# Internally Calabi–Yau algebras and cluster-tilting objects

Matthew Pressland  Matthew Pressland
Max-Planck-Institut für Mathematik
Vivatsgasse 7
53111 Bonn
Germany<EMAIL_ADDRESS>
(Date: December 28, 2016)

###### Abstract.

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

###### Key words and phrases:

Calabi–Yau algebra, cluster algebra, cluster-tilting object, Frobenius
category, Jacobian algebra, quiver with potential

###### 2010 Mathematics Subject Classification:

13F60, 16G20, 16G50, 18E30

##  1\. Introduction

Cluster categories, first introduced in special cases by
Buan–Marsh–Reineke–Reiten–Todorov [buantilting] and later generalised by Amiot
[amiotcluster], are certain $\operatorname{Hom}$-finite $2$-Calabi–Yau
triangulated categories that model the combinatorics of cluster algebras
without frozen variables. In particular, a cluster category $\mathcal{C}$
contains cluster-tilting objects, which are objects $T$ satisfying

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

Basic cluster-tilting objects, whose summands in any direct sum decomposition
are pairwise non-isomorphic, model the clusters of the cluster algebra; from
now on, any time we refer to a cluster-tilting object, we assume it to be
basic.

The main reason that the cluster-tilting objects of $\mathcal{C}$ can be said
to model the clusters of a cluster algebra is that, as with a cluster, it is
possible to pass from one cluster-tilting object to another by a process of
mutation. For any indecomposable summand $T_{i}$ of a cluster-tilting object
$T$, there exists a unique indecomposable $T_{i}^{\prime}\in\mathcal{C}$, not
isomorphic to $T_{i}$, such that $T/T_{i}\mathbin{\oplus}T_{i}^{\prime}$ is a
cluster-tilting object. Moreover, $T_{i}^{\prime}$ can be computed by either
of the exchange triangles

${T_{i}}$${X_{i}}$${T_{i}^{\prime}}$${T_{i}[1],}$${T_{i}^{\prime}}$${Y_{i}}$${T_{i}}$${T_{i}^{\prime}[1],}$$\scriptstyle{f}$$\scriptstyle{g}$

in which $f$ is a minimal left $\operatorname{add}{T/T_{i}}$-approximation of
$T_{i}$, and $g$ is a minimal right
$\operatorname{add}{T/T_{i}}$-approximation of $T_{i}$. Choosing an initial
cluster-tilting object $T^{0}=\bigoplus_{i=1}^{n}T_{i}^{0}$ of $\mathcal{C}$
yields a cluster character [calderocluster], [palucluster]
$\varphi\colon\mathcal{C}\to\mathscr{A}(Q,\mathbf{x})$ from the objects of
$\mathcal{C}$ to the cluster algebra $\mathscr{A}(Q,\mathbf{x})$ with initial
seed given by the quiver $Q$ of
$\operatorname{End}_{\mathcal{C}}(T)^{\mathrm{op}}$ with cluster variables
$x_{i}=\varphi_{T_{i}^{0}}$. The exchange triangles correspond to the exchange
relations

$\varphi_{T_{i}}\varphi_{T_{i}^{\prime}}=\varphi_{X_{i}}+\varphi_{Y_{i}}$

in this cluster algebra. The cluster variables of $\mathscr{A}(Q,\mathbf{x})$
are precisely the elements of the form $\varphi_{M}$ for $M$ an indecomposable
reachable rigid object of $\mathcal{C}$, where rigid means that
$\operatorname{Ext}^{1}_{\mathcal{C}}(M,M)=0$, and reachable means that $M$ is
a summand of a reachable cluster-tilting object, i.e. one obtained from
$T^{0}$ by a finite sequence of mutations. The clusters are the sets of the
form $\\{\varphi_{T_{1}},\dotsc,\varphi_{T_{n}}\\}$ for
$T=\bigoplus_{i=1}^{n}T_{i}$ reachable cluster-tilting in $\mathcal{C}$.

For a more thorough introduction to the theory of cluster algebras and their
categorification, we recommend Keller’s survey [kellercluster].

The categorification of cluster algebras by cluster categories has proved to
be very useful in studying their combinatorics, since the cluster category
$\mathcal{C}$ can be considered more globally than the associated cluster
algebra $\mathscr{A}(Q,\mathbf{x})$. For example, to identify clusters or
cluster variables of $\mathscr{A}(Q,\mathbf{x})$, one usually has to find a
sequence of mutations from a known cluster, which is a highly computationally
intensive procedure. By contrast, cluster-tilting or rigid objects of
$\mathcal{C}$ are characterised intrinsically.

For this reason, it would be extremely useful to be able to more readily
categorify cluster algebras that do have frozen variables, particularly as
most of the examples occurring in nature, such as those on the coordinate
rings of partial flag varieties and their unipotent cells, as studied by
Geiß–Leclerc–Schröer [geisspartial], are of this type. The natural candidate
for such a categorification is a stably $2$-Calabi–Yau Frobenius category, as
we now describe.

A Frobenius category is an exact category with enough projective and injective
objects, such that these two classes of objects coincide. If $\mathcal{E}$ is
a Frobenius category, then the stable category
$\underline{\mathcal{E}}=\mathcal{E}/\operatorname{proj}{\mathcal{E}}$ is
triangulated by a famous result of Happel [happeltriangulated]*§I.2. It is
immediate from the definition that if $T\in\mathcal{E}$ is cluster-tilting,
then $\operatorname{proj}{\mathcal{E}}\subset\operatorname{add}{T}$. In this
case, we must have $\operatorname{proj}{\mathcal{E}}=\operatorname{add}{P}$
for some object $P=\bigoplus_{i=r+1}^{n}T_{i}$; the intention is that the
objects $T_{i}$ for $r<i\leq n$, which occur as summands of every cluster-
tilting object of $\mathcal{E}$, will correspond to the frozen variables of a
cluster algebra. Factoring out $\operatorname{proj}{\mathcal{E}}$ corresponds
to setting these frozen variables to $1$ in the cluster algebra, to recover a
cluster algebra without frozen variables. Thus the stable category
$\underline{\mathcal{E}}$ should be a cluster category—in particular it should
be $2$-Calabi–Yau.

Such categorifications have been described for certain cluster algebras
relating to partial flag varieties, for example by Geiss–Leclerc–Schröer
[geisspartial], Jensen–King–Su [jensencategorification] and Demonet–Iyama
[demonetlifting], and we will recall some of these constructions in Section 3.
Nájera Chávez [najerachavez2calabiyau] has also categorified finite type
cluster algebras with ‘universal’ coefficients. However, in all cases, the
construction of the category depends on having at least a partial
understanding of the overall structure of the cluster algebra. Thus the
methods of these papers cannot be easily abstracted to produce
categorifications for more general cluster algebras with frozen variables.

The main aim of this paper is to consider how one might be able to produce a
categorification of a cluster algebra with frozen variables without
understanding this global structure, instead starting only from the data of a
single seed, which is how a cluster algebra is usually specified. This is
analogous to Amiot’s construction of cluster categories in the case that there
are no frozen variables; given a seed, one has to find a rigid potential on
the quiver of the seed such that the resulting Jacobian algebra (see Section
5) is finite dimensional, and then Amiot provides a general recipe for
constructing a categorification of the cluster algebra. Our construction in
the case that there are frozen variables is similar, but requires more data
satisfying more conditions. Given the seed of a cluster algebra with frozen
variables, we take its quiver, and aim to add arrows between frozen variables
and choose a potential such that the resulting frozen Jacobian algebra
satisfies a number of conditions, most importantly that of being bimodule
internally $3$-Calabi–Yau. If this can be achieved, the general machinery
developed in this paper can take over to produce the desired Frobenius
category, from which the frozen Jacobian algebra can be recovered as the
endomorphism algebra of a cluster-tilting object.

For most of the paper, we will in fact work in a higher level of generality,
and construct stably $d$-Calabi–Yau Frobenius categories admitting
$d$-cluster-tilting objects (Definition 3.2); setting $d=2$ recovers the
definition of cluster-tilting given above.

The structure of the paper is as follows. In Section 2 we say what it means
for an algebra to be internally $d$-Calabi–Yau with respect to an idempotent
$e$ (Definition 2.1), and also make a stronger, more symmetric, definition of
bimodule internally $d$-Calabi–Yau with respect to $e$ (Definition 2.4). An
algebra $A$ satisfying either of these definitions has, in particular, finite
global dimension, and a duality

$\mathrm{D}\operatorname{Ext}_{A}^{i}(M,N)=\operatorname{Ext}_{A}^{d-i}(N,M)$

for any pair $M$ and $N$ of $A$-modules such that $M$ is finite dimensional,
$eM=0$ and both $M$ and $N$ are perfect when considered as stalk complexes in
the bounded derived category of $A$. In much of the paper we will also assume
that $A$ is Noetherian, in which case this last condition reduces to $M$ and
$N$ being finitely generated.

In Section 3, we introduce the class of Frobenius $m$-cluster categories for
$m\geq 1$, and exploit a result of Keller–Reiten [kellerclustertilted] to show
that the endomorphism algebra of an $m$-cluster-tilting object in a Frobenius
$m$-cluster category is internally $(m+1)$-Calabi–Yau. A Morita-type theorem
of Iyama–Kalck–Wemyss–Yang [iyamafrobenius] implies that any Frobenius
$m$-cluster category admitting an $m$-cluster-tilting object with Noetherian
endomorphism algebra is equivalent to the category of Gorenstein projective
modules over some Iwanaga–Gorenstein ring (Definition 3.8), and so we pay
special attention to categories of this form. We also give brief descriptions
of some important families of Frobenius $m$-cluster categories already
appearing in the literature.

In Section 4, we show that an algebra $A$ that is internally $d$-Calabi–Yau
with respect to an idempotent $e$ (on both sides) for some $d\geq 2$, and
satisfies mild additional assumptions, necessarily arises as the endomorphism
algebra of a $(d-1)$-cluster-tilting object in some Frobenius category
determined by $A$ and $e$. Precisely, we prove the following theorem.

######  Theorem 1 (Theorem 4.1).

Let $A$ be a Noetherian algebra, and let $e\in A$ be an idempotent such that
$A/\langle e\rangle$ is finite dimensional and both $A$ and $A^{\mathrm{op}}$
are internally $d$-Calabi–Yau with respect to $e$. Write $B=eAe$ and
$\underline{A}=A/\langle e\rangle$. Then

  * (i)

$B$ is Iwanaga–Gorenstein with Gorenstein dimension at most $d$, so

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

is a Frobenius category,

  * (ii)

$eA$ is $(d-1)$-cluster-tilting in $\operatorname{GP}(B)$, and

  * (iii)

there are natural isomorphisms
$\operatorname{End}_{B}(eA)^{\mathrm{op}}\stackrel{{\scriptstyle\sim}}{{\to}}A$
and
$\operatorname{End}_{\operatorname{\underline{GP}}(B)}(eA)^{\mathrm{op}}\stackrel{{\scriptstyle\sim}}{{\to}}\underline{A}$.

Under the stronger assumption that $A$ is bimodule internally $d$-Calabi–Yau
with respect to $e$, we can show more.

######  Theorem 2 (Theorem 4.10).

Let $A$ be a Noetherian algebra and let $e\in A$ be an idempotent such that
$A/\langle e\rangle$ is finite dimensional, and $A$ is bimodule internally
$d$-Calabi–Yau with respect to $e$. Write $B=eAe$. Then all of the conclusions
of Theorem 1 hold, and moreover $\operatorname{\underline{GP}}(B)$ is
$(d-1)$-Calabi–Yau.

While, in general, checking that an algebra is internally $d$-Calabi–Yau with
respect to an idempotent can be very difficult, there is more hope in the case
that $A$ is a frozen Jacobian algebra (Definition 5.1). Such an algebra is
presented via a quiver with relations, in which the relations are dual to some
of the arrows; the arrows which do not have any corresponding relations are
called frozen, and their end-points are frozen vertices. In Section 5, we show
that the required Calabi–Yau symmetry can be deduced from the exactness of a
combinatorially defined complex $\mathbf{P}(A)\to A$ (Definition 5.4),
generalising work of Ginzburg [ginzburgcalabiyau] for Jacobian algebras. More
precisely, we prove the following.

######  Theorem 3 (Theorem 5.6).

If $A$ is a frozen Jacobian algebra such that $\mathbf{P}(A)$ is quasi-
isomorphic to $A$, then $A$ is bimodule internally $3$-Calabi–Yau with respect
to the idempotent $e=\sum_{v\in F_{0}}e_{v}$, where $F_{0}$ denotes the set of
frozen vertices.

The results of Section 5 are inspired by work of Broomhead [broomheaddimer] on
consistent dimer models (also known as brane tilings or bipartite field
theories) on closed surfaces, which has applications to theoretical physics.
We expect our results to have consequences for the more recent theory of dimer
models on surfaces with boundary, studied for example by Franco
[francobipartite], and which has already appeared in the context of cluster
categorification for the Grassmannian in work of Baur–King–Marsh [baurdimer].
Dimer models on surfaces with boundary have also appeared, under the name
‘plabic graphs’, in work of Postnikov [postnikovtotal] on the positive
Grassmannian, and in recent work of Goncharov [goncharovideal].

## Notation and conventions

Throughout, we fix a field $\mathbb{K}$, and assume all categories are
$\mathbb{K}$-linear and all algebras are associative $\mathbb{K}$-algebras
with unit. If $V$ is a $\mathbb{K}$-vector space, we write
$\mathrm{D}{V}=\operatorname{Hom}_{\mathbb{K}}(V,\mathbb{K})$ for the dual
space. All modules are unital, and are left modules unless otherwise
indicated. Given an algebra $A$, we denote by $\operatorname{Mod}{A}$ the
category of all $A$-modules, and by $\operatorname{mod}{A}$ the category of
finitely generated $A$-modules. Given an object $X$ of an exact category
$\mathcal{E}$, we denote by $\operatorname{add}{X}$ the full subcategory of
$\mathcal{E}$ with objects isomorphic to finite direct sums of direct summands
of $X$. We say an algebra $A$ is Noetherian if it is Noetherian as both a left
and right module over itself; this is stronger than requiring it to be
Noetherian as an $A$-bimodule. We denote by $\mathcal{D}{A}$ and
$\mathcal{D}^{b}{A}$ the derived and bounded derived categories of $A$, and by
$\mathcal{D}_{\mathrm{fd}}(A)$ the full subcategory of $\mathcal{D}^{b}{A}$
consisting of objects with finite dimensional total cohomology. We denote by
$\operatorname{per}{A}$ the thick triangulated subcategory of
$\mathcal{D}^{b}{A}$ generated by $A$ (or equivalently by the finitely
generated projective $A$-modules). Objects of $\operatorname{per}{A}$ are
called perfect, and we will say that an $A$-module $M$ is perfect if the stalk
complex with $M$ in degree $0$ is perfect.

## Acknowledgements

This work is based on part of my Ph.D. thesis [presslandfrobenius]*§5, which I
worked on at the University of Bath under the supervision of Alastair King,
and with the support of an Engineering and Physical Sciences Research Council
studentship. I am very grateful to Alastair for his support and encouragement.
I also thank Charlie Beil, Martin Kalck, Joe Karmazyn and Dong Yang for useful
conversations. This paper was revised during a stay at Universität Bielefeld
funded by Sonderforschungsbereich 701; I am grateful to Henning Krause and the
rest of the representation theory group in Bielefeld for their hospitality.
After helpful comments from the referee, it was further revised during a stay
at the Max-Planck-Institut für Mathematik in Bonn, funded by the Max-Planck-
Gesellschaft.

##  2\. Internally $d$-Calabi–Yau algebras

This section introduces our main definitions.

######  Definition 2.1.

Let $A$ be a $\mathbb{K}$-algebra, $e$ an idempotent of $A$, and $d$ a non-
negative integer. We say $A$ is _internally $d$-Calabi–Yau_ with respect to
$e$ if

  * (i)

$\operatorname{gl.dim}{A}\leq d$, and

  * (ii)

for each $i\in\mathbb{Z}$, there is a functorial duality

$\mathrm{D}\operatorname{Ext}_{A}^{i}(M,N)=\operatorname{Ext}_{A}^{d-i}(N,M)$

where $M$ and $N$ are perfect $A$-modules such that $M$ is also a finite
dimensional $A/\langle e\rangle$-module, where $\langle e\rangle=AeA$ is the
two-sided ideal of $A$ generated by $e$.

Recall that a module $M$ is perfect if the stalk complex with $M$ in degree
$0$ is perfect, i.e. quasi-isomorphic to a bounded complex of finitely
generated projective $A$-modules. Thus a perfect module is always both
finitely generated and finitely presented, but the converse statements need
not hold. If $A$ is Noetherian with finite global dimension (which will be the
case for much of the paper, including the main theorems in Section 4), then an
$A$-module is perfect if and only it is finitely generated. We will also
consider other situations in which perfectness is equivalent to a more
familiar notion, and will try to indicate these as they arise.

Note the lack of symmetry in the finiteness conditions on $M$ and $N$; cf.
[kellercalabiyau]*Lem. 4.1. These conditions are imposed in order to force the
space $\operatorname{Ext}_{A}^{d-i}(N,M)$ on the right-hand side of the
duality formula to be finite dimensional over $\mathbb{K}$.

######  Remark 2.2.

If $A$ is internally $d$-Calabi–Yau with respect to $e$ then it is internally
$d$-Calabi–Yau with respect to $e+e^{\prime}$ for any idempotent
$e^{\prime}\in A$ orthogonal to $e$. An algebra $A$ is $d$-Calabi–Yau if and
only if it is internally $d$-Calabi–Yau with respect to $0$, and hence with
respect to every idempotent. At the other extreme, $A$ is internally
$d$-Calabi–Yau with respect to $1$ if and only if
$\operatorname{gl.dim}{A}\leq d$.

######  Remark 2.3.

A finite dimensional algebra $A$ is internally $d$-Calabi–Yau with respect to
$e$ if and only if the same is true of $A^{\mathrm{op}}$; since $A$ is finite
dimensional, it is Noetherian, so
$\operatorname{gl.dim}{A}=\operatorname{gl.dim}{A^{\mathrm{op}}}$
[weibelintroduction]*Ex. 4.1.1, and
$\mathrm{D}=\operatorname{Hom}_{\mathbb{K}}(-,\mathbb{K})$ induces an
equivalence
$\operatorname{mod}{A^{\mathrm{op}}}\stackrel{{\scriptstyle\sim}}{{\to}}(\operatorname{mod}{A})^{\mathrm{op}}$
yielding the required functorial duality for $A^{\mathrm{op}}$.

Definition 2.1 is not necessarily left-right symmetric in this way if $A$ is
infinite dimensional, so we will also make a stronger definition that does
have this property. Denote by
${A}^{\varepsilon}=A\mathbin{\otimes}_{\mathbb{K}}A^{\mathrm{op}}$ the
enveloping algebra of $A$, so that an $A$-bimodule is the same as an
${A}^{\varepsilon}$-module. Write
$\Omega_{A}=\operatorname{\mathbf{R}Hom}_{{A}^{\varepsilon}}(A,{A}^{\varepsilon})$.
We view $\Omega_{A}$ as a complex in $\mathcal{D}{{A}^{\varepsilon}}$ via the
‘inner’ multiplication on ${A}^{\varepsilon}$; for any homomorphism $f\colon
M\to{A}^{\varepsilon}$ of $A$-bimodules such that $f(m)=u\mathbin{\otimes}v$
and any $x\mathbin{\otimes}y\in{A}^{\varepsilon}$, let
$xfy(m)=uy\mathbin{\otimes}xv$.

Recall [amiotstable]*Defn. 2.1 that $A$ is said to be _bimodule
$d$-Calabi–Yau_ if $A\in\operatorname{per}{{A}^{\varepsilon}}$ (i.e. $A$ is
quasi-isomorphic to a bounded complex of finitely generated projective
$A$-bimodules) and there is an isomorphism
$A\stackrel{{\scriptstyle\sim}}{{\to}}\Omega_{A}[d]$ in
$\mathcal{D}{{A}^{\varepsilon}}$. This definition is slightly weaker than that
of Ginzburg [ginzburgcalabiyau]*3.2.5, as we will not need to impose any
‘self-duality’ condition on the isomorphism.

If $A$ is an algebra with quotient $\underline{A}$, write
$\mathcal{D}_{\underline{A}}(A)$ for the full subcategory of $\mathcal{D}{A}$
consisting of complexes with homology groups in
$\operatorname{Mod}{\underline{A}}$, and
$\mathcal{D}_{\mathrm{fd},\underline{A}}(A)$ for the full subcategory of
$\mathcal{D}_{\underline{A}}(A)$ consisting of objects with finite dimensional
total cohomology.

######  Definition 2.4.

An algebra $A$ is _bimodule internally $d$-Calabi–Yau_ with respect to an
idempotent $e\in A$ if

  * (i)

$\operatorname{p.dim}_{{A}^{\varepsilon}}{A}\leq d$,

  * (ii)

$A\in\operatorname{per}{{A}^{\varepsilon}}$, and

  * (iii)

there exists a triangle

${A}$${\Omega_{A}[d]}$${C}$${A[1]}$$\scriptstyle{\psi}$

in $\mathcal{D}{{A}^{\varepsilon}}$, such that

$\operatorname{\mathbf{R}Hom}_{A}(C,M)=0=\operatorname{\mathbf{R}Hom}_{A^{\mathrm{op}}}(C,N)$

for any $M\in\mathcal{D}_{\mathrm{fd},\underline{A}}(A)$ and
$N\in\mathcal{D}_{\mathrm{fd},\underline{A}^{\mathrm{op}}}(A^{\mathrm{op}})$,
where $\underline{A}=A/\langle e\rangle$.

######  Remark 2.5.

An algebra $A$ is bimodule internally $d$-Calabi–Yau with respect to $0$ if
and only if $\psi$ can be chosen to be a quasi-isomorphism, or equivalently if
$A$ is bimodule $d$-Calabi–Yau. In this case, (i) follows from (ii) and (iii)
[amiotstable]*Prop. 2.5(b). When $e\neq 0$, this implication does not hold,
and so we must make the stronger condition part of the definition. Note also
that (i) does not imply (ii), since it asserts only that $A$ has a finite
resolution by projective ${A}^{\varepsilon}$-modules, whereas (ii) requires
additionally that these modules are finitely generated.

An algebra $A$ is bimodule internally $d$-Calabi–Yau with respect to $1$ if
and only if $\operatorname{p.dim}_{{A}^{\varepsilon}}{A}\leq d$ and
$A\in\operatorname{per}{{A}^{\varepsilon}}$; in this case $\underline{A}=0$,
so condition (iii) is satisfied for any $\psi$.

######  Remark 2.6.

There is an isomorphism
${A}^{\varepsilon}\stackrel{{\scriptstyle\sim}}{{\to}}{(A^{\mathrm{op}})}^{\varepsilon}$
given by reversing the order of the tensor product. The resulting equivalence
$\operatorname{mod}{{A}^{\varepsilon}}\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{mod}{{(A^{\mathrm{op}})}^{\varepsilon}}$
takes $A$ to $A^{\mathrm{op}}$ (and $\underline{A}$ to
$\underline{A}^{\mathrm{op}}$). As a result, Definition 2.4 is left-right
symmetric, meaning that $A$ is bimodule internally $d$-Calabi–Yau with respect
to $e$ if and only if the same is true of $A^{\mathrm{op}}$.

The following lemma, due to Keller, allows us to recover dualities of
extension groups between $A$-modules from bimodule properties of $A$.

######  Lemma 2.7 ([kellercalabiyau]*Lem. 4.1).

Assume $A\in\operatorname{per}{{A}^{\varepsilon}}$. For all objects
$M,N\in\mathcal{D}{A}$ such that $M$ has finite dimensional total cohomology,
there is a functorial isomorphism

$\mathrm{D}\operatorname{Hom}_{\mathcal{D}{A}}(M,N)\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{Hom}_{\mathcal{D}{A}}(\Omega_{A}\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{A}N,M).$

If $A$ is bimodule $d$-Calabi–Yau, then $\Omega_{A}\cong A[-d]$ in
$\mathcal{D}{{A}^{\varepsilon}}$. It then follows from Lemma 2.7 that for any
$M,N\in\operatorname{Mod}{A}$, with $M$ finite dimensional, we have

$\displaystyle\mathrm{D}\operatorname{Ext}^{i}_{A}(M,N)$
$\displaystyle=\mathrm{D}\operatorname{Hom}_{\mathcal{D}{A}}(M,N[i])$
$\displaystyle\cong\operatorname{Hom}_{\mathcal{D}{A}}(\Omega_{A}\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{A}N[i],M)$
$\displaystyle\cong\operatorname{Hom}_{\mathcal{D}{A}}(N[i-d],M)$
$\displaystyle=\operatorname{Ext}^{d-i}_{A}(N,M).$

We now use Lemma 2.7 to prove a similar result for bimodule internally
$d$-Calabi–Yau algebras.

######  Theorem 2.8.

Let $A$ be bimodule internally $d$-Calabi–Yau with respect to $e$, and let
$\underline{A}=A/\langle e\rangle$. Then for any $N\in\mathcal{D}{A}$ and any
$M\in\mathcal{D}_{\mathrm{fd},\underline{A}}(A)$, we have a functorial
isomorphism

$\mathrm{D}\operatorname{Hom}_{\mathcal{D}{A}}(M,N)=\operatorname{Hom}_{\mathcal{D}{A}}(N[-d],M).$

###### Proof.

Pick a triangle

${A[-d]}$${\Omega_{A}}$${C}$${A[1-d]}$$\scriptstyle{\psi}$

by applying $[-d]$ to a triangle as in Definition 2.4. Applying
$-\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{A}N$ yields a
triangle

${N[-d]}$${\Omega_{A}\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{A}N}$${C\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{A}N}$${N[1-d]}$

in $\mathcal{D}{A}$. Now apply $\operatorname{\mathbf{R}Hom}_{A}(-,M)$, to get
a triangle

${\operatorname{\mathbf{R}Hom}_{A}(\Omega_{A}\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{A}N,M)}$${\operatorname{\mathbf{R}Hom}_{A}(N[-d],M)}$${\operatorname{\mathbf{R}Hom}_{A}(C\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{A}N,M)}$$\scriptstyle{[1]}$

Since $M\in\mathcal{D}_{\mathrm{fd},\underline{A}}(A)$, we have
$\operatorname{\mathbf{R}Hom}_{A}(C,M)=0$ by definition, and so

$\operatorname{\mathbf{R}Hom}_{A}(C\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{A}N,M)=\operatorname{\mathbf{R}Hom}_{A}(N,\operatorname{\mathbf{R}Hom}_{A}(C,M))=0.$

Thus
$\operatorname{\mathbf{R}Hom}_{A}(\Omega_{A}\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{A}N,M)\cong\operatorname{\mathbf{R}Hom}_{A}(N[-d],M)$.
We obtain the desired result by taking $0$-th cohomology and applying Lemma
2.7. ∎

######  Corollary 2.9.

If $A$ is bimodule internally $d$-Calabi–Yau with respect to $e$, then it is
internally $d$-Calabi–Yau with respect to $e$, and moreover we have a
functorial isomorphism

$\mathrm{D}\operatorname{Ext}_{A}^{i}(M,N)=\operatorname{Ext}_{A}^{d-i}(N,M)$

for any finite dimensional $M\in\operatorname{mod}{A/\langle e\rangle}$ and
any $N\in\operatorname{Mod}{A}$, which need not be perfect as $A$-modules.

###### Proof.

Since $\operatorname{p.dim}_{{A}^{\varepsilon}}{A}\leq d$, there is an exact
sequence

${0}$${P_{d}}$${\dotsb}$${P_{1}}$${P_{0}}$${A}$${0}$

of $A$-bimodules, in which each $P_{i}$ is a projective $A$-bimodule. If $X$
is any $A$-module, then $P_{i}\mathbin{\otimes}_{A}X$ is a projective
$A$-module, and so applying $-\mathbin{\otimes}_{A}X$ to the above sequence
gives a projective resolution

${0}$${P_{d}\mathbin{\otimes}_{A}X}$${\dotsb}$${P_{1}\mathbin{\otimes}_{A}X}$${P_{0}\mathbin{\otimes}_{A}X}$${X}$${0}$

of $X$. It follows that $\operatorname{gl.dim}{A}\leq d$.

Now by Theorem 2.8, if $N\in\operatorname{Mod}{A}$ and
$M\in\operatorname{mod}{A/\langle e\rangle}$ is finite dimensional, we have

$\displaystyle\mathrm{D}\operatorname{Ext}^{i}_{A}(M,N)$
$\displaystyle=\mathrm{D}\operatorname{Hom}_{\mathcal{D}{A}}(M,N[i])$
$\displaystyle=\operatorname{Hom}_{\mathcal{D}{A}}(N[i-d],M)$
$\displaystyle=\operatorname{Ext}^{d-i}_{A}(N,M).$

In particular, this duality applies when $M$ and $N$ are perfect $A$-modules,
so $A$ is internally $d$-Calabi–Yau with respect to $e$. ∎

While Theorem 2.8 and 2.9 show that bimodule internally Calabi–Yau algebras
have stronger properties than those required by the definition of internally
Calabi–Yau, with the duality formula applying to a wider class of objects, we
do not currently know of any examples of internally Calabi–Yau algebras which
are not bimodule internally Calabi–Yau. It seems unlikely that the two classes
of algebras coincide, but it is tempting to speculate based on work of
Bocklandt [bocklandtgraded] that if $A$ is an internally Calabi–Yau algebra
admitting a suitable grading, then it is also bimodule internally Calabi–Yau.

######  Example 2.10.

Consider the algebra $A=\mathbb{K}Q/I$ given by the quiver

$Q=\mathord{\leavevmode\hbox to68.57pt{\vbox
to46.15pt{\pgfpicture\makeatletter\hbox{\hskip 5.83301pt\lower-20.7816pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{11.00417pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{54.40552pt}{11.00417pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.95276pt}{-17.4486pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}{}{
{}{}{}}{}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{
{\pgfsys@beginscope\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@miterjoin{}
{} {}
\pgfsys@moveto{-2.19997pt}{-2.39996pt}\pgfsys@lineto{0.2pt}{0.0pt}\pgfsys@lineto{-2.19997pt}{2.39996pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{6.033pt}{14.22638pt}\pgfsys@lineto{50.38972pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{50.38972pt}{14.22638pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.23381pt}{19.02248pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\alpha_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{}{}}{}{
{}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{50.87251pt}{8.19348pt}\pgfsys@lineto{34.82712pt}{-7.85292pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.70708}{-0.70712}{0.70712}{-0.70708}{34.82713pt}{-7.85292pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{46.21214pt}{-6.5473pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\alpha_{2}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{}{}}{}{
{}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{22.41975pt}{-8.19348pt}\pgfsys@lineto{6.37436pt}{7.85292pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.70708}{0.70712}{-0.70712}{-0.70708}{6.37437pt}{7.85292pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.25548pt}{-6.54648pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\alpha_{3}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$

with ideal of relations
$I=\langle\alpha_{2}\alpha_{1},\alpha_{1}\alpha_{3}\rangle$. This is an
example of a frozen Jacobian algebra; see Definition 5.1 and Example 5.3. One
can check (for example, by Theorem 5.6 below) that $A$ is bimodule internally
$3$-Calabi–Yau with respect to $e_{1}+e_{2}$, and so both $A$ and
$A^{\mathrm{op}}$ are internally $3$-Calabi–Yau with respect to this
idempotent.

Similarly, the algebra $A^{\prime}=\mathbb{K}Q^{\prime}/I^{\prime}$ given by
the quiver

$Q^{\prime}=\mathord{\leavevmode\hbox to125.48pt{\vbox
to60.38pt{\pgfpicture\makeatletter\hbox{\hskip
62.73853pt\lower-32.48256pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-59.40552pt}{-24.56178pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{18.11736pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{54.40552pt}{-24.56178pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-24.56178pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}{}{
{}{}{}}{}{ {}{}{}}{}{
{}{}{}}{}\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{-50.87251pt}{-16.81537pt}\pgfsys@lineto{-6.41922pt}{16.5278pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.79997}{0.60004}{-0.60004}{0.79997}{-6.41924pt}{16.5278pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\alpha_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{}{}}{}{
{}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{14.58435pt}\pgfsys@lineto{0.0pt}{-14.10156pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
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}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
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}\hbox{\scriptsize{$\alpha_{3}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
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}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\alpha_{4}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{}{}}{}{
{}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{6.033pt}{-21.33957pt}\pgfsys@lineto{50.38972pt}{-21.33957pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{50.38972pt}{-21.33957pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.23381pt}{-27.88646pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\alpha_{5}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{}{}}{}{
{}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
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}\pgfsys@transformcm{-0.79997}{0.60004}{-0.60004}{-0.79997}{6.41924pt}{16.5278pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.98576pt}{4.79715pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\alpha_{2}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$

with ideal of relations
$I^{\prime}=\langle\alpha_{3}\alpha_{1},\alpha_{1}\alpha_{4}-\alpha_{2}\alpha_{5},\alpha_{3}\alpha_{2}\rangle$
is bimodule internally $3$-Calabi–Yau with respect to $e_{1}+e_{2}+e_{3}$.

##  3\. Frobenius $m$-cluster categories

This section is devoted to describing a class of categories, which we term
Frobenius $m$-cluster categories, providing us with a rich source of
internally $(m+1)$-Calabi–Yau algebras. Indeed, certain categories in this
class motivated Definition 2.1.

Recall that an exact category $\mathcal{E}$ is Frobenius if it has enough
projective objects and enough injective objects, and
$\operatorname{proj}{\mathcal{E}}=\operatorname{inj}{\mathcal{E}}$. By a
famous result of Happel [happeltriangulated]*§I.2, the stable category
$\underline{\mathcal{E}}=\mathcal{E}/\operatorname{proj}{\mathcal{E}}$ is
triangulated, with suspension given by the inverse syzygy $\Omega^{-1}$,
taking an object to the cokernel of an injective hull.

######  Definition 3.1.

Let $\mathcal{E}$ be a Frobenius category, and let $m\geq 1$. We say that
$\mathcal{E}$ is _stably $m$-Calabi–Yau_ if its stable category
$\underline{\mathcal{E}}$ is $m$-Calabi–Yau, meaning that
$\underline{\mathcal{E}}$ is $\operatorname{Hom}$-finite, and there is a
functorial duality

$\mathrm{D}\operatorname{\underline{Hom}}_{\mathcal{E}}(X,Y)=\operatorname{\underline{Hom}}_{\mathcal{E}}(Y,\Omega^{-m}X)$

for all $X,Y\in\mathcal{E}$. Here
$\operatorname{\underline{Hom}}_{\mathcal{E}}(X,Y)$ denotes the space of
morphisms $X$ to $Y$ in $\underline{\mathcal{E}}$.

######  Definition 3.2.

Let $\mathcal{E}$ be an exact category, and let
$\mathcal{T}\subset\mathcal{E}$ be a full and functorially finite subcategory
closed under direct sums and direct summands. We say $\mathcal{T}$ is an _$m$
-cluster-tilting subcategory_ if

$\\{X\in\mathcal{E}:\operatorname{Ext}^{i}_{\mathcal{E}}(X,\mathcal{T})=0,\>0<i<m\\}=\mathcal{T}=\\{X\in\mathcal{E}:\operatorname{Ext}^{i}_{\mathcal{E}}(\mathcal{T},X)=0,\>0<i<m\\}.$

Here ‘$\operatorname{Ext}^{i}_{\mathcal{E}}(X,\mathcal{T})=0$’ is taken to
mean ‘$\operatorname{Ext}^{i}_{\mathcal{E}}(X,T)=0$ for all
$T\in\mathcal{T}$’. We say an object $T\in\mathcal{E}$ is an _$m$ -cluster-
tilting object_ if $\operatorname{add}{T}$ is an $m$-cluster-tilting
subcategory.

If $\mathcal{E}$ is a Frobenius category, then for any $X,Y\in\mathcal{E}$ and
$i>0$, we have

$\operatorname{Ext}^{i}_{\mathcal{E}}(X,Y)=\operatorname{\underline{Hom}}_{\mathcal{E}}(X,\Omega^{-i}Y).$

Thus if $\mathcal{E}$ is stably $m$-Calabi–Yau, the two equalities appearing
in Definition 3.2 are equivalent to one another.

######  Definition 3.3.

Let $\mathcal{E}$ be a Frobenius category, and $m\geq 1$. Then $\mathcal{E}$
is called a _Frobenius $m$-cluster category_ if it is idempotent complete,
stably $m$-Calabi–Yau, and
$\operatorname{gl.dim}{\operatorname{End}_{\mathcal{E}}(T)^{\mathrm{op}}}\leq
m+1$ for any $m$-cluster-tilting object $T$, of which there is at least one.
If these properties hold for $m=2$, then $\mathcal{E}$ will be called simply a
_Frobenius cluster category_.

Note that while $\underline{\mathcal{E}}$ is $\operatorname{Hom}$-finite for
any Frobenius $m$-cluster category $\mathcal{E}$, we do not assume that
$\mathcal{E}$ itself is $\operatorname{Hom}$-finite (cf. Example 3.12). By the
following result, which follows immediately from a theorem of Keller–Reiten
[kellerclustertilted]*5.4, Frobenius $m$-cluster categories provide a rich
source of examples of internally Calabi–Yau algebras.

######  Theorem 3.4.

Let $\mathcal{E}$ be a Frobenius $m$-cluster category, let $T\in\mathcal{E}$
be a basic $m$-cluster-tilting object, and write
$A=\operatorname{End}_{\mathcal{E}}(T)^{\mathrm{op}}$. Let $P$ be a maximal
projective-injective summand of $T$, and let $e\in A$ be the idempotent given
by projection onto $P$. Then $A$ is internally $(m+1)$-Calabi–Yau with respect
to $e$.

###### Proof.

Let $M$ and $N$ be perfect $A$-modules, and assume that $M$ is a finite
dimensional $A/\langle e\rangle$-module. Then by [kellerclustertilted]*5.4,
there is a functorial duality

$\displaystyle\operatorname{Ext}^{i}_{A}(M,N)$
$\displaystyle=\operatorname{Hom}_{\operatorname{per}{A}}(M[-i],N)$
$\displaystyle=\mathrm{D}\operatorname{Hom}_{\operatorname{per}{A}}(N,M[m+1-i])=\mathrm{D}\operatorname{Ext}^{m+1-i}_{A}(N,M)$

for all $i$. Since $\operatorname{gl.dim}{A}\leq m+1$ by assumption, we have
the desired result. ∎

######  Remark 3.5.

If $\mathcal{E}$ is a Frobenius $m$-cluster category such that
$A=\operatorname{End}_{\mathcal{E}}(T)^{\mathrm{op}}$ is Noetherian for any
$m$-cluster tilting object $T$, then $\mathcal{E}^{\mathrm{op}}$ is a
Frobenius $m$-cluster category with the same $m$-cluster-tilting objects as
$\mathcal{E}$; the extra assumption is used to ensure that

$\operatorname{gl.dim}A^{\mathrm{op}}=\operatorname{gl.dim}{A}\leq m+1.$

Thus, under these circumstances, $A^{\mathrm{op}}$ is also internally
$(m+1)$-Calabi–Yau with respect to $e$.

In the context of Theorem 3.4, we have a more straightforward characterisation
of the $A/\langle e\rangle$-modules that are perfect as $A$-modules, again
coming from Keller–Reiten’s work.

######  Proposition 3.6.

Let $\mathcal{E}$ be a Frobenius $m$-cluster category, let $T\in\mathcal{E}$
be a basic $m$-cluster-tilting object, and write
$A=\operatorname{End}_{\mathcal{E}}(T)^{\mathrm{op}}$. Let $P$ be a maximal
projective-injective summand of $T$, let $e\in A$ be the idempotent given by
projection onto $P$ and write $\underline{A}=A/\langle e\rangle$. Then an
$\underline{A}$-module $M$ is perfect as an $A$-module if and only if it is
finitely generated.

###### Proof.

Let $M\in\operatorname{mod}{\underline{A}}$. If $M$ is perfect over $A$, then
it is in particular finitely generated over $A$, and thus (by the same
generating set) over $\underline{A}$. Since
$\underline{A}=\operatorname{\underline{End}}_{\mathcal{E}}(T)^{\mathrm{op}}$
and $\underline{\mathcal{E}}$ is $\operatorname{Hom}$-finite, $\underline{A}$
is finite dimensional. It follows that any finitely generated
$\underline{A}$-module is finitely presented over $\underline{A}$, and thus
perfect over $A$ by [kellerclustertilted]*5.4(c). ∎

The following proposition, which is based on work of Iyama
[iyamaauslander]*§2, [iyamahigher]*Thm. 3.6.2, gives sufficient conditions on
a Frobenius category $\mathcal{E}$ for us to conclude that
$\operatorname{gl.dim}{\operatorname{End}_{\mathcal{E}}(T)^{\mathrm{op}}}\leq
m+1$ for any $m$-cluster-tilting object $T\in\mathcal{E}$, as well as
providing a more straightforward characterisation of the perfect modules over
this endomorphism algebra.

######  Proposition 3.7.

Let $\mathcal{E}\subseteq\mathcal{A}$ be a full, extension closed, Frobenius
subcategory of an abelian category $\mathcal{A}$ such that

  * (i)

$\mathcal{A}$ has enough projectives,

  * (ii)

$\mathcal{E}$ contains $\operatorname{proj}{\mathcal{A}}$,

  * (iii)

$\mathcal{E}$ is closed under kernels of epimorphisms, and

  * (iv)

for any $X\in\mathcal{A}$, and any exact sequence

${0}$${Y}$${P_{m}}$${\cdots}$${P_{0}}$${X}$${0}$

with $P_{i}\in\operatorname{proj}{\mathcal{A}}$, we have $Y\in\mathcal{E}$.

Let $T\in\mathcal{E}$ be $m$-cluster-tilting, and write
$A=\operatorname{End}_{\mathcal{E}}(T)^{\mathrm{op}}$. Then an $A$-module $M$
is perfect if and only if it is finitely presented, and for such $M$ we have
$\operatorname{p.dim}_{A}{M}\leq m+1$. It follows that if $A$ is Noetherian,
then $\operatorname{gl.dim}{A}\leq m+1$.

###### Proof.

As already noted, any perfect $A$-module is finitely presented. For the
converse, let $M$ be a finitely presented $A$-module with presentation

${\operatorname{Hom}_{\mathcal{E}}(T,T_{1})}$${\operatorname{Hom}_{\mathcal{E}}(T,T_{0})}$${M}$${0,}$$\scriptstyle{f_{*}}$

for some $T_{0},T_{1}\in\operatorname{add}{T}$, and form the exact sequence

${0}$${K_{1}}$${T_{1}}$${T_{0}}$$\scriptstyle{f}$

in $\mathcal{A}$. By [iyamaauslander]*Prop. 2.6, the subcategory $\mathcal{E}$
is contravariantly finite in $\mathcal{A}$, and hence so is
$\operatorname{add}{T}$. Working inductively for $1\leq j\leq m-1$, let
$r_{j}\colon T_{j+1}\to K_{j}$ be a right
$\operatorname{add}{T}$-approximation of $K_{j}$, and let $K_{j+1}$ be its
kernel (in $\mathcal{A}$). By additionally defining
$K_{0}=\operatorname{im}{f}$ and $K_{-1}=\operatorname{coker}{f}$, we obtain
exact sequences

${0}$${K_{j}}$${T_{j}}$${K_{j-1}}$${0}$

for all $0\leq j\leq m$. Combining these to form the exact sequence

${0}$${K_{m}}$${T_{m}}$${\cdots}$${T_{0}}$${K_{-1}}$${0,}$

we may use [iyamaauslander]*Prop. 2.6 again to see that $K_{m}\in\mathcal{E}$.

Next we check that $K_{m}\in\operatorname{add}{T}$, which we do by checking
that $\operatorname{Ext}^{i}_{\mathcal{E}}(T,K_{m})=0$ for $0<i<m$. For any
$j$, we may apply $\operatorname{Hom}_{\mathcal{E}}(T,-)$ to the short exact
sequence

${0}$${K_{j+1}}$${T_{j+1}}$${K_{j}}$${0}$

to find, using that $\operatorname{Ext}^{i}_{\mathcal{E}}(T,T_{i})=0$ for
$0<i<m$, that
$\operatorname{Ext}^{i}_{\mathcal{E}}(T,K_{j+1})=\operatorname{Ext}^{i-1}_{\mathcal{E}}(T,K_{j})$
for $1<i<m$. Moreover, if $j\geq 1$, the map $T_{j+1}\to K_{j}$ in the above
sequence is the right $\operatorname{add}{T}$-approximation $r_{j}$, for which
$\operatorname{Hom}_{\mathcal{E}}(T,r_{j})$ is surjective, so
$\operatorname{Ext}^{1}_{\mathcal{E}}(T,K_{j+1})=0$. It follows that for
$0<i<m$, we have

$\operatorname{Ext}^{i}_{\mathcal{E}}(T,K_{m})=\operatorname{Ext}^{i-1}_{\mathcal{E}}(T,K_{m-1})=\dotsb=\operatorname{Ext}^{1}_{\mathcal{E}}(T,K_{m-i+1})=0,$

as required.

It follows from the above calculations that the sequence

${0}$${\operatorname{Hom}_{\mathcal{E}}(T,K_{j+1})}$${\operatorname{Hom}_{\mathcal{E}}(T,T_{j+1})}$${\operatorname{Hom}_{\mathcal{E}}(T,K_{j})}$${0}$

is exact for all $j\geq 1$. Thus, writing
$T_{m+1}=K_{m}\in\operatorname{add}{T}$, applying
$\operatorname{Hom}_{\mathcal{E}}(T,-)$ to the sequence

${0}$${T_{m+1}}$${T_{m}}$${\cdots}$${T_{1}}$${T_{0}}$$\scriptstyle{f}$

yields a projective resolution of $M$ by finitely generated projective
$A$-modules, and so $M$ is perfect with $\operatorname{p.dim}_{A}{M}\leq m+1$.

For the final statement, if $A$ is Noetherian, then every finitely generated
$A$-module is also finitely presented, and thus has projective dimension at
most $m+1$, as required. ∎

We now recall some work of Iyama–Kalck–Wemyss–Yang [iyamafrobenius], which
gives a normal form for a Frobenius $m$-cluster category that is ‘small
enough’, in the sense it admits an $m$-cluster-tilting object with Noetherian
endomorphism algebra. It will follow from this description that any such
Frobenius $m$-cluster category may be embedded into an abelian category in
such a way that all of the assumptions of Proposition 3.7 hold. We begin with
the following definitions.

######  Definition 3.8.

An algebra $B$ is _Iwanaga–Gorenstein_ if it is Noetherian and has finite
injective dimension as both a left and right module over itself. In this case,
the left and right injective dimensions coincide, and are called the
_Gorenstein dimension_ of $B$. For brevity, an Iwanaga–Gorenstein algebra with
Gorenstein dimension $d$ will be called _$d$ -Iwanaga–Gorenstein_. If $B$ is
such an algebra, we write

$\operatorname{GP}(B)=\\{X\in\operatorname{mod}{B}:\operatorname{Ext}^{i}_{B}(X,B)=0,\
i>0\\}=\Omega^{d}(\operatorname{mod}{B})$

for the category of _Gorenstein projective_ $B$-modules. This is a full,
extension closed subcategory of $\operatorname{mod}{B}$, and is a Frobenius
category under the inherited exact structure [buchweitzmaximal]*§4.8. This
category is sometimes [amiotstable], [jensencategorification] denoted by
$\operatorname{CM}(B)$, and the objects called maximal Cohen–Macaulay
$B$-modules, but we do not do this since for certain choices of $B$ there
exist definitions of Cohen–Macaulay not coinciding with this one
[iyamafrobenius]*Rem. 3.3.

If $B$ is an $(m+1)$-Iwanaga–Gorenstein algebra, then the assumptions of
Proposition 3.7 hold for the exact subcategory
$\mathcal{E}=\operatorname{GP}(B)$ of the abelian category
$\mathcal{A}=\operatorname{mod}{B}$. Thus the endomorphism algebra of an
$m$-cluster-tilting object of $\operatorname{GP}(B)$ has global dimension at
most $m+1$ whenever it is Noetherian.

######  Theorem 3.9 ([iyamafrobenius]*Thm. 2.7).

Let $\mathcal{E}$ be an idempotent complete Frobenius category such that
$\operatorname{proj}{\mathcal{E}}=\operatorname{add}{P}$ for some
$P\in\mathcal{E}$. Assume there exists $M\in\mathcal{E}$ such that
$P\in\operatorname{add}{M}$, the endomorphism algebra
$A=\operatorname{End}_{\mathcal{E}}(M)^{\mathrm{op}}$ is Noetherian, and
$\operatorname{gl.dim}{A}=d<\infty$. Then
$B=\operatorname{End}_{\mathcal{E}}(P)^{\mathrm{op}}$ is Iwanaga–Gorenstein of
Gorenstein dimension at most $d$, and there is an equivalence

$\operatorname{Hom}_{\mathcal{E}}(P,-)\colon\mathcal{E}\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{GP}(B).$

######  Corollary 3.10.

Let $\mathcal{E}$ be a Frobenius $m$-cluster category and let
$T\in\mathcal{E}$ be an $m$-cluster-tilting object such that
$\operatorname{End}_{\mathcal{E}}(T)^{\mathrm{op}}$ is Noetherian. Let $P$ be
a maximal projective summand of $T$, and write
$B=\operatorname{End}_{\mathcal{E}}(P)^{\mathrm{op}}$. Then $B$ is
Iwanaga–Gorenstein of Gorenstein dimension at most $m+1$, and
$\mathcal{E}\simeq\operatorname{GP}(B)$.

###### Proof.

By definition, $\mathcal{E}$ is idempotent complete and
$\operatorname{gl.dim}{\operatorname{End}_{\mathcal{E}}(T)^{\mathrm{op}}}\leq
m+1$. Since $T$ is $m$-cluster-tilting,
$\operatorname{proj}{\mathcal{E}}\subseteq\operatorname{add}{T}$, and so we
have $\operatorname{proj}{\mathcal{E}}=\operatorname{add}{P}$. Now the result
follows from Theorem 3.9. ∎

Under the notation and assumptions of 3.10, let $e$ be the idempotent of
$A=\operatorname{End}_{\mathcal{E}}(T)^{\mathrm{op}}$ given by projection onto
$P$, with respect to which $A$ is internally $(m+1)$-Calabi–Yau by Theorem
3.4. Then the algebra $B=\operatorname{End}_{\mathcal{E}}(P)^{\mathrm{op}}$ is
the idempotent subalgebra $eAe$. We will see below that the Gorenstein
dimension of $B$ may be strictly less than $m+1$. An interesting question
suggested by 3.10, to which we have no good answer at this stage, is the
following: can one find reasonable conditions on an Iwanaga–Gorenstein algebra
$B$ under which $\operatorname{GP}(B)$ is a Frobenius $m$-cluster category?

The remainder of the section is devoted to examples. We describe two families
of examples of Frobenius cluster categories, one $\operatorname{Hom}$-finite
and the other $\operatorname{Hom}$-infinite, to which Theorem 3.4 applies to
show that the endomorphism algebras of cluster-tilting objects are internally
$3$-Calabi–Yau with respect to projection onto a maximal projective-injective
summand. We also give a family of examples of Frobenius $1$-cluster categories
arising as part of the algebraic McKay correspondence.

######  Example 3.11.

Buan–Iyama–Reiten–Scott [buancluster] construct a family of
$\operatorname{Hom}$-finite stably $2$-Calabi–Yau Frobenius categories
$\operatorname{Sub}{\Pi_{\omega}}$. Here $\Pi=\Pi(\Delta)$ is the
preprojective algebra associated to a graph $\Delta$, and $\omega$ is a finite
product of simple reflections in the Weyl group of $\Delta$. The algebra
$\Pi_{\omega}$ is a (finite dimensional) quotient of $\Pi$, and
$\operatorname{Sub}{\Pi_{\omega}}$ is the full subcategory of
$\operatorname{mod}{\Pi_{\omega}}$ given by objects isomorphic to submodules
of direct sums of copies of $\Pi_{\omega}$. Then
$\operatorname{Sub}{\Pi_{\omega}}$ is closed under extensions and subobjects
(in particular under kernels of epimorphisms), and contains
$\operatorname{proj}{\Pi_{\omega}}$ and
$\Omega(\operatorname{mod}{\Pi_{\omega}})$. Since
$\operatorname{Sub}{\Pi_{\omega}}$ is $\operatorname{Hom}$-finite, it follows
from Proposition 3.7 that it is a Frobenius cluster category.

We note that the categories $\operatorname{Sub}{\Pi_{\omega}}$ constructed by
Buan–Iyama–Reiten–Scott contain an important class of categories considered by
Geiß–Leclerc–Schröer [geisspartial], which we will also describe. For $\Delta$
a Dynkin diagram, let $\Pi$ be the preprojective algebra of type $\Delta$. If
$j$ is a node of $\Delta$, write $Q_{j}$ for the injective $\Pi$-module with
socle at $j$. Then for any subset $J$ of the nodes of $\Delta$, we may write
$Q_{J}=\bigoplus_{j\in J}Q_{j}$, and consider the category
$\operatorname{Sub}{Q_{J}}$ of $\Pi$-modules isomorphic to a submodule of a
direct sum of copies of $Q_{J}$, or equivalently of those $\Pi$-modules with
socle supported on $J$. The category $\operatorname{Sub}{Q_{J}}$ models a
cluster algebra structure on the coordinate ring of a dense open subset of the
partial flag variety attached to the data of $\Delta$ and $J$. For example,
when $\Delta$ is of type $\mathsf{A}_{n}$ and $J$ consists of a single node,
this partial flag variety is a Grassmannian. If $\omega_{0}$ is the longest
word in the Weyl group of type $\Delta$, and $\omega_{0}^{K}$ is the longest
word in the subgroup generated by simple reflections at nodes not in $J$, then
[geisskacmoody]*Lem. 17.2 we have

$\operatorname{Sub}{Q_{J}}=\operatorname{Sub}{\Pi_{\omega_{0}^{K}\omega_{0}}}.$

In particular, the categories $\operatorname{Sub}{Q_{J}}$ are Frobenius
cluster categories.

If $\Pi$ is the preprojective algebra of Dynkin type $\Delta$, then we have
$\operatorname{mod}{\Pi}=\operatorname{Sub}{\Pi}=\operatorname{Sub}{\Pi_{\omega_{0}}}$,
where $\omega_{0}$ is the longest word in the Weyl group of type $\Delta$, so
$\operatorname{mod}{\Pi}$ is a Frobenius cluster category. The algebra $A$
appearing in Example 2.10 is isomorphic to the endomorphism algebra of a
cluster-tilting object in $\operatorname{mod}{\Pi}$ for $\Pi$ the
preprojective algebra of type $\mathsf{A}_{2}$, and is thus internally
$3$-Calabi–Yau by Theorem 3.4. Similarly, the algebra $A^{\prime}$ from
Example 2.10 is isomorphic to the endomorphism algebra of a cluster-tilting
object in
$\operatorname{Sub}{\Pi_{s_{2}s_{1}s_{3}s_{2}}}=\operatorname{Sub}{Q_{2}}$,
where $\Pi$ is the preprojective algebra of type $\mathsf{A}_{3}$ and $Q_{2}$
is the indecomposable injective module with socle at the bivalent vertex $2$,
and so $A^{\prime}$ is also internally $3$-Calabi–Yau.

The category of projective objects of $\operatorname{Sub}{\Pi_{\omega}}$ is
given by $\operatorname{add}{\Pi_{\omega}}$. Since
$\operatorname{Sub}{\Pi_{\omega}}$ is a $\operatorname{Hom}$-finite Frobenius
cluster category, it follows from 3.10 that
$\operatorname{Sub}{\Pi_{\omega}}\simeq\operatorname{GP}(\Pi_{\omega})$. Since
$\Pi_{w}$ has Gorenstein dimension at most $1$ [buancluster]*Prop. III.2.2, we
even have $\operatorname{Sub}{\Pi_{\omega}}=\operatorname{GP}(\Pi_{\omega})$
as full subcategories of $\operatorname{mod}{\Pi_{w}}$. Note that the
Gorenstein dimension of $\Pi_{w}$ is strictly smaller than the bound provided
by 3.10.

######  Example 3.12.

Our second family of examples was introduced by Jensen–King–Su
[jensencategorification] to categorify the cluster algebra structure on the
homogeneous coordinate ring of the Grassmannian $G_{k}^{n}$ of $k$-planes in
$\mathbb{C}^{n}$. Each category in this family is of the form
$\operatorname{CM}(B)$ for a Gorenstein order $B$ (depending on positive
integers $1<k<n$) over $Z=\mathbb{C}[[t]]$. One description of $B$ is as
follows. Let $\Delta$ be the graph (of affine type
$\widetilde{\mathsf{A}}_{n-1}$) with vertex set given by the cyclic group
$\mathbb{Z}_{n}$, and edges between vertices $i$ and $i+1$ for all $i$. Let
$\Pi$ be the completion of the preprojective algebra on $\Delta$ with respect
to the arrow ideal. Write $x$ for the sum of ‘clockwise’ arrows $i\to i+1$,
and $y$ for the sum of ‘anti-clockwise’ arrows $i\to i-1$. Then we have

$B=\Pi/\langle x^{k}-y^{n-k}\rangle.$

In this description, $Z$ may be identified with the centre $\mathbb{C}[[xy]]$
of $B$.

Objects of $\operatorname{CM}(B)$ are $B$-modules that are free and finitely
generated over $Z$. Since $Z$ is a principal ideal domain, and hence
Noetherian, any submodule of a free and finitely generated $Z$-module is also
free and finitely generated, and so $\operatorname{CM}(B)$ is closed under
subobjects. In particular, $\operatorname{CM}(B)$ is closed under kernels of
epimorphisms. Moreover [jensencategorification]*Cor. 3.7,
$B\in\operatorname{CM}(B)$, and so
$\Omega(\operatorname{mod}{B})\subseteq\operatorname{CM}(B)$.

As a $Z$-module, any object $M\in\operatorname{CM}(B)$ is isomorphic to
$Z^{k}$ for some $k$, and so $\operatorname{End}_{Z}(M)^{\mathrm{op}}\cong
Z^{k^{2}}$ is a finitely generated $Z$-module. Since $Z$ is Noetherian, the
algebra
$\operatorname{End}_{B}(M)^{\mathrm{op}}\subseteq\operatorname{End}_{Z}(M)^{\mathrm{op}}$
is also finitely generated as a $Z$-module. Thus
$\operatorname{End}_{B}(M)^{\mathrm{op}}$ is Noetherian, as it is finitely
generated as a module over the commutative Noetherian ring $Z$. We may now
apply Proposition 3.7 to see that any cluster-tilting object
$T\in\operatorname{CM}(B)$ satisfies
$\operatorname{gl.dim}{\operatorname{End}_{B}(T)^{\mathrm{op}}}\leq 3$.
Moreover [jensencategorification]*Cor. 4.6,
$\operatorname{\underline{CM}}(B)=\operatorname{\underline{Sub}}{Q_{k}}$,
where $Q_{k}$ is an indecomposable injective module for the preprojective
algebra of type $\mathsf{A}_{n-1}$ (see Example 3.11), so
$\operatorname{\underline{CM}}(B)$ is $2$-Calabi–Yau, and
$\operatorname{CM}(B)$ is a Frobenius cluster category. This category is not
$\operatorname{Hom}$-finite, unlike the categories
$\operatorname{Sub}{\Pi_{\omega}}$.

The algebra $B$ is $1$-Iwanaga–Gorenstein, so the Gorenstein dimension is
again strictly smaller than the bound in 3.10, and we have equalities
$\operatorname{CM}(B)=\operatorname{GP}(B)=\operatorname{Sub}{B}$
[jensencategorification]*Cor. 3.7.

Baur–King–Marsh [baurdimer]*Thm. 10.3 show that for certain cluster-tilting
objects $T\in\operatorname{CM}(B)$, the endomorphism algebra
$\operatorname{End}_{B}(T)^{\mathrm{op}}$ is isomorphic to a frozen Jacobian
algebra (Definition 5.1) associated to a dimer model on a disk, with the
projection onto a maximal projective-injective summand corresponding to the
sum of idempotents at the frozen vertices. By Theorem 3.4, these dimer
algebras, which satisfy a natural consistency condition [baurdimer]*§5, are
internally $3$-Calabi–Yau with respect to their boundary idempotent; cf.
Broomhead [broomheaddimer]*§7, who shows that consistent dimer models on
closed surfaces give rise to $3$-Calabi–Yau Jacobian algebras.

######  Example 3.13.

The algebraic McKay correspondence provides many examples of Frobenius
$1$-cluster categories. Let the special linear group
$\operatorname{SL}_{2}(\mathbb{C})$ act on $\mathbb{C}[[x,y]]$ in the natural
way. Let $G$ be a finite subgroup of $\operatorname{SL}_{2}(\mathbb{C})$, and
consider the invariant ring $R=\mathbb{C}[[x,y]]^{G}$. For example, if $G$ is
cyclic of order $n$, generated by

$\begin{pmatrix}\omega&0\\\ 0&\omega^{-1}\end{pmatrix}$

for some primitive $n$-th root of unity $\omega$, then
$R=\mathbb{C}[[x^{n},xy,y^{n}]]$.

A well-known result of Herzog [herzogringe] shows that $\mathbb{C}[[x,y]]$ is
an additive generator (or equivalently, a $1$-cluster-tilting object) of the
Frobenius category $\operatorname{CM}(R)$ of maximal Cohen–Macaulay
$R$-modules. By computing Auslander–Reiten sequences in
$\operatorname{CM}(R)$, as in Leuschke–Wiegand [leuschkecohenmacaulay]*Prop.
13.22, one can see that the Auslander–Reiten translation on
$\operatorname{CM}(R)$ is trivial, and so $\operatorname{CM}(R)$ is stably
$1$-Calabi–Yau. Let $T$ be a basic $R$-module such that
$\operatorname{add}_{R}{T}=\operatorname{add}_{R}{\mathbb{C}[[x,y]]}$, so that
$T$ is the unique (up to isomorphism) basic $1$-cluster-tilting object of
$\operatorname{CM}(R)$. By Auslander’s Theorem [auslanderrational] and a
result of Reiten–Van den Bergh [reitentwodimensional], there are isomorphisms

$\displaystyle\operatorname{End}_{R}(T)^{\mathrm{op}}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\to}}\Pi(\widetilde{\Delta}),$
$\displaystyle\operatorname{\underline{End}}_{R}(T)^{\mathrm{op}}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\to}}\Pi(\Delta),$

where $\widetilde{\Delta}$ is the extended Dynkin diagram given by the McKay
graph of $G$, and $\Delta$ is its unextended counterpart. It is well-known
that $\Pi(\Delta)$ is finite dimensional, so
$\operatorname{\underline{CM}}(R)$ is $\operatorname{Hom}$-finite, and that
$\operatorname{gl.dim}{\Pi(\widetilde{\Delta})}\leq 2$. Thus
$\operatorname{CM}(R)$ is a Frobenius $1$-cluster category.

The algebra
$\operatorname{End}_{R}(T)^{\mathrm{op}}\cong\Pi(\widetilde{\Delta})$ is
bimodule $2$-Calabi–Yau, which is consistent with (but stronger than) the
conclusion of Theorem 3.4.

##  4\. From internally $d$-Calabi–Yau algebras to $d$-cluster-tilting objects

Theorem 3.4 shows how internally $(m+1)$-Calabi–Yau algebras arise as
endomorphism algebras of cluster-tilting objects in Frobenius $m$-cluster
categories. In this section we work in the opposite direction, and show how to
construct a Frobenius category admitting a $(d-1)$-cluster-tilting object from
the data of an internally $d$-Calabi–Yau algebra, thus generalising a result
of Amiot–Iyama–Reiten [amiotstable]*Thm. 2.3 on bimodule $d$-Calabi–Yau
algebras. Since we will work only with Noetherian algebras, by 3.10 we should
expect to produce categories of the form $\operatorname{GP}(B)$ for some
Iwanaga–Gorenstein algebra $B$, and indeed this is what we shall do. Our main
result is the following.

######  Theorem 4.1 (cf. [amiotstable]*Thm. 2.3).

Let $A$ be a Noetherian algebra and let $e\in A$ be an idempotent such that
$A/\langle e\rangle$ is finite dimensional, and both $A$ and $A^{\mathrm{op}}$
are internally $d$-Calabi–Yau with respect to $e$. Write $B=eAe$ and
$\underline{A}=A/\langle e\rangle$. Then

  * (i)

$B$ is Iwanaga–Gorenstein with Gorenstein dimension at most $d$, so
$\operatorname{GP}(B)$ is a Frobenius category,

  * (ii)

$eA$ is $(d-1)$-cluster-tilting in $\operatorname{GP}(B)$, and

  * (iii)

there are natural isomorphisms
$\operatorname{End}_{B}(eA)^{\mathrm{op}}\stackrel{{\scriptstyle\sim}}{{\to}}A$
and
$\operatorname{End}_{\operatorname{\underline{GP}}(B)}(eA)^{\mathrm{op}}\stackrel{{\scriptstyle\sim}}{{\to}}\underline{A}$.

######  Remark 4.2.

While all of the conclusions of Theorem 4.1, except for $B$ being
Iwanaga–Gorenstein, refer only to left $B$-modules, the proof we will give
uses the assumption that $A^{\mathrm{op}}$ is internally $d$-Calabi–Yau to
draw conclusions about right $A$-modules. This applies in particular to
showing that the right $A$-module $eA$ is cluster-tilting in the category of
Gorenstein projective $B$-modules; see Lemmas 4.8 and 4.9.

Since the assumptions of Theorem 4.1 are symmetric in $A$ and
$A^{\mathrm{op}}$, we may also conclude that
$\operatorname{GP}(B^{\mathrm{op}})$ is a Frobenius category in which $Ae$ is
a $(d-1)$-cluster-tilting object, and there are natural isomorphisms
$\operatorname{End}_{B^{\mathrm{op}}}(Ae)^{\mathrm{op}}\stackrel{{\scriptstyle\sim}}{{\to}}A^{\mathrm{op}}$
and
$\operatorname{End}_{\operatorname{\underline{GP}}(B^{\mathrm{op}})}(Ae)^{\mathrm{op}}\stackrel{{\scriptstyle\sim}}{{\to}}\underline{A}^{\mathrm{op}}$.

We emphasise two cases in which the assumptions of Theorem 4.1 may be made to
appear one-sided. Firstly, as in Remark 2.3, if $A$ is a finite dimensional
algebra then it is internally $d$-Calabi–Yau with respect to $e$ if and only
if the same is true of $A^{\mathrm{op}}$. Secondly, if $A$ is bimodule
internally $d$-Calabi–Yau with respect to $e$, then (Remark 2.6) so is
$A^{\mathrm{op}}$, and therefore both $A$ and $A^{\mathrm{op}}$ are internally
$d$-Calabi–Yau with respect to $e$ by 2.9.

We note that Amiot–Iyama–Reiten’s result [amiotstable]*Thm. 2.3 is a special
case of our Theorem 4.1. To obtain the same conclusions, they assume that $A$
is Noetherian, $A/\langle e\rangle$ is finite dimensional, and that $A$ is
bimodule $d$-Calabi–Yau. By 2.9, this means that both $A$ and
$A^{\mathrm{op}}$ are internally $d$-Calabi–Yau with respect to any
idempotent, in particular with respect to $e$.

The rest of the section is largely devoted to proving Theorem 4.1, so we let
$A$, $e$, $\underline{A}$ and $B$ be as in the assumptions of this theorem. We
begin with the following straightforward observation.

######  Proposition 4.3.

The algebra $B$ is Noetherian.

###### Proof.

Any left ideal $I$ of $B$ is of the form $e\widetilde{I}$ for a left ideal
$\widetilde{I}=AI$ of $A$. So any ascending chain of left ideals of $B$
determines and is determined by such a chain of ideals of $A$, which
stabilises as $A$ is Noetherian. A similar argument shows that $B$ is right
Noetherian. ∎

######  Proposition 4.4 (cf. [amiotstable]*Lem. 2.6).

For any $X\in\operatorname{mod}{\underline{A}}$, we have

  * (i)

$\operatorname{Ext}^{i}_{A}(X,A)=0$ for $i\neq d$, and

  * (ii)

$\operatorname{Ext}^{i}_{A}(X,Ae)=0$ for any $i\in\mathbb{Z}$.

###### Proof.

Both $A$ and $Ae$ are finitely generated projective $A$-modules, and so are in
particular perfect. Since $\underline{A}$, and therefore $X$, is finite
dimensional, $X$ is a finitely generated $A$-module, and thus perfect since
$A$ is Noetherian of finite global dimension. Now we can use the internal
Calabi–Yau duality of $A$ to deduce that

$\operatorname{Ext}^{i}_{A}(X,A)=\mathrm{D}{\operatorname{Ext}^{d-i}_{A}(A,X)}=0$

and

$\operatorname{Ext}^{i}_{A}(X,Ae)=\mathrm{D}\operatorname{Ext}^{d-i}_{A}(Ae,X)=0$

for $i\neq d$, since $A$ and $Ae$ are projective. Since
$X\in\operatorname{mod}{\underline{A}}$, we have $eX=0$, and so

$\operatorname{Ext}^{d}_{A}(X,Ae)=\mathrm{D}{\operatorname{Hom}_{A}(Ae,X)}=\mathrm{D}(eX)=0.\qed$

The assumption of part (i) of Proposition 4.4 is slightly more restrictive
than that of [amiotstable]*Lem. 2.6(a). This is necessary for the result to
hold in our setting, since our $A$ is only internally $d$-Calabi–Yau. However,
this stronger assumption is satisfied whenever [amiotstable]*Lem. 2.6(a) is
used in the proof of [amiotstable]*Thm. 2.3.

The following results (Propositions 4.5, 4.6, 4.8 and 4.9) are now close
analogues of [amiotstable]*Prop. 2.7, Lem. 2.9–2.11, with very similar proofs.
For the convenience of the reader, and to make the paper more self-contained,
we give a complete argument using our notation and conventions.

######  Proposition 4.5 (cf. [amiotstable]*Prop. 2.7).

We have isomorphisms

$\operatorname{Ext}^{i}_{B}(eA,B)\cong\begin{cases}Ae,&i=0,\\\ 0,&i\neq
0\end{cases}$

of $A\mathbin{\otimes}_{\mathbb{K}}B^{\mathrm{op}}$-modules, and isomorphisms

$\operatorname{Ext}^{i}_{B}(eA,eA)\cong\begin{cases}A^{\mathrm{op}},&i=0,\\\
0,&0<i<d-1,\end{cases}$

of vector spaces, the isomorphism in case $i=0$ being additionally an
isomorphism of algebras.

###### Proof.

We can compute $\operatorname{Ext}^{i}_{B}(eA,B)$ as the cohomology of

$\operatorname{\mathbf{R}Hom}_{B}(eA,B)\cong\operatorname{\mathbf{R}Hom}_{B}(eA,\operatorname{\mathbf{R}Hom}_{A}(Ae,Ae))\cong\operatorname{\mathbf{R}Hom}_{A}(Ae\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{B}eA,Ae),$

and wish to show that this is isomorphic to the cohomology of
$\operatorname{\mathbf{R}Hom}_{A}(A,Ae)$. To do this, we show that

$\operatorname{\mathbf{R}Hom}_{A}(Ae\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{B}eA,Ae)\cong\operatorname{\mathbf{R}Hom}_{A}(A,Ae).$

Let $f$ be the composition of the natural map

$Ae\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{B}eA\to\mathrm{H}^{0}(Ae\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{B}eA)=Ae\mathbin{\otimes}_{B}eA$

with the multiplication map $Ae\mathbin{\otimes}_{B}eA\to A$, and let $X$ be
the mapping cone of $f$, so we have a triangle

(1)
${Ae\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{B}eA}$${A}$${X}$${Ae\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{B}eA[1]}$$\scriptstyle{f}$

in $\mathcal{D}{A}$. The map $eA\mathbin{\otimes}_{A}f$ is the natural
isomorphism
$B\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{B}eA\stackrel{{\scriptstyle\sim}}{{\to}}eA$,
so $eA\mathbin{\otimes}_{A}X=0$. It follows that $e\mathrm{H}^{i}(X)=0$, and
hence $\mathrm{H}^{i}(X)\in\operatorname{mod}{\underline{A}}$ for all
$i\in\mathbb{Z}$. Thus, by Proposition 4.4,
$\operatorname{Ext}^{j}_{A}(\mathrm{H}^{i}(X),Ae)=0$ for all
$i,j\in\mathbb{Z}$.

We can compute $\mathrm{H}^{k}(\operatorname{\mathbf{R}Hom}_{A}(X,Ae))$ via a
hypercohomology spectral sequence ${}^{II}E^{ij}_{r}$
[weibelintroduction]*§5.7.9, see also Defn. 5.6.2, in which

${}^{II}E^{ij}_{2}=\operatorname{Ext}^{j}_{A}(\mathrm{H}^{i}(X),Ae)=0$

as above. It follows that
$\mathrm{H}^{k}(\operatorname{\mathbf{R}Hom}_{A}(X,Ae))=0$ for all $k$, and so
$\operatorname{\mathbf{R}Hom}_{A}(X,Ae)=0$. Now applying
$\operatorname{\mathbf{R}Hom}_{A}(-,Ae)$ to the triangle (1) yields the
required isomorphism

$\operatorname{\mathbf{R}Hom}_{A}(Ae\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{B}eA,Ae)\cong\operatorname{\mathbf{R}Hom}_{A}(A,Ae)$

in $\mathcal{D}{A\mathbin{\otimes}_{\mathbb{K}}B^{\mathrm{op}}}$, from which
the first assertion follows by our initial calculations.

Similarly, we have isomorphisms

$\operatorname{\mathbf{R}Hom}_{B}(eA,eA)\cong\operatorname{\mathbf{R}Hom}_{B}(eA,\operatorname{\mathbf{R}Hom}_{A}(Ae,A))\cong\operatorname{\mathbf{R}Hom}_{A}(Ae\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{B}eA,A),$

and so to obtain the second assertion we wish to show that

$\operatorname{\mathbf{R}Hom}_{A}(Ae\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{B}eA,A)\cong\operatorname{\mathbf{R}Hom}_{A}(A,A).$

Both $Ae$ and $eA$ are concentrated in degree $0$, so by triangle (1) we have
$\mathrm{H}^{i}(Ae\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{B}eA)=0$
for $i>0$, and so $\mathrm{H}^{i}(X)=0$ for $i>0$. Recalling that
$\mathrm{H}^{i}(X)\in\operatorname{mod}{\underline{A}}$, it follows from
Proposition 4.4 that
$\operatorname{Ext}^{j}_{A}(\mathrm{H}^{i}(X),A)=\operatorname{Hom}_{\mathcal{D}{A}}(X,A[i])=0$
for $j\neq d$. By an analogous spectral sequence argument to above,
$\mathrm{H}^{i}(\operatorname{\mathbf{R}Hom}_{A}(X,A))=0$ for $i<d$.

From (1), we obtain the long exact sequence

${\cdots}$${\operatorname{Hom}_{\mathcal{D}{A}}(X,A[i])}$${\operatorname{Hom}_{\mathcal{D}{A}}(A,A[i])}$${\operatorname{Hom}_{\mathcal{D}{A}}(Ae\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{B}eA,A[i])}$${\cdots}$

As $\operatorname{Hom}_{\mathcal{D}{A}}(X,A[i])=0$ for $i<d$ as above, it
follows from our initial calculations that

$\operatorname{Ext}^{i}_{B}(eA,eA)\cong\operatorname{Hom}_{\mathcal{D}{A}}(Ae\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{B}eA,A[i])\cong\operatorname{Hom}_{\mathcal{D}{A}}(A,A[i])\cong\begin{cases}A^{\mathrm{op}},&i=0,\\\
0,&0<i<d-1,\end{cases}$

as required. ∎

######  Lemma 4.6 (cf. [amiotstable]*Lem. 2.9).

For any $X\in\operatorname{mod}{B}$, we have

$\operatorname{p.dim}_{A^{\mathrm{op}}}\operatorname{Hom}_{B}(X,eA)\leq d-2.$

###### Proof.

Since $B$ is Noetherian by Proposition 4.3, $X$ has a projective presentation
$P_{1}\to P_{0}\to X\to 0$ with $P_{0}$ and $P_{1}$ finitely generated.
Applying $\operatorname{Hom}_{B}(-,eA)$ gives the exact sequence

${0}$${\operatorname{Hom}_{B}(X,eA)}$${\operatorname{Hom}_{B}(P_{0},eA)}$${\operatorname{Hom}_{B}(P_{1},eA)}$

of $A^{\mathrm{op}}$-modules. Since $\operatorname{Hom}_{B}(P_{i},eA)$ is a
projective $A^{\mathrm{op}}$-module, the above sequence shows that
$\operatorname{Hom}_{B}(X,eA)$ is a second syzygy module. Since
$\operatorname{gl.dim}{A^{\mathrm{op}}}\leq d$ by the assumption that
$A^{\mathrm{op}}$ is internally $d$-Calabi–Yau, it follows that
$\operatorname{p.dim}_{A^{\mathrm{op}}}\operatorname{Hom}_{B}(X,eA)\leq d-2$.
∎

######  Remark 4.7.

We can obtain the statement that $\operatorname{gl.dim}{A^{\mathrm{op}}}\leq
d$ needed in the proof of Lemma 4.6 without assuming that $A^{\mathrm{op}}$ is
internally $d$-Calabi–Yau. By Noetherianity of $A$, we have
$\operatorname{gl.dim}{A^{\mathrm{op}}}=\operatorname{gl.dim}{A}$, and
$\operatorname{gl.dim}{A}\leq d$ since $A$ is internally $d$-Calabi–Yau.
However, the next two results, Lemmas 4.8 and 4.9, will use this assumption on
$A^{\mathrm{op}}$ in a more fundamental way.

######  Lemma 4.8 (cf. [amiotstable]*Lem. 2.10).

If $X\in\operatorname{GP}(B)$ and $\operatorname{Ext}^{i}_{B}(X,eA)=0$ for all
$0<i<d-1$, then $X\in\operatorname{add}{{}_{B}(eA)}$.

###### Proof.

Pick an exact sequence

${0}$${Y}$${P_{d-3}}$${\cdots}$${P_{0}}$${X}$${0}$

in which each $P_{i}$ is a finitely generated projective $B$-module. By the
assumption on the vanishing of $\operatorname{Ext}^{i}_{B}(X,eA)$, we can
apply $\operatorname{Hom}_{B}(-,eA)$ to obtain an exact sequence

${0}$${\operatorname{Hom}_{B}(X,eA)}$${\operatorname{Hom}_{B}(P_{0},eA)}$${\cdots}$${\operatorname{Hom}_{B}(P_{d-3},eA)}$${\operatorname{Hom}_{B}(Y,eA)}$${0}$

of $A^{\mathrm{op}}$-modules. Each $\operatorname{Hom}_{B}(P_{i},eA)$ is a
projective $A^{\mathrm{op}}$-module, and by Lemma 4.6 we have
$\operatorname{p.dim}_{A^{\mathrm{op}}}\operatorname{Hom}_{B}(Y,eA)\leq d-2$,
so $\operatorname{Hom}_{B}(X,eA)$ is also a projective
$A^{\mathrm{op}}$-module. It follows that
$\operatorname{Hom}_{B}(X,B)=\operatorname{Hom}_{B}(X,eA)e\in\operatorname{add}_{B^{\mathrm{op}}}(Ae)$.
By [amiotstable]*Prop. 1.3(b) there are quasi-inverse dualities

$\displaystyle\operatorname{Hom}_{B}(-,B)$
$\displaystyle\colon\operatorname{GP}(B)\to\operatorname{GP}(B^{\mathrm{op}}),$
$\displaystyle\operatorname{Hom}_{B^{\mathrm{op}}}(-,B)$
$\displaystyle\colon\operatorname{GP}(B^{\mathrm{op}})\to\operatorname{GP}(B).$

Since we are assuming $A^{\mathrm{op}}$ is also internally $d$-Calabi–Yau with
respect to $e$, we can apply Proposition 4.5 to $A^{\mathrm{op}}$ to obtain an
isomorphism
$\operatorname{Hom}_{B^{\mathrm{op}}}(Ae,B)\stackrel{{\scriptstyle\sim}}{{\to}}eA$
of $B$-modules. Therefore

$X\cong\operatorname{Hom}_{B^{\mathrm{op}}}(\operatorname{Hom}_{B}(X,B),B)\in\operatorname{add}{{}_{B}(\operatorname{Hom}_{B^{\mathrm{op}}}(Ae,B))}=\operatorname{add}{{}_{B}(eA)}$

as required. ∎

######  Lemma 4.9 (cf. [amiotstable]*Lem. 2.11).

If $X\in\operatorname{GP}(B)$ and $\operatorname{Ext}^{i}_{B}(eA,X)=0$ for all
$0<i<d-1$, then $X\in\operatorname{add}{{}_{B}(eA)}$.

###### Proof.

The quasi-inverse dualities

$\displaystyle\operatorname{Hom}_{B}(-,B)$
$\displaystyle\colon\operatorname{GP}(B)\to\operatorname{GP}(B^{\mathrm{op}}),$
$\displaystyle\operatorname{Hom}_{B^{\mathrm{op}}}(-,B)$
$\displaystyle\colon\operatorname{GP}(B^{\mathrm{op}})\to\operatorname{GP}(B)$

from [amiotstable]*Prop. 1.3(b) preserve extension groups. Since
$\operatorname{Hom}_{B}(eA,B)\cong Ae$ by Proposition 4.5, it follows that
$\operatorname{Ext}^{i}_{B^{\mathrm{op}}}(\operatorname{Hom}_{B}(X,B),Ae)=0$
for all $0<i<d-1$. Thus by applying Lemma 4.8 to $A^{\mathrm{op}}$ and
$\operatorname{Hom}_{B}(X,B)\in\operatorname{GP}(A^{\mathrm{op}})$, we find
that
$\operatorname{Hom}_{B}(X,B)\in\operatorname{add}{{}_{B^{\mathrm{op}}}(Ae)}$.
Then, as in Lemma 4.8, applying $\operatorname{Hom}_{B^{\mathrm{op}}}(-,B)$
gives $X\in\operatorname{add}{{}_{B}(eA)}$. ∎

We are now ready to prove Theorem 4.1.

###### Proof of Theorem 4.1.

  * (i)

We have already shown in Proposition 4.3 that $B$ is Noetherian, so it remains
to show that $B$ has injective dimension at most $d$ on each side. First we
show that $\operatorname{Ext}^{d+1}_{B}(X,B)=0$ for all
$X\in\operatorname{mod}{B}$. Given such an $X$, let
$Y=Ae\mathbin{\otimes}_{B}X$, and let $\mathbf{P}$ be a projective resolution
of $Y$. Then $e\mathbf{P}$ is a bounded complex in the full subcategory
$\operatorname{add}(eA)$ of $\operatorname{mod}{B}$, quasi-isomorphic to
$eY=X$. By Proposition 4.5, $\operatorname{Ext}^{i}_{B}(eA,B)=0$ for $i>0$, so
another spectral sequence argument (now using ${}^{I}E_{pq}^{r}$ from
[weibelintroduction]*Defn. 5.6.1) shows that

$\operatorname{Ext}^{d+1}_{B}(X,B)\cong\mathrm{H}^{d+1}(\operatorname{Hom}_{B}(e\mathbf{P},B)),$

where $\operatorname{Hom}_{B}(e\mathbf{P},B)$ denotes the complex obtained by
applying $\operatorname{Hom}_{B}(-,B)$ to $e\mathbf{P}$. Since

$\displaystyle\operatorname{Hom}_{B}(e\mathbf{P},B)$
$\displaystyle=\operatorname{Hom}_{B}(eA\mathbin{\otimes}_{A}\mathbf{P},B)$
$\displaystyle=\operatorname{Hom}_{A}(\mathbf{P},\operatorname{Hom}_{B}(eA,B))\cong\operatorname{Hom}_{A}(\mathbf{P},Ae),$

with the final isomorphism coming from Proposition 4.5, it follows that

$\operatorname{Ext}^{d+1}_{B}(X,B)\cong\mathrm{H}^{d+1}(\operatorname{Hom}_{A}(\mathbf{P},Ae))\cong\operatorname{Ext}^{d+1}_{A}(Y,Ae)=0$

since $\operatorname{gl.dim}{A}\leq d$ by assumption. A dual argument, using
that $A^{\mathrm{op}}$ is internally $d$-Calabi–Yau with respect to $e$, shows
that $\operatorname{Ext}^{d+1}_{B^{\mathrm{op}}}(X,B)=0$ for all
$X\in\operatorname{mod}{B^{\mathrm{op}}}$. It follows that $B$ is
Iwanaga–Gorenstein of dimension at most $d$, and so $\operatorname{GP}(B)$ is
Frobenius [buchweitzmaximal]*§4.8.

  * (ii)

Since $A$ is Noetherian, the left ideal $\langle e\rangle=AeA$ is finitely
generated. Thus it has a finite generating set contained in $eA$, which must
generate $eA\subseteq AeA$ as a $B$-module, so $eA\in\operatorname{mod}{B}$.
Now $eA\in\operatorname{GP}(B)$ and $\operatorname{Ext}^{i}_{B}(eA,eA)=0$ for
$0<i<d-1$ by Proposition 4.5. This, together with Lemmas 4.8 and 4.9, shows
that $eA$ is $(d-1)$-cluster-tilting in $\operatorname{GP}(B)$.

  * (iii)

We have $\operatorname{End}_{B}(eA)^{\mathrm{op}}\cong A$ by Proposition 4.5,
and thus we have an equivalence

$\operatorname{Hom}_{B}(eA,-)\colon\operatorname{add}{{}_{B}(eA)}\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{add}{{}_{A}A}.$

By Proposition 4.5 again, $\operatorname{Hom}_{B}(eA,B)\cong Ae$. It follows
that

$\displaystyle\operatorname{End}_{\operatorname{\underline{GP}}(B)}(eA)^{\mathrm{op}}$
$\displaystyle=\operatorname{End}_{B}(eA)^{\mathrm{op}}/\langle\operatorname{add}{{}_{B}B}\rangle$
$\displaystyle\cong\operatorname{End}_{A}(A)^{\mathrm{op}}/\langle\operatorname{add}{{}_{A}(Ae)}\rangle\cong
A/\langle e\rangle=\underline{A}$

where $\langle\mathcal{C}\rangle$ denotes the ideal of maps factoring through
the subcategory $\mathcal{C}$. ∎

In the setting of Theorem 4.1, we would also like to conclude that
$\operatorname{GP}(B)$ is stably $(d-1)$-Calabi–Yau. We now show, using
descriptions by Kalck–Yang [kalckrelative] of
$\operatorname{\underline{GP}}(B)$ in terms of (complexes of) $A$-modules,
that this category is $(d-1)$-Calabi–Yau when we strengthen the assumptions of
Theorem 4.1 to require that $A$ is bimodule internally $d$-Calabi–Yau.

######  Theorem 4.10.

Let $A$ be a Noetherian algebra and let $e\in A$ be an idempotent such that
$A/\langle e\rangle$ is finite dimensional, and $A$ is bimodule internally
$d$-Calabi–Yau with respect to $e$. Write $B=eAe$. Then all of the conclusions
of Theorem 4.1 hold, and moreover $\operatorname{\underline{GP}}(B)$ is
$(d-1)$-Calabi–Yau.

###### Proof.

By 2.9, $A$ and $A^{\mathrm{op}}$ are internally $d$-Calabi–Yau with respect
to $e$, so our assumptions imply those of Theorem 4.1. It remains to check
that $\operatorname{\underline{GP}}(B)$ is $(d-1)$-Calabi–Yau.

By [kalckrelative]*Prop. 2.10 (see also [dwyernoncommutative]*§2–3), there
exists a dg-algebra $C$ and a dg-algebra homomorphism $A\to C$, where $A$ is
considered as a dg-algebra concentrated in degree $0$, such that $C$ fits into
a recollement

${\mathcal{D}{C}}$${\mathcal{D}{A}}$${\mathcal{D}{B}.}$

Moreover, $C^{i}=0$ for $i>0$ and $\mathrm{H}^{0}(C)=A/\langle
e\rangle=\underline{A}$. Since $\underline{A}$ is finite dimensional and
$\operatorname{gl.dim}{A}\leq d$, it follows from [kalckrelative]*Cor. 2.13
that $\dim\mathrm{H}^{i}(C)<\infty$ for all $i$, and so
$\operatorname{per}{C}$ is $\operatorname{Hom}$-finite. The proof of this
corollary also shows that
$\mathcal{D}_{\mathrm{fd}}(C)\subseteq\operatorname{per}{C}$.

By [kalckrelative]*Proof of Cor. 2.12, the functor
$i_{*}=\operatorname{\mathbf{R}Hom}_{C}(C,-)\colon\mathcal{D}{C}\to\mathcal{D}{A}$
induces a triangle equivalence
$\mathcal{D}{C}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{D}_{\underline{A}}(A)$,
which restricts to a triangle equivalence
$\mathcal{D}_{\mathrm{fd}}(C)\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{D}_{\mathrm{fd},\underline{A}}(A)$.
Thus for any $M\in\mathcal{D}_{\mathrm{fd}}(C)$ and
$N\in\operatorname{per}{C}$, we have functorial isomorphisms

$\mathrm{D}\operatorname{Hom}_{\mathcal{D}{C}}(M,N)=\mathrm{D}\operatorname{Hom}_{\mathcal{D}{A}}(i_{*}M,i_{*}N)=\operatorname{Hom}_{\mathcal{D}{A}}(i_{*}N,i_{*}M[d])=\operatorname{Hom}_{\mathcal{D}{C}}(N,M[d]),$

the second coming from Theorem 2.8, using the assumption that $A$ is bimodule
internally $d$-Calabi–Yau with respect to $e$. Thus
$(\operatorname{per}{C},\mathcal{D}_{\mathrm{fd}}(C),\operatorname{add}{C})$
is a $d$-Calabi–Yau triple in the sense of Iyama–Yang [iyamasilting]*§5.1, and
so it follows from [iyamasilting]*Thm. 5.8(a) (see also [amiotcluster]*§1,
[guocluster]*§2) that $\operatorname{per}{C}/\mathcal{D}_{\mathrm{fd}}(C)$ is
$(d-1)$-Calabi–Yau.

We complete the proof by showing that $\operatorname{\underline{GP}}(B)$ is
equivalent to a full triangulated subcategory of
$\operatorname{per}{C}/\mathcal{D}_{\mathrm{fd}}(C)$, and so is also
$(d-1)$-Calabi–Yau. Since $\operatorname{gl.dim}{A}\leq d$,
[kalckrelative]*Cor. 2.12a tells us that
$i^{*}=C\mathbin{\stackrel{{\scriptstyle\mathbf{L}}}{{\otimes}}}_{A}-$
provides a triangle equivalence between the idempotent completion of
$\mathcal{D}^{b}{A}/\operatorname{thick}{Ae}$, denoted by
$(\mathcal{D}^{b}{A}/\operatorname{thick}{Ae})^{\omega}$, and
$\operatorname{per}{C}$. Moreover, [kalckrelative]*Proof of Cor. 2.12 shows
that $i_{*}=\operatorname{\mathbf{R}Hom}_{C}(C,-)$ induces a triangle
equivalence
$\mathcal{D}_{\mathrm{fd}}(C)\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{D}_{\mathrm{fd},\underline{A}}(A)$,
and that the codomain of this equivalence coincides with
$\operatorname{thick}(\operatorname{mod}{\underline{A}})$. Since
$i^{*}i_{*}\simeq 1_{\mathcal{D}{C}}$, we see that $i^{*}$ restricts to an
equivalence
$\operatorname{thick}(\operatorname{mod}{\underline{A}})\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{D}_{\mathrm{fd}}(C)$,
and so induces an equivalence

$\frac{(\mathcal{D}^{b}{A}/\operatorname{thick}{Ae})^{\omega}}{q(\operatorname{thick}(\operatorname{mod}{\underline{A}}))}\stackrel{{\scriptstyle\sim}}{{\to}}\frac{\operatorname{per}{C}}{\mathcal{D}_{\mathrm{fd}}(C)},$

where $q$ denotes the projection
$\mathcal{D}^{b}{A}\to\mathcal{D}^{b}{A}/\operatorname{thick}{Ae}$, which
restricts to an equivalence on
$\operatorname{thick}(\operatorname{mod}{\underline{A}})$ by the above
observations. We also have equivalences

$\frac{\mathcal{D}^{b}{A}/\operatorname{thick}{Ae}}{q(\operatorname{thick}(\operatorname{mod}{\underline{A}}))}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{D}^{b}{B}/\operatorname{per}{B}\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{\underline{GP}}(B),$

the first from [kalckrelative]*Prop. 3.3 and the second from a result of
Buchweitz [buchweitzmaximal]*Thm. 4.4.1. Since
$\mathcal{D}^{b}{A}/\operatorname{thick}{Ae}$ is a full triangulated
subcategory of its idempotent completion, combining the above equivalences
gives a triangle equivalence of $\operatorname{\underline{GP}}(B)$ with a full
triangulated subcategory of the $(d-1)$-Calabi–Yau triangulated category
$\operatorname{per}{C}/\mathcal{D}_{\mathrm{fd}}(C)$, from which it follows
that $\operatorname{\underline{GP}}(B)$ is itself $(d-1)$-Calabi–Yau. ∎

It was necessary in the proof of Theorem 4.10 to use the bimodule internal
Calabi–Yau symmetry of $A$ to obtain a duality between spaces of maps of
complexes of $A$-modules, so it is unclear whether the conclusion that
$\operatorname{\underline{GP}}(B)$ is $(d-1)$-Calabi–Yau might hold only under
the weaker assumptions of Theorem 4.1. As already stated, we do not currently
have any examples of internally Calabi–Yau algebras that are not bimodule
internally Calabi–Yau, but it seems unlikely that the two classes coincide.

Under the assumptions of Theorem 4.10, we would like to be able to conclude
that the Frobenius category $\operatorname{GP}(B)$ is in fact a Frobenius
$(d-1)$-cluster category in the sense of Definition 3.3. It remains to check
that $\operatorname{gl.dim}{\operatorname{End}_{B}(T)^{\mathrm{op}}}\leq d$
for any $(d-1)$-cluster-tilting object $T\in\operatorname{GP}(B)$; a priori,
we only know this for the $(d-1)$-cluster-tilting object $eA$. Whenever
$\operatorname{End}_{B}(T)^{\mathrm{op}}$ is Noetherian, we can apply
Proposition 3.7 to get the desired conclusion. While this Noetherianity is
automatic in some situations, such as if $B$ is finite dimensional over
$\mathbb{K}$, in general it appears to be a more subtle issue.

######  Remark 4.11.

As indicated in the introduction, Theorems 4.1 and 4.10 are motivated by the
problem of constructing Frobenius categories modelling cluster algebras. Given
the seed of a cluster algebra with frozen variables, we can look for an
algebra $A$ with the same quiver (up to the addition of arrows between frozen
vertices), satisfying the conditions of Theorem 4.10 for $d=3$, and then apply
this theorem to obtain the Frobenius category $\operatorname{GP}(B)$. Since
constructing such an $A$ can be very difficult, we wish to comment on the
degree to which the conditions we are imposing are necessary.

Firstly, we consider it likely that the condition that $A$ is Noetherian can
be dropped, up to finding an appropriate replacement for the category
$\operatorname{GP}(B)$. While $B$ may not be Noetherian if $A$ fails to be, it
will still have injective dimension at most $d$ on each side, so there should
be a ‘good’ theory of Gorenstein projective modules over $B$. For our methods
to work, we would need the analogues of [amiotstable]*Prop. 1.3 to hold in
this setting. We would also hope for a more general version of the
Iyama–Kalck–Wemyss–Yang equivalence stated here as Theorem 3.9, without the
Noetherianity assumption, which would then apply to arbitrary Frobenius
$m$-cluster categories, and Buchweitz’s description of the stable category
[buchweitzmaximal]*Thm. 4.4.1.

The other conditions are more essential; if
$A=\operatorname{End}_{\mathcal{E}}(T)^{\mathrm{op}}$ for $T$ a cluster-
tilting object in a Frobenius cluster category $\mathcal{E}$, and $e$ is the
idempotent given by projecting onto a maximal projective summand of $T$, then
$A/\langle e\rangle$ must be finite dimensional since
$\underline{\mathcal{E}}$ is Hom-finite, and $A$ is internally $3$-Calabi–Yau
with respect to $e$ by Theorem 3.4. On the other hand, it may not be necessary
for $A$ to be bimodule internally $3$-Calabi–Yau.

##  5\. A bimodule complex for frozen Jacobian algebras

Given a Frobenius cluster category $\mathcal{E}$ and a cluster-tilting object
$T\in\mathcal{E}$, it is often the case that
$A=\operatorname{End}_{\mathcal{E}}(T)^{\mathrm{op}}$ takes the form of a
frozen Jacobian algebra (see Definition 5.1 below). Indeed, this is the case
for at least some cluster-tilting objects in the families of Frobenius cluster
categories we described in Examples 3.11 and 3.12; see [buanmutation]*Thm.
6.6, [baurdimer]*Thm. 10.3. Thus these algebras, which also come with a
preferred ‘frozen’ idempotent, are ideal candidates for constructing stably
$2$-Calabi–Yau Frobenius categories via the methods of Theorem 4.1 and Theorem
4.10. Moreover, $3$-Calabi–Yau properties of ordinary Jacobian algebras have
been widely studied, for example by Bocklandt [bocklandtgraded] and, in the
context of dimer models on closed surfaces, by Broomhead [broomheaddimer].

With this in mind, the main result of this section, Theorem 5.6, shows that a
frozen Jacobian algebra admitting a particular bimodule resolution (analogous
to one defined by Ginzburg [ginzburgcalabiyau]*5.1.5) is bimodule internally
$3$-Calabi–Yau with respect to its frozen idempotent.

######  Definition 5.1 (cf. [buanmutation]*Defn. 1.1,
[demoneticequivers1]*§2.1, [francobipartite]*§6.1).

An _ice quiver_ $(Q,F)$ consists of a finite quiver $Q$ without loops and a
(not necessarily full) subquiver $F$ of $Q$. Denote by $\widehat{\mathbb{K}Q}$
the completion of the path algebra of $Q$ over $\mathbb{K}$ with respect to
the arrow ideal. A _potential_ on $Q$ is a linear combination $W$ of cycles of
$Q$. A vertex or arrow of $Q$ is called _frozen_ if it is a vertex or arrow of
$F$, and _mutable_ or _unfrozen_ otherwise. For brevity, we write
$Q_{0}^{\mathrm{m}}=Q_{0}\setminus F_{0}$ and
$Q_{1}^{\mathrm{m}}=Q_{1}\setminus F_{1}$ for the sets of mutable vertices and
unfrozen arrows respectively. For $\alpha\in Q_{1}$ and
$\alpha_{n}\dotsm\alpha_{1}$ a cycle in $Q$, write

$\partial_{\alpha}{\alpha_{n}\dotsm\alpha_{1}}=\sum_{\alpha_{i}=\alpha}\alpha_{i-1}\dotsm\alpha_{1}\alpha_{n}\dotsm\alpha_{i+1}$

and extend linearly. The ideal $\langle\partial_{\alpha}{W}:\alpha\in
Q_{1}^{\mathrm{m}}\rangle$ of $\widehat{\mathbb{K}Q}$ is called the _Jacobian
ideal_ , and we may take its closure
$\overline{\langle\partial_{\alpha}{W}:\alpha\in Q_{1}^{\mathrm{m}}\rangle}$
since $\widehat{\mathbb{K}Q}$ is a topological algebra. We define the _frozen
Jacobian algebra_ associated to $(Q,F,W)$ by

$\mathcal{J}(Q,F,W)=\widehat{\mathbb{K}Q}/\overline{\langle\partial_{\alpha}{W}:\alpha\in
Q_{1}^{\mathrm{m}}\rangle}.$

Write $A=\mathcal{J}(Q,F,W)$. The above presentation of $A$ suggests a
preferred idempotent $e=\sum_{v\in F_{0}}e_{v}$, which we call the _frozen
idempotent_. We will call $B=eAe$ the _boundary algebra_ of $A$.

######  Remark 5.2.

If $F=\varnothing$, then $\mathcal{J}(Q,\varnothing,W)=:\mathcal{J}(Q,W)$ is
the usual Jacobian algebra.

######  Example 5.3.

Consider the ice quiver with potential $(Q,F,W)$, where

$Q=\mathord{\leavevmode\hbox to68.57pt{\vbox
to46.15pt{\pgfpicture\makeatletter\hbox{\hskip 5.83301pt\lower-20.7816pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{11.00417pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\boxed{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{54.40552pt}{11.00417pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\boxed{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{
{}{}{}}{}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{}{}}{}{
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.70708}{-0.70712}{0.70712}{-0.70708}{34.82713pt}{-7.85292pt}\pgfsys@invoke{
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}\hbox{\scriptsize{$\alpha_{2}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.70708}{0.70712}{-0.70712}{-0.70708}{6.37437pt}{7.85292pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\alpha_{3}$}} }}\pgfsys@invoke{\lxSVG@closescope
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\hbox{\scriptsize{$\alpha_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$

the frozen subquiver $F$ is the full subquiver on vertices $1$ and $2$,
indicated by boxed vertices and a dashed arrow, and
$W=\alpha_{3}\alpha_{2}\alpha_{1}$. Then the frozen Jacobian algebra
$\mathcal{J}(Q,F,W)$ is the quotient of $\widehat{\mathbb{K}Q}$ by the
relations $\partial_{\alpha_{2}}W=\alpha_{1}\alpha_{3}$ and
$\partial_{\alpha_{3}}W=\alpha_{2}\alpha_{1}$, so $\mathcal{J}(Q,F,W)$ is the
algebra $A$ from Example 2.10. (In this case $A$ is finite dimensional, so it
agrees with the quotient of the ordinary path algebra $\mathbb{K}Q$ by the
same relations.) It will follow from Theorem 5.6 below that
$\mathcal{J}(Q,F,W)$ is bimodule internally $3$-Calabi–Yau with respect to the
idempotent $e_{1}+e_{2}$ given by summing the idempotents corresponding to
frozen vertices. The usual Jacobian algebra $\mathcal{J}(Q,W)$ has the
additional relation $\alpha_{3}\alpha_{2}=0$ and is not bimodule
$3$-Calabi–Yau; indeed, it has infinite global dimension.

Given a quiver with potential $(Q,W)$, Ginzburg [ginzburgcalabiyau]*5.1.5 (see
also [broomheaddimer]*§7) defines a complex of projective bimodules over the
associated Jacobian algebra. For $(Q,W)$ a quiver with potential determined by
a dimer model on a torus, Broomhead shows in [broomheaddimer]*Thm. 7.7 that if
the dimer model is consistent, then this complex is isomorphic to
$A=\mathcal{J}(Q,W)$ in $\mathcal{D}^{b}{{A}^{\varepsilon}}$, and thus
provides a projective bimodule resolution of $A$. It follows in this case that
$A$ is $3$-Calabi–Yau, with this property arising from a natural symmetry in
the bimodule resolution.

We will now define an analogous complex $\mathbf{P}(A)$ for a frozen Jacobian
algebra $A=\mathcal{J}(Q,F,W)$. Our main result (Theorem 5.6) will be that if
$\mathbf{P}(A)$ is isomorphic to $A$ in $\mathcal{D}^{b}{{A}^{\varepsilon}}$,
then $A$ is bimodule internally $3$-Calabi–Yau with respect to the frozen
idempotent $e=\sum_{v\in F_{0}}e_{v}$, in the sense of Definition 2.4. While
we will write $\mathbf{P}(A)$ for this complex in order to save space, the
definition depends not only on $A$ but on the ice quiver with potential
$(Q,F,W)$ giving the presentation of $A$ as $\mathcal{J}(Q,F,W)$.

Recall that we write $Q_{0}^{\mathrm{m}}=Q_{0}\setminus F_{0}$ for the set of
mutable vertices and $Q_{1}^{\mathrm{m}}=Q_{1}\setminus F_{1}$ for the set of
unfrozen arrows. We also write $v^{+}$ for the set of arrows with tail at $v$,
and $v^{-}$ for the set of arrows with head at $v$. Denote the arrow ideal of
$A$ by $\mathfrak{m}(A)$, and let $S=A/\mathfrak{m}(A)$. For the remainder of
this section, we write $\mathbin{\otimes}=\mathbin{\otimes}_{S}$.

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

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

via the formulae

$\displaystyle e_{v}\cdot e_{v}\cdot e_{v}$ $\displaystyle=e_{v},$
$\displaystyle e_{h\alpha}\cdot\alpha\cdot e_{t\alpha}$
$\displaystyle=\alpha,$ $\displaystyle e_{t\alpha}\cdot\rho_{\alpha}\cdot
e_{h\alpha}$ $\displaystyle=\rho_{\alpha},$ $\displaystyle
e_{v}\cdot\omega_{v}\cdot e_{v}$ $\displaystyle=\omega_{v},$

where $h\alpha$ and $t\alpha$ denote the head and tail of the arrow $\alpha$.
For each $i$, the $S$-bimodule $\mathbb{K}Q_{i}$ splits as the direct sum

$\mathbb{K}Q_{i}=\mathbb{K}Q_{i}^{\mathrm{m}}\mathbin{\oplus}\mathbb{K}F_{i}.$

Since $\mathbb{K}Q_{0}\cong S$, the $A$-bimodule
$A\mathbin{\otimes}\mathbb{K}Q_{0}\mathbin{\otimes}A$ is canonically
isomorphic to $A\mathbin{\otimes}A$, and we will use the two descriptions
interchangeably.

We define maps $\bar{\mu}_{i}\colon
A\mathbin{\otimes}\mathbb{K}Q_{i}\mathbin{\otimes}A\to
A\mathbin{\otimes}\mathbb{K}Q_{i-1}\mathbin{\otimes}A$ for $1\leq i\leq 3$.
The map $\bar{\mu}_{1}$ is defined by

$\bar{\mu}_{1}(x\mathbin{\otimes}\alpha\mathbin{\otimes}y)=x\mathbin{\otimes}e_{h\alpha}\mathbin{\otimes}\alpha
y-x\alpha\mathbin{\otimes}e_{t\alpha}\mathbin{\otimes}y,$

or, composing with the natural isomorphism
$A\mathbin{\otimes}\mathbb{K}Q_{0}\mathbin{\otimes}A\stackrel{{\scriptstyle\sim}}{{\to}}A\mathbin{\otimes}A$,
by

$\bar{\mu}_{1}(x\mathbin{\otimes}\alpha\mathbin{\otimes}y)=x\mathbin{\otimes}\alpha
y-x\alpha\mathbin{\otimes}y.$

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

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

and extend by linearity to obtain a map
$\Delta_{\alpha}\colon\widehat{\mathbb{K}Q}\to
A\mathbin{\otimes}\mathbb{K}Q_{1}\mathbin{\otimes}A$. We then define

$\bar{\mu}_{2}(x\mathbin{\otimes}\rho_{\alpha}\mathbin{\otimes}y)=\sum_{\beta\in
Q_{1}}x\Delta_{\beta}(\partial_{\alpha}{W})y.$

Finally, let

$\bar{\mu}_{3}(x\mathbin{\otimes}\omega_{v}\mathbin{\otimes}y)=\sum_{\alpha\in
v^{+}}x\mathbin{\otimes}\rho_{\alpha}\mathbin{\otimes}\alpha y-\sum_{\beta\in
v^{-}}x\beta\mathbin{\otimes}\rho_{\beta}\mathbin{\otimes}y.$

######  Definition 5.4.

For $A=\mathcal{J}(Q,F,W)$, let $\mathbf{P}(A)$ be the sequence

${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}}$

of $A$-bimodules, 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
$A\mathbin{\otimes}\mathbb{K}Q_{2}^{\mathrm{m}}\mathbin{\otimes}A$ and
$A\mathbin{\otimes}\mathbb{K}Q_{3}^{\mathrm{m}}\mathbin{\otimes}A$
respectively. As $v^{+}\cup v^{-}\subseteq Q_{1}^{\mathrm{m}}$ for any $v\in
Q_{0}^{\mathrm{m}}$, the map $\mu_{3}$ takes values in
$A\mathbin{\otimes}\mathbb{K}Q_{2}^{\mathrm{m}}\mathbin{\otimes}A$ as claimed.

If $F=\varnothing$, then $\mathbf{P}(A)$ is the complex associated to $(Q,W)$
by Ginzburg [ginzburgcalabiyau]*5.1.5 and Broomhead [broomheaddimer]*§7. In
the general case, $\mathbf{P}(A)$ has already appeared in work of
Amiot–Reiten–Todorov [amiotubiquity]*Proof of Prop. 2.2.

######  Lemma 5.5.

For a frozen Jacobian algebra $A=\mathcal{J}(Q,F,W)$, the sequence
$\mathbf{P}(A)$ in Definition 5.4 is a complex of finitely generated
projective $A$-bimodules, and there is a morphism

${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}$${A\mathbin{\otimes}A}$${0}$${0}$${0}$${A}$$\scriptstyle{\mu_{3}}$$\scriptstyle{\mu_{2}}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\mu_{0}}$

from $\mathbf{P}(A)$ to $A$, where $\mu_{0}\colon A\mathbin{\otimes}A\to A$ is
the multiplication in $A$. Moreover, the complex $0\to\mathbf{P}(A)\to A\to 0$
is exact at $A$, $A\mathbin{\otimes}A$ and
$A\mathbin{\otimes}\mathbb{K}Q_{1}\mathbin{\otimes}A$.

###### Proof.

Each term of $\mathbf{P}(A)$ is a projective $A$-bimodule since $A$ is a
projective $A$-module on each side, and is finitely generated by finiteness of
$Q$. By standard results on presentations of algebras, see for example
Butler–King [butlerminimal]*1.2, $\mu_{0}$ is surjective,
$\operatorname{im}{\mu_{1}}=\ker{\mu_{0}}$ and
$\operatorname{im}{\mu_{2}}=\ker{\mu_{1}}$. Thus the only statement left to
check is that $\mu_{2}\circ\mu_{3}=0$.

Let $v\in Q_{0}$, and write

$W_{v}=\sum_{\alpha\in v^{+}}(\partial_{\alpha}{W})\alpha=\sum_{\beta\in
v^{-}}\beta(\partial_{\beta}{W}).$

We can calculate $\sum_{\gamma\in Q_{1}}\Delta_{\gamma}(W_{v})$ using each of
the two expressions, to get

$\displaystyle\sum_{\gamma\in Q_{1}}\Delta_{\gamma}(W_{v})$
$\displaystyle=\sum_{\alpha\in v^{+}}\sum_{\gamma\in
Q_{1}}\Delta_{\gamma}(\partial_{\alpha}{W})\alpha+\sum_{\alpha\in
v^{+}}\partial_{\alpha}{W}\mathbin{\otimes}\alpha\mathbin{\otimes}1,$
$\displaystyle\sum_{\gamma\in Q_{1}}\Delta_{\gamma}(W_{v})$
$\displaystyle=\sum_{\beta\in v^{-}}\sum_{\gamma\in
Q_{1}}\beta\Delta_{\gamma}(\partial_{\beta}{W})+\sum_{\beta\in
v^{-}}1\mathbin{\otimes}\beta\mathbin{\otimes}\partial_{\beta}{W}.$

If $v\in Q_{0}^{\mathrm{m}}$, then all arrows incident with $v$ are unfrozen,
and so $\partial_{\alpha}{W}=0=\partial_{\beta}{W}$ in $A$ for any $\alpha\in
v^{+}$ and $\beta\in v^{-}$. Thus in this case we have

$\sum_{\alpha\in v^{+}}\sum_{\gamma\in
Q_{1}}\Delta_{\gamma}(\partial_{\alpha}{W})\alpha=\sum_{\gamma\in
Q_{1}}\Delta_{\gamma}(W_{v})=\sum_{\beta\in v^{-}}\sum_{\gamma\in
Q_{1}}\beta\Delta_{\gamma}(\partial_{\beta}{W}).$

It follows that

$\displaystyle\mu_{2}(\mu_{3}(1\mathbin{\otimes}\omega_{v}\mathbin{\otimes}1))$
$\displaystyle=\mu_{2}\left(\sum_{\alpha\in
v^{+}}1\mathbin{\otimes}\rho_{\alpha}\mathbin{\otimes}\alpha-\sum_{\beta\in
v^{-}}\beta\mathbin{\otimes}\rho_{\beta}\mathbin{\otimes}1\right)$
$\displaystyle=\sum_{\alpha\in v^{+}}\sum_{\gamma\in
Q_{1}}\Delta_{\gamma}(\partial_{\alpha}{W})\alpha-\sum_{\beta\in
v^{-}}\sum_{\gamma\in Q_{1}}\beta\Delta_{\gamma}(\partial_{\beta}{W})$
$\displaystyle=0.$

This completes the proof. ∎

If the map

${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}$${A\mathbin{\otimes}A}$${0}$${0}$${0}$${A}$$\scriptstyle{\mu_{3}}$$\scriptstyle{\mu_{2}}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\mu_{0}}$

from Lemma 5.5 is a quasi-isomorphism, then $\mathbf{P}(A)$ is a projective
bimodule resolution of $A$. This means that, for the presentation of $A$ as a
frozen Jacobian algebra, with relations given by certain derivatives of the
superpotential, the first syzygies are dual to the mutable vertices, and there
are no higher syzygies. In particular, $\operatorname{gl.dim}{A}\leq 3$. By
Lemma 5.5, this map is a quasi-isomorphism if and only if the cohomology of
$\mathbf{P}(A)$ vanishes at
$A\mathbin{\otimes}\mathbb{K}Q_{2}^{\mathrm{m}}\mathbin{\otimes}A$ and
$A\mathbin{\otimes}\mathbb{K}Q_{3}^{\mathrm{m}}\mathbin{\otimes}A$ (cf.
[broomheaddimer]*Rem. 7.4). We will usually abuse notation and denote the map
$\mathbf{P}(A)\to A$ from Lemma 5.5 by $\mu_{0}$.

If $F=\varnothing$, the map $\mu_{0}\colon\mathbf{P}(A)\to A$ being a quasi-
isomorphism implies that $A$ is $3$-Calabi–Yau [ginzburgcalabiyau]*Cor. 5.3.3;
[broomheaddimer]*Thm. 7.7. We now show that, in the general case, $\mu_{0}$
being a quasi-isomorphism implies that $A$ is bimodule internally
$3$-Calabi–Yau with respect to $e$.

######  Theorem 5.6.

If $A$ is a frozen Jacobian algebra such that $\mu_{0}\colon\mathbf{P}(A)\to
A$ is a quasi-isomorphism, then $A$ is bimodule internally $3$-Calabi–Yau with
respect to the frozen idempotent $e=\sum_{v\in F_{0}}e_{v}$.

###### Proof.

Since $\mathbf{P}(A)\in\operatorname{per}{{A}^{\varepsilon}}$, the quasi-
isomorphism $\mu_{0}\colon\mathbf{P}(A)\stackrel{{\scriptstyle\sim}}{{\to}}A$
makes $\mathbf{P}(A)$ into a projective resolution of $A$, implying
immediately that $\operatorname{p.dim}_{{A}^{\varepsilon}}A\leq 3$ and
$A\in\operatorname{per}{{A}^{\varepsilon}}$. It remains to check condition
(iii) from Definition 2.4.

We begin by describing
$\Omega_{A}=\operatorname{\mathbf{R}Hom}_{{A}^{\varepsilon}}(A,{A}^{\varepsilon})\in\mathcal{D}^{b}{{A}^{\varepsilon}}$.
Denoting $\operatorname{Hom}_{{A}^{\varepsilon}}(-,-)$ by $(-,-)$, the complex
$\Omega_{A}$ is given by

${(A\mathbin{\otimes}A,{A}^{\varepsilon})}$${(A\mathbin{\otimes}\mathbb{K}Q_{1}\mathbin{\otimes}A,{A}^{\varepsilon})}$${(A\mathbin{\otimes}\mathbb{K}Q_{2}^{\mathrm{m}}\mathbin{\otimes}{A},{A}^{\varepsilon})}$${(A\mathbin{\otimes}\mathbb{K}Q_{3}^{\mathrm{m}}\mathbin{\otimes}A,{A}^{\varepsilon})}$$\scriptstyle{-\mu_{1}^{*}}$$\scriptstyle{\mu_{2}^{*}}$$\scriptstyle{-\mu_{3}^{*}}$

with $\mu_{i}^{*}\colon f\mapsto f\circ\mu_{i}$; see Keller
[kellercalabiyau]*§2.7 for the signs on the differentials.

There are $A$-bimodule isomorphisms $A\mathbin{\otimes}A\cong\bigoplus_{v\in
Q_{0}}Ae_{v}\mathbin{\otimes}_{\mathbb{K}}e_{v}A$ and ${A}^{\varepsilon}\cong
A\mathbin{\otimes}_{\mathbb{K}}A$. Introducing the shorthand notation

$\mathbf{x}\mathbin{\otimes}\mathbf{y}=\sum_{i=1}^{k}x^{i}\mathbin{\otimes}y^{i}$

for elements of $A\mathbin{\otimes}_{\mathbb{K}}A$, a homomorphism
$f_{0}\colon A\mathbin{\otimes}A\to{A}^{\varepsilon}$ is uniquely determined
by the values

$f_{0}(1\mathbin{\otimes}e_{v}\mathbin{\otimes}1)=\mathbf{x}_{v}\mathbin{\otimes}\mathbf{y}_{v}$

for each $v\in Q_{0}$. Since
$1\mathbin{\otimes}e_{v}\mathbin{\otimes}1=e_{v}\mathbin{\otimes}e_{v}\mathbin{\otimes}e_{v}$,
we must have

$\mathbf{x}_{v}\mathbin{\otimes}\mathbf{y}_{v}=e_{v}\mathbf{x}_{v}\mathbin{\otimes}\mathbf{y}_{v}e_{v}\in
e_{v}A\mathbin{\otimes}_{\mathbb{K}}Ae_{v},$

but $\mathbf{x}_{v}$ and $\mathbf{y}_{v}$ may otherwise be chosen freely. If
follows that we have an isomorphism

$\displaystyle(A\mathbin{\otimes}A,{A}^{\varepsilon})$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\to}}A\mathbin{\otimes}\mathbb{K}Q_{3}\mathbin{\otimes}A,$
$\displaystyle f_{0}$ $\displaystyle\mapsto\sum_{v\in
Q_{0}}\mathbf{y}_{v}\mathbin{\otimes}\omega_{v}\mathbin{\otimes}\mathbf{x}_{v}$
of $A$-bimodules. Similar arguments yield explicit isomorphisms
$\displaystyle(A\mathbin{\otimes}\mathbb{K}Q_{1}\mathbin{\otimes}A,{A}^{\varepsilon})$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\to}}A\mathbin{\otimes}\mathbb{K}Q_{2}\mathbin{\otimes}A,$
$\displaystyle f_{1}$ $\displaystyle\mapsto\sum_{\alpha\in
Q_{1}}\mathbf{y}_{\alpha}\mathbin{\otimes}\rho_{\alpha}\mathbin{\otimes}\mathbf{x}_{\alpha},$
$\displaystyle(A\mathbin{\otimes}\mathbb{K}Q_{2}^{\mathrm{m}}\mathbin{\otimes}A,{A}^{\varepsilon})$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\to}}A\mathbin{\otimes}\mathbb{K}Q_{1}^{\mathrm{m}}\mathbin{\otimes}A,$
$\displaystyle f_{2}$ $\displaystyle\mapsto\sum_{\alpha\in
Q_{1}^{\mathrm{m}}}\mathbf{y}^{\prime}_{\alpha}\mathbin{\otimes}\alpha\mathbin{\otimes}\mathbf{x}^{\prime}_{\alpha},$
$\displaystyle(A\mathbin{\otimes}\mathbb{K}Q_{3}^{\mathrm{m}}\mathbin{\otimes}A,{A}^{\varepsilon})$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\to}}A\mathbin{\otimes}\mathbb{K}Q_{0}^{\mathrm{m}}\mathbin{\otimes}A,$
$\displaystyle f_{3}$ $\displaystyle\mapsto\sum_{v\in
Q_{0}^{\mathrm{m}}}\mathbf{y}^{\prime}_{v}\mathbin{\otimes}e_{v}\mathbin{\otimes}\mathbf{x}^{\prime}_{v},$

where the functions $f_{1}$, $f_{2}$ and $f_{3}$ are uniquely determined by
the values

$\displaystyle f_{1}(1\mathbin{\otimes}\alpha\mathbin{\otimes}1)$
$\displaystyle=\mathbf{x}_{\alpha}\mathbin{\otimes}\mathbf{y}_{\alpha}\in
e_{h\alpha}A\mathbin{\otimes}_{\mathbb{K}}Ae_{t\alpha},$ $\displaystyle
f_{2}(1\mathbin{\otimes}\rho_{\alpha}\mathbin{\otimes}1)$
$\displaystyle=\mathbf{x}^{\prime}_{\alpha}\mathbin{\otimes}\mathbf{y}^{\prime}_{\alpha}\in
e_{t\alpha}A\mathbin{\otimes}_{\mathbb{K}}Ae_{h\alpha},$ $\displaystyle
f_{3}(1\mathbin{\otimes}\omega_{v}\mathbin{\otimes}1)$
$\displaystyle=\mathbf{x}^{\prime}_{v}\mathbin{\otimes}\mathbf{y}^{\prime}_{v}\in
e_{v}A\mathbin{\otimes}_{\mathbb{K}}Ae_{v}.$

Since $\alpha\in F_{1}$ implies that $h\alpha,t\alpha\in F_{0}$, the map
$\bar{\mu}_{1}\colon A\mathbin{\otimes}\mathbb{K}Q_{1}\mathbin{\otimes}A\to
A\mathbin{\otimes}\mathbb{K}Q_{0}\mathbin{\otimes}A$ restricts to a map
$A\mathbin{\otimes}\mathbb{K}F_{1}\mathbin{\otimes}A\to
A\mathbin{\otimes}\mathbb{K}F_{0}\mathbin{\otimes}A$, and thus taking
quotients yields a map $\mu_{1}^{\vee}\colon
A\mathbin{\otimes}\mathbb{K}Q_{1}^{\mathrm{m}}\mathbin{\otimes}A\to
A\mathbin{\otimes}\mathbb{K}Q_{0}^{\mathrm{m}}\mathbin{\otimes}A$. Explicitly,
$\mu_{1}^{\vee}$ is given by

$\mu_{1}^{\vee}(1\otimes\alpha\otimes
1)=1\otimes(1-e)\alpha-\alpha(1-e)\otimes 1.$

Define $\mu_{2}^{\vee}$ to be the composition of $\bar{\mu}_{2}$ with the
projection $A\mathbin{\otimes}\mathbb{K}Q_{1}\mathbin{\otimes}A\to
A\mathbin{\otimes}\mathbb{K}Q_{1}^{\mathrm{m}}\mathbin{\otimes}A$; explicitly

$\mu_{2}^{\vee}(1\mathbin{\otimes}\rho_{\alpha}\mathbin{\otimes}1)=\sum_{\beta\in
Q_{1}^{\mathrm{m}}}\Delta_{\beta}(\partial_{\alpha}{W}).$

Finally, let $\mu_{3}^{\vee}=\bar{\mu}_{3}$. Then one can check that the
isomorphisms of $A$-bimodules defined above induce an isomorphism of
$\Omega_{A}$ with the complex

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

As an example to illustrate the necessary calculations, we show that we get an
isomorphism of
$\mu_{2}^{*}\colon(A\mathbin{\otimes}\mathbb{K}Q_{1}\mathbin{\otimes}A,{A}^{\varepsilon})\to(A\mathbin{\otimes}\mathbb{K}Q_{2}^{\mathrm{m}}\mathbin{\otimes}A,{A}^{\varepsilon})$
with $\mu_{2}^{\vee}\colon
A\mathbin{\otimes}\mathbb{K}Q_{2}\mathbin{\otimes}A\to
A\mathbin{\otimes}\mathbb{K}Q_{1}^{\mathrm{m}}\mathbin{\otimes}A$. It suffices
to check this on the generators
$1\mathbin{\otimes}\rho_{\alpha}\mathbin{\otimes}1$ of
$A\mathbin{\otimes}\mathbb{K}Q_{2}\mathbin{\otimes}A$. First observe that
under the isomorphism
$(A\mathbin{\otimes}\mathbb{K}Q_{1}\mathbin{\otimes}A,{A}^{\varepsilon})\stackrel{{\scriptstyle\sim}}{{\to}}A\mathbin{\otimes}\mathbb{K}Q_{2}\mathbin{\otimes}A$,
the preimage of
$1\mathbin{\otimes}\rho_{\alpha}\mathbin{\otimes}1=e_{t\alpha}\mathbin{\otimes}\rho_{\alpha}\mathbin{\otimes}e_{h\alpha}$
is the $A$-bimodule homomorphism $f_{\alpha}$ determined by

$f_{\alpha}(1\mathbin{\otimes}\beta\mathbin{\otimes}1)=\begin{cases}e_{h\alpha}\mathbin{\otimes}e_{t\alpha},&\beta=\alpha,\\\
0,&\text{otherwise}.\end{cases}$

We then calculate for each $\beta\in Q_{1}^{\mathrm{m}}$ that

$\displaystyle\mu_{2}^{*}(f_{\alpha})(1\mathbin{\otimes}\rho_{\beta}\mathbin{\otimes}1)$
$\displaystyle=f_{\alpha}\mu_{2}(1\mathbin{\otimes}\rho_{\beta}\mathbin{\otimes}1)$
$\displaystyle=f_{\alpha}\Big{(}\sum_{\gamma\in
Q_{1}}\Delta_{\gamma}(\partial_{\beta}{W})\Big{)}$
$\displaystyle=\mathbf{x}_{\beta}\mathbin{\otimes}\mathbf{y}_{\beta},$

where

$\Delta_{\alpha}(\partial_{\beta}{W})=\mathbf{x}_{\beta}\mathbin{\otimes}\alpha\mathbin{\otimes}\mathbf{y}_{\beta}.$

We must then have

$\Delta_{\beta}(\partial_{\alpha}{W})=\mathbf{y}_{\beta}\mathbin{\otimes}\beta\mathbin{\otimes}\mathbf{x}_{\beta},$

and so the isomorphism
$(A\mathbin{\otimes}\mathbb{K}Q_{2}^{\mathrm{m}}\mathbin{\otimes}A,{A}^{\varepsilon})\stackrel{{\scriptstyle\sim}}{{\to}}A\mathbin{\otimes}\mathbb{K}Q_{1}^{\mathrm{m}}\mathbin{\otimes}A$
takes $\mu_{2}^{*}(f_{\alpha})$ to

$\sum_{\beta\in
Q_{1}^{\mathrm{m}}}\mathbf{y}_{\beta}\mathbin{\otimes}\beta\mathbin{\otimes}\mathbf{x}_{\beta}=\sum_{\beta\in
Q_{1}^{\mathrm{m}}}\Delta_{\beta}(\partial_{\alpha}{W})=\mu^{\vee}_{2}(1\mathbin{\otimes}\rho_{\alpha}\mathbin{\otimes}1),$

as required.

Now consider the commutative diagram

(2)
${0}$${0}$${0}$${0}$${0}$${0}$${A\mathbin{\otimes}\mathbb{K}F_{1}\mathbin{\otimes}A}$${A\mathbin{\otimes}\mathbb{K}F_{0}\mathbin{\otimes}A}$${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}$${A\mathbin{\otimes}\mathbb{K}Q_{0}\mathbin{\otimes}A}$${A\mathbin{\otimes}\mathbb{K}Q_{3}\mathbin{\otimes}A}$${A\mathbin{\otimes}\mathbb{K}Q_{2}\mathbin{\otimes}A}$${A\mathbin{\otimes}\mathbb{K}Q_{1}^{\mathrm{m}}\mathbin{\otimes}A}$${A\mathbin{\otimes}\mathbb{K}Q_{0}^{\mathrm{m}}\mathbin{\otimes}A}$${A\mathbin{\otimes}\mathbb{K}F_{3}\mathbin{\otimes}A}$${A\mathbin{\otimes}\mathbb{K}F_{2}\mathbin{\otimes}A}$${0}$${0}$${0}$${0}$${0}$${0}$$\scriptstyle{\mu_{3}}$$\scriptstyle{+}$$\scriptstyle{\mu_{2}}$$\scriptstyle{-}$$\scriptstyle{\mu_{1}}$$\scriptstyle{+}$$\scriptstyle{-}$$\scriptstyle{-\mu_{3}^{\vee}}$$\scriptstyle{-\mu_{2}^{\vee}}$$\scriptstyle{-\mu_{1}^{\vee}}$

in which the columns are split exact, the second row is $\mathbf{P}(A)$, the
third row is isomorphic to $\Omega_{A}[3]$ by the preceding calculations, and
the signs on the vertical arrows indicate whether the corresponding map is the
inclusion or its negative.

The diagram (2) provides us with a map of complexes
$A\cong\mathbf{P}(A)\to\Omega_{A}[3]$ in $\mathcal{D}^{b}{{A}^{\varepsilon}}$,
and shows that the cone of this map has the form

${C=A\mathbin{\otimes}\mathbb{K}F_{3}\mathbin{\otimes}A}$${A\mathbin{\otimes}(\mathbb{K}F_{2}\mathbin{\oplus}\mathbb{K}F_{1})\mathbin{\otimes}A}$${A\mathbin{\otimes}\mathbb{K}F_{0}\mathbin{\otimes}A.}$

Let $M\in\mathcal{D}_{\underline{A}}(A)$ have finite dimensional total
cohomology. Since the cohomology of $M$ is concentrated in some interval,
$M\in\mathcal{D}^{b}{A}$. We pick a bounded representative $M^{\bullet}$ of
the quasi-isomorphism class of $M$, allowing us to compute the complex
$\operatorname{\mathbf{R}Hom}_{A}(C,M)$ as the total complex of the double
complex with terms

$\operatorname{Hom}_{A}(A\mathbin{\otimes}V_{i}\mathbin{\otimes}A,M^{j}),$

where $V_{1}=\mathbb{K}F_{3}$,
$V_{2}=\mathbb{K}F_{2}\mathbin{\oplus}\mathbb{K}F_{1}$,
$V_{3}=\mathbb{K}F_{0}$ and $V_{i}=0$ for all other $i$. Since each
$S$-bimodule $V_{i}$ has the property that $eV_{i}e=V_{i}$, we have

$A\mathbin{\otimes}V_{i}\mathbin{\otimes}A=Ae\mathbin{\otimes}V_{i}\mathbin{\otimes}eA,$

so the terms of the relevant double complex are isomorphic to

$\operatorname{Hom}_{S}(V_{i}\mathbin{\otimes}eA,\operatorname{Hom}_{A}(Ae,M^{j}))=\operatorname{Hom}_{S}(V_{i}\mathbin{\otimes}eA,eM^{j}).$

Since $M\in\mathcal{D}_{\underline{A}}(A)$, the complex $eM^{\bullet}$ is
acyclic. Moreover, since $S$ is semi-simple,
$\operatorname{Hom}_{S}(V_{i}\mathbin{\otimes}eA,-)$ preserves acyclicity, so
the vertical cohomology of the double complex vanishes. It follows that
$\operatorname{\mathbf{R}Hom}_{A}(C,M)=0$. A similar argument shows that
$\operatorname{\mathbf{R}Hom}_{A^{\mathrm{op}}}(C,N)=0$ for all
$N\in\mathcal{D}_{\underline{A}^{\mathrm{op}}}(A^{\mathrm{op}})$, so we
conclude that $A$ is bimodule internally $3$-Calabi–Yau with respect to $e$. ∎

######  Example 5.7.

Since the frozen Jacobian algebra $A$ in Examples 2.10 and 5.3 is finite
dimensional (of small dimension!) it is possible to check that
$\mathbf{P}(A)\stackrel{{\scriptstyle\sim}}{{\to}}A$ directly by choosing a
basis of $A$, whence this algebra is bimodule internally $3$-Calabi–Yau with
respect to this frozen idempotent. The algebra $A^{\prime}$ in Example 2.10
may also be presented as a frozen Jacobian algebra, and again the fact that
$A^{\prime}$ is finite dimensional allows us to check directly that
$\mathbf{P}(A^{\prime})\stackrel{{\scriptstyle\sim}}{{\to}}A^{\prime}$, giving
the bimodule internally $3$-Calabi–Yau property.

However, we also have many more examples, some of which are infinite
dimensional. For example, let

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