr{A}_{D}$ associated to $D$ to the ring
of rational functions on $\widehat{\Pi}^{\circ}(\mathcal{P}_{D})$, sending
each initial cluster variable to the restricted Plücker coordinate
corresponding to the same quiver vertex. A long-standing conjecture,
formalised by Muller–Speyer [36, Conj. 3.4] and recently verified by Galashin
and Lam [17] (see also [47] for the special case of Schubert varieties), is
the following.

######  Theorem 1.2 ([17]).

The specialisation map
$\mathscr{A}_{D}\to\mathbb{C}(\widehat{\Pi}^{\circ}(\mathcal{P}_{D}))$ above
is injective and its image is precisely the coordinate ring
$\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]$.

Thus the coordinate ring $\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]$
has a cluster algebra structure, with initial seed $(Q_{D},\mathcal{C}_{D})$,
isomorphic to the cluster algebra we categorify in Theorem 1. While
$\Pi^{\circ}(\mathcal{P}_{D})$ is an affine variety, and so it is slightly
unusual to consider its affine cone, it is this cone on which restricted
Plücker coordinates are functions. By contrast, the coordinate ring
$\mathbb{C}[\Pi^{\circ}(\mathcal{P}_{D})]$ is the degree zero part of
$\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]$ in the
$\mathbb{Z}$-grading in which Plücker coordinates have degree $1$. Said
differently, the grading determines a $\mathbb{C}^{\times}$-action on
$\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]$ by $\lambda\cdot
f=\lambda^{d}f$ when $f$ has degree $d$, and
$\mathbb{C}[\Pi^{\circ}(\mathcal{P}_{D})]$ is the invariant subring
$\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]^{\mathbb{C}^{\times}}$ for
this action.

######  Remark 1.3.

For Theorem 1.2 it is necessary to adopt the convention that the frozen
variables of $\mathscr{A}_{D}$ are invertible, since the images of these
frozen variables in $\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]$ are
precisely the Plücker coordinates $\Delta_{I}$ for $I\in\mathcal{I}_{D}$,
which are non-zero on $\widehat{\Pi}^{\circ}(\mathcal{P}_{D})$ by definition.
This convention is not relevant to Theorem 1 however, and so we will freely
compare our results with those of authors who do not invert frozen variables,
such as [27].

As already observed, the positroid $\mathcal{P}_{D}$ and its corresponding
variety $\Pi^{\circ}(\mathcal{P}_{D})$ do not determine $D$ uniquely; choosing
a different Postnikov diagram with positroid $\mathcal{P}_{D}$ amounts to
choosing a different initial seed in the cluster algebra structure on
$\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]$. This is reflected in our
categorical picture by the fact (Corollary 4.7) that if
$\mathcal{P}_{D}=\mathcal{P}_{D^{\prime}}$ for two connected Postnikov
diagrams $D$ and $D^{\prime}$, then the two algebras $B_{D}$ and
$B_{D}^{\prime}$ are isomorphic, and hence the categories
$\operatorname{GP}(B_{D})$ and $\operatorname{GP}(B_{D^{\prime}})$ from
Theorem 1 are equivalent.

The best understood case of these constructions is that in which
$\mathcal{P}_{D}=\binom{[n]}{k}$ is the _uniform positroid_ , in which the
notion of a maximal weakly separated collection from [38] coincides with that
of Leclerc–Zelevinsky [33]. In this case we have
$\Pi(\mathcal{P}_{D})=\mathrm{Gr}_{k}^{n}$, the open subvariety
$\Pi^{\circ}(\mathcal{P}_{D})$ is determined by the non-vanishing of those
Plücker coordinates labelled by cyclic intervals, and Muller–Speyer’s
conjecture was verified much earlier by Scott [46]. (The case $k=2$ even
appears in Fomin–Zelevinsky’s first paper on cluster algebras [15].) Scott in
fact proves more, namely that a completely analogous specialisation map to
that of Theorem 1.2 is an isomorphism from the cluster algebra with non-
invertible frozen variables determined by $Q_{D}$ to the homogeneous
coordinate ring of $\mathrm{Gr}_{k}^{n}$. However, this depends on special
properties of the uniform positroid, and there is no corresponding result for
arbitrary Postnikov diagrams.

###  1.3. The Grassmannian cluster category

When $\mathcal{P}_{D}$ is the uniform positroid, a categorification of
$\mathscr{A}_{D}$ is provided by Jensen–King–Su [27] (see also Demonet–Luo
[13] for $k=2$). This _Grassmannian cluster category_ , denoted
$\operatorname{CM}(C)$, consists of Cohen–Macaulay modules over a certain
algebra $C$ (depending on $k$ and $n$), defined directly via a quiver with
relations. Some of the results in [27] rely on the categorification by
Geiß–Leclerc–Schröer [18] of a cluster algebra obtained from $\mathscr{A}_{D}$
by specialising a single frozen variable to $1$, or equivalently by removing a
frozen vertex and all adjacent arrows from the quiver.

Jensen–King–Su’s categorification is connected back to the combinatorics of
Postnikov diagrams by work of Baur–King–Marsh [2], which is our main
inspiration. They show [2, Cor. 10.4] that for any Postnikov diagram $D$ for
which $\mathcal{P}_{D}$ is the uniform positroid, there is an isomorphism
$B_{D}\cong C$ between the boundary algebra of $D$, with which we describe our
categorification, and the algebra $C$ describing the Grassmannian cluster
category. Since $C$ has the property that
$\operatorname{CM}(C)=\operatorname{GP}(C)$, our categorification coincides
with Jensen–King–Su’s in this case.

In our more general situation, because $B_{D}$ is defined as an idempotent
subalgebra of the dimer algebra $A_{D}$, we do not typically know a
presentation of $B_{D}$ via a quiver with relations. Our approach therefore
differs significantly from that of Jensen–King–Su, since we must deduce
properties of the category $\operatorname{GP}(B_{D})$ from the combinatorics
of the diagram $D$ and the representation theory of $A_{D}$, for which we do
have such a presentation. This is done primarily via the internal Calabi–Yau
property of Theorem 3.7 which, in the full strength established here, is also
a new result in the case that $\mathcal{P}_{D}$ is the uniform positroid.

For more general Postnikov diagrams $D$, a categorification analogous to that
of Geiß–Leclerc–Schröer’s in the uniform case is provided by Leclerc [32] in
work on Richardson varieties; again he categorifies a cluster algebra obtained
from $\mathscr{A}_{D}$ by specialising a frozen variable to $1$. Leclerc’s
category plays a crucial role in the construction of a cluster structure on
$\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]$ in [47, 17]. In this
paper we will always consider the full quiver $Q_{D}$ and its associated
cluster algebra—in particular, this means that in the case of the uniform
positroid Theorem 1 recovers Jensen–King–Su’s category [27], not
Geiß–Leclerc–Schröer’s [18].

Our category $\operatorname{GP}(B_{D})$ is closely connected to the coordinate
ring $\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]$ (and its cluster
algebra structure). To demonstrate this, we exhibit a cluster character on
$\operatorname{GP}(B_{D})$, in the sense of Fu–Keller [16], providing a
bijection between reachable rigid indecomposable objects of
$\operatorname{GP}(B_{D})$ and cluster variables of
$\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]$, and a bijection between
reachable cluster-tilting objects of $\operatorname{GP}(B_{D})$ and clusters
of $\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]$. We also recall
results from joint work with Çanakçı and King [11] demonstrating even stronger
combinatorial connections between the representation theory of $B_{D}$ and the
geometry of $\widehat{\Pi}^{\circ}(\mathcal{P}_{D})$. Precisely, there is a
fully faithful functor
$\rho\colon\operatorname{GP}(B_{D})\to\operatorname{CM}(C)$ from the
categorification described here to the Grassmannian cluster category (for the
appropriate Grassmannian). For each $I\in\binom{[n]}{k}$, there is a
combinatorially defined module $M_{I}$ corresponding to the Plücker coordinate
$\Delta_{I}$, and we show (Theorem 7.6) that if $\rho(M)=M_{I}$, then the
cluster character on $\operatorname{GP}(B_{D})$ takes $M$ to the restriction
of $\Delta_{I}$ to $\widehat{\Pi}^{\circ}(\mathcal{P}_{D})$.

######  Remark 1.4.

Leclerc’s categories [32] can be used to categorify a cluster algebra
structure on $\mathbb{C}[\Pi^{\circ}(\mathcal{P}_{D})]$, i.e. on the
coordinate ring of the open positroid variety itself rather than its affine
cone, but in a way that depends on choosing a frozen variable $\Delta_{I}$
with $I\in\mathcal{I}_{D}$. Indeed, the construction involves an isomorphism
of $\Pi^{\circ}(\mathcal{P}_{D})$ with the subvariety of
$\widehat{\Pi}^{\circ}(\mathcal{P}_{D})$ on which $\Delta_{I}=1$; this is the
graph of a global section of the line bundle
$\widehat{\Pi}^{\circ}(\mathcal{P}_{D})\to\Pi^{\circ}(\mathcal{P}_{D})$. In
ring-theoretic terms, this isomorphism corresponds to splitting the inclusion

$\mathbb{C}[\Pi^{\circ}(\mathcal{P}_{D})]=\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]^{\mathbb{C}^{\times}}\subseteq\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]$

using the map taking $f\in\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]$
of degree $d$ to the $\mathbb{C}^{\times}$-invariant function
$f\Delta_{I}^{-d}$, recalling that $\Delta_{I}$ is invertible on
$\Pi^{\circ}(\mathcal{P}_{D})$.

By avoiding this arbitrary choice, our categorification better reflects the
symmetries of $\Pi^{\circ}(\mathcal{P}_{D})$. For example, in the case of the
uniform positroid, both $\Pi^{\circ}(\mathcal{P}_{D})$ and our category (which
coincides with Jensen–King–Su’s [27] in this case) carry an action of the
order $n$ cyclic group coming from natural action of this group on the
cyclically ordered set $[n]$, and by extension on the set $\binom{[n]}{k}$
labelling Plücker coordinates. Since this action does not fix the frozen
variables, but rather cyclically permutes them, choosing a frozen vertex
breaks the symmetry, which indeed is no longer present in
Geiß–Leclerc–Schröer’s category [18].

In the case of the uniform positroid, the distinction between Jensen–King–Su’s
category and Geiß–Leclerc–Schröer’s is even more critical, since in this case
the homogeneous coordinate ring
$\mathbb{C}[\widehat{\Pi}(\mathcal{P}_{D})]=\mathbb{C}[\widehat{\mathrm{Gr}}{\vphantom{\mathrm{Gr}}}_{k}^{n}]$
of the closed positroid variety carries its own cluster algebra structure, in
which the $n$ frozen variables are non-invertible. For more general positroids
$\mathcal{P}$, the ring $\mathbb{C}[\widehat{\Pi}(\mathcal{P})]$ is not a
cluster algebra, but we still hope that the avoidance of symmetry-breaking in
our constructions will allow us to use the algebra $B_{D}$ constructed here,
and its category of Cohen–Macaulay modules, to study the closed varieties
$\Pi(\mathcal{P})$ and $\widehat{\Pi}(\mathcal{P})$—see [11, Rem. 8.6] for a
first step in this direction.

The structure of the paper is as follows. In Section 2 we recall the
definition of a Postnikov diagram and the construction of its dimer algebra.
The necessary Calabi–Yau property of this algebra is established in Section 3,
and used to construct the categorification $\operatorname{GP}(B_{D})$ in
Section 4. In Section 5 we show that $\operatorname{GP}(B_{D})$ has a mutation
class of cluster-tilting objects, again determined by $D$, whose quivers have
no loops or $2$-cycles. This allows us to show in Section 6 that there is a
cluster character on $\operatorname{GP}(B_{D})$ inducing bijections between
the cluster-tilting objects in this class and the clusters of
$\mathscr{A}_{D}$, and between their indecomposable summands and the cluster
(and frozen) variables of $\mathscr{A}_{D}$. We close in Section 7 by
explaining how our categorification is related to Jensen–King–Su’s
Grassmannian cluster category [27], and how its cluster character relates to
Galashin and Lam’s isomorphism of $\mathscr{A}_{D}$ with the coordinate ring
$\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]$.

##  2\. Postnikov diagrams and their dimer algebras

In this section, we introduce Postnikov diagrams in the disc, and their
associated dimer algebras, which will be defined over a field $\mathbb{K}$ of
characteristic zero.

######  Definition 2.1 ([40, §14], see also [46, §3] and [2, §2]).

Let $\Sigma$ be an oriented disc, with $n\geq 3$ marked points on its
boundary. A _Postnikov diagram_ $D$ in $\Sigma$ consists of $n$ oriented
strands in $\Sigma$, starting and ending at the boundary marked points,
subject to the following constraints.

  * (P0)

Each boundary marked point is the start point of exactly one strand, and the
end point of exactly one strand.

  * (P1)

The strands intersect in finitely many points, and each such crossing is
transverse and involves only two strands.

  * (P2)

Moving along a strand, the signs of its crossings with other strands
alternate.

  * (P3)

A strand may not intersect itself in the interior of the disc.

  * (P4)

If two strands intersect twice, the strands are oriented in opposite
directions between these intersection points. In other words, the
configuration

$\cdots$$\cdots$

is not permitted, but the configuration obtained from this by reversing the
orientation of one of the strands is legal.

For axioms (P2) and (P4) the intersection of two strands at the boundary, with
its natural sign, is also treated as a crossing.

The diagram $D$ divides $\Sigma$ into _regions_ , the connected components of
the complement of the strands, which have three types; the orientations of the
strands around the boundary of such a region can be inconsistent (in which
case we call the region _alternating_), oriented anticlockwise (i.e.
consistent with the orientation of $\Sigma$) or oriented clockwise. We will
use the terminology _anticlockwise_ and _clockwise_ for regions with these
boundary orientations. The alternating regions are further subdivided into
_boundary regions_ , those incident with the boundary of $\Sigma$, and
_internal regions_.

We say that $D$ is _connected_ if the union of its strands is connected. This
forces each boundary region to meet the boundary of $\Sigma$ in a single arc,
and so there are exactly $n$ such regions.

There is an integer $k\in\\{1,\dotsc,n\\}$ such that each alternating region
of $D$ is on the left of exactly $k$ strands. We say that the _type_ of $D$ is
$(k,n)$, recalling that $n$ is the number of strands. One can also compute $k$
as the average number of boundary regions to the left of a strand. If $D$ has
the property that every strand has exactly $k$ boundary regions on its left,
then each strand must terminate at a marked point $k$ steps clockwise from its
source, and in this case $D$ is sometimes [2] called a _$(k,n)$ -diagram_.
Diagrams of this kind are always connected.

######  Remark 2.2.

It is possible to consider Postnikov diagrams on $n<3$ strands, but they do
not lead to interesting cluster algebras. For example, their associated ice
quivers (see Definition 2.4 below) have only frozen vertices, and so describe
a cluster algebra with only frozen variables, i.e. a polynomial ring. Thus the
assumption that $n\geq 3$, which will be useful in Section 5, only excludes a
small number of highly degenerate examples. Both Theorems 1 and 2 do hold when
$n<3$, but they reduce to the claim that $B_{D}$ is an algebra of global
dimension at most $3$ with $n$ indecomposable projective modules up to
isomorphism, which follows immediately from observing that $B_{D}$ is the
complete path algebra of an $n$-cycle in this case (or from the results of
Section 3, which do not use the assumption $n\geq 3$).

A Postnikov diagram in the disc has an associated algebra, called its _dimer
algebra_ , which, when $D$ is connected, is an instance of a frozen Jacobian
algebra defined via an ice quiver with potential. We first give the necessary
definitions for this general construction, and then explain how to read off
the relevant data from a Postnikov diagram.

######  Definition 2.3 (cf. [43, §2]).

An _ice quiver_ is a pair $(Q,F)$, where $Q$ is a quiver, and $F$ is a (not
necessarily full) subquiver of $Q$. We call $F_{0}$, $F_{1}$ and $F$ the
_frozen_ vertices, arrows and subquiver respectively. Vertices of $Q$ not in
$F_{0}$ will be called _mutable_ , whereas arrows of $Q$ not in $F_{1}$ will
be simply called _unfrozen_. We assume for simplicity that $Q$ has no loops.

Let $\mathbb{K}\langle\langle Q\rangle\rangle$ denote the complete path
algebra of $Q$ (see [43, Defn. 2.6]). This algebra is graded by path length,
which makes its quotient $\mathbb{K}\langle\langle
Q\rangle\rangle_{\mathrm{cyc}}$ by the vector subspace spanned by commutators
into a graded vector space. A _potential_ on $Q$ is an element
$W\in\mathbb{K}\langle\langle Q\rangle\rangle_{\mathrm{cyc}}$ expressible as a
linear combination of homogeneous elements of degree at least $2$. We usually
think of $W$ as a linear combination of cyclic paths in $Q$ (of length at
least $2$), considered up to cyclic equivalence, i.e. the finest equivalence
relation on such paths such that

$a_{n}a_{n-1}\dotsc a_{1}\sim a_{n-1}\dotsc a_{1}a_{n}.$

We call the triple $(Q,F,W)$ an _ice quiver with potential_.

Let $p=a_{n}\dotsm a_{1}$ be a cyclic path, with each $a_{i}\in Q_{1}$, and
let $a\in Q_{1}$ be any arrow. Then the _cyclic derivative_ of $p$ with
respect to $a$ is

$\partial_{a}{p}:=\sum_{a_{i}=a}a_{i-1}\dotsm a_{1}a_{n}\dotsm a_{i+1}.$

Extending $\partial_{a}{}$ by linearity and continuity, we obtain a map
$\mathbb{K}\langle\langle
Q\rangle\rangle_{\mathrm{cyc}}\to\mathbb{K}\langle\langle Q\rangle\rangle$.
For an ice quiver with potential $(Q,F,W)$, we define the _frozen Jacobian
algebra_

$\mathcal{J}(Q,F,W)=\mathbb{K}\langle\langle
Q\rangle\rangle/\overline{\langle\partial_{a}W:a\in Q_{1}\setminus
F_{1}\rangle}.$

Here the closure is taken in the $J$-adic topology on
$\mathbb{K}\langle\langle Q\rangle\rangle$, where $J$ is the ideal generated
by arrows.

######  Definition 2.4.

Let $D$ be a Postnikov diagram in the disc. We associate to $D$ an ice quiver
$(Q_{D},F_{D})$ as follows. The vertices of $Q_{D}$ are the alternating
regions of $D$, as in Definition 2.1. When the closures of two different
alternating regions $v_{1}$ and $v_{2}$ meet in a crossing point between
strands of $D$, or at one of the boundary marked points, we draw an arrow
between $v_{1}$ and $v_{2}$, oriented in a way consistent with these strands,
as shown in Figure 1.

$v_{1}$$v_{2}$

$v_{1}$$v_{2}$

Figure 1. Arrows associated to strand crossings.

The vertices of $F_{D}$ are the boundary regions, and the arrows of $F_{D}$
are those corresponding to boundary marked points, as on the right of Figure
1.

Each clockwise or anticlockwise region $f$ of $D$ determines a _fundamental
cycle_ $C_{f}$ in $Q$, by following the arrows corresponding to the crossing
points in the boundary of the region, as shown in Figure 2 for a quadrilateral
anticlockwise region.

$f$$v_{1}$$v_{2}$$v_{3}$$v_{4}$ Figure 2. A fundamental cycle around an
oriented region.

Thus we may define a potential

$W_{D}=\sum_{\text{$f$ anticlockwise}}C_{f}-\sum_{\text{$f$ clockwise}}C_{f}$

on $Q_{D}$, and we call $A_{D}:=\mathcal{J}(Q_{D},F_{D},W_{D})$ the _dimer
algebra_ of $D$.

######  Remark 2.5.

While it is straightforward to describe the defining relations of $A_{D}$
directly without using a potential (see e.g. [11, Def. 2.9]), results on
frozen Jacobian algebras will play a crucial role for us in Section 3, making
it important that $A_{D}$ has this structure.

######  Proposition 2.6 (cf. [2, Rem. 3.3]).

The quiver $Q_{D}$ is strongly connected, i.e. any two vertices are connected
by a directed path.

###### Proof.

Since the regions of $D$ cover the disc, the underlying graph of $Q_{D}$ is
connected. Moreover, every arrow $a$ of $Q_{D}$ is contained in at least one
fundamental cycle—exactly one if $a$ is frozen, and exactly two otherwise.
Thus every arrow has a path from its head to its tail, and so $Q_{D}$ is even
strongly connected. ∎

######  Remark 2.7.

Having now defined Postnikov diagrams and their associated quivers and
algebras more precisely, we may provide a few more details on the relationship
to positroid varieties discussed in §1.2.

Label the boundary marked points of $\Sigma$ by $\\{1,\dotsc,n\\}$, in the
clockwise direction. These labels can be transferred first to the strands, by
giving a strand the label of its source, and then to the alternating regions,
by applying the label of a strand to every alternating region on its left. By
the definition of $k$, this process labels each alternating region $v$ of $D$
by a _$k$ -label_ $I_{v}\in\binom{[n]}{k}$, and the collection
$\mathcal{C}_{D}$ of these labels is a maximal weakly separated collection in
the positroid $\mathcal{P}_{D}$. We note here that
$\mathcal{P}_{D}=\binom{[n]}{k}$ (i.e. $\mathcal{P}_{D}$ is the uniform
positroid) if and only if $D$ is a $(k,n)$-diagram.

The cluster algebra $\mathscr{A}_{D}$ has initial cluster variables $x_{v}$
indexed by the vertices of $Q_{D}$. By construction, these vertices are the
alternating regions of $Q_{D}$, and so each $v$ has an attached $k$-label
$I_{v}\in\binom{[n]}{k}$ and Plücker coordinate $\Delta_{I_{v}}$. The
assignment $x_{v}\mapsto\Delta_{I_{v}}$ induces Galashin and Lam’s isomorphism
$\mathscr{A}_{D}\to\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]$ [17].
(Strictly speaking, Galashin and Lam use the complementary set of
$(n-k)$-labels obtained by labelling regions to the right of each strand, so
they put a cluster structure on the coordinate ring of an isomorphic positroid
variety in $\mathrm{Gr}_{n-k}^{n}$. Our isomorphism is obtained from theirs
using the natural isomorphism
$\mathbb{C}[\mathrm{Gr}_{n-k}^{n}]\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{C}[\mathrm{Gr}_{k}^{n}]$
with $\Delta_{I}\mapsto\Delta_{I^{c}}$.)

We note that we could have obtained a different set of $k$-labels, and hence a
different specialisation map
$\mathscr{A}_{D}\to\mathbb{C}(\widehat{\Pi}^{\circ}(\mathcal{P}_{D}))$, by
instead giving the strands of $D$ the labels of their targets. This can
produce a different cluster algebra structure (in the sense that different
functions are cluster variables) on
$\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]$, isomorphic to the same
abstract cluster algebra $\mathscr{A}_{D}$. While this paper is mostly
concerned with categorifying $\mathscr{A}_{D}$, and so is insensitive to such
labelling conventions, they will reappear in Section 7, when we use results of
the author with Çanakçı and King [11] to relate our categories more directly
with the coordinate rings
$\mathbb{C}[\widehat{\Pi}^{\circ}(\mathcal{P}_{D})]$. In particular, it will
turn out to be the source labelling convention that is most compatible with
our categorification and its relationship to Jensen–King–Su’s Grassmannian
cluster category $\operatorname{CM}(C)$ [27].

######  Remark 2.8.

Given a Postnikov diagram $D$ in the disc $\Sigma$, we can construct from $D$
a bipartite graph $G$ as follows. The vertices of $G$ are given by the
anticlockwise and clockwise regions of $D$, and two such vertices are
connected by an edge if the the closures of the regions meet at a crossing of
strands in the interior of $\Sigma$. The graph $G$ is connected if and only if
$D$ is connected. Adding to $G$ the data of half-edges, which connect the
vertex corresponding to a region to any boundary marked points in its closure,
we obtain a dimer model in $\Sigma$ in the sense of [43, Ex. 2.5]. This
construction appears in [40, §14], where the dimer model is called a plabic
graph. The dimer algebra $A_{D}$ is precisely the dimer algebra of
$(\Sigma,G)$, as defined in [43, Ex. 2.12]. An example of a Postnikov diagram,
its corresponding dimer model, and the ice quiver of its dimer algebra, is
shown in Figure 3.

$\diamond$$\diamond$$\diamond$$\diamond$$\diamond$$\bullet$$\bullet$

Figure 3. A Postnikov diagram (left), its corresponding dimer model (centre)
and the ice quiver of its dimer algebra (right). The frozen vertices of this
ice quiver are shown as white diamonds, and the frozen arrows are drawn in
bold.

Given the data of a dimer model $G$ in the disc, or indeed in any oriented
surface $\Sigma$ with or without boundary, it is always possible to produce a
collection of strands, dividing $\Sigma$ into alternating, clockwise and
anticlockwise regions, from which $G$ may be recovered by the above rules.
These strands are usually called the _zig-zag paths_ or flows of $G$ [3,
§3.3]. Note that in general these strands can either travel between points on
the boundary of $\Sigma$, as in Definition 2.1, or be closed curves in the
interior. In the case that $\Sigma$ is a disc, these strands will always
satisfy conditions (P0)–(P2) from Definition 2.1. Requiring that they also
satisfy (P3) and (P4)—which in particular rules out any closed curves in the
interior—places an extra condition on the dimer model $G$, the analogue of
Broomhead’s geometric consistency [3, Prop. 3.12]. Since these conditions,
particularly (P4), will be important for us, we prefer to start from the data
of a Postnikov diagram.

Postnikov diagrams are typically considered up to isotopy fixing the boundary,
which does not affect the construction of the dimer algebra at all, and
certain twisting and untwisting moves shown in Figure 4. These moves do affect
the construction of the dimer algebra, in the sense that two Postnikov
diagrams related by such a move determine different ice quivers with
potential, as shown in Figure 5 (cf. [43, Ex. 4.5]). However, the frozen
Jacobian algebras of these two quivers with potential, i.e. the dimer algebras
of the two Postnikov diagrams, are isomorphic in a natural way. These moves
also do not affect the $k$-labels attached to the quiver vertices.

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Figure 4. Twisting (right-to-left) and untwisting (left-to-right) moves for a
Postnikov diagram, in the interior (above) and at the boundary (below). The
reflections of these figures in a horizontal line also show twisting and
untwisting moves.

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Figure 5. The effect on the quiver of applying a twisting or untwisting move
in the interior (left) or at the boundary (right). Bold arrows are frozen.

######  Proposition 2.9 (cf. [2, Lem. 12.1]).

Let $D$ and $D^{\prime}$ be Postnikov diagrams such that $D^{\prime}$ is
obtained from $D$ via an untwisting move. Then the dimer algebras $A_{D}$ and
$A_{D^{\prime}}$ are isomorphic.

###### Proof.

First we consider the case that the untwisting move happens in the interior,
as in the upper part of Figure 4. Then $Q_{D^{\prime}}$ is obtained from
$Q_{D}$ by removing two arrows $a$ and $b$ which form a fundamental cycle, as
in the left-hand side of Figure 5. Since these arrows are unfrozen, they are
each contained in a second fundamental cycle, and we write these cycles as
$ap$ and $bq$ for some paths $p$ and $q$ respectively. Since these paths do
not contain the arrows $a$ or $b$, they are also paths in $Q_{D^{\prime}}$.
Moreover, the relation $\partial_{a}{W_{D}}=0$ implies that $b=p$ in $A_{D}$,
and the relation $\partial_{b}{W_{D}}=0$ that $a=q$ in $A_{D}$. Define a map
$\varphi\colon\mathbb{K}\langle\langle
Q_{D}\rangle\rangle\to\mathbb{K}\langle\langle Q_{D^{\prime}}\rangle\rangle$
fixing all vertex idempotents (the two quivers having the same vertex set) and
all arrows different from $a$ and $b$, and with $\varphi(a)=q$ and
$\varphi(b)=p$. This map induces a homomorphism $A_{D}\to A_{D^{\prime}}$
inverse to that induced by the map $\mathbb{K}\langle\langle
Q_{D^{\prime}}\rangle\rangle\to\mathbb{K}\langle\langle Q_{D}\rangle\rangle$
arising from the inclusion map between arrow sets.

On the other hand, if the untwisting move takes place at the boundary, then
$Q_{D^{\prime}}$ is obtained from $Q_{D}$ by removing a frozen arrow $a$ lying
in a fundamental $2$-cycle, and $F_{D^{\prime}}$ is obtained from $F_{D}$ by
replacing $a$ by the other arrow $b$ in this $2$-cycle. As before, since $b$
is unfrozen it lies in a second fundamental cycle $bq$ for some path $q$ not
involving $a$, and it follows from the relation $\partial_{b}{W_{D}}=0$ that
$a=q$ in $A_{D}$. Thus defining $\varphi$ as in the first case except for
$\varphi(b)=b$ provides an isomorphism of dimer algebras in an analogous way.
∎

We call a Postnikov diagram _reduced_ if no untwisting moves can be applied to
it; any Postnikov diagram $D$ is equivalent under untwisting moves to a
reduced diagram $D^{\prime}$, unique up to isotopy fixing the boundary. The
reader is warned that this does not correspond to Postnikov’s definition of
reducedness for plabic graphs [40, §12], but rather is compatible with
terminology for quivers with potential: the ice quiver with potential
$(Q_{D^{\prime}},F_{D^{\prime}},W_{D^{\prime}})$ is the reduction of
$(Q_{D},F_{D},W_{D})$, in the sense of [43, §3]. Indeed, the above
isomorphisms of dimer algebras in the proof of Proposition 2.9 are local
versions of the reduction isomorphism given at the end of the proof of [43,
Thm. 3.6].

######  Proposition 2.10.

If $D$ is reduced, the quiver $Q_{D}$ has no $2$-cycles.

###### Proof.

A $2$-cycle in $Q_{D}$ containing at least one unfrozen arrow corresponds to a
possible untwisting move (see Figures 4 and 5), and so do not occur when $D$
is reduced. A $2$-cycle consisting only of frozen arrows would correspond to a
pair of boundary regions meeting at two points on the boundary of the disc,
which must then be the only boundary regions. But $D$ has at least $3$
boundary regions, so this is also impossible. ∎

Before moving on to the homological part of the paper, we discuss one
additional feature of the dimer algebra $A_{D}$ of a connected Postnikov
diagram $D$. For each vertex $v$ of $Q_{D}$, choose a path $t_{v}\colon v\to
v$ representing a fundamental cycle. Writing $t=\sum_{v\in Q_{0}}t_{v}\in
A_{D}$, it follows from connectedness of $D$ that $t$ does not depend on the
choices of paths $t_{v}$. Moreover, $t$ a central element of the algebra. Thus
$A_{D}$ has the structure of an algebra over $Z=\mathbb{K}[[t]]$; the abuse of
notation in identifying the abstract generator of $Z$ with the element $t\in
A_{D}$, thus making the $Z$-action clear, is justified by the following
result, which is also fundamental to the arguments in Section 3. The proof is
close to that of Baur–King–Marsh [2] for the case that $D$ is a
$(k,n)$-diagram, and we refer to their paper when the arguments apply without
change to our more general setting.

######  Proposition 2.11.

The dimer algebra $A_{D}$ of a connected Postnikov diagram $D$ is thin,
meaning that for any vertices $v$ and $w$ there is some $p\in e_{w}A_{D}e_{v}$
such that

$e_{w}A_{D}e_{v}=Zp,$

and moreover that $Zp$ is a free $Z$-module of rank $1$. Moreover, there is a
unique such $p$ expressible as the image of a path from $v$ to $w$ under the
projection $\mathbb{K}\langle\langle Q_{D}\rangle\rangle\to A_{D}$.

###### Proof.

As in [2, §4], we may weight the arrows of $Q=Q_{D}$ by elements of the set
$\left\\{1,\dotsc,n\right\\}$ labelling the boundary marked points of
$\Sigma$. An arrow $\alpha$ of $Q$ is crossed by two strands of $D$, say that
starting at marked point $i$ from right to left, and that starting at $j$ from
left to right. We then weight $\alpha$ by the (indicator function of) the
cyclic interval $[i,j-1]$ (cf. [2, Defn. 4.1]). A path in $Q$ is weighted by
the sum $\mathrm{wt}_{p}$ of weights of its arrows, and its total weight is
defined to be $\sum_{i=1}^{n}\mathrm{wt}_{p}(i)$, which is always at least $1$
if the path has non-zero length.

The proof of [2, Cor. 4.4], stated for $(k,n)$-Postnikov diagrams, remains
valid in our more general setting to show that every fundamental cycle has
constant weight $w(i)=1$ for all $1\leq i\leq n$. If $p_{+}=p_{-}$ is one of
the defining relations of $A_{D}$, then there is an arrow $\alpha\in Q_{1}$
such that both $\alpha p_{+}$ and $\alpha p_{-}$ are fundamental cycles, from
which it follows that the weights of $p_{+}$ and $p_{-}$ agree. It follows
that the weight, and hence the total weight, descends to a grading of $A$.

Now let $v,w\in Q_{0}$. Since $Q$ is strongly connected as in Proposition 2.6,
there is some path from $v$ to $w$ in $Q$, and we choose $p$ to be such a path
with minimal total weight. If $q$ is any other path from $v$ to $w$, then [2,
Prop. 9.3] applies to show that there is a path $r\colon v\to w$ and non-
negative integers $N_{p}$ and $N_{q}$ such that

$p=t^{N_{p}}r,\qquad q=t^{N_{q}}r$

in $A_{D}$. As before, this proposition is stated only in the case that $D$ is
a $(k,n)$-Postnikov diagram, but its proof is still valid under our weaker
assumptions—the key property of $D$ here is (P4).

Since the total weight of $t$ is non-zero, and $p$ has minimal total weight
among paths from $v$ to $w$, we must have $N_{p}=0$ and $p=r$ in $A_{D}$. Thus
$q=t^{N_{q}}p$, and we see that $e_{w}Ae_{v}$ is generated over $Z$ by $p$. It
is moreover freely generated since each element of $\\{t^{N}p:N\geq 0\\}$ has
a different total weight, which implies that these elements are linearly
independent in $A_{D}$. ∎

######  Definition 2.12.

We call a path $p\colon v\to w$ in $Q_{D}$ a minimal path if
$e_{w}A_{D}e_{v}=Zp$. Proposition 2.11 then states that there is a minimal
path between any pair of vertices in $Q_{D}$, and that any two such paths
define the same element of $A_{D}$. We write $p_{wv}$ for the class in $A_{D}$
of any minimal path from $v$ to $w$.

Thinness of $A_{D}$, as in Proposition 2.11, is the analogue of algebraic
consistency for dimer models on the torus, as described by Broomhead [3, Def.
5.12] (the analogue of geometric consistency, cf. [3, Prop. 3.12], being
conditions (P3) and (P4) in the definition of a Postnikov diagram). The
following observation about minimal paths is straightforward but useful.

######  Lemma 2.13.

Let $D$ be a connected Postnikov diagram, and let $p\colon u\to v$ and
$q\colon v\to w$ be paths in $Q_{D}$. If the composition $qp$ is a minimal
path, then both $p$ and $q$ are minimal paths.

###### Proof.

By Proposition 2.11, there are $m,n\geq 0$ such that $p=t^{m}p_{vu}$ and
$q=t^{n}p_{wv}$ in $A_{D}$. Thus $qp=t^{m+n}p_{wv}p_{vu}=t^{m+n+\delta}p_{wu}$
for some $\delta\geq 0$. But $qp$ is minimal, so by Proposition 2.11 again we
must have $m+n+\delta=0$, and so in particular $m=n=0$. Hence both $p$ and $q$
are minimal paths. ∎

##  3\. The Calabi–Yau property

Let $A=A_{D}$ be the dimer algebra of a connected Postnikov diagram $D$. In
this section we show that this algebra is bimodule internally $3$-Calabi–Yau,
in the sense of [42], with respect to the idempotent corresponding to the
boundary regions of $D$, thus proving Theorem 2. Since $A$ is presented as a
frozen Jacobian algebra, we will prove this statement via [42, Thm. 5.6],
which tells us that it is enough to check the exactness of a certain complex
of $A$-bimodules.

Throughout the proof, we will write

$\mathrm{H}_{v}=\left\\{a\in
Q_{1}:ha=v\right\\},\qquad\mathrm{T}_{v}=\left\\{a\in Q_{1}:ta=v\right\\}$

for the set of arrows with head, respectively tail, at $v\in Q_{0}$, and

$\mathrm{H}_{v}^{\mathrm{m}}=\mathrm{H}_{v}\cap
Q_{1}^{\mathrm{m}},\qquad\mathrm{T}_{v}^{\mathrm{m}}=\mathrm{T}_{v}\cap
Q_{1}^{\mathrm{m}}.$

As well as using the fact that $A$ is thin, as in Proposition 2.11, we will
also rely on the following gradability property of $A$, which allows us to
deduce exactness of the relevant complex of bimodules from the exactness of
certain complexes of (one-sided) $A$-modules.

######  Proposition 3.1.

The dimer algebra $A_{D}$ of a connected Postnikov diagram $D$ admits a
grading in which all arrows have positive degrees.

###### Proof.

The algebra $A_{D}$ may be graded by total weights of paths, as discussed in
the proof of Proposition 2.11. As already mentioned in that proof, each arrow
has positive total weight. ∎

Most of the arguments in this section will depend on the fact that $A$ is
thin, as in Proposition 2.11, so that each piece $e_{w}Ae_{v}$ is freely
generated over $Z=\mathbb{K}[[t]]$ by a minimal path. Before introducing the
complexes of $A$-modules whose exactness implies the required $3$-Calabi–Yau
property of $A$, we give one more lemma, concerning these paths.

######  Lemma 3.2.

Let $v,w\in Q_{0}$ with $v$ mutable, and consider a minimal path $p_{wv}$.
Then there is some $a\in\mathrm{H}_{v}$ such that $p_{wv}a=p_{w,ta}$ in $A$.

###### Proof.

We first observe that if $p\colon u\to w$ is a minimal path in $Q$ passing
through the vertex $v$, then $p$ includes our desired arrow as follows. Write
$p=p_{2}ap_{1}$ for some $a\in\mathrm{H}_{v}$. Then by Lemma 2.13 each subpath
of $p$ is also minimal, so in $A$ we have $p_{2}=p_{wv}$ and
$p_{2}a=p_{w,ta}$, and hence $a$ is our desired arrow.

Now for each $a\in\mathrm{H}_{v}$, choose a minimal path $m_{a}\colon ta\to
w$. If any of these paths passes through $v$, then we find our desired arrow
as above, so we assume the contrary. Since $v$ is mutable, each
$b\in\mathrm{T}_{v}$ is involved in two fundamental cycles $a_{+}pb$ and
$a_{-}qb$ with $a_{\pm}\in\mathrm{H}_{v}$, and we can write
$p^{\prime}=m_{a_{+}}p$ and $q^{\prime}=m_{a_{-}}q$. By our assumption on the
$m_{a}$, neither $p^{\prime}$ nor $q^{\prime}$ passes through $v$.

Using again that $v$ is mutable, the union of fundamental cycles containing
$v$ is a disc with $v$ in the interior, and so there must be some $b$ for
which the paths $p^{\prime}$ and $q^{\prime}$ are two sides of a digon
containing $v$ (cf. the argument in [3, Lem. 6.17]). From now on, we assume
that we are in this situation, shown in Figure 6.

${\bullet}$${\bullet}$${\bullet}$${v}$${w}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m_{a_{+}}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{+}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q}$$\scriptstyle{a_{-}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m_{a_{-}}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b}$
Figure 6. Fundamental cycles $a_{+}pb$ and $a_{-}qb$ together with minimal
paths $m_{a_{\pm}}\colon ta_{\pm}\to w$ such that $p^{\prime}=m_{a_{+}}p$ and
$q^{\prime}=m_{a_{-}}q$ form a digon containing $v$. Solid arrows represent
arrows in $Q$, whereas dashed arrows represent paths.

Observe that $p_{wv}a_{+}=t^{\delta}p_{w,ta_{+}}$ for some $\delta\geq 0$ by
Proposition 2.11. If $\delta=0$, then $a_{+}$ was our desired arrow, so assume
$\delta>0$. In a similar way, we find $m,n\geq 0$ such that
$p_{w,ta_{+}}p=t^{m}p_{w,hb}$ and $p_{w,hb}b=t^{n}p_{wv}$. Since $pba_{+}$ is
a fundamental cycle, we must have

$tp_{w,ta_{+}}=p_{w,ta_{+}}pba_{+}=t^{m}p_{w,hb}ba_{+}=t^{m+n}p_{wv}a_{+}=t^{m+n+\delta}p_{w,ta_{+}}.$

By Proposition 2.11 again, we must have $m+n+\delta=1$, and so we conclude
from positivity of $\delta$ that $m=n=0$ (and $\delta=1$). In particular, this
means that $p^{\prime}=m_{a_{+}}p$ is a minimal path, representing $p_{w,hb}$.
Repeating the argument with $a_{-}$ and $q$ replacing $a_{+}$ and $p$ we see
that either $a_{-}$ is our desired arrow or $q^{\prime}=m_{a_{-}}q$ is also a
minimal path representing $p_{w,hb}$.

In the latter case, $p^{\prime}$ and $q^{\prime}$ are paths from $hb$ to $w$
defining the same element of $A$, namely $p_{w,hb}$, and bounding a digon
containing $v$. It is then a consequence of [3, Lem. 6.17] that there is a
path $r\colon hb\to w$ passing through $v$, and also defining $p_{w,hb}$ in
$A$. In particular, $r$ is a minimal path passing through $v$, and so we may
find our desired arrow as in the first paragraph of the proof. ∎

We now turn to the main part of the argument. In [42, §5], it is explained how
an ice quiver with potential $(Q,F,W)$ determines a complex of projective
bimodules for the associated frozen Jacobian algebra $A=\mathcal{J}(Q,F,W)$;
we will denote this complex by $\mathbf{P}(A)$, although strictly it depends
on the presentation of $A$ determined by $(Q,F,W)$. For the convenience of the
reader, we repeat its definition here, following [41, §4].

We denote by $Q_{0}^{\mathrm{m}}=Q_{0}\setminus F_{0}$ and
$Q_{1}^{\mathrm{m}}=Q_{1}\setminus F_{1}$ the sets of mutable vertices and
unfrozen arrows of $Q$ respectively. Let $S=A/\overline{J}$, where $J$ is the
the ideal generated by arrows. As a left $A$-module, $S$ is the direct sum of
the vertex simple modules, and has a basis consisting of the vertex
idempotents $e_{v}$. For the remainder of this section, we write
$\mathbin{\otimes}=\mathbin{\otimes}_{S}$.

Introduce formal symbols $\rho_{a}$ for each $a\in Q_{1}$ and $\omega_{v}$ for
each $v\in Q_{0}$, and define $S$-bimodule structures on the vector spaces

$\displaystyle\mathbb{K}Q_{0}$ $\displaystyle=\bigoplus_{v\in
Q_{0}}\mathbb{K}e_{v},$ $\displaystyle\mathbb{K}Q_{1}$
$\displaystyle=\bigoplus_{a\in Q_{1}}\mathbb{K}a,$
$\displaystyle\mathbb{K}Q_{2}^{\mathrm{m}}$ $\displaystyle=\bigoplus_{a\in
Q_{1}^{\mathrm{m}}}\mathbb{K}\rho_{a},$
$\displaystyle\mathbb{K}Q_{3}^{\mathrm{m}}$ $\displaystyle=\bigoplus_{v\in
Q_{0}^{\mathrm{m}}}\mathbb{K}\omega_{v},$

via the formulae

$e_{v}\cdot e_{v}\cdot e_{v}=e_{v},\qquad e_{ha}\cdot a\cdot e_{ta}=a,\qquad
e_{ta}\cdot\rho_{a}\cdot e_{ha}=\rho_{a},\qquad e_{v}\cdot\omega_{v}\cdot
e_{v}=\omega_{v}.$

Since $\mathbb{K}Q_{0}$ is naturally isomorphic to $S$ as a bimodule, there is
a natural isomorphism

$A\mathbin{\otimes}\mathbb{K}Q_{0}\mathbin{\otimes}A\stackrel{{\scriptstyle\sim}}{{\to}}A\mathbin{\otimes}A,$

which we can compose with the multiplication map for $A$ to obtain a map
$\mu_{0}\colon A\mathbin{\otimes}\mathbb{K}Q_{0}\mathbin{\otimes}A\to A$.
Define $\mu_{1}\colon A\mathbin{\otimes}\mathbb{K}Q_{1}\mathbin{\otimes}A\to
A\mathbin{\otimes}\mathbb{K}Q_{0}\mathbin{\otimes}A$ by

$\mu_{1}(x\mathbin{\otimes}a\mathbin{\otimes}y)=x\mathbin{\otimes}e_{ha}\mathbin{\otimes}ay-
xa\mathbin{\otimes}e_{ta}\mathbin{\otimes}y.$

For any path $p=a_{m}\dotsm a_{1}$ in $Q$, we may define

$\Delta_{a}(p)=\sum_{a_{i}=a}a_{m}\dotsm
a_{i+1}\mathbin{\otimes}a_{i}\mathbin{\otimes}a_{i-1}\dotsm a_{1},$

and extend by linearity and continuity to obtain a map
$\Delta_{a}\colon\mathbb{K}\langle\langle Q\rangle\rangle\to
A\mathbin{\otimes}\mathbb{K}Q_{1}\mathbin{\otimes}A$. We then define
$\mu_{2}\colon
A\mathbin{\otimes}\mathbb{K}Q_{2}^{\mathrm{m}}\mathbin{\otimes}A\to
A\mathbin{\otimes}\mathbb{K}Q_{1}\mathbin{\otimes}A$ by

$\mu_{2}(x\mathbin{\otimes}\rho_{a}\mathbin{\otimes}y)=\sum_{b\in
Q_{1}}x\Delta_{b}(\partial_{a}{W})y.$

Finally, define $\mu_{3}\colon
A\mathbin{\otimes}\mathbb{K}Q_{3}^{\mathrm{m}}\mathbin{\otimes}A\to
A\mathbin{\otimes}\mathbb{K}Q_{2}^{\mathrm{m}}\mathbin{\otimes}A$ by

$\mu_{3}(x\mathbin{\otimes}\omega_{v}\mathbin{\otimes}y)=\sum_{a\in\mathrm{T}_{v}}x\mathbin{\otimes}\rho_{a}\mathbin{\otimes}ay-\sum_{b\in\mathrm{H}_{v}}xb\mathbin{\otimes}\rho_{b}\mathbin{\otimes}y.$

As $\mathrm{T}_{v}\cup\mathrm{H}_{v}\subseteq Q_{1}^{\mathrm{m}}$ for any
$v\in Q_{0}^{\mathrm{m}}$, this map has the claimed codomain.

######  Definition 3.3.

For an ice quiver with potential $(Q,F,W)$, with frozen Jacobian algebra
$A=\mathcal{J}(Q,F,W)$, let $\mathbf{P}(A)$ be the complex of $A$-bimodules
with non-zero terms

${A\mathbin{\otimes}\mathbb{K}Q_{3}^{\mathrm{m}}\mathbin{\otimes}A}$${A\mathbin{\otimes}\mathbb{K}Q_{2}^{\mathrm{m}}\mathbin{\otimes}A}$${A\mathbin{\otimes}\mathbb{K}Q_{1}\mathbin{\otimes}A}$${A\mathbin{\otimes}\mathbb{K}Q_{0}\mathbin{\otimes}A}$$\scriptstyle{\mu_{3}}$$\scriptstyle{\mu_{2}}$$\scriptstyle{\mu_{1}}$

and $A\mathbin{\otimes}\mathbb{K}Q_{0}\mathbin{\otimes}A$ in degree $0$.

Using this complex, we can give a sufficient condition for the frozen Jacobian
algebra to be bimodule internally $3$-Calabi–Yau [42, Defn. 2.4] with respect
to the idempotent determined by the vertices of $F$. This condition will
suffice as a definition for the purposes of the present paper.

######  Theorem 3.4 ([42, Thm. 5.7]).

If $A$ is a frozen Jacobian algebra such that

${0}$${\mathbf{P}(A)}$${A}$${0}$$\scriptstyle{\mu_{0}}$ (3.1)

is exact, then $A$ is bimodule internally $3$-Calabi–Yau with respect to the
idempotent $e=\sum_{v\in F_{0}}e_{v}$.

######  Remark 3.5.

By standard results on presentations of algebras, as in Butler–King [9], we
need only check exactness at the two left-most non-zero terms of
$\mathbf{P}(A)$ (in degrees $-3$ and $-2$), since the rest of the complex
(3.1) is the standard projective bimodule presentation of an algebra defined
by a quiver with relations.

######  Proposition 3.6.

For $A=A_{D}$ the dimer algebra of a connected Postnikov diagram, the complex
(3.1) is exact in degree $d$ if and only if the complex

${0}$${\mathbf{P}(A)\mathbin{\otimes}_{A}S}$${S}$${0}$$\scriptstyle{\mu_{0}\mathbin{\otimes}S}$
(3.2)

is exact in degree $d$, if and only if the complex

${0}$${\mathbf{P}(A)\mathbin{\otimes}_{A}S_{v}}$${S_{v}}$${0}$$\scriptstyle{\mu_{0}\mathbin{\otimes}S_{v}}$
(3.3)

is exact in degree $d$ for any $v\in Q_{0}$, where $S_{v}$ denotes the simple
left $A$-module at $v$.

###### Proof.

Exactness of (3.1) and of (3.2) are equivalent because of the existence of a
grading as in Proposition 3.1, by an argument that is essentially due to
Broomhead [3, Prop. 7.5]. The reader can find an explanation of how this
argument extends to the case of frozen Jacobian algebras in [41, §4]. The
second equivalence is simply the observation that the complex (3.2) is the
direct sum of the complexes (3.3). ∎

It will be useful to introduce some notation for elements of terms of
$\mathbf{P}(A)\mathbin{\otimes}_{A}S_{v}$, following [41, §4]. First note that
we have isomorphisms

$\displaystyle\begin{split}A\mathbin{\otimes}\mathbb{K}Q_{1}\mathbin{\otimes}A\mathbin{\otimes}_{A}S_{v}&\cong\bigoplus_{b\in\mathrm{T}_{v}}Ae_{hb},\\\
A\mathbin{\otimes}\mathbb{K}Q_{2}^{\mathrm{m}}\mathbin{\otimes}A\mathbin{\otimes}_{A}S_{v}&\cong\bigoplus_{a\in\mathrm{H}_{v}^{\mathrm{m}}}Ae_{ta},\\\
A\mathbin{\otimes}\mathbb{K}Q_{3}^{\mathrm{m}}\mathbin{\otimes}A\mathbin{\otimes}_{A}S_{v}&\cong{\begin{cases}Ae_{v},&v\in
Q_{0}^{\mathrm{m}},\\\ 0,&v\in F_{0}.\end{cases}}\end{split}$ (3.4)

In the first two cases, the right-hand sides are of the form
$\bigoplus_{\alpha\in S}Ae_{\delta a}$, where $S$ is a set of arrows, and
$\delta\colon S\to Q_{0}$. The map $Ae_{\delta a}\to\bigoplus_{a\in
S}Ae_{\delta a}$ 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 $\delta$ is not injective. As a consequence, a general element
of the direct sum is

$x=\sum_{a\in S}x_{a}\otimes[a]$ (3.5)

with $x_{a}\in Ae_{\delta a}$.

The most complicated map in the complex $\mathbf{P}(A)$ is $\mu_{2}$, so we
will spell out $\mu_{2}\mathbin{\otimes}_{A}S_{v}$ explicitly. Using the
isomorphisms from (3.4), and our notation for elements of the direct sums, we
have

$(\mu_{2}\mathbin{\otimes}_{A}S_{v})(x)=\sum_{b\in\mathrm{T}_{v}}\bigg{(}\sum_{a\in\mathrm{H}_{v}^{\mathrm{m}}}x_{a}\partial^{r}_{b}{\partial_{a}{W}}\bigg{)}\mathbin{\otimes}[b],$

where $\partial^{r}_{b}$, called the _right derivative_ with respect to $b$,
is defined on paths by

$\partial^{r}_{b}(a_{k}\dotsm a_{1})=\begin{cases}a_{k}\dotsm
a_{2},&a_{1}=b,\\\ 0,&a_{1}\neq b\end{cases}$ (3.6)

and extended linearly and continuously. Similarly, there is an _left
derivative_ , defined on paths by

$\partial^{l}_{b}(a_{k}\dotsm a_{1})=\begin{cases}a_{k-1}\dotsm
a_{1},&a_{k}=b,\\\ 0,&a_{k}\neq b.\end{cases}$ (3.7)

Given a Postnikov diagram $D$ and two arrows $a$ and $b$ of $Q_{D}$, we may
observe that

$\partial^{r}_{b}{\partial_{a}{W}}=\partial^{l}_{a}{\partial_{b}{W}}.$ (3.8)

Indeed, there are at most two fundamental cycles in $Q_{D}$ containing $a$,
and so the part of $W$ consisting of terms containing both $a$ and $b$ is of
the form $apb-aqb$ for some $p$ and $q$ which are either paths, not containing
the arrows $a$ or $b$, or are zero. Thus one can directly calculate each side
of (3.8), which both result in $p-q$.

We are now ready to prove our main theorem.

######  Theorem 3.7.

For $A=A_{D}$ the dimer algebra of a connected Postnikov diagram $D$, the
complex (3.1) is exact, and hence $A$ is bimodule internally $3$-Calabi–Yau
with respect to the idempotent determined by the boundary vertices.

###### Proof.

As already noted in Remark 3.5, the complex (3.1) can only fail to be exact in
degrees $-2$ and $-3$, and so, by Proposition 3.6, we need only check that the
complex (3.3) is exact at the terms
$A\mathbin{\otimes}\mathbb{K}Q_{3}^{\mathrm{m}}\mathbin{\otimes}S_{v}$ and
$A\mathbin{\otimes}\mathbb{K}Q_{2}^{\mathrm{m}}\mathbin{\otimes}S_{v}$ for
each $v\in Q_{0}$.

We first deal with exactness at
$A\mathbin{\otimes}\mathbb{K}Q_{3}^{\mathrm{m}}\mathbin{\otimes}S_{v}$. This
term is zero unless $v\in Q_{0}^{\mathrm{m}}$, so we may additionally assume
this. By the third isomorphism in (3.4), we then have
$A\mathbin{\otimes}\mathbb{K}Q_{3}^{\mathrm{m}}\mathbin{\otimes}S_{v}\cong
Ae_{v}$, so let $x\in Ae_{v}$. We calculate

$(\mu_{3}\otimes
S_{v})(x)=\sum_{a\in\mathrm{H}_{v}^{\mathrm{m}}}xa\otimes[a],$

using the notation (3.5) for elements of
$A\mathbin{\otimes}\mathbb{K}Q_{2}^{\mathrm{m}}\mathbin{\otimes}S_{v}\cong\bigoplus_{a\in\mathrm{H}_{v}^{\mathrm{m}}}Ae_{ta}$.
Thus $x\in\ker(\mu_{3}\otimes S_{v})$ if and only if $xa=0$ for all
$a\in\mathrm{H}_{v}^{\mathrm{m}}$. Note that since $v$ is mutable,
$\mathrm{H}_{v}^{\mathrm{m}}=\mathrm{H}_{v}\neq\varnothing$.

Let $w\in Q_{0}$, so $e_{w}x\in e_{w}Ae_{v}$. Since $A$ is thin by Proposition
2.11, we have $e_{w}Ae_{v}=Zp_{wv}$, so we may write
$e_{w}x=\sum_{n\in\mathbb{N}}z_{n}t^{n}\cdot p_{wv}$ for some sequence of
scalars $z_{n}\in\mathbb{K}$. Using thinness again, for any
$a\in\mathrm{H}_{v}^{\mathrm{m}}$ there is some $\delta_{a}\in\mathbb{N}$ such
that $p_{wv}a=t^{\delta_{a}}p_{w,ta}$. It then follows that

$e_{w}xa=\sum_{n\in\mathbb{N}}z_{n}t^{n}\cdot
p_{wv}a=\sum_{n\in\mathbb{N}}z_{n}t^{n+\delta_{a}}\cdot p_{w,ta}.$

Thus if $xa=0$, we have $z_{n}=0$ for all $n$, and so $e_{w}x=0$. Since $w$
was chosen arbitrarily, we conclude that $x=0$, and so the kernel of
$\mu_{3}\otimes S_{v}$ is trivial. This establishes exactness of (3.3) at the
term $A\mathbin{\otimes}\mathbb{K}Q_{3}^{\mathrm{m}}\mathbin{\otimes}S_{v}$.

We now move to the term
$A\mathbin{\otimes}\mathbb{K}Q_{2}^{\mathrm{m}}\mathbin{\otimes}S_{v}$. Using
the relevant isomorphisms in (3.4) and the notation of (3.5), a general
element of this term is of the form
$\phi=\sum_{a\in\mathrm{H}_{v}^{\mathrm{m}}}x_{a}\mathbin{\otimes}[a]$ for
$x\in Ae_{ta}$, and its image under $\mu_{2}\mathbin{\otimes}S_{v}$ is

$(\mu_{2}\mathbin{\otimes}S_{v})(\phi)=\sum_{b\in\mathrm{T}_{v}}\Bigg{(}\sum_{a\in\mathrm{H}_{v}^{\mathrm{m}}}x_{a}\partial^{r}_{b}{\partial_{a}{W}}\Bigg{)}\mathbin{\otimes}[b]=\sum_{b\in\mathrm{T}_{v}}\Bigg{(}\sum_{a\in\mathrm{H}_{v}^{\mathrm{m}}}x_{a}\partial^{l}_{a}{\partial_{b}{W}}\Bigg{)}\mathbin{\otimes}[b],$

where the second equality follows from (3.8). The reader is warned that, since
$b\in\mathrm{T}_{v}$ may be frozen, the derivative $\partial_{b}{W}$ is not
necessarily zero in $A$.

Now assume that $(\mu_{2}\mathbin{\otimes}S_{v})(\phi)=0$, or equivalently by
the above calculation that

$\sum_{a\in\mathrm{H}_{v}^{\mathrm{m}}}x_{a}\partial^{l}_{a}{\partial_{b}{W}}=0$
(3.9)

for all $b\in\mathrm{T}_{v}$. Picking $w\in Q_{0}$ and multiplying by $e_{w}$
on the left, we obtain elements $e_{w}x_{a}\in e_{w}Ae_{ta}$ for each
$a\in\mathrm{H}_{v}^{\mathrm{m}}$.

As in the first part of the proof, we will now use thinness of $A$ to write
the elements $e_{w}x_{a}$ as the product of a power series in $t$ with
$p_{w,ta}$, but in order to keep the notation clean in the subsequent
argument, we will do this in a slightly unusual way. First note that, having
fixed the vertex $w$, any path $p$ determines $\delta_{p}\in\mathbb{N}$ with
the property that $p_{w,hp}p=t^{\delta_{p}}p_{w,tp}$. Since this equality is
formulated in the algebra $A$, the quantity $\delta_{p}$ depends only on the
class of $p$ in the algebra.

Using this notation, we can write

$e_{w}x_{a}=t^{\delta_{a}}\sum_{n\in\mathbb{Z}}z_{n}^{w,a}t^{n}\cdot p_{w,ta}$
(3.10)

for some scalars $z_{n}^{w,a}\in\mathbb{K}$. Since the right-hand side of this
expression may only involve positive powers of $t$, we have $z_{n}^{w,a}=0$
whenever $n<-\delta_{a}$.

The key step in the argument is to show, under our assumption that
$(\mu_{2}\mathbin{\otimes}S_{v})(\phi)=0$, that the coefficient $z_{n}^{w,a}$
depends only on $n$ and $w$, and not on $a\in\mathrm{H}_{v}^{\mathrm{m}}$. We
consider the configuration

${v}$${\bullet}$${\bullet}$${\bullet}$${w}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}e_{w}x_{-}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{-}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p}$$\scriptstyle{a_{+}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}e_{w}x_{+}}$
(3.11)

Here solid arrows represent arrows in $Q_{1}$, dashed arrows represent paths
in $Q$, and dotted arrows represent linear combinations of paths. Our
assumption is that $a_{+}$ and $a_{-}$ are unfrozen arrows, with head at $v$
as depicted, which appear in the two fundamental cycles $a_{+}pb$ and
$a_{-}qb$ containing the arrow $b$, also unfrozen, with tail at $v$. As
indicated by the notation in (3.11), we will usually omit the letter $a$ when
$a_{+}$ or $a_{-}$ appear in subscripts or superscripts in the following
argument—for example this means that $x_{+}:=x_{a_{+}}$. Our aim is to show
that $z_{n}^{w,+}=z_{n}^{w,-}$ for all $n\in\mathbb{Z}$; it will then follow
from the way that the quiver $Q$ is embedded in the disc that $z_{n}^{w,a}$ is
independent of $a\in\mathrm{H}_{v}^{\mathrm{m}}$, since if
$a,a^{\prime}\in\mathrm{H}_{v}^{\mathrm{m}}$, we can find sequences
$a_{0},\dotsc,a_{k}$ and $b_{1}\cdots b_{k}$ such that $a_{0}=a$,
$a_{k}=a^{\prime}$, and for all $1\leq i\leq k$ the triple of arrows
$(a_{i-1},b_{i},a_{i})$ has the same configuration as $(a_{-},b,a_{+})$ in
(3.11) (or its mirror image), so that $z_{n}^{w,a_{i-1}}=z_{n}^{w,a_{i}}$.

So consider the configuration shown in (3.11). We see from the figure that
$\partial_{b}{W}=a_{+}p-a_{-}q$. While it can happen that $a_{+}=a_{-}$ (if
and only if $v$ is a bivalent vertex of $Q$), in this case there is nothing to
prove, so we may assume $a_{+}\neq a_{-}$. Then for
$a\in\mathrm{H}_{v}^{\mathrm{m}}$, we have

$\partial^{l}_{a}{\partial_{b}{W}}=\begin{cases}p,&a=a_{+},\\\ -q,&a=a_{-},\\\
0,&\text{otherwise}.\end{cases}$

Thus

$\sum_{a\in\mathrm{H}_{v}^{\mathrm{m}}}x_{a}\partial^{l}_{a}{\partial_{b}{W}}=x_{+}p-x_{-}q.$

Since $\phi=\sum_{a\in\mathrm{H}_{v}^{\mathrm{m}}}x_{a}\mathbin{\otimes}[a]$
is in the kernel of $\mu_{2}\mathbin{\otimes}S_{v}$, this quantity is zero by
(3.9), and so $x_{+}p=x_{-}q$. It follows that

$\displaystyle e_{w}x_{+}p$
$\displaystyle=t^{\delta_{+}}\sum_{n\in\mathbb{Z}}z_{n}^{w,+}t^{n}\cdot
p_{w,ta_{+}}p=t^{\delta_{+}+\delta_{p}}\sum_{n\in\mathbb{Z}}z_{n}^{w,+}t^{n}\cdot
p_{w,hb},$ $\displaystyle e_{w}x_{-}q$
$\displaystyle=t^{\delta_{-}}\sum_{n\in\mathbb{Z}}z_{n}^{w,-}t^{n}\cdot
p_{w,ta_{-}}q=t^{\delta_{-}+\delta_{q}}\sum_{n\in\mathbb{Z}}z_{n}^{w,-}t^{n}\cdot
p_{w,hb},$

must also be equal. Since $a_{+}p=a_{-}q$ in $A$ (because of the relation
$\partial_{b}{W}$), we must have
$\delta_{+}+\delta_{p}=\delta_{-}+\delta_{q}$, and so we can conclude, for any
$n\in\mathbb{Z}$, that $z_{n}^{w,+}=z_{n}^{w,-}$, and further that
$z_{n}^{w,a}$ is independent of $a\in\mathrm{H}_{v}^{\mathrm{m}}$, as
required. From now on, we abbreviate $z_{n}^{w}:=z_{n}^{w,a}$, and for any
$a\in\mathrm{H}_{v}^{\mathrm{m}}$ we have

$e_{w}x_{a}=t^{\delta_{a}}\sum_{n\in\mathbb{Z}}z_{n}^{w}t^{n}\cdot p_{w,ta}.$
(3.12)

We now return to our main purpose, to show that the complex (3.1) is exact. We
recall that we are assuming that
$\phi=\sum_{a\in\mathrm{H}_{v}^{\mathrm{m}}}x_{a}\mathbin{\otimes}[a]$
satisfies $(\mu_{2}\mathbin{\otimes}S_{v})(\phi)=0$, and wish to show that
$\phi$ is in the image of $\mu_{3}\mathbin{\otimes}S_{v}$.

First we deal with the case that $v$ is a frozen vertex, in which case
$\mu_{3}\mathbin{\otimes}S_{v}=0$, and we wish to conclude that $x_{a}=0$ for
all $a\in\mathrm{H}_{v}^{\mathrm{m}}$. This is equivalent to showing that the
complex numbers $z_{n}^{w}$ appearing in (3.12) are zero for all $n$ and all
$w$.

It follows from the construction of $(Q_{D},F_{D})$ that, since $v$ is frozen,
it is incident with a frozen arrow. If
$\mathrm{H}_{v}^{\mathrm{m}}=\varnothing$ then we have nothing to show, and
otherwise there is an arrow $a\in\mathrm{H}_{v}^{\mathrm{m}}$ as in one of the
following configurations

${\bullet}$${\bullet}$${\bullet}$${\diamond}$${v}$${\diamond}$${v}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}p}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}c}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b}$
(3.13)

in which $apb$ and $cqb$ are fundamental cycles. As earlier in the paper, the
bold arrows are frozen, and the dashed arrows represent paths. The vertices
indicated by diamonds are frozen, like the vertex $v$, whereas the others are
mutable.

In the two configurations in (3.13) we have $\partial_{b}{W}=\pm ap$ and
$\partial_{b}{W}=\pm(ap-cq)$ respectively. Thus in either case we calculate
for $a^{\prime}\in\mathrm{H}_{v}^{\mathrm{m}}$ that

$\partial^{l}_{a^{\prime}}{\partial_{b}{W}}=\begin{cases}\pm
p,&a^{\prime}=a,\\\ 0,&\text{otherwise}.\end{cases}$

Since $(\mu_{2}\mathbin{\otimes}S_{v})(\phi)=0$, we deduce from (3.9) that

$x_{a}p=\pm\sum_{a^{\prime}\in\mathrm{H}_{v}^{\mathrm{m}}}x_{a}\partial^{l}_{a^{\prime}}\partial_{b}{W}=0.$

Multiplying on the left by $e_{w}$ and using that $A$ is thin, we see that

$0=e_{w}x_{a}p=t^{\delta_{a}}\sum_{n\in\mathbb{Z}}z_{n}^{w}t^{n}\cdot
p_{w,ta}p=t^{\delta_{a}+\delta_{p}}\sum_{n\in\mathbb{Z}}z_{n}^{w}t^{n}\cdot{p_{w,hb}},$

and so $z_{n}^{w}=0$ for all $n$ and $w$, as required.

Now we treat the case that $v$ is a mutable vertex; in this case
$\mathrm{H}_{v}^{\mathrm{m}}=\mathrm{H}_{v}$ and we use the latter to keep the
notation simpler. In this case, we want to construct $y\in Ae_{v}$ such that

$(\mu_{3}\mathbin{\otimes}S_{v})(y)=\sum_{a\in\mathrm{H}_{v}}ya\mathbin{\otimes}[a]=\phi.$

Using the sequence of complex numbers $z_{n}^{w}$ appearing in (3.12), we
write

$y_{w}=\sum_{n\in\mathbb{Z}}z_{n}^{w}t^{n}\cdot p_{wv}.$

For this to be a well-defined element of $e_{w}Ae_{v}$, we need $z_{n}^{w}=0$
for all $n<0$. Choosing any $a\in\mathrm{H}_{v}$, we have that
$z_{n}^{w}=z_{n}^{w,a}$ is zero whenever $n<-\delta_{a}$, as in (3.10). But by
Lemma 3.2, there is some $a\in\mathrm{H}_{v}$ such that $\delta_{a}=0$, and so
$y_{w}\in e_{w}Ae_{v}$ as required.

Now, letting $y=\sum_{w\in Q_{0}}y_{w}\in Ae_{v}\cong
A\mathbin{\otimes}\mathbb{K}Q_{3}^{\mathrm{m}}\mathbin{\otimes}S_{v}$, we can
calculate

$\displaystyle(\mu_{3}\mathbin{\otimes}S_{v})(y)=\sum_{w\in
Q_{0}}\sum_{a\in\mathrm{H}_{v}}y_{w}a\mathbin{\otimes}[a]$
$\displaystyle=\sum_{w\in
Q_{0}}\sum_{a\in\mathrm{H}_{v}}\Bigg{(}\sum_{n\in\mathbb{N}}z_{n}^{w}t^{n}\cdot
p_{wv}a\Bigg{)}\mathbin{\otimes}[a]$ $\displaystyle=\sum_{w\in
Q_{0}}\sum_{a\in\mathrm{H}_{v}}\Bigg{(}\sum_{n\in\mathbb{N}}z_{n}^{w}t^{n+\delta_{a}}\cdot
p_{w,ta}\Bigg{)}\mathbin{\otimes}[a]$ $\displaystyle=\sum_{w\in
Q_{0}}\sum_{a\in\mathrm{H}_{v}}e_{w}x_{a}\mathbin{\otimes}[a]=\sum_{a\in\mathrm{H}_{v}}x_{a}\mathbin{\otimes}[a]=\phi,$

and so we conclude that
$\phi\in\operatorname{im}(\mu_{3}\mathbin{\otimes}S_{v})$, as required. ∎

##  4\. Categorification

Given a suitable algebra $A$ which is bimodule internally $3$-Calabi–Yau with
respect to an idempotent $e$, one can construct from $A$ and $e$ a Frobenius
category having many desirable properties from the point of view of
categorifying cluster algebras. This done via the main result of [42]
(specialised to the case that the Calabi–Yau dimension is $3$), which we will
recall after collecting the necessary algebraic definitions. Applied to the
dimer algebra $A_{D}$ of a connected Postnikov diagram $D$, this will allow us
to show (Theorem 4.5) that the category $\operatorname{GP}(B_{D})$ appearing
in Theorem 1 has many of the properties required of an additive
categorification of the cluster algebra $\mathscr{A}_{D}$. Further such
properties will be proved in Section 6.

######  Definition 4.1.

Let $\mathcal{E}$ be an exact $\mathbb{K}$-linear category, as in [8]. We do
not recall the full definition of this structure here, but simply note that
any full extension-closed subcategory $\mathcal{E}$ of an abelian category
$\mathcal{A}$ has a natural structure of an exact category, by taking the
class of short exact sequences to be the short exact sequences of
$\mathcal{A}$ with terms in $\mathcal{E}$.

We say $\mathcal{E}$ is a _Frobenius category_ if it has enough projective
objects and enough injective objects (meaning that every object of
$\mathcal{E}$ admits a projective cover and an injective envelope) and these
two classes of objects coincide. The _stable category_
$\underline{\mathcal{E}}$ is obtained from $\mathcal{E}$ by factoring out the
ideal of morphisms which factor over a projective (equivalently injective)
object, and has a natural triangulated structure [21, §I.2]. We say that
$\underline{\mathcal{E}}$ is _$2$ -Calabi–Yau_, and $\mathcal{E}$ is _stably
$2$-Calabi–Yau_, if $\underline{\mathcal{E}}$ is Hom-finite and there is a
functorial isomorphism

$\operatorname{Hom}_{\underline{\mathcal{E}}}(X,Y)=\mathrm{D}\operatorname{Hom}_{\underline{\mathcal{E}}}(Y,\Sigma^{2}X)$

for any two objects $X$ and $Y$ of $\underline{\mathcal{E}}$, where $\Sigma$
is the shift functor in the triangulated structure of
$\underline{\mathcal{E}}$ (computed by taking the cokernel of an injective
envelope) and $\mathrm{D}$ is duality over the ground field $\mathbb{K}$.

A full additive subcategory $\mathcal{T}\subseteq\mathcal{E}$ is called
_cluster-tilting_ if it is functorially finite and

$\mathcal{T}=\\{X\in\mathcal{E}:\operatorname{Ext}^{1}_{\mathcal{E}}(X,T)=0\
\forall\
T\in\mathcal{T}\\}=\\{X\in\mathcal{E}:\operatorname{Ext}^{1}_{\mathcal{E}}(T,X)\
\forall\ T\in\mathcal{T}\\}.$

We can make the same definition in $\underline{\mathcal{E}}$, using the usual
definition
$\operatorname{Ext}^{1}_{\underline{\mathcal{E}}}(M,N)=\operatorname{Hom}_{\underline{\mathcal{E}}}(M,\Sigma
N)$ for triangulated categories, and observe that
$\mathcal{T}\subseteq\mathcal{E}$ is cluster-tilting if and only if its image
in the stable category $\underline{\mathcal{E}}$ (i.e. the full subcategory of
$\underline{\mathcal{E}}$ on the same set of objects as $\mathcal{T}$) is
cluster-tilting. When $\mathcal{E}$ is stably $2$-Calabi–Yau, the second
equality in the definition of a cluster-tilting subcategory is automatic. An
object $T\in\mathcal{E}$ is a _cluster-tilting object_ if its additive closure
$\operatorname{add}{T}$ is a cluster-tilting subcategory.

An algebra $B$ is called _Iwanaga–Gorenstein_ if it is Noetherian and has
finite injective dimension both as a left and as a right module over itself.
In this case these two injective dimensions coincide [24], and we call this
number the _Gorenstein dimension_ of $B$. For an Iwanaga–Gorenstein algebra
$B$, we define the category of _Gorenstein projective_ $B$-modules to be the
full subcategory

$\operatorname{GP}(B)=\left\\{X\in\operatorname{mod}{B}:\text{$\operatorname{Ext}^{i}_{B}(X,B)=0$
for all $i>0$}\right\\}$

of the abelian category of finitely generated $B$-modules. This is a Frobenius
category; see for example [7, §4], where this category is denoted by
$\operatorname{MCM}(B)$ and its objects called maximal Cohen–Macaulay modules.

######  Remark 4.2.

The reader is warned that, when $B$ is free and finitely generated over
$Z=\mathbb{K}[[t]]$, as will be the case in our examples, one can define the
(maximal) Cohen–Macaulay $B$-modules to be those $B$-modules which are free
and finitely generated over $Z$, as in [27, §3]. While a module for the
algebra $C$ defining the Grassmannian cluster category is Gorenstein
projective if and only if it is Cohen–Macaulay [27, Cor. 3.7], this will not
be the case for $B$ in general (see Example 4.8). To avoid having two non-
equivalent definitions of Cohen–Macaulay $B$-modules, we prefer the
terminology ‘Gorenstein projective’ for the homological condition. The reader
is however further warned that Gorenstein projective modules are defined over
an arbitrary ring (as in [22, Defn. 2.1] for example), but in a way that need
not be equivalent to Definition 4.1 if the ring is not Iwanaga–Gorenstein.

######  Theorem 4.3 ([42, Thm. 4.1, Thm. 4.10]).

Let $A$ be an algebra, and $e\in A$ an idempotent. If $A$ is Noetherian,
$\underline{A}=A/\langle e\rangle$ is finite-dimensional, and $A$ is bimodule
internally $3$-Calabi–Yau with respect to $e$, then

  1. (1)

$B=eAe$ is Iwanaga–Gorenstein with Gorenstein dimension at most $3$,

  2. (2)

$eA$ is a cluster-tilting object in the Frobenius category
$\operatorname{GP}(B)$,

  3. (3)

the stable category $\operatorname{\underline{GP}}(B)$ is a $2$-Calabi–Yau
triangulated category, and

  4. (4)

the natural maps $A\to\operatorname{End}_{B}(eA)^{\mathrm{op}}$ and
$\underline{A}\to\operatorname{\underline{End}}_{B}(eA)^{\mathrm{op}}$ are
isomorphisms.

We already saw in Theorem 3.7 that the dimer algebra $A_{D}$ of a connected
Postnikov diagram $D$ is bimodule internally $3$-Calabi–Yau with respect to
its boundary idempotent. Checking the other conditions needed to apply Theorem
4.3 to $A_{D}$ is comparatively straightforward, although we note that we once
again rely heavily on Proposition 2.11.

######  Proposition 4.4.

Let $D$ be a connected Postnikov diagram with dimer algebra $A_{D}$, and let
$e$ be the boundary idempotent of this algebra. Then $A_{D}$ is Noetherian and
$A_{D}/\langle e\rangle$ is finite-dimensional.

###### Proof.

Since $A_{D}$ is thin, as in Proposition 2.11, it is finitely generated as a
module over the commutative Noetherian ring $\mathbb{K}[[t]]$, and so is
Noetherian.

By thinness again, $A_{D}$ has a basis $\\{t^{n}p_{wv}:v,w\in
Q_{0},n\in\mathbb{N}\\}$. To see that $A_{D}/\langle e\rangle$ is finite-
dimensional, we show that all but finitely many of these basis vectors are
zero in the quotient. Picking any $v,w\in Q_{0}$, let $c$ be any cycle at $w$
passing through a boundary vertex, which exists since $Q_{D}$ is strongly
connected as in Proposition 2.6. Then since $A_{D}$ is thin, we have
$cp_{wv}=t^{N}p_{wv}$ for some $N\in\mathbb{N}$. By construction, this element
is zero in the quotient $A_{D}/\langle e\rangle$, and thus $t^{n}p_{wv}=0$ in
this quotient algebra for all $n\geq N$. Running over all pairs $v,w\in Q_{0}$
completes the proof. ∎

Recall that we write $\mathscr{A}_{D}$ for the cluster algebra with frozen
variables determined by the ice quiver $(Q_{D},F_{D})$ of a Postnikov diagram
$D$. Note that arrows between frozen vertices play no role in the construction
of this cluster algebra. Since $Q_{D}$ may contain $2$-cycles, we are
committing a small abuse of notation here; really $\mathscr{A}_{D}$ is the
cluster algebra determined by the quiver $Q^{\prime}$ of the unique reduced
Postnikov diagram obtained from $D$ by untwisting moves, as in Figure 4.
Equivalently, $Q^{\prime}$ is obtained from $Q_{D}$ by removing a maximal
collection of $2$-cycles incident with unfrozen vertices. Using the
description via untwisting and Proposition 2.9, we see that $Q^{\prime}$ has a
potential $W^{\prime}$ such that
$\mathcal{J}(Q^{\prime},F_{D},W^{\prime})\cong A_{D}$ is the dimer algebra of
the original diagram $D$.

By Theorem 3.7 and Proposition 4.4, we may apply Theorem 4.3 to the dimer
algebra $A_{D}$ of a connected Postnikov diagram, to obtain the following.

######  Theorem 4.5.

Let $D$ be a connected Postnikov diagram, with dimer algebra $A_{D}$. Let $e$
be the boundary idempotent of this algebra, and write $B_{D}=eA_{D}e$. Then

  1. (1)

$B_{D}$ is Iwanaga–Gorenstein of Gorenstein dimension at most $3$,

  2. (2)

$\operatorname{GP}(B_{D})$ is a stably $2$-Calabi–Yau Frobenius category,

  3. (3)

$T_{D}=eA_{D}\in\operatorname{GP}(B_{D})$ is cluster-tilting, and

  4. (4)

$\operatorname{End}_{B_{D}}(T_{D})^{\mathrm{op}}\cong A_{D}$.

In particular, it follows from (4) that the Gabriel quiver of
$\operatorname{End}_{B_{D}}(T_{D})^{\mathrm{op}}$ is, up to arrows between
frozen vertices, the quiver of a seed in the cluster algebra
$\mathscr{A}_{D}$.

We note that if the full subquiver of $Q_{D}$ on the mutable vertices is,
after removing $2$-cycles, mutation-equivalent to an acyclic quiver
$\underline{Q}$, it follows from a result of Keller and Reiten [29, Thm. 2.1]
that the stable category $\operatorname{\underline{GP}}(B_{D})$ of our
categorification is equivalent to the cluster category
$\mathcal{C}_{\underline{Q}}$ [5]. This mutation-acyclicity assumption does
not hold for most Postnikov diagrams, however. We conjecture that
$\operatorname{\underline{GP}}(B_{D})$ is always equivalent to Amiot’s cluster
category [1] for the quiver with potential obtained from $(Q_{D},F_{D},W_{D})$
by deleting the frozen vertices and all incident arrows from $Q_{D}$ and
removing terms passing through these vertices from $W_{D}$.

If $D$ is a $(k,n)$-diagram, as in Definition 2.1, it follows from [2, Cor.
10.4] that the boundary algebra $B_{D}=eA_{D}e$ does not depend on the choice
of $D$ within this class, up to isomorphism, and the category
$\operatorname{GP}(B_{D})$ is equivalent to Jensen–King–Su’s categorification
of the cluster algebra structure on the whole Grassmannian [27].

$v_{0}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$

$v_{0}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$

Figure 7. A _geometric exchange_ of a Postnikov diagrams transforms the local
configuration of a quadrilateral alternating region, shown on the left, to
that shown on the right. The effect on the quiver with potential is a mutation
[14, §5] at the vertex corresponding to the quadrilateral alternating region
(cf. [48]).

In general, we can consider two Postnikov diagrams $D$ and $D^{\prime}$
related by geometric exchange as shown in Figure 7. Combining [40, Thm. 13.4]
and [40, Thm. 17.1], we see that these diagrams determine the same positroid,
and hence the same positroid varieties. Moreover, the quivers $Q_{D}$ and
$Q_{D^{\prime}}$ are related by a mutation, and so the cluster algebras
$\mathscr{A}_{D}$ and $\mathscr{A}_{Q_{D^{\prime}}}$ are isomorphic. Thus we
would like both $D$ and $D^{\prime}$ to give rise to the same category of
Gorenstein projective modules in Theorem 4.5. This is indeed the case, as
follows.

######  Proposition 4.6.

Let $D$ and $D^{\prime}$ be Postnikov diagrams related by geometric exchange.
Then the boundary algebras $B_{D}$ and $B_{D^{\prime}}$ are isomorphic, and
hence the categories $\operatorname{GP}(B_{D})$ and
$\operatorname{GP}(B_{D^{\prime}})$ are equivalent.

###### Proof.

This can be proved exactly as in [2, Prop. 12.2]; while this proposition is
stated only for $(k,n)$-diagrams, the proof does not use the extra assumption
on the strand permutation. ∎

######  Corollary 4.7.

Given a Postnikov diagram $D$, the algebra $B_{D}$ is determined up to
isomorphism by the positroid $\mathcal{P}_{D}$.

###### Proof.

Again combining [40, Thm. 13.4] and [40, Thm. 17.1], any two diagrams
determining the same positroid are connected by a sequence of geometric
exchanges, so the result follows from Proposition 4.6. ∎

One consequence of the general results in Theorem 4.3 on bimodule internally
$3$-Calabi–Yau algebras is that the Gorenstein dimension of $B_{D}$ is at most
$3$ for any connected Postnikov diagram $D$. In fact, this Gorenstein
dimension never exceeds $2$ by [11, Cor. 10.4]. When $D$ is a $(k,n)$-diagram,
$B_{D}$ even has Gorenstein dimension $1$, as in the proof of [27, Cor. 3.7].
Since $1$ is also the Krull dimension of $Z=\mathbb{K}[[t]]$, in this case the
category $\operatorname{GP}(B_{D})$ coincides with the category of
Cohen–Macaulay $B_{D}$-modules, meaning those $B_{D}$-modules free and
finitely generated over $Z$. However, the Gorenstein dimension of $B_{D}$ can
be equal to $2$, and the inclusion
$\operatorname{GP}(B_{D})\to\operatorname{CM}(B_{D})$ can be proper, as the
following example shows.

######  Example 4.8.

Consider the Postnikov diagram $D$ shown in Figure 8 and its dimer algebra
$A_{D}$, the quiver of which is shown in the same figure. Since every
alternating region of $D$ is on the boundary (equivalently, ever vertex of the
quiver is frozen), we have $B_{D}=A_{D}$, which has finite global dimension by
Theorem 3.7. In fact, since the left-most term of $\mathbf{P}(A_{D})$ is zero
in this case, again because all vertices are frozen, the global dimension of
$B_{D}$ is $2$.

$\diamond$$\diamond$$\diamond$$\diamond$

Figure 8. A Postnikov diagram $D$ of type $(2,4)$ giving rise to a boundary
algebra $B_{D}$ for which not all Cohen–Macaulay modules are Gorenstein
projective. The right-hand figure shows the quiver $Q_{D}$, with frozen arrows
marked in bold as usual, which is the Gabriel quiver of $A_{D}=B_{D}$.

As is the case for any algebra of finite global dimension, $B_{D}$ is
Iwanaga–Gorenstein (of Gorenstein dimension equal to the global dimension, so
in this case $2$), and a $B_{D}$-module is Gorenstein projective if and only
if it is projective. However, one can observe that the radical of the
projective $B_{D}$-module at the lowest vertex (the head of the unfrozen
arrow) is not projective, but it is free and finitely-generated over $Z$, i.e.
Cohen–Macaulay, since it is a submodule of a projective $B_{D}$-module, which
is Cohen–Macaulay by Proposition 2.11. Hence in this case the category of
Gorenstein projective modules for $B_{D}$ is a proper subcategory of the
category of Cohen–Macaulay modules.

##  5\. Loops and $2$-cycles

Let $B=B_{D}$ be the boundary algebra of a connected Postnikov diagram $D$,
and let $T_{D}=eA_{D}\in\operatorname{GP}(B)$ be the cluster-tilting object
from Theorem 4.5. In this section we show that the quiver of
$\operatorname{End}_{B}(T)^{\mathrm{op}}$ has no loops or $2$-cycles for any
cluster-tilting object $T\in\operatorname{GP}(B)$ mutation equivalent to
$T_{D}$—so far we know this only for the relatively small number of such
objects arising from the Postnikov diagrams related to $D$ by sequences of
geometric exchanges. This will allow us to complete the proof of Theorem 1 in
the next section, by giving us access to results of Fu and Keller [16]. The
statement requires our non-degeneracy assumption that $D$ has at least $3$
strands (cf. [43, Prop. 5.17]).

For the remainder of the section, we fix a cluster-tilting object
$T_{0}\in\operatorname{GP}(B)$, and write
$A_{0}=\operatorname{End}_{B}(T_{0})^{\mathrm{op}}$. We assume that

  1. (1)

$A_{0}$ is isomorphic to the frozen Jacobian algebra of an ice quiver with
potential $(Q,F,W)$, in such a way that the vertices of $F$ correspond to
projection onto indecomposable projective summands of $T_{0}$,

  2. (2)

the quiver $Q$ has no loops or $2$-cycles, and

  3. (3)

$A_{0}/\langle e_{i}\rangle$ is finite-dimensional whenever $i\in F_{0}$.

All of these assumptions hold if $T_{0}=eA_{D}$ is the initial cluster-tilting
object from Theorem 4.5—the absence of loops and $2$-cycles in $Q_{D}$ (after
replacing $D$ by a reduced diagram if necessary) follows from Proposition
2.10, and finite-dimensionality in (3) follows from Proposition 2.11 via an
exactly analogous argument to that in the proof of Proposition 4.4. Our goal
is to show that these properties are preserved by mutations, and so hold in
the entire mutation class of those cluster-tilting objects coming from
Postnikov diagrams.

With this in mind, we additionally fix for the rest of the section a cluster-
tilting object $T$ obtained from $T_{0}$ by mutation at a non-projective
indecomposable summand, and write $A=\operatorname{End}_{B}(T)^{\mathrm{op}}$.
By [43, Thm. 5.14], the algebra $A$ again satisfies (1), with the required ice
quiver with potential being the mutation