realises the bijection between $\mathbf{k}$ and the set of equivalence classes
of strong enhancements of
$(\operatorname{proj}(\Lambda),\Sigma,{\pentagon}_{\alpha})$ given by
Corollary 5.4.12.

###  5.5. Comments on the (non-)uniqueness of enhancements

We wish to illustrate the necessity of the assumptions in Theorems 5.1.3,
5.1.8 and 5.4.10 by analysing the case when $\Lambda=\mathbf{k}$ is an
arbitrary field, which is the most basic example of a periodic algebra of any
period $d\geq 1$. Thus, let $d\geq 1$ and $\sigma=\mathbf{1}_{\Lambda}$ so
that

$\Lambda(\sigma,d)=\mathbf{k}\langle\imath^{\pm}\rangle,\qquad|\imath|=-d,$

is the algebra of Laurent polynomials in a single variable of degree $-d$,
which we view as a DG algebra with vanishing differential (this algebra is
considered implicitly in [Lad, Ex. 3.30] and explicitly in [Lor, Ex. 6.7], see
also Remark 5.5.2).

By definition, a DG $\Lambda(\sigma,d)$-module $M=(V,\varphi)$ consists of a
cochain complex of vector spaces $V$ equipped with an invertible degree $-d$
cocycle
$\varphi\in\operatorname{Z}^{-d}(\operatorname{hom}_{\mathbf{k}}\\!\left(V,V\right))$
that we interpret as an isomorphism $\varphi\colon
V\stackrel{{\scriptstyle\sim}}{{\to}}V[-d]$. Similarly, a morphism of DG
$\Lambda(\sigma,d)$-modules $f\colon(V,\varphi)\to(W,\psi)$ is a morphism of
cochain complexes of vector spaces $f\colon V\to W$ such that the diagram

${V}$${V[-d]}$${W}$${W[-d]}$$\scriptstyle{f}$$\scriptstyle{\varphi}$$\scriptstyle{f[-d]}$$\scriptstyle{\psi}$

commutes. In particular, such an $f$ is a quasi-isomorphism if and only if the
induced linear map

$H^{i}(f)\colon H^{i}(V)\longrightarrow H^{i}(W)$

is an isomorphism for all $0\leq i<d$. We claim that functor

$\operatorname{D}(\Lambda(\sigma,d))\longrightarrow\prod_{i=0}^{d-1}\operatorname{Mod}(\mathbf{k}),\qquad
M\longmapsto(H^{0}(M),H^{1}(M),\dots,H^{d-1}(M)),$

is an equivalence of (ordinary) categories (although this fact is well known
to experts, we give a proof for the convenience of the reader). For this,
notice that there is an apparent fully faithful functor

$\prod_{i=0}^{d-1}\operatorname{Mod}(\mathbf{k})\longrightarrow\operatorname{dgMod}(\Lambda(\sigma,d))$

which sends a tuple $(V^{0},V^{1},\dots,V^{d-1})$ to the DG
$\Lambda(\sigma,d)$-module with trivial differential $(V,\mathbf{1}_{V})$
where

${V\colon}$${\cdots}$${V^{d-1}}$${V^{0}}$${V^{1}}$${\cdots}$${V^{d-1}}$${V^{0}}$${\cdots}$

Notice that the composite

$\prod_{i=0}^{d-1}\operatorname{Mod}(\mathbf{k})\longrightarrow\operatorname{dgMod}(\Lambda(\sigma,d))\longrightarrow
H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\Lambda(\sigma,d)))$

is also fully faithful. To complete the proof, it is enough to observe the
following:

  * •

Every DG $\Lambda(\sigma,d)$-module is homotopy equivalent to its cohomology
(viewed as a DG $\Lambda(\sigma,d)$-module in the obvious way).

  * •

A morphism of DG $\Lambda(\sigma,d)$-modules is a quasi-isomorphism if and
only if it is a homotopy equivalence and therefore

$H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\Lambda(\sigma,d)))=\operatorname{D}(\Lambda(\sigma,d)).$

  * •

The composite

$\prod_{i=0}^{d-1}\operatorname{Mod}(\mathbf{k})\longrightarrow\operatorname{D}(\Lambda(\sigma,d))\longrightarrow\prod_{i=0}^{d-1}\operatorname{Mod}(\mathbf{k})$

is isomorphic to the identity functor.

The first claim is obvious and the second and third claims can be shown
exactly as in the case of cochain complexes of vector spaces, observing that
the category $\operatorname{dgMod}(\Lambda(\sigma,d))$ is equivalent to the
category of $d$-periodic cochain complexes of vector spaces. Under the above
equivalence, the shift functor on $\operatorname{D}(\Lambda(\sigma,d))$
corresponds to the automorphism

$(V^{0},V^{1},\dots,V^{d-1})\longmapsto(V^{1},\dots,V^{d-1},V^{0}).$

of $\prod_{i=0}^{d-1}\operatorname{Mod}(\mathbf{k})$. Since the graded algebra
$H^{\bullet}(\Lambda(\sigma,d))=\Lambda(\sigma,d)$ is finite-dimensional in
each degree and

$H^{i}(\Lambda(\sigma,d){[-j]})=\Lambda(\sigma,d)^{i-j}=\begin{cases}\mathbf{k}&i=j,\\\
0&i\neq j,\end{cases}\qquad 0\leq i,j<d,$

it is clear that the functor

$\operatorname{D}^{\mathrm{c}}(\Lambda(\sigma,d))\longrightarrow\prod_{i=0}^{d-1}\operatorname{mod}(\mathbf{k}),\qquad
M\longmapsto(H^{0}(M),H^{1}(M),\dots,H^{d-1}(M)),$

is also an equivalence of (ordinary) categories. Finally, the free DG module
${\Lambda(\sigma,d)\in\operatorname{D}^{\mathrm{c}}(\Lambda(\sigma,d))}$ is a
$d\mathbb{Z}$-cluster tilting object, since this is obviously the case for its
image

$(\mathbf{k},0,\dots,0)\in\prod_{i=0}^{d-1}\operatorname{mod}(\mathbf{k}).$

######  Remark 5.5.1.

According to Theorem 2.2.1, the additive category

$\operatorname{mod}(\mathbf{k})\simeq\operatorname{add}(\Lambda(\sigma,d))\subseteq\operatorname{D}^{\mathrm{c}}(\Lambda(\sigma,d))$

admits a $(d+2)$-angulated structure with suspension functor
$\Sigma=\mathbf{1}$ (up to natural isomorphism, the only autoequivalence of
$\operatorname{mod}(\mathbf{k})$) whose $(d+2)$-angles are induced by the
triangles in $\operatorname{D}^{\mathrm{c}}(\Lambda(\sigma,d))$. In fact,
$\operatorname{mod}(\mathbf{k})$ admits a unique $(d+2)$-angulation: A
$(d+2)$-angle

$V_{1}\to V_{2}\to\cdots\to V_{d+2}\to V_{1}$

in $\operatorname{mod}(\mathbf{k})$ must yield a $(d+2)$-periodic exact
sequence

$\cdots\to V_{d+2}\to V_{1}\to V_{2}\to\cdots\to V_{d+2}\to V_{1}\to\cdots$

(see [GKO13, Prop. 2.5]) and, conversely, every $(d+2)$-angulation must
contain all such sequences for these are isomorphic to a finite direct sum of
rotations of the trivial sequence

$\mathbf{k}\xrightarrow{\mathbf{1}missing}\mathbf{k}\to 0\to\cdots\to
0\to\mathbf{k}$

with $d$ zeroes. Moreover, $\operatorname{mod}(\mathbf{k})$ has a unique DG-
enhancement, namely the graded category
$\operatorname{add}(\Lambda(\sigma,d))$ endowed with the trivial differential.

######  Remark 5.5.2.

The above discussion and Theorem 5.1.6 show that the triangulated category
$\operatorname{D}^{\mathrm{c}}(\Lambda(\sigma,d))$ is equivalent to the
$d$-cluster category of $\mathbf{k}$, that is the orbit category [Kel05a]

$\mathcal{C}_{d}(\mathbf{k})\coloneqq\operatorname{D}^{\mathrm{c}}(\operatorname{mod}(\mathbf{k}))/[d].$

Indeed, $\mathcal{C}_{d}(\mathbf{k})$ is an algebraic triangulated category
with $[d]=\mathbf{1}$, finite-dimensional morphism spaces, split idempotents
and $\mathbf{k}\in\mathcal{C}_{d}(\mathbf{k})$ is a $d\mathbb{Z}$-cluster
tilting object with endomorphism algebra isomorphic to $\mathbf{k}$
(alternatively, one can also invoke the Recognition Theorem of Keller and
Reiten [KR08]).

######  Remark 5.5.3.

We record the following observation for the sake of completeness. The
triangulated category $\operatorname{D}^{\mathrm{c}}(\Lambda(\sigma,d))$
admits an additive generator that, under the identification
$\operatorname{D}^{\mathrm{c}}(\Lambda(\sigma,d))\simeq\prod_{i=0}^{d-1}\operatorname{mod}(\mathbf{k})$,
is given by the direct sum $\bigoplus_{i=0}^{d-1}\mathbf{k}(i)$, where

$\mathbf{k}(i)\coloneqq(0,\dots,0,\mathbf{k},0,\dots,0)$

with $\mathbf{k}$ placed in the $i$-th coordinate. It is straightforward to
verify that the derived endomorphism algebra of this additive generator is
quasi-isomorphic to the graded algebra $\mathbf{k}Q/I$ where $Q$ is the graded
quiver

$0$$1$$2$$\cdots$$d-1$

with arrows $u\colon i\to i+1$ of degree $-1$ and $v\colon i+1\to i$ of degree
$1$, and $I$ is the graded ideal generated by the relations $vu=e_{i}$ and
$uv=e_{i+1}$ (with all indices modulo $d$). We leave the details to the
reader.

####  5.5.1. Non-unique enhancements in the $\operatorname{Hom}$-infinite case

Let $\ell$ be a field and $\mathbf{k}=\ell(x_{0},x_{1},\dots,x_{d-1})$; notice
that $\mathbf{k}$ is perfect if $\ell$ has characteristic $0$. Rizzardo and
Van den Bergh [RVdB19] show that
$\operatorname{D}^{\mathrm{c}}(\Lambda(\sigma,d))$, viewed as an $\ell$-linear
(!) algebraic triangulated category, admits two inequivalent enhancements.
This does not contradict A, Theorem 5.1.8 or Theorem 5.4.10: as an
$\ell$-linear category, the morphism spaces in
$\operatorname{D}^{\mathrm{c}}(\Lambda(\sigma,d))$ are infinite dimensional
and therefore the hypotheses in A and Theorems 5.1.8 and 5.4.10 are not
satisfied. Thus, this example explains why it is necessary to restrict to
triangulated categories with finite-dimensional morphism spaces.

####  5.5.2. Non-unique enhancements for non-separable algebras

Although A requires the ground field $\mathbf{k}$ to be perfect, in Theorems
5.1.3, 5.1.6 and 5.4.10 it is enough to assume that $\Lambda/J_{\Lambda}$ is
separable. We shall see that the latter assumption is essential. Indeed, let
$p>0$ be a prime number and $\mathbf{k}=\mathbb{F}_{p}(x)$; thus, $\mathbf{k}$
is not perfect. Let

$\Lambda\coloneqq\mathbb{F}_{p}(x^{\frac{1}{p}})=\frac{\mathbf{k}[X]}{(X^{p}-x)},$

which is a purely inseparable field extension of $\mathbf{k}$ of degree $p$
with trivial Galois group; thus, in particular, $\Lambda=\Lambda/J_{\Lambda}$
is not separable over $\mathbf{k}$. Notice also that the algebras $\mathbf{k}$
and $\Lambda$ are isomorphic over $\mathbb{F}_{p}$, but not over $\mathbf{k}$.
Moreover, there is an isomorphism of $\mathbf{k}$-algebras

$\Lambda^{e}=\Lambda\otimes_{\mathbf{k}}\Lambda^{\mathrm{op}}\cong\frac{\mathbf{k}[X,Y]}{(X^{p}-x,Y^{p}-x)}\cong\frac{\Lambda[Y]}{(Y-X)^{p}};$

we treat the above isomorphism as an identification. Over $\Lambda$, we have
seen that the semisimple category $\operatorname{mod}(\Lambda)$ admits a
unique $(d+2)$-angulated structure (Remark 5.5.1), and the latter admits a
unique enhancement, represented by the graded category
$\operatorname{add}(\Lambda(\sigma,d))$ with trivial differential. We shall
see that $\operatorname{mod}(\Lambda)$ admits two inequivalent enhancements
over the field $\mathbf{k}$. Indeed, notice that the diagonal
$\Lambda$-bimodule has infinite projective dimension and admits a $2$-periodic
projective resolution

${\cdots}$${\Lambda^{e}}$${\Lambda^{e}}$${\Lambda^{e}}$${\Lambda^{e}}$${0}$$\scriptstyle{(Y-X)^{p-1}}$$\scriptstyle{Y-X}$$\scriptstyle{(Y-X)^{p-1}}$$\scriptstyle{Y-X}$

that is in fact $1$-periodic if $p=2$. Thus, for even $d$ (or every $d\geq 1$
if $p=2$), we obtain a non-trivial extension of length $d+2$

${0}$${\Lambda}$${\Lambda^{e}}$${\cdots}$${\Lambda^{e}}$${\Lambda^{e}}$${\Lambda}$${0}$$\scriptstyle{Y-X}$$\scriptstyle{(Y-X)^{p-1}}$$\scriptstyle{Y-X}$

and we may consider the AL $(d+2)$-angulation associated to this extension
(Theorem 2.2.15). By Theorem 5.1.3, this AL $(d+2)$-angulation admits an
enhancement $\mathcal{B}$ whose restricted universal Massey product is
represented, up to sign, by the class of the above non-trivial extension; in
particular, $\mathcal{B}$ cannot be quasi-isomorphic to
$H^{\bullet}(\mathcal{B})$ as DG categories over $\mathbf{k}$ and, therefore,
$\mathcal{B}$ cannot be quasi-equivalent to the enhancement
$\operatorname{add}(\Lambda(\sigma,d))$ of $\operatorname{mod}(\Lambda)$. This
does not contradict Theorem 5.1.3: the algebra $\Lambda=\Lambda/J_{\Lambda}$
is not separable as a $\mathbf{k}$-algebra, and hence the hypotheses in
Theorem 5.1.3 are not satisfied.

Moreover, just as in [RVdB20], $\operatorname{D}^{\mathrm{c}}(\mathcal{B})$ is
equivalent to $\operatorname{D}^{\mathrm{c}}(\Lambda(\sigma,d))$ as
triangulated categories, but the DG enhancements $\mathcal{B}$ and
$\operatorname{add}(\Lambda(\sigma,d))$ cannot be Morita equivalent. Indeed,
if they were, the arguments in the proof of Theorem 5.4.10 would show that
they also have to be quasi-equivalent, and we have already argued that they
are not. This proves that the separability hypothesis is necessary in Theorems
5.1.8 and 5.4.10.

####  5.5.3. $d\mathbb{Z}$-cluster tilting does not guarantee twisted
$(d+2)$-periodicity

According to Proposition 2.2.10, the endomorphism algebra $\Lambda$ of a
$d\mathbb{Z}$-cluster tilting object in a triangulated category with finite-
dimensional morphism spaces is twisted $(d+2)$-periodic, provided that
$\Lambda/J_{\Lambda}$ is separable. This need not be the case if the algebra
$\Lambda/J_{\Lambda}$ is not separable, even if the ambient triangulated
category is algebraic. Indeed, as in Section 5.5.2 above, let $p>0$ be a prime
number, $\mathbf{k}=\mathbb{F}_{p}(x)$ and
$\Lambda=\mathbb{F}_{p}(x^{\frac{1}{p}})$. As explained in Section 5.5, the
triangulated category

$\operatorname{D}^{\mathrm{c}}(\Lambda\langle\imath^{\pm}\rangle),\qquad|\imath|=-d,$

is algebraic and admits a $d\mathbb{Z}$-cluster tilting object with
endomorphism algebra isomorphic to $\Lambda$ (since $\Lambda$ is itself a
field). However, if $p\neq 2$ and $d$ is odd (in particular if $d=1$), then
$\Lambda$ is not twisted $(d+2)$-periodic as a $\mathbf{k}$-algebra and,
therefore, the restricted universal Massey product associated to the canonical
$\mathbf{k}$-linear (!) enhancement of
$\operatorname{D}^{\mathrm{c}}(\Lambda\langle\imath^{\pm}\rangle)$ cannot be
represented by a unit in the Hochschild–Tate cohomology of $\Lambda$ (viewed
as a $\mathbf{k}$-algebra). This does not contradict Proposition 2.2.10 nor
the implication (1)$\Rightarrow$(3) in Corollary 4.5.17, since
$\Lambda=\Lambda/J_{\Lambda}$ is not separable as a $\mathbf{k}$-algebra.

####  5.5.4. Differential graded enhancements vs topological enhancements

Our results do not give any information on the (non-)existence of exotic
_topoplogical_ enhancements, that is enhancements given in terms of spectral
categories, say. We illustrate this point with the following example from
[SS03, Rmk. 3.2.1]. Fix an arbitrary prime number $p$ and an integer $n\geq
1$. We consider the perfect field $\mathbf{k}=\mathbb{F}_{p}$ and set
$d=2p^{n}-2$ and $\Lambda=\mathbf{k}$. The graded algebra

$\Lambda\langle\imath^{\pm 1}\rangle=\mathbf{k}\langle\imath^{\pm
1}\rangle=\mathbb{F}_{p}[\imath^{\pm 1}],\qquad|\imath|=-d=-(2p^{n}-2),$

is isomorphic to the graded homotopy ring $\pi_{*}(K(n))$ of the Morava
$K$-theory spectrum $K(n)$ of height $n$ at the prime $p$. Moreover, there is
an equivalence of triangulated categories

$\operatorname{D}(K(n))\simeq\operatorname{D}(\mathbb{F}_{p}[\imath^{\pm
1}]),$

where $\operatorname{D}(K(n))$ is the derived category of $K(n)$-module
spectra; the above equivalence of triangulated categories implies that
$K(n)\in\operatorname{D}^{\mathrm{c}}(K(n))\simeq\operatorname{D}^{\mathrm{c}}(\mathbb{F}_{p}[\imath^{\pm
1}])$ is a $d\mathbb{Z}$-cluster tilting object in the compact derived
category of $K(n)$. By construction, the triangulated category on the left-
hand side is equipped with a topological enhancement while the triangulated
category on the right-hand side is equipped with a DG ehancement; however,
identifying DG $\mathbf{k}$-algebras with algebra spectra over the
Eilenberg–MacLane commutative ring spectrum $H\mathbf{k}$ of $\mathbf{k}$
[Shi07], these two enhancements are inequivalent (the reason being that the
spectrum $K(n)$ is not weakly equivalent to the the Eilenberg–MacLane spectrum
of $\mathbb{F}_{p}[\imath^{\pm 1}]$). This does not contradict A since the
theorem only pertains to $\mathbf{k}$-linear DG enhancements and DG
$\mathbf{k}$-algebras (equivalently, spectral enhancements over $H\mathbf{k}$
and $H\mathbf{k}$-algebra spectra).

##  6\. Recognition theorems

In this section we discuss several recognition theorems for algebraic
triangulated categories of interest in representation theory and algebraic and
symplectic geometry that are immediate consequences of A (given the existent
knowledge about these categories). Except in special cases, the triangulated
categories considered below do no admit an additive generator and therefore
the results in [Mur22] generally do not apply to them.

######  Setting 6.0.1.

For simplicity, we assume that $\mathbf{k}$ is a perfect field throughout this
section.

Excluding Theorem 6.4.3, all of the applications of A that we discuss in this
section hold, more generally, when the endomorphism algebra $\Lambda$ of the
(basic) $d\mathbb{Z}$-cluster tilting objects under consideration is such that
the quotient $\Lambda/J_{\Lambda}$ is a separable algebra over the ground
field. This assumption is satisfied automatically in the following cases:

  * •

When the ground field is perfect (for example if it is algebraically closed or
it has characteristic $0$).

  * •

When the ground field is arbitrary and $\Lambda\cong\mathbf{k}Q/I$ is
(isomorphic to) the bounded path algebra of a finite quiver. Indeed, in this
case there is an isomorphism $\Lambda/J_{\Lambda}\cong\mathbf{k}^{Q_{0}}$,
where $Q_{0}$ denotes the set of vertices of $Q$. The analogous statement
holds when the finite-dimensional algebra $\Lambda$ is the (complete) Jacobian
algebra of a quiver with potential.

Under this more general assumption, references to A should be replaced by
references to Theorems 5.1.6 and 5.1.8. Similarly, non-separable algebras
should be considered in place of non-semisimple algebras when appropriate
(since these notions coincide for perfect fields but not for arbitrary
fields).

###  6.1. Amiot–Guo–Keller cluster categories

We begin by recalling the following theorem, proven by Amiot in the case $d=2$
and by Guo in the general case, both using results of Keller [Kel05a, Kel11]
in an essential way.

Recall that a triangulated category $\mathcal{T}$ with finite-dimensional
morphism spaces is _$n$ -Calabi–Yau_ if there exists a natural isomorphism

$\mathcal{T}(y,x[n])\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}D\mathcal{T}(x,y),\qquad
x,y\in\mathcal{T}.$

where $V\mapsto DV$ denotes the passage to the $\mathbf{k}$-linear dual.

Given a DG algebra $\Gamma$, we denote by
$\operatorname{D}^{\mathrm{fd}}(\Gamma)$ the full triangulated subcategory of
$\operatorname{D}(\Gamma)$ spanned by the DG $\Gamma$-modules $M$ such that

$\sum_{i\in\mathbb{Z}}\dim H^{i}(M)<\infty.$

Following Kontsevich, we say that $\Gamma$ is _homologically smooth_ if the
diagonal bimodule $\Gamma$ is perfect as a DG $\Gamma$-bimodule, that is
$\Gamma\in\operatorname{D}^{\mathrm{c}}(\Gamma^{e})$. Also, given an integer
$n$, the DG algebra $\Gamma$ is _bimodule $n$-Calabi–Yau_ if there is an
isomorphism

$\operatorname{RHom}_{\Gamma^{e}}\\!\left(\Gamma,\Gamma^{e}\right)\cong\Gamma[-n]$

in the derived category of DG $\Gamma$-bimodules. If $\Gamma$ is homologically
smooth and bimodule $n$-Calabi–Yau, then the triangulated category
$\operatorname{D}^{\mathrm{fd}}(\Gamma)$ is $n$-Calabi–Yau, see [Kel08, Lemma
4.1].

######  Theorem 6.1.1 ([Ami09, Thm. 2.1] and [Guo11, Thm. 2.2], see also
[IY18, Thm. 5.8] for the case $d=1$).

Let $d\geq 1$ and $\Gamma$ a DG algebra that satisfies the following
properties:

  * •

The DG algebra $\Gamma$ is homologically smooth.

  * •

The cohomology of $\Gamma$ is concentrated in non-positive degrees.

  * •

The (ordinary) algebra $H^{0}(\Gamma)$ is finite-dimensional.

  * •

The DG algebra $\Gamma$ is bimodule $(d+1)$-Calabi–Yau.

Then, the following statements hold:

  1. (1)

The Verdier quotient

$\mathcal{C}(\Gamma)=\operatorname{D}^{\mathrm{c}}(\Gamma)/\operatorname{D}^{\mathrm{fd}}(\Gamma)$

is well defined and is an (algebraic) $d$-Calabi–Yau triangulated category
with split idempotents. In particular, $\mathcal{C}(\Gamma)$ has finite-
dimensional morphism spaces.

  2. (2)

[Lad, Lemma 2.7] There are isomorphisms of vector spaces

$\mathcal{C}(\Gamma)(\Gamma[i],\Gamma)=H^{-i}(\Gamma),\qquad 0\leq i<d.$

  3. (3)

The image of $\Gamma$ in $\mathcal{C}(\Gamma)$ is a $d$-cluster tilting object
whose endomorphism algebra is isomorphic to $H^{0}(\Gamma)$.

######  Definition 6.1.2.

Let $\Gamma$ be a DG algebra that satisfies the conditions in Theorem 6.1.1
for some $d\geq 1$. Following [IY18], we call $\mathcal{C}(\Gamma)$ the
_Amiot–Guo–Keller (AGK) cluster category of $\Gamma$_.

In the context of Theorem 6.1.1, the following result of Iyama and Oppermann
gives necessary and sufficient conditions for the image of $\Gamma$ in
$\mathcal{C}(\Gamma)$ to be a $d\mathbb{Z}$-cluster tilting object. We
introduce the following definition, simply as a matter of convenience, compare
with the ‘vosnex property’ in [IO13, Not. 3.5].

######  Definition 6.1.3.

Let $\Gamma$ be a DG algebra and $d\geq 1$. We say that the cohomology of
$\Gamma$ _vanishes in small negative degrees (relative to $d$)_, if

$\forall 0<i<d-1,\qquad H^{-i}(\Gamma)=0.$

(notice that this condition is in fact vacuous for $d=1,2$).

######  Proposition 6.1.4 ([IO11, Prop. 3.6]).

Let $\Gamma$ be a DG algebra that satisfies the conditions in Theorem 6.1.1
for some $d\geq 1$. The following statements are equivalent:

  1. (1)

The image of $\Gamma$ in $\mathcal{C}(\Gamma)$ is a $d\mathbb{Z}$-cluster
tilting object.

  2. (2)

The following conditions are satisfied:

     * •

The finite-dimensional algebra $H^{0}(\Gamma)$ is self-injective.

     * •

The cohomology of $\Gamma$ vanishes in small negative degrees (relative to
$d$).

Our main result yields the following recognition theorem for the AGK cluster
category of a DG algebra with vanishing cohomology in small negative degrees.

######  Theorem 6.1.5.

Let $\Gamma$ be a DG algebra that satisfies the conditions in Theorem 6.1.1
for some $d\geq 1$. Assume that

  * •

The finite-dimensional algebra $H^{0}(\Gamma)$ is self-injective.

  * •

The cohomology of $\Gamma$ vanishes in small negative degrees (relative to
$d$).

(equivalently, the image of $\Gamma$ in $\mathcal{C}(\Gamma)$ is a
$d\mathbb{Z}$-cluster tilting object). Then, the following statements hold:

  1. (1)

The $d$-Calabi–Yau AGK cluster category $\mathcal{C}(\Gamma)$ admits a unique
DG enhancement.

  2. (2)

Suppose, moreover, that the algebra $H^{0}(\Gamma)$ is connected and non-
semisimple. Let $\mathcal{T}$ be an algebraic triangulated category with
finite-dimensional morphism spaces and split idempotents. If there exists a
$d\mathbb{Z}$-cluster tilting object $c\in\mathcal{T}$ such that the algebras
$\mathcal{T}(c,c)$ and $H^{0}(\Gamma)$ are isomorphic, then

$\mathcal{T}\simeq\mathcal{C}(\Gamma)$

as triangulated categories.

###### Proof.

In view of Theorem 6.1.1 and Proposition 6.1.4, both claims are immediate
consequences of A and Proposition 2.2.6. ∎

######  Remark 6.1.6.

In the literature, there are several theorems (both in the Calabi–Yau case and
in the general case) that exhibit an algebraic triangulated category with a
$d$-cluster tilting object (or even a more general type of generator) as the
AGK cluster category of a homologically smooth DG algebra, see [Tab07b], [KY,
Thm.4.1] and [Han22, Thm. 3.3]. These theorems rely on the choice of an
enhancement and do not address the question of _uniqueness_ of the
enhancement. Notice, however, that Theorem 6.1.5 includes a stringent
assumption on the DG algebra being considered: Namely, its cohomology is
required to vanish in small negative degrees, see also Remark 6.1.7 below.

######  Remark 6.1.7.

Suppose that $\mathbf{k}$ is an algebraically closed field and let $d>2$.
Given an _arbitrary_ basic finite-dimensional algebra $A$, Ladkani constructs
a DG algebra $\Gamma$ with $H^{0}(\Gamma)\cong A$ and that satisfies the
conditions in Theorem 6.1.1, see [Lad, Thm. 3.26]. In fact, Ladkani’s methods
permit to construct, for certain algebras $A$, DG algebras $\Gamma$ and
$\Gamma^{\prime}$ such that

$H^{0}(\Gamma)\cong A\cong H^{0}(\Gamma^{\prime}),$

but such that $\mathcal{C}(\Gamma)$ and $\mathcal{C}(\Gamma^{\prime})$ are
_not_ equivalent as triangulated categories, see [Lad, Ex. 3.30 and Prop.
3.31]. However, Ladkani also shows that the cohomology of $\Gamma$ vanishes in
small negative degrees only when $A$ is hereditary and, as it turns out,
$\mathcal{C}(\Gamma)$ is equivalent as a triangulated category to the
$d$-Calabi–Yau cluster category of $A$ [Kel05a, Sec. 6] when this is the case,
see [Lad, Prop. 3.18 and Ex. 3.30]. In particular, in view of the equivalent
conditions in Theorem 6.1.1, Ladkani’s methods yield a $d\mathbb{Z}$-cluster
tilting object $\Gamma\in\mathcal{C}(\Gamma)$ only when $A$ is semisimple
(since $A$ must be both hereditary and self-injective).

###  6.2. The AGK cluster category of a $d$-representation finite algebra

######  Setting 6.2.1.

We assume that $\mathbf{k}$ is a perfect field throughout this subsection.

We highlight a class of DG algebras that satisfy the conditions in Theorem
6.1.5, and hence their associated AGK cluster category is essentially unique.
These DG algebras arise naturally in the context of Iyama’s higher-dimensional
Auslander–Reiten theory [Iya07a, Iya07b].

The following class of algebras was introduced by Iyama and Oppermann. Our
choice of terminology is inspired by [HIO14, Def. 3.2 and Thm. 3.4].

######  Definition 6.2.2 ([IO11, Def. 2.2]).

Let $A$ be a finite-dimensional algebra. We say that $A$ is _$d$
-representation-finite ($d$-hereditary)_ if the following conditions are
satisfied:

  1. (1)

The algebra $A$ has global dimension at most $d$.

  2. (2)

There exists a _$d$ -cluster tilting $A$-module_ $M\in\operatorname{mod}A$,
that is such that the following conditions are satisfied:

    1. (a)

An $A$-module $L$ lies in $\operatorname{add}(M)$ if and only if

$\forall 0<i<d,\qquad\operatorname{Ext}_{A}^{i}\\!\left(L,M\right)=0.$

    2. (b)

An $A$-module $N$ lies in $\operatorname{add}(M)$ if and only if

$\forall 0<i<d,\qquad\operatorname{Ext}_{A}^{i}\\!\left(M,N\right)=0.$

######  Remark 6.2.3.

Notice that the $1$-representation finite algebras are precisely the
hereditary finite-dimensional algebras of finite representation type. Thus,
$d$-representation finite algebras can be regarded as ‘higher-dimensional
analogues’ of the latter class of algebras.

######  Remark 6.2.4.

In the context of Definition 6.2.2, the $d$-cluster tilting module turns out
to be unique, up to multiplicity of its indecomposable direct summands [Iya11,
Prop. 1.3].

Following Keller [Kel11], given a homologically smooth DG algebra $A$ and an
integer $n$, we consider its _derived n-preprojective algebra
$\mathbf{\Pi}_{n}(A)$_, which is defined as the tensor DG algebra of the
shifted inverse dualising DG $A$-bimodule

$\operatorname{RHom}_{A^{e}}\\!\left(A,A^{e}\right){[n-1]}.$

Crucially, the DG algebra $\mathbf{\Pi}_{n}(A)$ is homologically smooth and
bimodule $n$-Calabi–Yau [Kel11, Thm. 4.8].

Notice that we may apply the above construction when $A$ is a finite-
dimensional algebra of global dimension at most $d$, viewed as a DG algebra
concentrated in degree $0$ (recall that we assume $\mathbf{k}$ to be a perfect
field). In this case, we consider the derived $(d+1)$-preprojective algebra
$\mathbf{\Pi}_{d+1}(A)$ and, following Iyama and Oppermann, we also consider
the (non-derived) _(d+1)-preprojective algebra_

$\Pi_{d+1}(A)\coloneqq H^{0}(\mathbf{\Pi}_{d+1}(A)),$

compare with [IO13, Def. 2.11 and Rmk. 2.12]. As explained in the paragraph
before [Guo11, Cor. 4.8], the cohomology of $\mathbf{\Pi}_{d+1}(A)$ is
concentrated in non-positive degrees.

######  Remark 6.2.5.

Let $A$ be a finite-dimensional algebra of global dimension at most $d$ and
$\mathbf{\Pi}_{d+1}(A)$ its derived $(d+1)$-preprojective algebra. Then, the
$(d+1)$-preprojective algebra $\Pi_{d+1}(A)$ is isomorphic to the tensor
algebra of the $A$-bimodule $\operatorname{Ext}_{A}^{d}\\!\left(DA,A\right)$,
see [IO11, Prop. 2.12] and compare with [Guo11, Lemma 2.7] and the discussion
afterwards.

In the context of this article, our interest in $d$-representation finite
algebras stems from the following theorem, see also the proof of [CDIM, Prop.
8.6].

######  Theorem 6.2.6 ([IO13, Lemma 2.13 and Cor. 3.7]).

Let $A$ be a $d$-representation finite algebra. Then, its derived
$(d+1)$-preprojective algebra $\mathbf{\Pi}_{d+1}(A)$ satisfies the equivalent
conditions in Proposition 6.1.4.

We obtain the following recognition theorem for the AGK cluster category of a
$d$-representation finite algebra.

######  Theorem 6.2.7.

Let $A$ be a $d$-representation finite algebra and $\mathbf{\Pi}_{d+1}(A)$ its
derived $(d+1)$-preprojective algebra. Then, the following statements hold:

  1. (1)

The $d$-Calabi–Yau AGK cluster category $\mathcal{C}(\mathbf{\Pi}_{d+1}(A))$
admits a unique DG enhancement.

  2. (2)

Suppose, moreover, that the algebra $\Pi_{d+1}(A)$ is connected and non-
semisimple. Let $\mathcal{T}$ be an algebraic triangulated category with
finite-dimensional morphism spaces and split idempotents. If there exists a
$d\mathbb{Z}$-cluster tilting object $c\in\mathcal{T}$ such that the algebra
$\mathcal{T}(c,c)$ is isomorphic to the (non-derived) $(d+1)$-preprojective
algebra $\Pi_{d+1}(A)$, then

$\mathcal{T}\simeq\mathcal{C}(\mathbf{\Pi}_{d+1}(A))$

as triangulated categories.

###### Proof.

In view of Theorem 6.2.6, the claims is immediate from Theorem 6.1.5 and
Proposition 2.2.10. ∎

###  6.3. The Amiot cluster category of a self-injective quiver with potential

Recall that a _quiver with potential_ is a pair $(Q,W)$ consisting of a finite
quiver and a possibly-infinite linear combination $W$ of (oriented) cycles in
$Q$. To these data, one associates its _(completed) Jacobian algebra
$J(Q,W)$_, see [DWZ08, Def. 3.1] for the precise definition. Following
Herschend and Iyama [HI11], we say that $(Q,W)$ is _self-injective_ if its
completed Jacobian algebra is a finite-dimensional self-injective algebra.

Given a quiver with potential $(Q,W)$, we are also interested in the
_completed ( $3$-Calabi–Yau) Ginzburg DG algebra $\Gamma(Q,W)$ of $(Q,W)$_,
see [Gin] and [KY11, Sec. 2.6]. Replacing DG algebras and DG modules by their
pseudo-compact counter-parts, Theorem 6.1.1 holds _mutatis mutandis_ for the
pseudo-compact DG algebra $\Gamma(Q,W)$, see [KY11, Thm. A.27]. In particular,
associated to $(Q,W)$ there is a well-defined $2$-Calabi–Yau triangulated
category

$\mathcal{C}(Q,W)=\mathcal{C}(\Gamma(Q,W)),$

which we call the _Amiot cluster category of $(Q,W)$_, and the image of
$\Gamma$ in $\mathcal{C}(\Gamma)$ is a $2$-cluster tilting object whose
endomorphism algebra is isomorphic to the Jacobian algebra of $(Q,W)$, see
also [Ami09]. Consequently, we obtain the following recognition theorem for
the Amiot cluster category of a self-injective quiver with potential.

######  Theorem 6.3.1.

Let $\mathbf{k}$ be an arbitrary field. Let $(Q,W)$ be a quiver with potential
whose Jacobian algebra is finite-dimensional and self-injective. Then, the
following statements hold:

  1. (1)

The (2-Calabi–Yau) Amiot cluster category $\mathcal{C}(Q,W)$ admits a unique
DG enhancement.

  2. (2)

Suppose, moreover, that the Jacobian algebra $J(Q,W)$ is connected and non-
semisimple. Let $\mathcal{T}$ be an algebraic triangulated category with
finite-dimensional morphism spaces and split idempotents. If there exists a
$2\mathbb{Z}$-cluster tilting object $c\in\mathcal{T}$ such that the algebrsa
$\mathcal{T}(c,c)$ and $J(Q,W)$ are isomorphic, then

$\mathcal{T}\simeq\mathcal{C}(Q,W)$

as triangulated categories.

###### Proof.

In view of the previous discussion, both claims are immediate consequences of
Theorems 5.1.6 and 5.1.8 and Proposition 2.2.6, keeping in mind that, for
$d=2$, the condition on the cohomology of $\Gamma$ vanishing in small negative
degrees is vacuous.∎

######  Remark 6.3.2.

Suppose that $\mathbf{k}$ is an algebraically closed field. In this case,
Theorem 6.3.1 can be seen as an instance of Theorem 6.2.7 for completed
derived higher preprojective algebras. Indeed, the completed Ginzburg DG
algebra of a self-injective quiver with potential is quasi-isomorphic to the
completed derived $3$-preprojective algebra of some $2$-representation finite
algebra, see the proof of [HI11, Thm. 3.11(a)] and [HI11, Prop. 2.4].

######  Remark 6.3.3.

Classifying all quivers with potential whose Jacobian algebra is self-
injective seems to be an intractable problem. Families of such quivers with
potential are constructed in [HI11, Pas20], see also [Jas15, Thm. 1.3] where
the self-injective cluster tilted algebras of canonical type are classified.

######  Remark 6.3.4.

In relation to Theorem 6.3.1, we note that Keller and Liu have recently
announced [KL23] a proof of the following (modified) conjecture of Amiot
[Ami08]: Suppose that the ground field $\mathbf{k}$ is algebraically closed
and of characteristic $0$. Given a Karoubian pre-triangulated DG category
$\mathcal{A}$ (enriched in cochain complexes of pseudo-compact vector spaces)
equipped with a right $2$-Calabi–Yau structure in the sense of [KS06] which
admits a basic $2$-cluster tilting object $c\in H^{0}(\mathcal{A})$ with
finite-dimensional endomorphism algebra, there exists a quiver with potential
$(Q,W)$ and an equivalence of triangulated categories

$\mathcal{C}(Q,W)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}H^{0}(\mathcal{A}),\qquad\Gamma(Q,W)\longmapsto
c.$

In particular, $H^{0}(\mathcal{A}(c,c))\cong H^{0}(\Gamma(Q,W))=J(Q,W)$.
Notice, however, the explicit assumption on the enhancement. On the other
hand, Keller’s theorem is more general than Theorem 6.1.5 in that the object
$c$ need not be $2\mathbb{Z}$-cluster tilting and its endomorphism algebra is
known to be isomorphic to a completed Jacobian algebra only _a posteriori_ ,
and this is one place where the existence of a right $2$-Calabi–Yau structure
plays an important role in Keller’s proof (compare with [VdB15]).

###  6.4. Cohen–Macaulay modules for one-dimensional hypersurface
singularities

Let $(R,\mathfrak{m})$ be a local complete $d$-dimensional commutative
noetherian Gorenstein isolated singularity. We are interested in the category

$\operatorname{CM}(R)\coloneqq\left\\{M\in\operatorname{mod}(R)\right|\operatorname{depth}M=\dim
R\\}$

of _maximal Cohen–Macaulay $R$-modules_; this is a Frobenius exact category
whose projective objects are the projective $R$-modules. Moreover, it is well
known that there is an equivalence of triangulated categories [Buc21]

$\underline{\operatorname{CM}}(R)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{D}_{\mathrm{sing}}(R)\coloneqq\operatorname{D}^{\mathrm{b}}(R)/\operatorname{D}^{\mathrm{c}}(R)$

between the stable category of maximal Cohen–Macaulay $R$-modules and the
singularity category of $R$. The following result of Burban, Iyama, Keller and
Reiten gives a large class of one-dimensional hypersurface singularities whose
stable category of maximal Cohen–Macaulay modules admits a
$2\mathbb{Z}$-cluster tilting object.

######  Theorem 6.4.1 ([BIKR08, Thm. 1.5]).

Let $\mathbf{k}$ be an algebraically closed field of characteristic zero. Let
$R=\mathbf{k}\llbracket x,y\rrbracket/(f)$ be a one-dimensional reduced
hypersurface singularity. The following statements are equivalent:

  1. (1)

The stable category $\underline{\operatorname{CM}}(R)$ of Cohen–Macaulay
$R$-modules admits a $2\mathbb{Z}$-cluster tilting object.

  2. (2)

$f$ is a product $f=f_{1}f_{2}\cdots f_{n}$ with $f_{i}\not\in(x,y)^{2}$.

######  Remark 6.4.2.

In [BIKR08], the authors only discuss $2$-cluster tilting objects. However,
for hypersurface singularities the stable category of Cohen–Macaulay modules
is $2$-periodic, that is the shift functor squares to the identity up to
natural isomorphism [Eis80] (see also [BIKR08, Sec. 1]). In particular, all
$2$-cluster tilting objects are also $2\mathbb{Z}$-cluster tilting.

######  Theorem 6.4.3.

Let $\mathbf{k}$ be an algebraically closed field of characteristic zero. Let
$R=\mathbf{k}\llbracket x,y\rrbracket/(f)$ be a one-dimensional reduced
hypersurface singularity such that $f$ is a product $f=f_{1}f_{2}\cdots f_{n}$
with $f_{i}\not\in(x,y)^{2}$. Then, the stable category
$\underline{\operatorname{CM}}(R)$ of Cohen–Macaulay $R$-modules admits a
unique enhancement.

###### Proof.

In view of Theorem 6.4.1, the claim follows immediately from A. ∎

###  6.5. Stable categories of self-injective higher Nakayama algebras

Given integers $n\geq 1$ and $\ell\geq 2$, using results of Darpö and Iyama
[DI20], Külshammer and the first-named author introduced in [JK19] a family of
finite-dimensional algebras

$\widetilde{A}_{n-1,\ell}^{(d)},\qquad n,\ell\geq 1,$

called the _self-injective $d$-Nakayama algebras_. For example, if $d=1$, then
$\widetilde{A}_{n-1,\ell}^{(1)}$ is the classical (connected) Nakayama algebra
with $n$-simples and indecomposable projective modules of Loewy length $\ell$.
On the other hand, if $d=2$ and $n=1$, then $\widetilde{A}_{0,\ell}^{(2)}$ is
isomorphic to the Gelfand–Ponomarev preprojective algebra of Dynkin type
$\mathbb{A}_{\ell}$. More generally, if $d\geq 2$ and $n=1$, then
$\widetilde{A}_{0,\ell}^{(d)}$ is isomorphic to the _$d$ -preprojective
algebra of Dynkin type $\mathbb{A}_{\ell}$_ first investigated by Iyama and
Oppermann in [IO11, Sec. 5] (see also [GKO13, 6.3 and 6.5]).

######  Theorem 6.5.1 ([JK19, Thm. 4.10]).

Let $n\geq 1$ and $\ell\geq 2$ and $A=\widetilde{A}_{n-1,\ell}^{(d)}$ the
corresponding self-injective $d$-Nakayama algebra. The following statements
hold:

  1. (1)

The algebra $A$ is indeed a finite-dimensional self-injective algebra.

  2. (2)

There exists a distinguished $d\mathbb{Z}$-cluster tilting $A$-module
$M=M_{n-1,\ell}^{(d)}$.

  3. (3)

The triangulated category $\operatorname{\underline{mod}}(A)$ admits a
$d\mathbb{Z}$-cluster tilting object (namely, the image of $M$).

We obtain the following recognition theorem for the stable module category of
a self-injective $d$-Nakayama algebra.

######  Theorem 6.5.2.

Let $\mathbf{k}$ be an arbitrary field. Let $n\geq 1$ and $\ell\geq 2$ and
$A=\widetilde{A}_{n-1,\ell}^{(d)}$ the corresponding self-injective
$d$-Nakayama algebra. The following statements hold:

  1. (1)

The stable module category $\operatorname{\underline{mod}}(A)$ admits a unique
enhancement.

  2. (2)

Let $\mathcal{T}$ be an algebraic triangulated category with finite-
dimensional morphism spaces and split idempotents. If there exists a
$d\mathbb{Z}$-cluster tilting object $c\in\mathcal{T}$ such that the algebra
$\mathcal{T}(c,c)$ is isomorphic to the stable endomorphism algebra
$\underline{\operatorname{End}}_{A}(M)$ of the distinguished
$d\mathbb{Z}$-cluster tilting $A$-module $M$. Then,

$\mathcal{T}\simeq\operatorname{\underline{mod}}(A)$

as triangulated categories.

###### Proof.

In view of Theorem 6.5.1, both claims are immediate from A. Indeed, using the
explicit description of the endomorphism algebra of the distinguished
$d\mathbb{Z}$-cluster tilting $A$-module $M$ given in [JK19, Def. 4.9], it is
easy to verify that there is an isomorphism of algebras

$\underline{\operatorname{End}}_{A}(M)\cong A_{n-1,\ell-1}^{(d+1)},$

that is the length parameter $\ell$ is reduced by $1$ while the dimension
parameter $d$ increases by $1$. In particular,
$\underline{\operatorname{End}}_{A}(M)$ is isomorphic to the bounded path
algebra of a finite quiver. ∎

###  6.6. Singularity categories of Iwanaga–Gorenstein algebras with a
$d\mathbb{Z}$-cluster tilting module

Recall that a finite-dimensional algebra $A$ is _Iwanaga–Gorenstein_ if

$\operatorname{inj.dim}A_{A}<\infty\qquad\text{and}\qquad\operatorname{inj.dim}{}_{A}A<\infty.$

We are interested in the category

$\operatorname{Gproj}(A)\coloneqq\left\\{M\in\operatorname{mod}(A)\right|\forall
i>0,\ \operatorname{Ext}_{A}^{i}\\!\left(M,A\right)=0\\}$

of (finite-dimensional) _Gorenstein projective $A$-modules_; this is a
Frobenius exact category whose projective objects are the projective
$A$-modules. Moreover, it is well known that there is an equivalence of
triangulated categories [Buc21]

$\underline{\operatorname{Gproj}}(A)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{D}_{\mathrm{sing}}(A)\coloneqq\operatorname{D}^{\mathrm{b}}(A)/\operatorname{D}^{\mathrm{c}}(A)$

between the stable category of Gorenstein projective $A$-modules and the
singularity category of $A$. We recall the following theorem of Kvamme.

######  Theorem 6.6.1 ([Kva21, Cor. 1.3]).

Let $A$ be a finite-dimensional Iwanaga–Gorenstein algebra. Suppose that there
exists $d\mathbb{Z}$-cluster tilting $A$-module $M\in\operatorname{mod}(A)$
and let $\overline{M}$ be the largest direct summand of $M$ that is Gorenstein
projective. Then, $\overline{M}$ is a $d\mathbb{Z}$-cluster tilting object in
$\underline{\operatorname{Gproj}}(A)\simeq\operatorname{D}_{\mathrm{sing}}(A)$.

We obtain the following recognition theorem for the singularity category of an
Iwanaga–Gorenstein algebra with a $d\mathbb{Z}$-cluster tilting module.

######  Theorem 6.6.2.

Let $\mathbf{k}$ be an arbitrary field. Let $A$ be a finite-dimensional
Iwanaga–Gorenstein algebra. Suppose that there exists a $d\mathbb{Z}$-cluster
tilting $A$-module $M\in\operatorname{mod}(A)$ and let $\overline{M}$ be the
largest direct summand of $M$ that is Gorenstein projective. The following
statements hold:

  1. (1)

The singularity category $\operatorname{D}_{\mathrm{sing}}(A)$ admits a unique
enhancement.

  2. (2)

Suppose, moreover, that the algebra
$\underline{\operatorname{End}}_{A}(\overline{M})$ is connected and non-
semisimple. Let $\mathcal{T}$ be an algebraic triangulated category with
finite-dimensional morphism spaces and split idempotents. If there exists a
$d\mathbb{Z}$-cluster tilting object $c\in\mathcal{T}$ such that the algebra
$\mathcal{T}(c,c)$ is isomorphic to the stable endomorphism algebra
$\underline{\operatorname{End}}_{A}(\overline{M})$, then

$\mathcal{T}\simeq\operatorname{D}_{\mathrm{sing}}(A)$

as triangulated categories.

###### Proof.

In view of Theorem 6.6.1, both claims are immediate from A. ∎

For instances where Theorem 6.6.2 can be applied see [McM20, Xin23] for
example.

##  Appendix A The Donovan–Wemyss conjecture, by Bernhard Keller

We work over the field of complex numbers $\mathbf{k}=\mathbb{C}$. A compound
Du Val (=cDV) singularity is a singularity of the form

$R=\mathbb{C}[[u,v,x,y]]/(f(u,v,x)+yg(u,v,x,y))$

such that $\mathbb{C}[[u,v,x]]/(f(u,v,x))$ is a Kleinian surface singularity,
cf. Definition 4.2 in [Aug20]. These singularities were introduced by Miles
Reid [Rei83] at the beginning of the eighties and play an important role in
the minimal model program in birational geometry. We refer to the introduction
of [Aug19] as well as to [Wem23] for excellent introductory surveys on this
subject.

From now on, we make the blanket assumption that all the cDV singularities we
consider are isolated and admit a _(smooth) resolution_ because the conjecture
only applies to these.

Let us fix a cDV singularity $R$. Before stating the conjecture, let us list
the most important properties of its singularity category

$\mathrm{sg}(R)=\operatorname{D}_{\mathrm{sing}}(R)=\operatorname{D}^{\mathrm{b}}(R)/\operatorname{D}^{\mathrm{c}}(R).$

Notice first that since $R$ is a hypersurface, it is Gorenstein and the
canonical functor

$\operatorname{CM}(R)\to\operatorname{mod}R\to\operatorname{D}^{\mathrm{b}}(R)$

induces an equivalence from the stable category of Cohen-Macauley modules
$\underline{\operatorname{CM}}(R)$ to $\mathrm{sg}(R)$, cf. [Buc21]. Since the
singularity is isolated, the category $\mathrm{sg}(R)$ is
$\mathrm{Hom}$-finite. Moreover, it is Krull–Schmidt with split idempotents
since $R$ is complete. By definition, it is algebraic. Since $R$ is a
hypersurface, we can describe $\mathrm{sg}(R)$ using matrix factorisations and
thus, the square $\Sigma^{2}$ of the suspension functor is isomorphic to the
identity functor, i.e. the category $\mathrm{sg}(R)$ is $2$-periodic. Finally,
it is $2$-Calabi–Yau since it is the stable category of Cohen–Macauley modules
over a Gorenstein ring of Krull dimension $3$, cf. Prop. 1.3 in Ch. III of
[Aus78].

A contraction algebra for $R$ is [DW16, Wem18] the stable endomorphism algebra
of a $2$-cluster-tilting object $T$ of the singularity category
$\mathrm{sg}(R)$. Now let $R_{1}$ and $R_{2}$ be two cDV singularities and let
$A_{i}$ be a contraction algebra for $R_{i}$, $i=1,2$. In its original form
the Donovan–Wemyss conjecture states

######  Conjecture A.1 ([DW16]).

Suppose that $A_{1}$ and $A_{2}$ are local. Then $A_{1}$ is isomorphic to
$A_{2}$ if and only if $R_{1}$ is isomorphic to $R_{2}$.

The sufficiency was shown in [DW16]. In type A, the conjecture follows from
the work of Reid [Rei83]. In type D, evidence is given in [BW18, vG, Kaw].
Further evidence comes from [Hua18, HT18, HK, Boo21], where the authors use
enhancements of the contraction algebra.

Later Donovan–Wemyss generalized their conjecture as follows for not
necessarily local contraction algebras:

######  Conjecture A.2.

The contraction algebras $A_{1}$ and $A_{2}$ are derived equivalent if and
only if the singularities $R_{1}$ and $R_{2}$ are isomorphic.

In this form, the conjecture appears as Conjecture 1.3 in [Aug20]. The
sufficiency follows by combining results of [Wem18] with [Dug15]. We will now
deduce the necessity from A. Let us emphasize that our strategy of proof has
been known to the experts since the appearance of [HK] on the arXiv in October
2018. The missing puzzle piece was precisely (a special case of) A.

Let $R_{i}$ be cDV singularities with contraction algebra $A_{i}$, $i=1,2$,
and suppose that $A_{1}$ is derived equivalent to $A_{2}$. Then, by Theorem
1.6 of [Aug20], there is a contraction algebra $A^{\prime}_{2}$ for $R_{2}$
which is isomophic to $A_{1}$. Thus, we may and will assume that $A_{1}$ and
$A_{2}$ are isomorphic. By definition, the algebras $A_{i}$ are endomorphism
algebras of $2$-cluster-tilting objects $T_{i}$ in $\mathrm{sg}(R_{i})$. By
$2$-periodicity, these are in fact $2\mathbb{Z}$-cluster-tilting objects. The
$2$-periodicity also yields the commutative square

${\mathrm{sg}(R_{i})}$${\operatorname{add}(T_{i})}$${\operatorname{proj}(A_{i})}$${\mathrm{sg}(R_{i})}$${\operatorname{add}(T_{i})}$${\operatorname{proj}(A_{i})}$$\scriptstyle{\Sigma^{2}}$$\scriptstyle{\mathrm{Hom}(T_{i},?)}$$\scriptstyle{?\otimes_{A_{i}}\mbox{}_{\sigma_{i}}A_{i}}$$\scriptstyle{\mathrm{Hom}(T_{i},?)}$

where $\sigma_{i}$ is the identity automorphism of $A_{i}$. Now clearly the
given isomorphism $A_{1}\stackrel{{{}_{\sim}}}{{\rightarrow}}A_{2}$ yields an
equivalence between the pairs $(A_{1},\sigma_{1})$ and $(A_{2},\sigma_{2})$.
Thus, by A applied in dimension $d=2$, we obtain a triangle equivalence

${\mathrm{sg}(R_{1})}$${\mathrm{sg}(R_{2}).}$∼

Moreover, by the uniqueness of the dg enhancement in A , we even obtain an
isomorphism in the homotopy category of dg categories

${\mathrm{sg}_{dg}(R_{1})}$${\mathrm{sg}_{dg}(R_{2})\,,}$∼

where $\mathrm{sg}_{dg}(R_{i})$ denotes the canonically dg enhanced
singularity category. Using Theorem 5.9 of [HK], we deduce that there is an
isomorphism $R_{1}\cong R_{2}$.

## References

  * [AI12] Takuma Aihara and Osamu Iyama.  Silting mutation in triangulated categories.  J. Lond. Math. Soc. (2), 85(3):633–668, 2012.
  * [AIR15] Claire Amiot, Osamu Iyama, and Idun Reiten.  Stable categories of Cohen-Macaulay modules and cluster categories.  Amer. J. Math., 137(3):813–857, 2015.
  * [Ami07] Claire Amiot.  On the structure of triangulated categories with finitely many indecomposables.  Bull. Soc. Math. France, 135(3):435–474, 2007.
  * [Ami08] Claire Amiot.  Sur les petites catégories triangulées.  PhD thesis, Université Paris 7, July 2008.
  * [Ami09] Claire Amiot.  Cluster categories for algebras of global dimension 2 and quivers with potential.  Ann. Inst. Fourier (Grenoble), 59(6):2525–2590, 2009.
  * [Ang11] Vigleik Angeltveit.  Uniqueness of Morava _K_ -theory.  Compos. Math., 147(2):633–648, 2011.
  * [Ant] Benjamin Antieau.  On the uniqueness of infinity-categorical enhancements of triangulated categories, arXiv:1812.01526 [math.AG].
  * [AR75] Maurice Auslander and Idun Reiten.  Representation theory of Artin algebras. III. Almost split sequences.  Comm. Algebra, 3:239–294, 1975.
  * [AR78] Maurice Auslander and Idun Reiten.  Representation theory of Artin algebras. VI. A functorial approach to almost split sequences.  Comm. Algebra, 6(3):257–300, 1978.
  * [AR91] Maurice Auslander and Idun Reiten.  On a theorem of E. Green on the dual of the transpose.  In Representations of finite-dimensional algebras (Tsukuba, 1990), volume 11 of CMS Conf. Proc., pages 53–65. Amer. Math. Soc., Providence, RI, 1991.
  * [AR96] Maurice Auslander and Idun Reiten.  $D$tr-periodic modules and functors.  In Representation theory of algebras (Cocoyoc, 1994), volume 18 of CMS Conf. Proc., pages 39–50. Amer. Math. Soc., Providence, RI, 1996.
  * [AS80] M. Auslander and Sverre O. Smalø.  Preprojective modules over Artin algebras.  J. Algebra, 66(1):61–122, 1980.
  * [AS81] M. Auslander and Sverre O. Smalø.  Almost split sequences in subcategories.  J. Algebra, 69(2):426–454, 1981.
  * [AS93] M. Auslander and Ø. Solberg.  Gorenstein algebras and algebras with dominant dimension at least $2$.  Comm. Algebra, 21(11):3897–3934, 1993.
  * [Aug19] Jenny August.  The tilting theory of contraction algebras.  Ph. D. thesis, University of Edinburgh, 2019.
  * [Aug20] Jenny August.  On the finiteness of the derived equivalence classes of some stable endomorphism rings.  Math. Z., 296(3-4):1157–1183, 2020.
  * [Aus71] Maurice Auslander.  Representation dimension of Artin algebras.  Lecture Notes. Queen Mary College, 1971.
  * [Aus78] Maurice Auslander.  Functors and morphisms determined by objects.  In Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976), Lecture Notes in Pure Appl. Math., Vol. 37, pages 1–244. Dekker, New York, 1978.
  * [Bal02] P. Balmer.  Triangulated categories with several triangulations.  Available online at the author’s website: https://www.math.ucla.edu/~balmer/Pubfile/TriangulationS.pdf, 2002.
  * [BD89] Hans Joachim Baues and Winfried Dreckmann.  The cohomology of homotopy categories and the general linear group.  $K$-Theory, 3(4):307–338, 1989.
  * [Bel15] Apostolos Beligiannis.  Relative homology, higher cluster-tilting theory and categorified Auslander-Iyama correspondence.  J. Algebra, 444:367–503, 2015.
  * [BGL87] Dagmar Baer, Werner Geigle, and Helmut Lenzing.  The preprojective algebra of a tame hereditary Artin algebra.  Comm. Algebra, 15(1-2):425–457, 1987.
  * [BIKR08] Igor Burban, Osamu Iyama, Bernhard Keller, and Idun Reiten.  Cluster tilting for one-dimensional hypersurface singularities.  Adv. Math., 217(6):2443–2484, 2008.
  * [BIRS09] A. B. Buan, O. Iyama, I. Reiten, and J. Scott.  Cluster structures for 2-Calabi-Yau categories and unipotent groups.  Compos. Math., 145(4):1035–1079, 2009.
  * [BK72] A. K. Bousfield and D. M. Kan.  Homotopy limits, completions and localizations.  Lecture Notes in Mathematics, Vol. 304. Springer-Verlag, Berlin-New York, 1972.
  * [BK89] A. I. Bondal and M. M. Kapranov.  Representable functors, Serre functors, and reconstructions.  Izv. Akad. Nauk SSSR Ser. Mat., 53(6):1183–1205, 1337, 1989.
  * [BK90] A. I. Bondal and M. M. Kapranov.  Framed triangulated categories.  Mat. Sb., 181(5):669–683, 1990.
  * [BKS04] David Benson, Henning Krause, and Stefan Schwede.  Realizability of modules over Tate cohomology.  Trans. Amer. Math. Soc., 356(9):3621–3668, 2004.
  * [BLM08] Yu. Bespalov, V. Lyubashenko, and O. Manzyuk.  Pretriangulated $A_{\infty}$-categories, volume 76 of Proceedings of Institute of Mathematics of NAS of Ukraine. Mathematics and its Applications.  Institute of Mathematics of NAS of Ukraine, Kiev, 2008.
  * [BMFM20] Urtzi Buijs, José M. Moreno-Fernández, and Aniceto Murillo.  $A_{\infty}$ structures and Massey products.  Mediterr. J. Math., 17(1):Paper No. 31, 15, 2020.
  * [BMR+06] Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov.  Tilting theory and cluster combinatorics.  Adv. Math., 204(2):572–618, 2006.
  * [BO01] Alexei Bondal and Dmitri Orlov.  Reconstruction of a variety from the derived category and groups of autoequivalences.  Compositio Math., 125(3):327–344, 2001.
  * [Bod14] Agnieszka Bodzenta-Skibińska.  DG Categories and Derived Categories of Coherent Sheaves.  PhD thesis, University of Warsaw, 2014.
  * [Bol84] Michael L. Bolla.  Isomorphisms between endomorphism rings of progenerators.  J. Algebra, 87(1):261–281, 1984.
  * [Boo21] Matt Booth.  Singularity categories via the derived quotient.  Adv. Math., 381:Paper No. 107631, 56, 2021.
  * [BS01] Paul Balmer and Marco Schlichting.  Idempotent completion of triangulated categories.  J. Algebra, 236(2):819–834, 2001.
  * [BT13] Petter Andreas Bergh and Marius Thaule.  The axioms for $n$-angulated categories.  Algebr. Geom. Topol., 13(4):2405–2428, 2013.
  * [Buc21] Ragnar-Olaf Buchweitz.  Maximal Cohen-Macaulay modules and Tate cohomology, volume 262 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, [2021] ©2021.  With appendices and an introduction by Luchezar L. Avramov, Benjamin Briggs, Srikanth B. Iyengar and Janina C. Letz.
  * [BW18] Gavin Brown and Michael Wemyss.  Gopakumar-Vafa invariants do not determine flops.  Comm. Math. Phys., 361(1):143–154, 2018.
  * [CDIM] Aaron Chan, Erik Darpö, Osamu Iyama, and René Marczinzik.  Periodic trivial extension algebras and fractionally Calabi–Yau algebras, arXiv:2012.11927 [math.RT].
  * [CF17] J. Daniel Christensen and Martin Frankland.  Higher Toda brackets and the Adams spectral sequence in triangulated categories.  Algebr. Geom. Topol., 17(5):2687–2735, 2017.
  * [Che21] Justin Chen.  Surjections of unit groups and semi-inverses.  J. Commut. Algebra, 13(3):323–331, 2021.
  * [CNS22] Alberto Canonaco, Amnon Neeman, and Paolo Stellari.  Uniqueness of enhancements for derived and geometric categories.  Forum Math. Sigma, 10:Paper No. e92, 65, 2022.
  * [Coh68] Joel M. Cohen.  The decomposition of stable homotopy.  Ann. of Math. (2), 87:305–320, 1968.
  * [CS18] Alberto Canonaco and Paolo Stellari.  Uniqueness of dg enhancements for the derived category of a Grothendieck category.  J. Eur. Math. Soc. (JEMS), 20(11):2607–2641, 2018.
  * [DI20] Erik Darpö and Osamu Iyama.  $d$-representation-finite self-injective algebras.  Adv. Math., 362:106932, 50, 2020.
  * [DJL21] Tobias Dyckerhoff, Gustavo Jasso, and Yankı Lekili.  The symplectic geometry of higher Auslander algebras: Symmetric products of disks.  Forum of Mathematics, Sigma, 9:e10, 2021.
  * [DK08] Raika Dehy and Bernhard Keller.  On the combinatorics of rigid objects in 2-Calabi-Yau categories.  Int. Math. Res. Not. IMRN, (11):Art. ID rnn029, 17, 2008.
  * [DS07] Daniel Dugger and Brooke Shipley.  Topological equivalences for differential graded algebras.  Adv. Math., 212(1):37–61, 2007.
  * [Dug10] Alex S. Dugas.  Periodic resolutions and self-injective algebras of finite type.  J. Pure Appl. Algebra, 214(6):990–1000, 2010.
  * [Dug12] Alex Dugas.  Periodicity of $d$-cluster-tilted algebras.  J. Algebra, 368:40–52, 2012.
  * [Dug15] Alex Dugas.  A construction of derived equivalent pairs of symmetric algebras.  Proc. Amer. Math. Soc., 143(6):2281–2300, 2015.
  * [DW16] Will Donovan and Michael Wemyss.  Noncommutative deformations and flops.  Duke Math. J., 165(8):1397–1474, 2016.
  * [DWZ08] Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky.  Quivers with potentials and their representations. I. Mutations.  Selecta Math. (N.S.), 14(1):59–119, 2008.
  * [Eis80] David Eisenbud.  Homological algebra on a complete intersection, with an application to group representations.  Trans. Amer. Math. Soc., 260(1):35–64, 1980.
  * [EKMM07] A. Elmendorf, I. Kriz, M. Mandell, and J. May.  Rings, Modules, and Algebras in Stable Homotopy Theory, volume 47 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, Rhode Island, April 2007\.
  * [Eno81] Edgar E. Enochs.  Injective and flat covers, envelopes and resolvents.  Israel J. Math., 39(3):189–209, 1981.
  * [ERZ57] Samuel Eilenberg, Alex Rosenberg, and Daniel Zelinsky.  On the dimension of modules and algebras. VIII. Dimension of tensor products.  Nagoya Math. J., 12:71–93, 1957.
  * [ES06] Karin Erdmann and Andrzej Skowroński.  The stable Calabi-Yau dimension of tame symmetric algebras.  J. Math. Soc. Japan, 58(1):97–128, 2006.
  * [ES08] Karin Erdmann and Andrzej Skowroński.  Periodic algebras.  In Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., pages 201–251. Eur. Math. Soc., Zürich, 2008\.
  * [Fed19] Francesca Fedele.  Auslander-Reiten $(d+2)$-angles in subcategories and a $(d+2)$-angulated generalisation of a theorem by Brüning.  J. Pure Appl. Algebra, 223(8):3554–3580, 2019.
  * [Fre66] Peter Freyd.  Stable homotopy.  In Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), pages 121–172. Springer, New York, 1966.
  * [Fre09] Benoit Fresse.  Operadic cobar constructions, cylinder objects and homotopy morphisms of algebras over operads, volume 504 of Contemp. Math., page 125–188.  Amer. Math. Soc., Providence, RI, 2009.
  * [FZ02] Sergey Fomin and Andrei Zelevinsky.  Cluster algebras. I. Foundations.  J. Amer. Math. Soc., 15(2):497–529, 2002.
  * [Ger63] Murray Gerstenhaber.  The cohomology structure of an associative ring.  Ann. of Math. (2), 78:267–288, 1963.
  * [GHW21] Eugene Gorsky, Matthew Hogancamp, and Paul Wedrich.  Derived Traces of Soergel Categories.  International Mathematics Research Notices, 04 2021.  rnab019.
  * [Gin] Victor Ginzburg.  Calabi–Yau algebras, math/0612139.
  * [GKO13] Christof Geiss, Bernhard Keller, and Steffen Oppermann.  $n$-angulated categories.  J. Reine Angew. Math., 675:101–120, 2013.
  * [GLS06] Christof Geiss, Bernard Leclerc, and Jan Schröer.  Rigid modules over preprojective algebras.  Invent. Math., 165(3):589–632, 2006.
  * [GP79] I. M. Gelfand and V. A. Ponomarev.  Model algebras and representations of graphs.  Funktsional. Anal. i Prilozhen., 13(3):1–12, 1979.
  * [GSS03] Edward L. Green, Nicole Snashall, and Øyvind Solberg.  The Hochschild cohomology ring of a selfinjective algebra of finite representation type.  Proc. Amer. Math. Soc., 131(11):3387–3393, 2003.
  * [Guo11] Lingyan Guo.  Cluster tilting objects in generalized higher cluster categories.  J. Pure Appl. Algebra, 215(9):2055–2071, 2011.
  * [Han20] Norihiro Hanihara.  Auslander correspondence for triangulated categories.  Algebra Number Theory, 14(8):2037–2058, 2020.
  * [Han22] Norihiro Hanihara.  Morita theorem for hereditary Calabi-Yau categories.  Adv. Math., 395:Paper No. 108092, 60, 2022.
  * [Hap88] Dieter Happel.  Triangulated categories in the representation theory of finite-dimensional algebras, volume 119 of London Mathematical Society Lecture Note Series.  Cambridge University Press, Cambridge, 1988.
  * [Hel68] Alex Heller.  Stable homotopy categories.  Bull. Amer. Math. Soc., 74:28–63, 1968.
  * [Her16] Stephen Hermes.  Minimal model of Ginzburg algebras.  J. Algebra, 459:389–436, 2016.
  * [HI11] Martin Herschend and Osamu Iyama.  Selfinjective quivers with potential and 2-representation-finite algebras.  Compos. Math., 147(6):1885–1920, 2011.
  * [HIMO] Martin Herschend, Osamu Iyama, Hiroyuki Minamoto, and Steffen Oppermann.  Representation theory of Geigle–Lenzing complete intersections, 1409.0668.  To appear in Mem. Amer. Math. Soc.
  * [HIO14] Martin Herschend, Osamu Iyama, and Steffen Oppermann.  $n$-representation infinite algebras.  Adv. Math., 252:292–342, 2014.
  * [HJR10] Thorsten Holm, Peter Jørgensen, and Raphaël Rouquier, editors.  Triangulated categories, volume 375 of London Mathematical Society Lecture Note Series.  Cambridge University Press, Cambridge, 2010.
  * [HK] Zheng Hua and Bernhard Keller.  Cluster categories and rational curves, arXiv:1810.00749 [math.AG]. Accepted for publication in Geom. Topol.
  * [HKK17] F. Haiden, L. Katzarkov, and M. Kontsevich.  Flat surfaces and stability structures.  Publ. Math. Inst. Hautes Études Sci., 126(1):247–318, 2017.
  * [Hov99] Mark Hovey.  Model categories, volume 63 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 1999.
  * [HPS97] Mark Hovey, John H. Palmieri, and Neil P. Strickland.  Axiomatic stable homotopy theory.  Mem. Amer. Math. Soc., 128(610):x+114, 1997.
  * [HT18] Zheng Hua and Yukinobu Toda.  Contraction algebra and invariants of singularities.  Int. Math. Res. Not. IMRN, (10):3173–3198, 2018.
  * [Hua18] Zheng Hua.  Contraction algebra and singularity of three-dimensional flopping contraction.  Math. Z., 290(1-2):431–443, 2018.
  * [IJ17] Osamu Iyama and Gustavo Jasso.  Higher Auslander correspondence for dualizing $R$-varieties.  Algebr. Represent. Theory, 20(2):335–354, 2017.
  * [IO11] Osamu Iyama and Steffen Oppermann.  $n$-representation-finite algebras and $n$-APR tilting.  Trans. Amer. Math. Soc., 363(12):6575–6614, 2011.
  * [IO13] Osamu Iyama and Steffen Oppermann.  Stable categories of higher preprojective algebras.  Adv. Math., 244:23–68, 2013.
  * [IS18] Osamu Iyama and Øyvind Solberg.  Auslander-Gorenstein algebras and precluster tilting.  Adv. Math., 326:200–240, 2018.
  * [IT13] Osamu Iyama and Ryo Takahashi.  Tilting and cluster tilting for quotient singularities.  Math. Ann., 356(3):1065–1105, 2013.
  * [IV14] S. O. Ivanov and Y. V. Volkov.  Stable Calabi-Yau dimension of self-injective algebras of finite type.  J. Algebra, 413:72–99, 2014.
  * [IW14] Osamu Iyama and Michael Wemyss.  Maximal modifications and Auslander-Reiten duality for non-isolated singularities.  Invent. Math., 197(3):521–586, 2014.
  * [IY08] Osamu Iyama and Yuji Yoshino.  Mutation in triangulated categories and rigid Cohen-Macaulay modules.  Invent. Math., 172(1):117–168, 2008.
  * [IY18] Osamu Iyama and Dong Yang.  Silting reduction and Calabi-Yau reduction of triangulated categories.  Trans. Amer. Math. Soc., 370(11):7861–7898, 2018.
  * [Iya07a] Osamu Iyama.  Auslander correspondence.  Adv. Math., 210(1):51–82, 2007.
  * [Iya07b] Osamu Iyama.  Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories.  Adv. Math., 210(1):22–50, 2007.
  * [Iya11] Osamu Iyama.  Cluster tilting for higher Auslander algebras.  Adv. Math., 226(1):1–61, 2011.
  * [Jas15] Gustavo Jasso.  $\tau^{2}$-stable tilting complexes over weighted projective lines.  Adv. Math., 273:1–31, 2015.
  * [Jas16] Gustavo Jasso.  $n$-abelian and $n$-exact categories.  Math. Z., 283(3-4):703–759, 2016.
  * [JK19] Gustavo Jasso and Julian Külshammer.  Higher Nakayama algebras I: Construction.  Adv. Math., 351:1139–1200, 2019.
  * [Kad80] T. V. Kadeishvili.  On the theory of homology of fiber spaces.  Uspekhi Mat. Nauk, 35(3(213)):183–188, 1980.  International Topology Conference (Moscow State Univ., Moscow, 1979).
  * [Kad82] T. V. Kadeishvili.  The algebraic structure in the homology of an $A(\infty)$-algebra.  Soobshch. Akad. Nauk Gruzin. SSR, 108(2):249–252 (1983), 1982.
  * [Kad88] T. V. Kadeishvili.  The structure of the $A(\infty)$-algebra, and the Hochschild and Harrison cohomologies.  Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR, 91:19–27, 1988.
  * [Kaj13] Hiroshige Kajiura.  On _A_ $\infty$-enhancements for triangulated categories.  J. Pure Appl. Algebra, 217(8):1476–1503, 2013.
  * [Kaw] Yujiro Kawamata.  Non-commutative deformations of perverse coherent sheaves and rational curves, arXiv:2006.09547 [math.AG].
  * [Kel65] G. M. Kelly.  Chain maps inducing zero homology maps.  Proc. Cambridge Philos. Soc., 61:847–854, 1965.
  * [Kel94] Bernhard Keller.  Deriving DG categories.  Ann. Sci. École Norm. Sup. (4), 27(1):63–102, 1994.
  * [Kel01] Bernhard Keller.  Introduction to $A$-infinity algebras and modules.  Homology Homotopy Appl., 3(1):1–35, 2001.
  * [Kel05a] Bernhard Keller.  On triangulated orbit categories.  Doc. Math., 10:551–581, 2005.
  * [Kel05b] G. M. Kelly.  Basic concepts of enriched category theory.  Repr. Theory Appl. Categ., (10):vi+137, 2005.
  * [Kel06] Bernhard Keller.  On differential graded categories.  In International Congress of Mathematicians. Vol. II, pages 151–190. Eur. Math. Soc., Zürich, 2006.
  * [Kel08] Bernhard Keller.  Calabi-Yau triangulated categories, 2008.
  * [Kel11] Bernhard Keller.  Deformed Calabi-Yau completions.  J. Reine Angew. Math., 654:125–180, 2011.  With an appendix by Michel Van den Bergh.
  * [Kel13] Bernhard Keller.  The periodicity conjecture for pairs of Dynkin diagrams.  Ann. of Math. (2), 177(1):111–170, 2013.
  * [KL23] Bernhard Keller and Junyang Liu.  Calabi-yau structures on drinfeld quotients and amiot’s conjecture.  February 2023, 2302.03681.
  * [KM98] János Kollár and Shigefumi Mori.  Birational geometry of algebraic varieties, volume 134 of Cambridge Tracts in Mathematics.  Cambridge University Press, Cambridge, 1998.  With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original.
  * [KR08] Bernhard Keller and Idun Reiten.  Acyclic Calabi-Yau categories.  Compos. Math., 144(5):1332–1348, 2008.  With an appendix by Michel Van den Bergh.
  * [Kra15] Henning Krause.  Krull-Schmidt categories and projective covers.  Expo. Math., 33(4):535–549, 2015.
  * [KS06] M. Kontsevich and Y. Soibelman.  Notes on $A_{\infty}$-algebras, $A_{\infty}$-categories and non-commutative geometry.  In Homological mirror symmetry, volume 757 of Lecture Notes in Phys., pages 153–219. Springer, Berlin, 2006.
  * [KV87] Bernhard Keller and Dieter Vossieck.  Sous les catégories dérivées.  C. R. Acad. Sci. Paris Sér. I Math., 305(6):225–228, 1987.
  * [Kva21] Sondre Kvamme.  $d\mathbb{Z}$-cluster tilting subcategories of singularity categories.  Math. Z., 297(1-2):803–825, 2021.
  * [KY] Martin Kalck and Dong Yang.  Relative singularity categories III: Cluster resolutions, arXiv:2006.09733 [math.RT].
  * [KY11] Bernhard Keller and Dong Yang.  Derived equivalences from mutations of quivers with potential.  Adv. Math., 226(3):2118–2168, 2011.
  * [Lad] Sefi Ladkani.  Finite-dimensional algebras are $(m>2)$-Calabi-Yau-tilted, arXiv:1603.09709 [math.RT].
  * [LH] Kenji Lefèvre-Hasegawa.  Sur les ${A}$-infini catégories, math/0310337.
  * [Lin19] Zengqiang Lin.  A general construction of $n$-angulated categories using periodic injective resolutions.  J. Pure Appl. Algebra, 223(7):3129–3149, 2019.
  * [Lin21] Zengqiang Lin.  Idempotent completion of $n$-angulated categories.  Appl. Categ. Structures, 29(6):1063–1071, 2021.
  * [LO10] Valery A. Lunts and Dmitri O. Orlov.  Uniqueness of enhancement for triangulated categories.  J. Amer. Math. Soc., 23(3):853–908, 2010.
  * [Lor] Antonio Lorenzin.  Formality and strongly unique enhancements, arXiv:2204.09527 [math.KT].
  * [LP11] Yankı Lekili and Timothy Perutz.  Fukaya categories of the torus and Dehn surgery.  Proc. Natl. Acad. Sci. USA, 108(20):8106–8113, 2011.
  * [Lur17] Jacob Lurie.  Higher algebra.  Available online at the author’s webpage: https://www.math.ias.edu/~lurie/papers/HA.pdf, May 2017.
  * [LV12] Jean-Louis Loday and Bruno Vallette.  Algebraic operads, volume 346 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer, Heidelberg, 2012.
  * [Lyu11] Volodymyr Lyubashenko.  Homotopy unital ${A}_{\infty}$-algebras.  J. Algebra, 329:190–212, 2011.
  * [May69] J. Peter May.  Matric Massey products.  J. Algebra, 12:533–568, 1969.
  * [McM20] Jordan McMahon.  Fabric idempotents and higher Auslander-Reiten theory.  J. Pure Appl. Algebra, 224(8):106343, 19, 2020.
  * [Mit72] Barry Mitchell.  Rings with several objects.  Adv. Math., 8:1–161, 1972.
  * [MSS07] Fernando Muro, Stefan Schwede, and Neil Strickland.  Triangulated categories without models.  Invent. Math., 170(2):231–241, 2007.
  * [Mur06] Fernando Muro.  On the functoriality of cohomology of categories.  J. Pure Appl. Algebra, 204(3):455–472, 2006.
  * [Mur11] Fernando Muro.  Homotopy theory of nonsymmetric operads.  Algebr. Geom. Topol., 11(3):1541–1599, 2011.
  * [Mur14] Fernando Muro.  Moduli spaces of algebras over nonsymmetric operads.  Algebr. Geom. Topol., 14(3):1489–1539, 2014.
  * [Mur16] Fernando Muro.  Cylinders for non-symmetric dg-operads via homological perturbation theory.  Journal of Pure and Applied Algebra, 220(9):3248–3281, 2016.
  * [Mur20a] Fernando Muro.  Enhanced $A_{\infty}$-obstruction theory.  J. Homotopy Relat. Struct., 15(1):61–112, 2020.
  * [Mur20b] Fernando Muro.  The first obstructions to enhancing a triangulated category.  Math. Z., 296(1-2):719–759, 2020.
  * [Mur22] Fernando Muro.  Enhanced Finite Triangulated Categories.  J. Inst. Math. Jussieu, 21(3):741–783, 2022.
  * [Nee01] Amnon Neeman.  Triangulated categories, volume 148 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 2001.
  * [Pal08] Yann Palu.  Cluster characters for 2-Calabi-Yau triangulated categories.  Ann. Inst. Fourier (Grenoble), 58(6):2221–2248, 2008.
  * [Pas20] Andrea Pasquali.  Self-injective Jacobian algebras from Postnikov diagrams.  Algebr. Represent. Theory, 23(3):1197–1235, 2020.
  * [Pup62] Dieter Puppe.  On the formal structure of stable homotopy theory.  In Colloquium on algebraic topology, pages 65–71. Aarhus Universitet, 1962.
  * [Rei83] Miles Reid.  Minimal models of canonical $3$-folds.  In Algebraic varieties and analytic varieties (Tokyo, 1981), volume 1 of Adv. Stud. Pure Math., pages 131–180. North-Holland, Amsterdam, 1983.
  * [Rou08] Raphaël Rouquier.  Dimensions of triangulated categories.  J. K-Theory, 1(2):193–256, 2008.
  * [RVdB02] I. Reiten and M. Van den Bergh.  Noetherian hereditary abelian categories satisfying Serre duality.  J. Amer. Math. Soc., 15(2):295–366, 2002.
  * [RVdB19] Alice Rizzardo and Michel Van den Bergh.  A note on non-unique enhancements.  Proc. Amer. Math. Soc., 147(2):451–453, 2019.
  * [RVdB20] Alice Rizzardo and Michel Van den Bergh.  A $k$-linear triangulated category without a model.  Ann. of Math. (2), 191(2):393–437, 2020.
  * [Sag08] Steffen Sagave.  Universal Toda brackets of ring spectra.  Trans. Amer. Math. Soc., 360(5):2767–2808, 2008.
  * [Sai20] Shunya Saito.  Tilting objects in periodic triangulated categories, 2020, arXiv:2011.14096 [math.RT].
  * [Sch01] Stefan Schwede.  The stable homotopy category has a unique model at the prime 2.  Adv. Math., 164(1):24–40, 2001.
  * [Sch02] Marco Schlichting.  A note on $K$-theory and triangulated categories.  Invent. Math., 150(1):111–116, 2002.
  * [Sch07] Stefan Schwede.  The stable homotopy category is rigid.  Ann. of Math. (2), 166(3):837–863, 2007.
  * [Sch10] Stefan Schwede.  Algebraic versus topological triangulated categories.  In Triangulated categories, volume 375 of London Math. Soc. Lecture Note Ser., pages 389–407. Cambridge Univ. Press, Cambridge, 2010\.
  * [Sei08] Paul Seidel.  Fukaya categories and Picard-Lefschetz theory.  Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008.
  * [Sei15] Paul Seidel.  Homological mirror symmetry for the quartic surface.  Mem. Amer. Math. Soc., 236(1116):vi+129, 2015.
  * [Shi02] Brooke Shipley.  An algebraic model for rational $S^{1}$-equivariant stable homotopy theory.  Q. J. Math., 53(1):87–110, 2002.
  * [Shi07] Brooke Shipley.  $H\mathbb{Z}$-algebra spectra are differential graded algebras.  Amer. J. Math., 129(2):351–379, 2007.
  * [SS02] Stefan Schwede and Brooke Shipley.  A uniqueness theorem for stable homotopy theory.  Math. Z., 239(4):803–828, 2002.
  * [SS03] Stefan Schwede and Brooke Shipley.  Stable model categories are categories of modules.  Topology, 42(1):103–153, 2003.
  * [ST01] Paul Seidel and Richard Thomas.  Braid group actions on derived categories of coherent sheaves.  Duke Math. J., 108(1):37–108, 2001.
  * [Tab05] Gonçalo Tabuada.  Invariants additifs de DG-catégories.  Int. Math. Res. Not., (53):3309–3339, 2005.
  * [Tab06] Gonçalo Tabuada.  Addendum à Invariants Additifs de dg-Catégories.  Int. Math. Res. Not., 2006:Art. ID 75853, 3, 2006.
  * [Tab07a] Gonçalo Tabuada.  Corrections à Invariants Additifs de DG-catégories.  Int. Math. Res. Not. IMRN, 2007(24):Art. ID rnm149, 17, 2007.
  * [Tab07b] Gonçalo Tabuada.  On the structure of Calabi-Yau categories with a cluster tilting subcategory.  Doc. Math., 12:193–213, 2007.
  * [Tac64] Hiroyuki Tachikawa.  On dominant dimensions of QF-3 algebras.  Trans. Amer. Math. Soc., 112:249–266, 1964.
  * [Toë11] Bertrand Toën.  Lectures on dg-categories.  In Topics in algebraic and topological $K$-theory, volume 2008 of Lecture Notes in Math., pages 243–302. Springer, Berlin, 2011.
  * [TV08] Bertrand Toën and Gabriele Vezzosi.  Homotopical algebraic geometry. II. Geometric stacks and applications.  Mem. Amer. Math. Soc., 193(902):x+224, 2008.
  * [Val14] Bruno Vallette.  Algebra + homotopy = operad.  In Symplectic, Poisson, and noncommutative geometry, volume 62 of Math. Sci. Res. Inst. Publ., pages 229–290. Cambridge Univ. Press, New York, 2014.
  * [VdB15] Michel Van den Bergh.  Calabi-Yau algebras and superpotentials.  Selecta Math. (N.S.), 21(2):555–603, 2015.
  * [Ver96] Jean-Louis Verdier.  Des catégories dérivées des catégories abéliennes.  Astérisque, (239):xii+253 pp. (1997), 1996.  With a preface by Luc Illusie, Edited and with a note by Georges Maltsiniotis.
  * [vG] Okke van Garderen.  Donaldson–Thomas invariants of length 2 flops, arXiv:2008.02591 [math.AG].
  * [Wei94] Charles A. Weibel.  An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 1994.
  * [Wem18] Michael Wemyss.  Flops and clusters in the homological minimal model programme.  Invent. Math., 211(2):435–521, 2018.
  * [Wem23] Michael Wemyss.  A lockdown survey on cDV singularities.  In Yukari Ito, Akira Ishii, and Osamu Iyama, editors, McKay Correspondence, Mutation and Related Topics, volume 88 of Adv. Stud. Pure Math., pages 47–93, 2023.
  * [Xin23] Wei Xing.  Singularity categories of higher nakayama algebras, 2023, arXiv:2306.07006.
  * [Zho22] Panyue Zhou.  Higher-dimensional Auslander-Reiten theory on $(d+2)$-angulated categories.  Glasg. Math. J., 64(3):527–547, 2022.
  * [Zim14] Alexander Zimmermann.  Representation theory, volume 19 of Algebra and Applications.  Springer, Cham, 2014.  A homological algebra point of view.
  * [ZZ] Yu Zhou and Bin Zhu.  Cotorsion pairs and t-structures in a $2-$Calabi-Yau triangulated category, arXiv:1210.6424 [math.RT].