Non-commutative crepant resolutions 1Michel Van den BerghM. Van den Bergh
1Free University of Brussels
Pleinlaan 2
1050 Brussel 1Hasselt University
Martelarenlaan 42
3500 Hasselt 1Research Foundation - Flanders
Egmontstraat 5
1000 Brussel 14E30, 18E30]14A22

# Non-commutative crepant resolutions, an overview

###### Abstract

Non-commutative crepant resolutions (NCCRs) are non-commutative analogues of
the usual crepant resolutions that appear in algebraic geometry. In this paper
we survey some results around NCCRs.

###### :

[

###### keywords:

Non-commutative crepant resolutions, tilting objects, quotient singularities

††volume: ?

##  1 Introduction

In this paper we will give an introduction to non-commutative crepant
resolutions with some emphasis on our joint work with Špela Špenko about
quotient singularities of reductive groups. Other surveys are [Leu12, Wem16,
Spe21].

###  1.1 Notation and conventions

We fix a few notations and definitions which are mostly self explanatory. For
simplicity we assume throughout that $k$ is an algebraically closed field of
characteristic zero, although this is often not necessary. In §5 we put
$k={\mathbb{C}}$ when invoking Hodge theory. For us an _algebraic variety_ is
a possibly singular integral separated scheme of finite type over $k$.

Modules over rings or sheaves of rings are left modules. Right modules are
indicated by $(-)^{\circ}$. If $\Lambda$ is a ring then we denote by
$D(\Lambda)$ the unbounded derived category of complexes of $\Lambda$-modules
and by $\operatorname{Perf}(\Lambda)$ its full subcategory of perfect
$\Lambda$-complexes. If $\Lambda$ is noetherian then we write
$\operatorname{mod}(\Lambda)$ for the category of finitely generated
$\Lambda$-modules. We also put
${\mathcal{D}}(\Lambda)=D^{b}(\operatorname{mod}(\Lambda))$. We use similar
notations in the geometric context. If ${\mathcal{X}}$ is an Artin stack and
$\Lambda$ is a quasi-coherent sheaf of rings on ${\mathcal{X}}$ then
$D_{\operatorname{Qch}}(\Lambda)$ is the unbounded derived category of
complexes of left $\Lambda$-modules with quasi-coherent cohomology. The
category of perfect $\Lambda$-complexes is denoted by
$\operatorname{Perf}(\Lambda)$ and we also put
${\mathcal{D}}(\Lambda)=D^{b}(\mathop{\text{{coh}}}(\Lambda))$ when $\Lambda$
is noetherian. If $\Lambda={\mathcal{O}}_{{\mathcal{X}}}$ then we replace
$\Lambda$ in the notations by ${\mathcal{X}}$.

A finitely generated $R$-module $M$ over a normal noetherian domain is said to
_reflexive_ if the canonical map $M\mapsto M^{\vee\vee}$ is an isomorphism
where $M^{\vee}=\operatorname{Hom}_{R}(M,R)$. This implies in particular that
$M$ is torsion free. If $R$ a commutative Noetherian domain then a _maximal
Cohen-Macaulay_ $R$-module is an $R$-module $M$ such that $M_{m}$ is maximal
Cohen-Macaulay as $R_{m}$-module for every maximal ideal $m$. If $R$ is has
finite injective dimension then we say that $R$ is _Gorenstein_. This implies
that $R$ is maximal Cohen-Macaulay.

###  1.2 Crepant resolutions and derived equivalences

Let $X$ be a normal algebraic variety with Gorenstein singularities. A
resolution of singularities $\pi:Y\rightarrow X$ is said to _crepant_ if
$\pi^{\ast}\omega_{X}=\omega_{Y}$. In some sense a crepant resolution is the
tightest possible smooth approximation of an algebraic variety. Such crepant
resolutions need not exist however. For starters, their existence implies that
$X$ has rational singularities [KM98, Corollary 5.24] and this already strong
restriction is far from sufficient. For example the three-dimensional
hypersurface singularities

$x^{2}+y^{2}+z^{2}+w^{n}=0\qquad(n\geq 2)$ (1.1)

have crepant resolutions if and only if $n$ is even [Rei83, Corollary 1.16].
When crepant resolution do exist they are generally not unique. E.g.

$xy-zw=0,$ (1.2)

which corresponds to $n=2$ in (1.1), has two distinct crepant resolutions
given by blowing up $(x,z)$ and $(x,w)$. This is the so-called “Atiyah flop”.

Nonetheless experience has shown that such different crepant resolutions are
strongly related. In particular we have the following result:

######  Theorem 1.1 ([Kon95, Bat98], see also §5 below).

Assume that $X$ has canonical Gorenstein singularities. Then the Hodge numbers
of $Y$ for a crepant resolution $Y\rightarrow X$ are independent of $Y$.

Kawamata and independently Bondal and Orlov in their lecture at ICM2002
conjectured an analogous categorical result of which we state a variant111We
have omitted the projectivity hypotheses which appear in the original context
and we have extended the conjecture to DM-stacks which is the natural context
as will become clear below. below.

######  Conjecture 1.2 ([BO02, Conjecture 4.4], [Kaw02, Conjecture 1.2]).

Assume $X$ is a normal algebraic variety with Gorenstein singularities and
$\pi_{i}:Y_{i}\rightarrow X$ for $i=1,2$ are two crepant resolutions (by
schemes or DM-stacks). Then there is an equivalence of triangulated categories
$F:{\mathcal{D}}(Y_{1})\cong{\mathcal{D}}(Y_{2})$, _linear over $X$_ (cfr
Remark 1.5 below).

The conjecture is known (under some probably unnecessary projectivity
hypotheses) in dimension $\leq 3$, by the work of Bridgeland [Bri02] (see §1.4
below), and for toric varieties, by the work of Kawamata [Kaw16]. For
symplectic singularities [Bea00] it is true, up to an étale covering of $X$,
by [Kal08, Theorem 1.6]. Furthermore it is known for many specific crepant
resolutions, e.g. those related by variation of GIT [HLS16, BFK19, HLS20] (see
also §4.2 below).

######  Remark 1.3.

Conjecture 1.2 makes no statement about the nature of the equivalence
${\mathcal{D}}(Y_{1})\cong{\mathcal{D}}(Y_{2})$. In the case of the Atiyah
flop one possible equivalence is given by the Fourier-Mukai functor for the
“fiber product kernel” ${\mathcal{O}}_{Y_{1}\times_{X}Y_{2}}$ [BO95, Theorem
3.6] (see also [BB15]) but this is far from the only possibility. Furthermore,
${\mathcal{O}}_{Y_{1}\times_{X}Y_{2}}$ does not always work as Example 1.4
below shows.

It is now understood, thanks to intuition from mirror symmetry, that the
equivalences in Conjecture 1.2 should be canonically associated to paths
connecting two points in a topological space called the “stringy Kähler moduli
space” (SKMS). In the case of the Atiyah flop the SKMS is given by
${\mathbb{P}}^{1}-\\{0,1,\infty\\}$ [Don19, HLS20]. See also [HW19] and §4.3
below. The fact that the asserted equivalence in Conjecture 1.2 is expected to
be non-canonical by itself might be the reason that the conjecture seems
difficult to prove.

Below $\operatorname{Gr}(d,n)$ is the Grassmannian of $d$-dimensional
subspaces of the $n$-dimensional vector space $k^{n}$.

######  Example 1.4.

The cotangent bundles $T^{\ast}\operatorname{Gr}(d,n)$ and
$T^{\ast}\operatorname{Gr}(n-d,n)$, for complementary Grassmannians with
$d\leq n/2$ are crepant resolutions of $\overline{B(d)}:=\\{X\in M_{n}(k)\mid
X^{2}=0,\operatorname{rk}X\leq d\\}$ (e.g. [CKL13, §6.1]). According to
[CKL13, §6] there is an equivalence
$F:{\mathcal{D}}(T^{\ast}\operatorname{Gr}(d,n))\rightarrow{\mathcal{D}}(T^{\ast}\operatorname{Gr}(n-d,n))$
but it is not given by the fiber product kernel (see [Nam04, Kaw06] for the
case $(k,n)=(2,4)$).

######  Remark 1.5.

As said one requires the derived equivalence $F$ in Conjecture 1.2 to be
linear over $X$. On the most basic level this means the following: let
$\operatorname{Perf}(X)$ be the category of perfect complexes on $X$. Then
${\mathcal{D}}(Y_{1})$, ${\mathcal{D}}(Y_{2})$ are
$\operatorname{Perf}(X)$-modules, where $A\in\operatorname{Perf}(X)$ acts as
$L\pi_{i}^{\ast}A\overset{L}{\otimes}_{Y_{i}}-$, for $i=1,2$, and we want
natural isomorphisms $F(L\pi^{\ast}_{1}A\overset{L}{\otimes}_{Y_{1}}-)\cong
L\pi_{2}^{\ast}A\overset{L}{\otimes}_{Y_{2}}F(-)$ satisfying the appropriate
compatibilities. _To simplify the exposition we will implicitly assume in the
rest of this paper that all constructions satisfy the appropriate linearity
hypotheses._

###  1.3 Non-commutative rings

Most of the results below will be based on the interplay between algebraic
geometry and non-commutative rings. The relation between those subjects was
first observed by Beilinson [Bei78]. The connection is via tilting complexes.

######  Definition 1.6.

Let $Y$ be a Noetherian scheme. A _partial tilting complex_ ${\mathcal{T}}$ on
$Y$ is a perfect complex such that
$\operatorname{Ext}^{i}_{Y}({\mathcal{T}},{\mathcal{T}})=0$ for $i\neq 0$. A
_tilting complex_ is a partial tilting complex that _generates_
$D_{\operatorname{Qch}}(Y)$ in the sense that its right orthogonal is zero,
i.e. $\operatorname{RHom}_{Y}({\mathcal{T}},{\mathcal{F}})=0$ implies
${\mathcal{F}}=0$. A _(partial) tilting bundle_ is a (partial) tilting complex
which is a vector bundle.

Below we will also use tilting complexes in slightly more general contexts
(e.g. DM-stacks). Very general results concerning tilting complexes are [KK20,
Theorems 1,2 ]. For simplicity we state a slightly dumbed down version of
them, although below we will sometimes silently rely on the stronger results
in loc. cit. See also [Ric89, Bon90].

######  Theorem 1.7 ([KK20, Theorems 1,2 ]).

If ${\mathcal{T}}$ is a tilting complex on a noetherian scheme $Y$ then
$\operatorname{RHom}_{Y}({\mathcal{T}},-)$ defines an equivalence of
categories between $D_{\operatorname{Qch}}(Y)$ and $D(\Lambda^{\circ})$ for
$\Lambda=\operatorname{End}_{Y}({\mathcal{T}})$. Moreover, if $Y$ is regular
then $\Lambda$ has finite global dimension. If furthermore $\Lambda$ is right
noetherian then $\operatorname{RHom}_{Y}({\mathcal{T}},-)$ restricts to an
equivalence of categories
${\mathcal{D}}(Y)\cong{\mathcal{D}}(\Lambda^{\circ})$.

So a tilting complex reduces the homological algebra of $Y$ to the usually
non-commutative ring $\Lambda=\operatorname{End}_{Y}({\mathcal{T}})$. In the
case of projective space ${\mathbb{P}}^{n}$ one can take
${\mathcal{T}}={\mathcal{O}}\oplus{\mathcal{O}}(1)\oplus\cdots\oplus{\mathcal{O}}(n)$
[Bei78].

###  1.4 Bridgeland’s result

####  1.4.1 Flops

Let us return to Conjecture 1.2. In the absence of any specific conjectural
construction of the asserted derived equivalence (see Remark 1.3) one may try
to use the fact that if $\pi_{1}$, $\pi_{2}$ are projective then $Y_{1}$,
$Y_{2}$ are connected by a sequence of “flops” [Kaw08, Theorem 1], so that it
is then sufficient to prove the conjecture for flops. Recall that crepant
resolutions $\pi_{1}:Y_{1}\rightarrow X$, $\pi_{2}:Y_{2}\rightarrow X$ form a
flop if $X$ has terminal singularities [KM98, Definition 2.12] and there is a
line bundle ${\mathcal{L}}$ on $Y_{1}$, relatively ample for $\pi_{1}$, such
that the corresponding222This makes sense since $Y_{1}$ and $Y_{2}$ are
isomorphic in codimension one. line bundle ${\mathcal{L}}^{\prime}$ on $Y_{2}$
is anti-ample.

In [Bri02] Bridgeland proves that Conjecture 1.2 is true for three-dimensional
flops (see also [Che02]). The key point is that the fibers of $\pi_{1}$,
$\pi_{2}$ have dimension $\leq 1$. In the next section we explain a
reinterpretation of Bridgeland’s proof, following [VdB04b].

####  1.4.2 Maps with fibers of dimension $\leq 1$

Assume that $\pi:Y\rightarrow X$ is a projective map between noetherian
schemes. We impose the following conditions:

  1. 1.

$R\pi_{\ast}{\mathcal{O}}_{Y}={\mathcal{O}}_{X}$.

  2. 2.

The fibers of $\pi$ have dimension $\leq 1$.

To simplify the discussion we will restrict ourselves furthermore to the case
that $X=\operatorname{Spec}R$ is affine.333In [VdB04b] $X$ is assumed to be
quasi-projective. It turns out that in this case $\mathop{\text{{coh}}}(Y)$
contains a tilting bundle which is of the form
${\mathcal{T}}:={\mathcal{O}}_{Y}\oplus{\mathcal{T}}_{0}$ where
${\mathcal{T}}_{0}$ is obtained as an extension

$0\rightarrow{\mathcal{O}}_{Y}^{r}\rightarrow{\mathcal{T}}_{0}\rightarrow{\mathcal{L}}\rightarrow
0$ (1.3)

where ${\mathcal{L}}$ is an ample line bundle on ${\mathcal{O}}_{Y}$ generated
by global sections and (1.3) is associated to an arbitrary finite set of
generators of $H^{1}(Y,{\mathcal{L}}^{-1})$ as $R$-module (see [VdB04b,
(3.1)]).

######  Remark 1.8.

Note that by hypotheses (1), ${\mathcal{O}}_{X}$, ${\mathcal{L}}$ are partial
tilting bundles on $Y$ such that ${\mathcal{O}}_{X}\oplus{\mathcal{L}}$
generates $D_{\operatorname{Qch}}(Y)$ [VdB04b, Lemma 3.2.2]. Moreover (2) and
the fact that ${\mathcal{L}}$ is generated by global sections implies
$\operatorname{Ext}^{>0}_{Y}({\mathcal{O}}_{Y},{\mathcal{L}})=0$. Likewise (2)
implies $\operatorname{Ext}^{>1}_{Y}({\mathcal{L}},{\mathcal{O}}_{Y})=0$. The
construction of the tilting bundle ${\mathcal{T}}$ is based on the principle
of “killing the remaining backward $\operatorname{Ext}^{1}$” in the sequence
$({\mathcal{O}}_{X},{\mathcal{L}})$ by a so-called “semi-universal extension”.
This principle extends to longer sequences. See e.g. [Har21, Lemma 2.4],
[HP14, Lemma 3.1]. See also §3.4 below for another application.

So if we put $\Lambda=\operatorname{End}_{Y}({\mathcal{T}})$, then we have
$\operatorname{End}_{Y}({\mathcal{T}}^{\vee})=\Lambda^{\circ}$ and from
Theorem 1.7 we obtain equivalences

$\operatorname{RHom}_{Y}({\mathcal{T}},-):{\mathcal{D}}(Y)\cong{\mathcal{D}}(\Lambda^{\circ}),\qquad\operatorname{RHom}_{Y}({\mathcal{T}}^{\vee},-):{\mathcal{D}}(Y)\cong{\mathcal{D}}(\Lambda).$
(1.4)

To understand (1.4) we can ask what $\Lambda$ looks like.

######  Example 1.9.

Consider again the Atiyah flop (1.2). In this case $R=k[x,y,z,w]/(xy-zw)$.
This is a toric singularity and one can check that its class group is
${\mathbb{Z}}$ with generator $I=(x,z)$. The inverse of $I$ is the fractional
ideal $I^{-1}=x^{-1}(x,w)$. The ring $\Lambda$ turns out to be the same (up to
isomorphism) for both crepant resolutions of $\operatorname{Spec}R$:

$\Lambda=\begin{pmatrix}R&I\\\ I^{-1}&R\end{pmatrix}.$ (1.5)

Interestingly $\Lambda$ is built up from the three indecomposable graded
maximal Cohen-Macaulay $R$-modules: $R$, $I$ and $I^{-1}$. In particular
$\Lambda$ is itself Cohen-Macaulay as $R$-module. This last fact turns out to
be true more generally.

######  Theorem 1.10.

Assume that $X=\operatorname{Spec}R$ is a normal Gorenstein variety. Assume
that there exists a projective crepant resolution of singularities
$\pi:Y\rightarrow X$ such that the dimensions of the fibers of $\pi$ are $\leq
1$. Let ${\mathcal{T}}$ be the tilting bundle defined above444As we have
stated in §1.2, the fact that $X$ has a crepant resolution implies that it has
rational singularities by [KM98, Corollary 5.24]. Thus in particular
$R\pi_{\ast}{\mathcal{O}}_{Y}={\mathcal{O}}_{X}$. and put
$T=\Gamma(Y,{\mathcal{T}})$. Then
$\Lambda=\operatorname{End}_{Y}({\mathcal{T}})=\operatorname{End}_{R}(T)$.
Furthermore $\Lambda$ and $T$ are maximal Cohen-Macaulay $R$-modules.

###### Proof.

The fact that $\Lambda=\operatorname{End}_{Y}({\mathcal{T}})$ is maximal
Cohen-Macaulay follows from [VdB04b, Lemma 3.2.9] (see also [IW14b, Theorem
1.5], stated as Theorem 2.6 below). $T$ is maximal Cohen-Macaulay because it
is a direct summand of $\Lambda$ as $R$-modules. Functoriality yields a map
$i:\Lambda\rightarrow\operatorname{End}_{R}(T)$ which is an isomorphism in
codimension one (since the singular locus of $X$ has codimension $\geq 2$, as
$X$ is normal). Since $\Lambda$ is maximal Cohen-Macaulay, it is reflexive and
hence $i$ must be an isomorphism. ∎

This result applies in particular if $X$ has dimension $2$ or if it is of
dimension 3 with terminal singularities since then the condition on the
dimension of the fibers is automatic.

Let us now assume that $X$ in Conjecture 1.2 is 3-dimensional and $\pi_{1}$,
$\pi_{2}$ form a flop (see §1.4.1). We will still be assuming that
$X=\operatorname{Spec}R$ is affine for simplicity. For $i=1,2$ we then have
tilting bundles ${\mathcal{T}}_{i}$ on $Y_{i}$ defined via (1.3), using
${\mathcal{L}}$ on $Y_{1}$ and $({\mathcal{L}}^{\prime})^{-1}$ on $Y_{2}$ (see
§1.4.1 for ${\mathcal{L}},{\mathcal{L}}^{\prime}$). Let
$(\Lambda_{i})_{i=1,2}$ be the corresponding endomorphism rings. In this case
Conjecture 1.2 follows from

${\mathcal{D}}(Y_{1})\overset{\eqref{eq:equivalences}}{\cong}{\mathcal{D}}(\Lambda_{1}^{\circ}),\qquad{\mathcal{D}}(Y_{2})\overset{\eqref{eq:equivalences}}{\cong}{\mathcal{D}}(\Lambda_{2}),\qquad\Lambda_{1}^{\circ}\overset{\text{Morita}}{\cong}\Lambda_{2}.$

The asserted Morita equivalence is obtained in [VdB04b, §4.4] using the local
structure of 3-dimensional terminal singularities (see [Kol89, Example 2.3]).
Nowadays we may use [IR08, Corollary 8.8] (see also [IW13, Theorem 1.5])
combined with [IW14b, Theorem 1.5] (stated as Theorem 2.6 below) to obtain
that in any case $\Lambda_{1}$, $\Lambda_{2}^{\circ}$ are derived equivalent.

At the end of the day we find that the two crepant resolutions $Y_{1}$,
$Y_{2}$ of $X$ are derived equivalent to the same non-commutative ring (either
$\Lambda^{\circ}_{1}$ or $\Lambda_{2}$). It turns out to be fruitful to think
of this intermediate non-commutative ring as a _third crepant resolution_ of
$X$, or of $R$, namely a _non-commutative crepant resolution._

##  2 Non-commutative (crepant) resolutions

###  2.1 Generalities

Below $R$ is a normal noetherian domain with quotient field $K$. We denote by
$\operatorname{ref}(R)$ the category of reflexive $R$-modules and if $\Lambda$
is a reflexive $R$-algebra then $\operatorname{ref}(\Lambda)$ is the category
of $\Lambda$-modules which are reflexive as $R$-modules. A _reflexive Azumaya
algebra_ [LB89] $\Lambda$ is a reflexive $R$-algebra which is Azumaya in
codimension one. A reflexive Azumaya algebra $\Lambda$ is said to be trivial
if it is of the form $\operatorname{End}_{R}(M)$ for $M$ a reflexive
$R$-module. In that case $\operatorname{ref}(R)$ and
$\operatorname{ref}(\Lambda)$ are equivalent. This is a particular case of
“reflexive Morita equivalence” which is defined in the obvious way.

######  Definition 2.1.

A _twisted non-commutative resolution_ of $R$ is a reflexive Azumaya algebra
$\Lambda$ over $R$ such that $\operatorname{gl\,dim}\Lambda<\infty$. If
$\Lambda$ is trivial then $\Lambda$ is said to be a _non-commutative
resolution (NCR)_ of $R$.

######  Definition 2.2.

Assume that $R$ is Gorenstein. A _twisted non-commutative crepant resolution_
$\Lambda$ of $R$ is a twisted NCR of $R$ which is in addition a Cohen-Macaulay
$R$-module. If $\Lambda$ is an NCR then such a $\Lambda$ is said to be a _non-
commutative crepant resolution (NCCR)_ of $R$.

The point of these definitions is that they provide reasonable non-commutative
substitutes for “regularity”, “birationality” and “crepancy”. This is
explained in more detail in [VdB04b, §4].

######  Remark 2.3.

We will sometimes use the concepts introduced in Definitions 2.1, 2.2 for
schemes, possibly non-affine. It is then understood that they reduce to the
affine concepts, when restricting to open affine subschemes.

######  Remark 2.4.

In the sequel we will be mostly concerned with NCCRs and thus the other
definitions are mainly provided for context. Twisted NCCRs are natural
generalizations of NCCRs, but the good properties of NCCRs (sometimes
conjectural) are usually not shared by twisted NCCRs. See e.g. Example 5.11
below. The definition of a (twisted) NCR is more tentative. In particular the
normality and reflexivity hypotheses do not seem very relevant. For example
there is a nice theory of non-commutative resolutions of non-normal
singularities in dimension one [Leu12].

######  Example 2.5.

It follows from Theorem 1.10 and Theorem 1.7 that if there exists a projective
crepant resolution of singularities $\pi:Y\rightarrow X$ such that the
dimensions of the fibers of $\pi$ are $\leq 1$ then $R$ has an NCCR.

We mention the following theorem which gives another indication that the
definition of an NCCR is the “correct one”.

######  Theorem 2.6 ([IW14b, Theorem 1.5]).

Let $f:Y\rightarrow\operatorname{Spec}R$ be a projective birational morphism
between Gorenstein varieties. Suppose that $Y$ is derived equivalent to some
ring $\Lambda$, then $f$ is a crepant resolution if and only if $\Lambda$ is
an NCCR of $R$.

The following conjecture is a natural extension of Conjecture 1.2.

######  Conjecture 2.7 ([VdB04a, Conjecture 4.6]).

All crepant resolutions of $X$ (commutative as well as non-commutative) are
derived equivalent.

We have the following result which is proved in the same way as the
3-dimensional McKay correspondence [BKR01].

######  Proposition 2.8 ([VdB04a, Theorem 6.3.1, Proposition 6.2.1]).

If $X$ has three-dimensional Gorenstein singularities and it has an NCCR
$\Lambda$, then it has a projective crepant resolution $Y\rightarrow X$ such
that $\Lambda$ and $Y$ are derived equivalent.

######  Proposition 2.9.

Conjecture 2.7 is true if $X$ has dimension three, if we restrict to
projective crepant resolutions.

###### Proof.

If $X$ has an NCCR $\Lambda$ then by Proposition 2.8 $\Lambda$ is derived
equivalent to a crepant resolution. Hence we are reduced to Bridgeland’s
result (see §1.4.1). Alternatively, to have a very nice direct argument that
any two NCCRs are derived equivalent in dimension three we may use [IR08,
Corollary 8.8] (see also [IW13, Theorem 1.5]). ∎

Proposition 2.8 is false for arbitrary three-dimensional Gorenstein
singularities as was shown by Dao [Dao10].

######  Proposition 2.10 ([Dao10, Theorem 3.1, Remark 3.2]).

Assume $S$ is a regular local ring which is equicharacteristic or unramified,
$0\neq f\in S$ and $R=S/(f)$ is normal. If $\dim R=3$ and $R$ is factorial
then $R$ has no NCCR.

######  Example 2.11.

It turns out that there are 3-dimensional factorial hypersurface singularities
that admit a crepant resolution. A concrete example is given by
$R=k[[x_{0},x_{1},x_{2},x_{3}]]/(x_{0}^{4}+x_{1}^{3}+x_{2}^{3}+x_{3}^{3})$
[Lin02, Theorem A,B]. In particular a crepant resolution of such $R$ does not
admit a tilting complex by Theorem 2.6.

If $X$ is a normal Gorenstein algebraic variety with a crepant resolution then
it has rational singularities [KM98, Corollary 5.24]. A similar result is true
for NCCRs.

######  Theorem 2.12 ([SVdB08, Theorem 1.1]).

Let $R$ be a normal finitely generated Gorenstein $k$-algebra. If $R$ has a
twisted NCCR then it has rational singularities.

The actual result proved in loc. cit. applies in a more general context and
this has been further exploited in [IY13a, IY13b] (see also [DITW20, Corollary
1.7]).

######  Remark 2.13.

In order to deal with singularities with a singular minimal model Iyama and
Wemyss generalize the definition of an NCCR [IW14a, IW14b, Wem18] to certain
rings, of possibly infinite global dimension, called _maximal modification
algebras_ (MMAs). Remarkably, many of the results about NCCRs extend to MMAs.
However in this overview we will restrict ourselves for simplicity to NCCRs.

###  2.2 Relation with crepant categorical resolutions

We conjecture that non-commutative crepant resolutions are examples of
“strongly crepant categorical resolutions” as introduced by Kuznetsov in
[Kuz08b]. However we can only prove this in special cases.

Let $X$ be an algebraic variety. A _categorical_ resolution [Kuz08b] of
${\mathcal{D}}(X)$ is a “smooth” triangulated category $\tilde{{\mathcal{D}}}$
together with functors

$\pi_{\ast}:\tilde{{\mathcal{D}}}\rightarrow{\mathcal{D}}(X),\qquad\pi^{\ast}:\operatorname{Perf}(X)\rightarrow\tilde{{\mathcal{D}}}$

which are adjoint (i.e.
$\operatorname{Hom}_{\tilde{{\mathcal{D}}}}(\pi^{\ast}A,B)\cong\operatorname{Hom}_{{\mathcal{D}}(X)}(A,\pi_{\ast}B)$
for $A\in\operatorname{Perf}(X)$, $B\in\tilde{{\mathcal{D}}}$) such that the
natural transformation
${\operatorname{id}}_{\operatorname{Perf}(X)}\rightarrow\pi_{\ast}\pi^{\ast}$,
obtained by putting $B=\pi^{\ast}A$, is an isomorphism. This implies in
particular that $\pi^{\ast}$ is fully faithful. There is some variation
possible in the definition of smoothness. For us it means that
$\tilde{{\mathcal{D}}}$ is equivalent to the derived category of perfect
modules over a smooth DG-algebra [KKP08, Definition 2.23].

######  Remark 2.14.

If $\pi:Y\rightarrow X$ is a resolution of singularities of $X$ then
$({\mathcal{D}}(Y),R\pi_{\ast},L\pi^{\ast})$ is a categorical resolution of
${\mathcal{D}}(X)$ if and only if $X$ has rational singularities. Remarkably
however, it has been shown in [KL15] that ${\mathcal{D}}(Y)$ can be suitably
enlarged to yield a categorical resolution. On the other hand, this result
cannot be extended to more general dg-categories [Efi20].

Following [Kuz08b] we say that a categorical resolution
$(\tilde{{\mathcal{D}}},\pi_{\ast},\pi^{\ast})$ of ${\mathcal{D}}(X)$ is
_weakly crepant_ if $\pi^{\ast}$ is both a left and a right adjoint to
$\pi_{\ast}$.

There is also a notion of a _strongly crepant categorical_ resolution for
which we need the notion of a _relative Serre functor_. To define this assume
that $X$ is Gorenstein and that $\tilde{{\mathcal{E}}}$ is a smooth
triangulated category which is a $\operatorname{Perf}(X)$-module. We will
denote the action of $A\in\operatorname{Perf}(X)$ on
$B\in\tilde{{\mathcal{E}}}$ as $A\otimes_{X}B$ and we assume that $-\otimes-$
is exact in both arguments. We also assume that the functor
$\operatorname{Perf}(X)\rightarrow\tilde{{\mathcal{E}}}:A\mapsto
A\otimes_{X}B$ has a right adjoint
$\tilde{{\mathcal{E}}}\rightarrow{\mathcal{D}}(X)$ which we denote by
$\operatorname{R\mathcal{H}\mathit{om}}_{\tilde{{\mathcal{E}}}/X}(B,-)$. I.e.
for $C\in\tilde{{\mathcal{E}}}$ we have functorial isomorphisms

$\operatorname{Hom}_{\tilde{{\mathcal{E}}}}(A\otimes_{X}B,C)\cong\operatorname{Hom}_{X}(A,\operatorname{R\mathcal{H}\mathit{om}}_{\tilde{{\mathcal{E}}}/X}(B,C)).$

An autoequivalence
$S_{\tilde{{\mathcal{E}}}/X}:\tilde{{\mathcal{E}}}\rightarrow\tilde{{\mathcal{E}}}$
is said to be a _relative Serre functor_ for $\tilde{{\mathcal{E}}}/X$ if
there are functorial isomorphisms

$\operatorname{R\mathcal{H}\mathit{om}}_{X}(\operatorname{R\mathcal{H}\mathit{om}}_{\tilde{{\mathcal{E}}}/X}(B,C),{\mathcal{O}}_{X})\cong\operatorname{R\mathcal{H}\mathit{om}}_{\tilde{{\mathcal{E}}}/X}(C,S_{\tilde{{\mathcal{E}}}/X}B)$

for $B,C\in\tilde{{\mathcal{E}}}$. We say that $\tilde{{\mathcal{E}}}/X$ is
_strongly crepant_ if the identity functor
$\tilde{{\mathcal{E}}}\rightarrow\tilde{{\mathcal{E}}}$ is a relative Serre
functor.

A _strongly crepant categorical resolution_ of $X$ is a quadruple
$(\tilde{{\mathcal{D}}},\pi_{\ast},\pi^{\ast},\otimes_{X})$ such that
$(\tilde{{\mathcal{D}}},\pi_{\ast},\pi^{\ast})$ is a categorical resolution of
$X$, $-\otimes_{X}-$ is a $\operatorname{Perf}(X)$-module structure on
$\tilde{{\mathcal{D}}}$ such that $\tilde{{\mathcal{D}}}/X$ is strongly
crepant and $\pi^{\ast}$ is $\otimes_{X}$-linear. The last condition means
that for $A,B\in\operatorname{Perf}(X)$ we have functorial isomorphisms
$A\otimes_{X}\pi^{\ast}B\cong\pi^{\ast}(A\otimes_{X}B)$ satisfying the
appropriate compatibilities.

It is shown in [Kuz08b, §3] that a strongly crepant categorical resolution is
weakly crepant, and moreover that if $\pi:Y\rightarrow X$ is a crepant
resolution in the usual sense then
$({\mathcal{D}}(Y),R\pi_{\ast},L\pi^{\ast},L\pi^{\ast}(-)\otimes_{Y}-)$ is a
strongly crepant categorical resolution of $X$.

The following easy lemma, which is an extension of [Lun10, Example 5.3], shows
that, under suitable conditions, rings of the form $\operatorname{End}_{R}(M)$
form crepant categorical resolutions. If $M$ is an $R$-module then
$\operatorname{add}(M)$ is the category spanned by modules which are direct
summands of some $M^{\oplus n}$.

######  Lemma 2.2.1.

Assume that $X=\operatorname{Spec}R$ is an algebraic variety and let $M$ be a
finitely generated $R$-module such that $\Lambda=\operatorname{End}_{R}(M)$
has finite global dimension. Then ${\mathcal{D}}(\Lambda)$ is smooth. Assume
in addition that $R\in\operatorname{add}(M)$. Then

$\operatorname{Perf}(R)\rightarrow{\mathcal{D}}(\Lambda):N\mapsto
M\overset{L}{\otimes}_{R}N$ (2.1)

yields a categorical resolution of singularities of $X$ (since
$\operatorname{Perf}(R)\cong\operatorname{Perf}(X)$). Moreover, assuming
furthermore that $R$ is normal Gorenstein:

  1. 1.

if $M$ is maximal Cohen-Macaulay then this categorical resolution is weakly
crepant;

  2. 2.

if $\Lambda$ is an NCCR then this categorical resolution is strongly crepant.

Note that if (2) holds then $M$ is maximal Cohen-Macaulay since we have
assumed that $R\in\operatorname{add}(M)$.

The hypotheses of Lemma 2.2.1 are actually too strong. For example an NCCR is
always a strongly crepant categorical resolution in dimension $\leq 3$. This
follows from Proposition 2.15 below which can be proved using the methods of
[IR08, IW13].

######  Proposition 2.15.

Assume that $\Lambda=\operatorname{End}_{R}(M)$ is an NCCR and $\dim R\leq 3$
then

$\operatorname{Ext}^{i}_{\Lambda}(M,M)=0\qquad\text{for $i>0$}.$ (2.2)

###### Proof.

For the benefit of the reader we give a proof. We may assume that $R$ is local
of dimension 3 (the case $\dim\leq 2$ is easy). By the Auslander Buchsbaum
formula [IR08, Proposition 2.3] $\Lambda$ has global dimension $3$. Since $M$
is reflexive, it has depth $\geq 2$, and hence, again by the Auslander-
Buchsbaum formula, it has projective dimension $\leq 1$ over $\Lambda$ and
moreover it is projective over $\Lambda$ in codimension $2$.

Hence we have a projective resolution of $M$ as $\Lambda$-module:

$0\rightarrow P_{1}\rightarrow P_{0}\rightarrow M\rightarrow 0.$

Applying $\operatorname{Hom}_{A}(-,M)$ we get a long exact sequence of
$R$-modules

$0\rightarrow
R\rightarrow\operatorname{Hom}_{\Lambda}(P_{0},M)\rightarrow\operatorname{Hom}_{\Lambda}(P_{1},M)\rightarrow\operatorname{Ext}^{1}_{\Lambda}(M,M)\rightarrow
0.$ (2.3)

Assume $\operatorname{Ext}^{1}_{\Lambda}(M,M)\neq 0$. Since $M$ is projective
over $\Lambda$ in codimension two, $\operatorname{Ext}^{1}_{\Lambda}(M,M)$ is
finite dimensional and hence it has depth 0 as $R$-module. On the other hand,
since $\operatorname{Hom}_{\Lambda}(P_{i},M)$ is reflexive as $R$-module, it
has $\operatorname{depth}\geq 2$. Finally $R$ being maximal Cohen-Macaulay has
depth 3. One may verify that these depth restrictions are incompatible with
(2.3). ∎

It seems too much to hope for that (2.2) would always be true, but lack of
time has prevented us from seriously looking for a counterexample. On the
other hand we are sufficiently optimistic to make the following conjecture.

######  Conjecture 2.16.

If $X=\operatorname{Spec}R$ is a normal algebraic variety with Gorenstein
singularities then an NCCR of $R$ always yields a strongly crepant categorical
resolution of $X$.

To prove this conjecture one would have to construct for an NCCR $\Lambda$ of
$R$ a partial tiling complex $P^{\bullet}$ of $\Lambda$-modules such that
$\operatorname{RHom}_{\Lambda}(P^{\bullet},P^{\bullet})=R$.

######  Remark 2.17.

The strongly crepantness of ${\mathcal{E}}/X$ as defined above is independent
of the resolution property. One may check that if $\Lambda/R$ is a twisted
NCCR then ${\mathcal{D}}(\Lambda)$ is strongly crepant over
$\operatorname{Spec}R$. But one may check that it is not a categorical
resolution.

##  3 Constructions of non-commutative crepant resolutions

###  3.1 Quotient singularities

Here we will restrict ourselves to quotient singularities for finite groups.
Quotient singularities for (non-finite) reductive groups will be covered in
§4.

If $G$ is a finite group and $W$ is a faithful finite dimensional unimodular
(i.e. $\det W=k$) representation of $G$ then the skew group ring
$\operatorname{Sym}(W)\\#G=\operatorname{End}_{\operatorname{Sym}(W)^{G}}(\operatorname{Sym}(W))$
is an NCCR for $R=\operatorname{Sym}(W)^{G}$ (which is Gorenstein because of
the unimodularity hypothesis).

In dimension $\leq 3$ such quotient singularities always have a crepant
resolution by the celebrated BKR-theorem [BKR01]. In higher dimension this is
not so. The simplest counterexample is given by ${\mathbb{Z}}_{2}$ acting with
weights $(-1,-1,-1,-1)$ on $W=k^{4}$ because in that case $R$ is
${\mathbb{Q}}$-factorial and terminal. See e.g. [Mo11].

###  3.2 Crepant resolutions with tilting complexes

In case $R$ is a normal Gorenstein domain and
$Y\rightarrow\operatorname{Spec}R$ is a crepant resolution and ${\mathcal{T}}$
is a tilting complex on $Y$ then
$\operatorname{\mathcal{E}\mathit{nd}}_{Y}({\mathcal{T}})$ is an NCCR of $R$
by Theorem 2.6. Conversely, assuming a crepant resolution exists, any NCCR has
to be of this form if we accept Conjecture 2.7 (${\mathcal{T}}$ is the dual of
the image of $\Lambda$ under the asserted derived equivalence
${\mathcal{D}}(\Lambda)\cong{\mathcal{D}}(Y)$).

This is a very general method for constructing NCCRs. Note however that even
in dimension three there may be crepant resolutions without tilting complex.
See Example 2.11. Furthermore as indicated in §3.1 there are normal Gorenstein
singularities that admit an NCCR but not a crepant resolution.

######  Example 3.1.

A textbook example where this method works very well is the case of
determinantal varieties [BLVdB10, BLVdB16]. Let $n\geq 1$ and $0\leq l<n$. Let
$X_{l,n}=\operatorname{Spec}R$ be the varieties of matrices in
$\operatorname{Hom}_{k}(k^{n},k^{n})=M_{n\times n}(k)$ which have rank $\leq
l$. It is a classical result that $X$ is Gorenstein. It is also well known
that $X$ has a crepant Springer type resolution given by

$Y=\\{(\phi,V)\mid
V\in\operatorname{Gr}(l,n),\phi\in\operatorname{Hom}_{k}(k^{n},V)\\}$

where $\pi:Y\rightarrow X$ sends $(\phi,V)$ to the composition of $\phi$ with
the inclusion $V\hookrightarrow k^{n}$. If ${\mathcal{R}}$ denotes the
universal subbundle on $\operatorname{Gr}(l,n)$ then $Y$ is the vector bundle
$\operatorname{Hom}(k^{n},{\mathcal{R}})$ (i.e.
$Y=\underline{\operatorname{Spec}}\operatorname{Sym}(({\mathcal{R}}^{\oplus
n})^{\vee})$). Using Bott’s theorem one computes that the Kapranov tilting
bundle on $\operatorname{Gr}(l,n)$ [Kap88] (see also [BLVdB15][Efi17] for the
case of finite characteristic), pulls back to a tilting bundle on $Y$, which
then gives an NCCR of $R$. For other approaches to this example see [DS14] and
Theorem 4.10 below.

Alas things are often more complicated. For determinantal varieties associated
to symmetric of skew-symmetric matrices, the Springer type resolutions are not
crepant so a tilting bundle on them only gives an NCR (see [WZ12]). NCCRs of
such generalized determinantal varieties will be obtained in §4 using a
different approach.

######  Example 3.2.

Another beautiful, and much deeper example [Bez06, BMR08] is given by
cotangent bundles of (partial) flag varieties $T^{\ast}(G/P)$. If $P$ is a
Borel subgroup of $G$ then this is a crepant resolution of the nilpotent cone
in $\mathop{\text{Lie}}(G)$. In general they are crepant resolutions of
closures of Richardson orbits [Nam09]. It is shown in [Bez06, BMR08] that
$T^{\ast}(G/P)$ has a tilting bundle but it is not obtained as the pullback of
a tilting bundle on $G/P$. In fact the construction of the tilting bundle is
highly non-trivial. To explain the construction it is useful to exhibit a
slightly different point of view on tilting bundles.

Let $Y$ be a noetherian scheme. If ${\mathcal{A}}$ is a quasi-coherent sheaf
of algebras on $Y$ and $A=\Gamma(Y,{\mathcal{A}})$ then we say that
${\mathcal{A}}$ is _derived affine_ if $A=R\Gamma(Y,{\mathcal{A}})$ and the
right orthogonal to ${\mathcal{A}}$ in $D_{\operatorname{Qch}}({\mathcal{A}})$
is zero. In that case $R\Gamma(Y,-)$ defines an equivalence of categories
between $D_{\operatorname{Qch}}({\mathcal{A}})$ and $D(A)$. It is not
difficult to see that a vector bundle ${\mathcal{T}}$ on $Y$ which has
everywhere non-zero rank is a tilting bundle provided
$\operatorname{\mathcal{E}\mathit{nd}}_{Y}({\mathcal{T}})$ is derived
affine.555To verify this it is useful to observe that ${\mathcal{T}}$ is
tilting if and only if ${\mathcal{T}}^{\vee}$ is tilting. The only non-trivial
part is the generation property. To this end one may use that ${\mathcal{T}}$
generates $D_{\operatorname{Qch}}(Y)$ if and only if $\operatorname{Perf}(Y)$
is the smallest épaisse subcategory of $D_{\operatorname{Qch}}(Y)$ containing
${\mathcal{T}}$ [Nee92, Lemma 2.2], together with the fact that $(-)^{\vee}$
is an autoequivalence of $\operatorname{Perf}(Y)$.

We say that $Y$ is derived ${\mathcal{D}}$-affine if ${\mathcal{D}}_{Y}$ is
derived affine where ${\mathcal{D}}_{Y}$ is the sheaf of differential
operators on $Y$. In characteristic $>0$ we mean by ${\mathcal{D}}_{Y}$ the
sheaf of _crystalline_ differential operators, i.e. differential operators
which may be expressed in terms of derivations, without using divided powers.

Now let $Z=G/P$. The Bernstein Beilinson theorem [BB81], valid in
characteristic zero, states that $Z$ is even “${\mathcal{D}}$-affine” meaning
that the equivalence $R\Gamma(Z,-)$ is also compatible with the natural
t-structures. This is false in characteristic $>0$. However $Z$ is still
derived ${\mathcal{D}}$-affine [BMR08, Theorem 3.2] whenever $p$ is strictly
bigger than the Coxeter number, which we will assume now.

We will give a rough sketch how this is used in [Bez06, BMR08] to construct a
tilting bundle on $Y=T^{\ast}Z$. Let us first assume that the characteristic
of $k$ is $p>0$. To indicate this we will adorn our notations with $(-)_{p}$.
In that case ${\mathcal{D}}_{Z_{p}}$ is coherent as a module over its center
which is equal to $(\operatorname{Sym}_{Z_{p}}{\mathcal{E}}_{p})^{(1)}$ where
$(-)^{(1)}$ denotes the Frobenius twist, and ${\mathcal{E}}_{p}$ is the
tangent bundle on $Z_{p}$. Hence we may view ${\mathcal{D}}_{Z_{p}}$ as a
sheaf of coherent algebras $\tilde{{\mathcal{D}}}$ on
$\underline{\operatorname{Spec}}(\operatorname{Sym}_{Z_{p}}{\mathcal{E}}_{p})^{(1)}=Y_{p}^{(1)}$
where $Y_{p}=T^{*}Z_{p}$. The sheaf $\tilde{{\mathcal{D}}}$ is still derived
affine.

Now $\tilde{{\mathcal{D}}}$ is not of the form
$\operatorname{\mathcal{E}\mathit{nd}}_{Y^{(1)}_{p}}({\mathcal{T}}^{(1)}_{p})$.
However if we let $\hat{Y}_{p}$ be the formal completion of $Y_{p}$ at the
zero section then it turns out that the restriction $\hat{{\mathcal{D}}}$ of
$\tilde{{\mathcal{D}}}_{p}$ to to $\hat{Y}^{(1)}_{p}$ is of the form
$\operatorname{\mathcal{E}\mathit{nd}}_{\hat{Y}_{p}}(\hat{{\mathcal{T}}}_{p})^{(1)}$
for a vector bundle $\hat{{\mathcal{T}}}_{p}$ on $\hat{Y}_{p}$. Moreover
$\hat{{\mathcal{D}}}$ is still derived affine and so $\hat{{\mathcal{T}}}_{p}$
is a tilting bundle on $\hat{Y}_{p}$. Then one uses the rigidity666Tilting
bundles have in particular vanishing $\operatorname{Ext}^{1,2}$. Hence by
classical deformation theory they are unobstructed and rigid.  of tilting
bundles to lift $\hat{{\mathcal{T}}}_{p}$ to a tilting bundle
$\hat{{\mathcal{T}}}$ in characteristic zero. Finally one may use the fact
that $Y=T^{\ast}Z$ (as a vector bundle) admits a nice $G_{m}$ action to
conclude by [Kal08, Theorem 1.8] that $\hat{{\mathcal{T}}}$ is actually the
completion of a tilting bundle ${\mathcal{T}}$ on $Y$.

Hidden behind this construction is the fact that ${\mathcal{D}}_{Z}$ is in
some sense a canonical non-commutative deformation of the symplectic variety
$T^{\ast}Z$. If $Y$ is a general symplectic variety then one may try to
construct a non-commutative deformation using Fedosov quantization. This
general idea has been used by Bezrukavnikov and Kaledin to prove an analogue
of the BKR theorem [BKR01] for crepant resolutions of symplectic quotient
singularities [BK04] and by Kaledin to prove a suitable version of Conjecture
1.2 [Kal08] for general symplectic singularities. To apply the method one
needs to be able to do Fedosov quantization in finite characteristic, a
problem which has been solved to some extent in [BK08].

###  3.3 Resolutions with partial tilting complexes

Assume $R$ is a normal Gorenstein domain with rational singularities and
$Y\rightarrow\operatorname{Spec}R$ is a resolution which is not crepant. A
strengthening of Conjecture 2.7 inspired by [Kuz08b] is that NCCRs are minimal
in a categorical sense, i.e. their derived category embeds inside
${\mathcal{D}}(Y)$. This means that they are obtained as
$\operatorname{\mathcal{E}\mathit{nd}}_{Y}({\mathcal{T}})$ for a partial
tilting complex ${\mathcal{T}}$ on $Y$. For a very general result in this
direction see [Kuz08b, Theorem 2]. We will restrict ourselves to a special
case which will be useful in §5 and which can be easily proved directly.

######  Proposition 3.3 ([Kuz08b]).

Let $Z$ be a smooth projective variety with ample line bundle
${\mathcal{O}}_{Z}(1)$ and let $X=\operatorname{Spec}R$ be the corresponding
cone. Assume $\omega_{Z}={\mathcal{O}}_{Z}(-n)$ for $n\geq 1$. Then $R$ is
Gorenstein. Moreover a resolution of singularities $\pi:Y\rightarrow X$ of $X$
is given by the line bundle over $Z$ associated to ${\mathcal{O}}_{Z}(1)$.
Assume ${\mathcal{E}}\in{\mathcal{D}}(Z)$ is such that:

  1. 1.

$\operatorname{Ext}^{i}_{Z}({\mathcal{E}},{\mathcal{E}}(m))=0$ for $i>0$ and
$m\geq 0$;

  2. 2.

$\operatorname{Ext}^{i}_{Z}({\mathcal{E}},{\mathcal{E}}(m))=0$ for $i\geq 0$
and $m\in\\{-1,\ldots,-n+1\\}$;

  3. 3.

${\mathcal{E}}\oplus{\mathcal{E}}(1)\oplus\ldots\oplus{\mathcal{E}}(n-1)$ is a
generator for $D_{\operatorname{Qch}}(Z)$.

Let $\gamma:Y\rightarrow Z$ be the projection map and put
${\mathcal{T}}=\gamma^{\ast}{\mathcal{E}}$. Then
$\operatorname{End}_{Y}({\mathcal{T}})$ is an NCCR of $R$.

###### Proof.

We write ${\mathcal{T}}(m)=\gamma^{\ast}({\mathcal{E}}(m))$. Then we have

$\operatorname{RHom}_{Y}({\mathcal{T}},{\mathcal{T}}(m))=\bigoplus_{l\geq
0}\operatorname{RHom}_{Z}({\mathcal{E}},{\mathcal{E}}(m+l)).$ (3.1)

Using (1,2) we deduce in particular that
$\bar{{\mathcal{T}}}:={\mathcal{T}}\oplus\cdots\oplus{\mathcal{T}}(-n+1)$ is
partial tilting (and hence this is also the case for ${\mathcal{T}}$).
Furthermore from (3) we obtain $\bar{{\mathcal{T}}}^{\perp}=0$. So
$\bar{{\mathcal{T}}}$ is in fact tilting. Put
$\Lambda=\operatorname{End}_{Y}({\mathcal{T}})$,
$\bar{\Lambda}=\operatorname{End}_{Y}(\bar{{\mathcal{T}}})$. By Theorem 1.7
$\bar{\Lambda}$ has finite global dimension.

Via the decomposition (3.1) $\Lambda$ is a ${\mathbb{N}}$-graded ring. Put
$\Lambda_{\geq u}=\bigoplus_{m\geq u}\Lambda_{m}$. Then (as ungraded rings) we
have

$\bar{\Lambda}=\begin{pmatrix}\Lambda&\Lambda_{\geq 1}&\cdots&\Lambda_{\geq
n-1}\\\ \Lambda&\Lambda&\cdots&\Lambda_{\geq n-2}\\\
\vdots&\vdots&\ddots&\vdots\\\ \Lambda&\Lambda&\cdots&\Lambda_{\geq 1}\\\
\Lambda&\Lambda&\cdots&\Lambda\end{pmatrix}.$

If we put $\Gamma=M_{n}(\Lambda)$ then $\bar{\Lambda}\subset\Gamma$ and
moreover $\Gamma$ is (left and right) projective over $\bar{\Lambda}$ and in
addition the multiplication map
$\Gamma\otimes_{\bar{\Lambda}}\Gamma\rightarrow\Gamma$ is an isomorphism (it
is a surjective map between projective $\Gamma$-modules of the same rank). We
claim that $\Gamma$ (and hence $\Lambda$) has finite global dimension. Indeed
if $M$ is a right $\Gamma$-module and $P^{\bullet}\rightarrow M$ is a finite
projective resolution of $M$ as $\bar{\Lambda}$-module (which exists since
$\operatorname{gl\,dim}\bar{\Lambda}<\infty$) then
$\Gamma\otimes_{\bar{\Lambda}}P^{\bullet}$ is a finite $\Gamma$-projective
resolution of
$\Gamma\otimes_{\bar{\Lambda}}M=\Gamma\otimes_{\bar{\Lambda}}\Gamma\otimes_{\Gamma}M\cong
M$.

Moreover for $i>0$:

$\displaystyle\operatorname{Ext}^{i}_{R}(\operatorname{End}_{Y}({\mathcal{T}}),\omega_{R})$
$\displaystyle=\operatorname{Ext}^{i}_{X}(\pi_{\ast}\operatorname{\mathcal{E}\mathit{nd}}_{Y}({\mathcal{T}}),\omega_{X})$
$\displaystyle=\operatorname{Ext}^{i}_{Y}(\operatorname{\mathcal{E}\mathit{nd}}_{Y}({\mathcal{T}}),\omega_{Y})$
$\displaystyle=\operatorname{Ext}^{i}_{Y}({\mathcal{T}},{\mathcal{T}}(-n+1))$
$\displaystyle=0$

where in the second line we have used Grothendieck duality, in the third line
the easily verified fact that $\omega_{Y}=\gamma^{\ast}(\omega_{Z}(1))$ and in
the fourth line (3.1) and (1,2). It follows that
$\operatorname{End}_{Y}({\mathcal{T}})$ is maximal Cohen-Macaulay over $R$. ∎

###  3.4 Three-dimensional affine toric varieties

For simplicity we define an affine toric variety as $X=\operatorname{Spec}R$
where $R=k[\sigma^{\vee}\cap M]$ where $M$ is a lattice and $\sigma^{\vee}$ is
a strongly convex full dimensional lattice cone in $M_{\mathbb{R}}$. Such an
$R$ is Gorenstein if there exist $m\in M$ such that $\sigma$ (the dual cone of
$\sigma^{\vee}$) is spanned by lattice vectors $x\in M^{\vee}$ satisfying
$\langle x,m\rangle=1$. The lattice polytope associated to $R$ is defined as
$P=\sigma\cap\langle m,-\rangle$.

In this case there is the following beautiful result by Broomhead [Bro12,
Theorem 8.6].

######  Theorem 3.4.

The coordinate ring of a 3-dimensional Gorenstein affine toric variety admits
a toric NCCR.

By a toric NCCR we mean that the reflexive module defining the NCCR is
isomorphic to a sum of ideals. Broomhead’s proof uses the theory of “dimer
models” which is possible thanks to the combinatorics [Gul08, IU15]. A proof
not using dimer models but using this combinatorics directly was given in
[SVdB20b].

A different method for constructing NCCRs for affine Gorenstein toric
varieties was given in [SVdB20a] and is based on a standard fact from toric
geometry:

######  Lemma 3.5.

A subdivision of $\sigma$ obtained by a regular triangulation of $P$ with no
extra vertices yields a projective crepant resolution of
$\operatorname{Spec}R$ by a toric Deligne Mumford stack [BCS05]. If $\dim
X\leq 3$ then such a crepant resolution has fibers of dimension $\leq 1$.

In dimension $\leq 3$ one may then, starting from a sequence of generating
line bundles, construct a tilting bundle using the principle of “killing
backward $\operatorname{Ext}^{1}$’s” (see Remark 1.8).

While this method yields an NCCR, it generally does not yield a toric one. On
the other hand it is also applicable to some higher dimensional toric
singularities which do not have a toric NCCR.

######  Example 3.6 ([SVdB17, §9.1],[SVdB20a, Example 6.4]).

Let $T=G^{2}_{m}$ be the two dimensional torus and (after the identifying the
character group $X(T)$ of $T$ with ${\mathbb{Z}}^{2}$) consider the vector
space $W$ with weights $(3,0)$, $(1,1)$, $(0,3)$, $(-1,0)$, $(-3,-3)$,
$(0,-1)$. Put
$R:=\operatorname{Sym}(W)^{T}=k[x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}]^{T}=k[x_{2}x_{4}x_{6},x_{1}x_{3}x_{5},x_{1}x_{4}^{3},x_{3}x_{6}^{3},x_{2}^{3}x_{5}]\cong
k[a,b,c,d,e]/(a^{3}b-cde)$. Clearly $R$ is the coordinate ring of a
4-dimensional affine toric variety, but it was shown in [SVdB17, §9.1] that
$R$ does not have a toric NCCR.

On the other by [SVdB20a, Proposition 6.1], $R$ does have a non-toric NCCR. In
[SVdB20a, Example 6.4] an explicit NCCR is constructed which is given by a
reflexive module which is the direct sum of 12 modules of rank 1 and 1 module
of rank 2.

We conjecture:

######  Conjecture 3.7.

An affine Gorenstein toric variety always has an NCCR.

Besides Theorem 3.4 this conjecture is also true for “quasi-symmetric GIT
quotients” for tori. See Corollary 4.7 below.

By [SVdB20a, Theorem A.1] the Grothendieck group of the DM-stack exhibited in
Lemma 3.5 has rank $\operatorname{Vol}(P)$. This suggests the following
conjecture:

######  Conjecture 3.8.

The number of indecomposable summands in the reflexive module defining an NCCR
of $R$ is equal to $\operatorname{Vol}(P)$.

###  3.5 Mutations

It follows from the minimal model program that the number of crepant
resolutions of an algebraic variety is finite777I thank Shinnosuke Okawa for
explaining to me how this follows from [BCHM10].. On the other hand NCCRs can
be modified by a process called “mutation” which is closely related to
flopping of crepant resolutions. The difference is that the mutation process
generally leads to an infinite number of different NCCRs (see however Example
3.12 below).

The following definitions and results are taken from [IW14a]. Let $R$ be a
normal Gorenstein ring. Let $M$ be a reflexive $R$-module such that
$\Lambda=\operatorname{End}_{R}(M)$ is an NCCR and let $0\neq
N\in\operatorname{add}(M)$. Let $K_{0}$ be defined by the short exact sequence

$0\rightarrow K_{0}\rightarrow N_{0}\rightarrow M$

where $N_{0}\in\operatorname{add}(N)$ is a _right approximation_ of $M$, i.e.
any other map $N^{\prime}_{0}\rightarrow M$ with
$N^{\prime}_{0}\in\operatorname{add}(N)$ factors through $N_{0}$. One defines
the _right mutation_ of $M$ at $N$ to be $\mu^{+}_{N}(M):=N\oplus K_{0}$. The
_left mutation_ of $M$ at $N$ is defined via duality as
$\mu^{-}_{N}(M)=(\mu^{+}_{N^{\vee}}(M^{\vee}))^{\vee}$. We also put
$\mu^{\pm}_{N}(\Lambda)=\operatorname{End}_{R}(\mu^{\pm}_{N}(M))$. Note
however that the passage from $\mu^{\pm}_{N}(M)$ to $\mu^{\pm}_{N}(\Lambda)$
loses some information.

######  Remark 3.9.

Needless to say that $\mu^{\pm}_{N}(M)$ is only determined up to additive
closure (i.e. up to taking $\operatorname{add}(-)$). However if $R$ is
complete local then we can make a minimal choice for $\mu^{+}_{N}(M)$ which we
will do silently.

######  Theorem 3.10 ([IW14a, Theorems 1.22, 1.23]).

Let $M,N,\Lambda$ be as above.

  1. 1.

$\mu^{\pm}_{N}(M)$ define NCCRs.

  2. 2.

$\Lambda$, $\mu^{+}_{N}(\Lambda)$ and $\mu^{-}_{N}(\Lambda)$ are all derived
equivalent.

  3. 3.

$\mu^{+}_{N}$ and $\mu^{-}_{N}$ are mutually inverse operations (this
statement makes sense since $N\in\operatorname{add}(\mu^{\pm}_{N}(M))$).

If $R$ is complete local of dimension 3, things simplify. Let us call a
reflexive $R$-module _basic_ if every indecomposable summand appears occurs
only once.

######  Theorem 3.11 ([IW14a, Theorems 1.25]).

Assume that $R$ is complete local of dimension 3. Let $M$ be a basic reflexive
$R$-module defining an NCCR, having at least two non-isomorphic indecomposable
summands and let $M_{i}$ be such an indecomposable summand. Then
$\mu_{M/M_{i}}^{+}(M)\cong\mu_{M/M_{i}}^{-}(M)$.

######  Example 3.12 ([HW19, IW21, Wem21]).

If $R$ is complete local ring with a 3-dimensional terminal Gorenstein
singularity then the basic reflexive modules yielding an NCCR correspond to
the maximal cells in an affine hyperplane arrangement of dimension
$\operatorname{rk}\operatorname{Cl}(R)$ with mutations at indecomposable
summands corresponding to wall crossings [Wem21, Theorem 4.4]. The group
$\operatorname{Cl(R)}$ acts by translation on this hyperplane arrangement and
the quotient consists of a finite number of cells which correspond to the
NCCRs of $R$. The number of such NCCRs is generally higher than the number of
crepant resolutions.

It is an interesting problem to understand this for other types of
3-dimensional singularities.

######  Remark 3.13.

If $\Lambda=\operatorname{End}_{R}(M)$ is an NCCR then because of the
reflexive Morita equivalence
$\operatorname{ref}(\Lambda)=\operatorname{ref}(R)$ the mutation procedure may
also be defined on the level of reflexive $\Lambda$-modules (see [IR08, §5]).
The resulting procedure also works for twisted NCCRs, where there is no
reflexive Morita equivalence.

We now describe a different point of view on mutations, taken from [DWZ08].
For $Q$ a quiver with $n$ vertices let $\widehat{kQ}$ be the completion of the
path algebra of $Q$ at path length. A _potential_ $w\in\widehat{kQ}$ is a
convergent sum of cycles considered up to rotation (or equivalently
$w\in\widehat{kQ}/[\widehat{kQ},\widehat{kQ}]\,\hat{}$). If $w$ is a potential
then $(\partial w)$ denotes the (completed) two sided ideal generated by the
cyclic derivatives $\partial_{x}w$ of $w$ with respect to the arrows in $Q$,
where for a cyclic path $m$ we have $\partial_{x}m:=\sum_{m=uxv}vu$ (note that
this is invariant under path rotation). The _completed Jacobi algebra_
associated to $(Q,w)$ is defined as $\hat{J}(Q,w):=\widehat{kQ}/(\partial w)$.
We say that $w$ is _reduced_ if it only contains cycles of length $\geq 3$. We
can also consider the uncompleted version $J(Q,w):=kQ/(\partial w)$, in case
$w$ is a finite sum. We have the following result.

######  Theorem 3.14 ([VdB15, Theorems A&B]).

If $\Lambda$ is a basic (i.e. $\Lambda/\operatorname{rad}\Lambda\cong
k^{\oplus n}$) twisted NCCR of a 3-dimensional normal Gorenstein complete
local ring then $\Lambda$ is a completed Jacobi algebra $\hat{J}(Q,w)$ with
$w$ reduced.

If $Q$ does not have loops or $2$-cycles then the mutations of
$\Lambda:=\hat{J}(Q,w)$ can be obtained by an alternative procedure described
in [DWZ08]. The procedure to mutate at a vertex $i$ of $Q$ yields a new Jacobi
algebra $\hat{J}(Q^{\prime},w^{\prime})$ defined as follows (see [KY11,
§2.4]).

  1. 1.

For each arrow $\beta$ with target $i$ and each arrow $\alpha$ with source
$i$, add a new arrow $[\alpha\beta]$ from the source of $\beta$ to the target
of $\alpha$.

  2. 2.

Replace each arrow $\alpha$ with source or target $i$ with an arrow
$\alpha^{\ast}$ in the opposite direction.

The new potential $w^{\prime}$ is the sum of two potentials $w_{1}^{\prime}$
and $w_{2}^{\prime}$ . The potential $w_{1}^{\prime}$ is obtained from $w$ by
replacing each composition $\alpha\beta$ (up to cyclic rotation) by
$[\alpha\beta]$, where $\beta$ is an arrow with target $i$. The potential
$w_{2}^{\prime}$ is given by

$w_{2}^{\prime}=\sum_{\alpha,\beta}[\alpha\beta]\beta^{\ast}\alpha^{\ast}$

where the sum ranges over all pairs of arrows $\alpha$ and $\beta$ such that
$\beta$ ends at $i$ and $\alpha$ starts at $i$. It follows from [KY11, Theorem
3.2] that this mutation coincides with the mutation defined in [IW14a] and
described above.

It may be that $w^{\prime}$ is not reduced, i.e. it contains $2$-cycles. In
that case the corresponding relations allow one to eliminate some arrows in
$Q^{\prime}$. By doing this we find that the Jacobi algebra
$\hat{J}(Q^{\prime},w^{\prime})$ can be more economically written as
$J((Q^{\prime})^{\text{red}},(w^{\prime})^{\text{red}})$ where
$(w^{\prime})^{\text{red}}$ is reduced.

If we are lucky that $(Q^{\prime})^{\text{red}}$ does not contain any 2-cycles
(it cannot contain loops) then we can repeat the mutation procedure at
arbitrary vertices. If we can keep doing this forever then we call the
original potential $w$ _non-degenerate_.

Note that if $(Q^{\prime})^{\text{red}}$ does not contain 2-cycles, it can be
obtained from $Q^{\prime}$ by simply deleting all $2$-cycles, so that the
mutation procedure becomes to some extent combinatorial [Kel]. For a non-
degenerate potential this nice property persists under iterated mutations. The
catch however is that in general it is not clear how to check that a potential
is non-degenerate. A useful criterion, based on the theory of graded mutations
[AO14], is given in [dTdVVdB13].

######  Theorem 3.15 ([dTdVVdB13, Corollary 1.3]).

Assume:

  1. 1.

$Q$ is a ${\mathbb{Z}}$-graded quiver such that $(kQ)_{\leq 0}$ is finite
dimensional.

  2. 2.

$Q$ has at least three vertices.

  3. 3.

$w$ is a homogeneous reduced potential of degree $r$ (in particular it is a
finite sum).

  4. 4.

$\Lambda=J(Q,w)$ is a twisted NCCR whose center is 3-dimensional with an
isolated singularity.

  5. 5.

$\Lambda/[\Lambda,\Lambda]$ does not contain any elements whose degree is in
the interval $[1,r/2]$.

Then $w$ is non-degenerate.

Note that (5) is automatic if $r=1$. This gives an alternative proof why the
potentials associated to “rolled up helix algebras” of Del Pezzo surfaces are
non-degenerate (see [BS10, Theorem 1.7][dTdVVdB13, Theorem 4.2.1]). Theorem
3.15 also applies to many skew group rings
$\Lambda=k[x,y,z]\\#({\mathbb{Z}}/n{\mathbb{Z}})$. For example $n=5$ and
$\bar{1}$ acting with weights $(1/5,2/5,2/5)$ (see [IR08, §7]).

##  4 Quotient singularities for reductive groups

###  4.1 NCCRs via modules of covariants

In this section we discuss some results from [SVdB17]. $G$ will always be a
reductive group. Let $S$ be the coordinate ring of a smooth affine $G$-variety
$X$. Then $S^{G}$ is the coordinate ring of the categorical quotient
$X/\mskip-5.40001mu/G$. We will be interested in constructing (twisted) NCCRs
for $S^{G}$. In the particular case that $G$ is finite and $X$ is a faithful
unimodular $G$-representation then this was discussed in §3.1. An NCCR for
$S^{G}$ is given by the skew group ring $\Lambda=S\\#G$. However $\Lambda$ can
be described in a different way. For $U$ a finite dimensional
$G$-representation put $M(U):=(U\otimes S)^{G}$. Then $M(U)$ is a reflexive
$S^{G}$ module (in fact it is maximal Cohen-Macaulay). If every irreducible
representation of $G$ occurs at least once in $U$ then $\Lambda$ is Morita
equivalent to $\operatorname{End}_{S^{G}}(M(U))$. Hence $M(U)$ defines an NCCR
of $S^{G}$.

The modules $M(U)$ we introduced are so-called _modules of covariants_ [Bri93]
and they make perfect sense for general reductive groups. A mild obstacle is
that modules of covariants do not have to be reflexive in general [Bri93].
This is not a serious problem, but if we want to avoid it anyway, we can
restrict the pairs $(G,X)$ we consider. We will say that $G$ acts
_generically_ on a smooth affine variety if the locus of points with closed
orbit and trivial stabilizer is non-empty and its complement has codimension
$\geq 2$. If $W$ is a $G$-representation then we will say that $(G,W)$ is
_generic_ if $G$ acts generically on
$\operatorname{Spec}\operatorname{Sym}W\cong W^{\ast}$. We then have in
particular

$\operatorname{End}_{S^{G}}(M(U))=M(\operatorname{End}(U)).$ (4.1)

It is reasonable to search for NC(C)Rs of the form
$\operatorname{End}_{S^{G}}(M(U))$. However if $G$ is not finite there are
non-trivial obstacles:

  1. 1.

There are an infinite number irreducible representations so we cannot just
take the sum of all of them. We need to make a careful selection.

  2. 2.

Modules of covariants are usually not Cohen-Macaulay and so demanding that
$\operatorname{End}_{S^{G}}(M(U))=M(\operatorname{End}(U))$ (cfr (4.1)) is
Cohen-Macaualay is a severe restriction on $U$.

The first issue is handled in [SVdB17, §10] where we construct certain nice
complexes relating different modules of covariants (see also [Wey03, Chapter
5]). The second issue is handled using results from [VdB91] (see also [VdB93,
VdB94, VdB95, VdB99]).

Before we discuss NCCRs let’s give a result on NCRs.

######  Proposition 4.1 ([SVdB17, Corollary 1.3.5]).

Assume that $(G,W)$ is generic. Then there exists a finite dimensional
$G$-representation $U$ containing the trivial representation such that
$\Lambda=\operatorname{End}_{S^{G}}(M(U))$ is an NCR for $S^{G}$.

######  Remark 4.2.

The fact that $U$ contains the trivial representation implies that $\Lambda$
defines a categorical resolution by Lemma 2.2.1. It turns out that NCRs are
easier to construct than NCCRs since it is sufficient to take $U$ big enough,
in a suitable sense.

To state our results about (twisted) NCCRs we need to introduce some notation.
Let $G$ be a connected888In [SVdB17] we also consider the non-connected case.
reductive group. Let $T\subset B\subset G$ be respectively a maximal torus and
a Borel subgroup of $G$ with ${\mathcal{W}}=N(T)/T$ being the corresponding
Weyl group. Put $X(T)=\operatorname{Hom}(T,G_{m})$ and let $\Phi\subset X(T)$
be the roots of $G$. By convention the roots of $B$ are the negative roots
$\Phi^{-}$ and $\Phi^{+}=\Phi-\Phi^{-}$ is the set of positive roots. We write
$\bar{\rho}\in X(T)_{\mathbb{R}}$ for half the sum of the positive roots. Let
$X(T)^{+}_{\mathbb{R}}$ be the dominant cone in $X(T)_{\mathbb{R}}$ and let
$X(T)^{+}=X(T)^{+}_{\mathbb{R}}\cap X(T)$ be the set of dominant weights. For
$\chi\in X(T)^{+}$ we denote the simple $G$-representation with highest weight
$\chi$ by $V(\chi)$.

Let $W$ be a finite dimensional $G$-representation of dimension $d$ and put
$S=\operatorname{Sym}(W)$,
$X=\operatorname{Spec}\operatorname{Sym}(W)=W^{\ast}$. Let
$(\beta_{i})_{i=1}^{d}\in X(T)$ be the $T$-weights of $W$.

Put

$\displaystyle\Sigma$ $\displaystyle=\left\\{\sum_{i}a_{i}\beta_{i}\mid
a_{i}\in]-1,0]\right\\}\subset X(T)_{\mathbb{R}}.$

The elements of the intersection $X(T)^{+}\cap(-2\bar{\rho}+\Sigma)$ are
called _strongly critical (dominant) weights_ for $G$.

######  Theorem 4.3 ([SVdB17, Theorem 3.4.3][VdB91]).

Assume that $X$ contains a point with closed orbit and finite stabilizer. Let
$\chi\in X(T)^{+}$ be a strongly critical weight and $U=V(\chi)$. Then
$M(U^{\ast})$ is a Cohen-Macaulay $S^{G}$-module.

If we look at (4.1) and observe that the weights of $\operatorname{End}(U)$
are very roughly speaking about twice those of $U$, then Theorem 4.3 suggests
that to construct an NCCR we should restrict ourselves to representations
whose highest weights are approximately contained in
$-\bar{\rho}+(1/2)\Sigma$. This idea works for the class of “quasi-symmetric”
representations, which includes the class of self dual representations.

We say that $W$ is _quasi-symmetric_ if for every line $\ell\subset
X(T)_{\mathbb{R}}$ through the origin we have
$\sum_{\beta_{i}\in\ell}\beta_{i}=0$. This implies in particular that $W$ is
unimodular and hence $S^{G}$ is Gorenstein if $W$ is generic by a result of
Knop [Kno86].

From now on we assume that $(G,W)$ is generic and $W$ is quasi-symmetric.
Deviating slightly from [SVdB17], following [HLS20], we introduce a certain
affine hyperplane arrangement on $X(T)^{{\mathcal{W}}}_{\mathbb{R}}$. Let
$\bar{{\mathcal{H}}}$ be the collection of affine hyperplanes spanned by the
facets of $-\bar{\rho}+(1/2)\bar{\Sigma}$. We consider the hyperplane
arrangement in999Note that $X(T)^{\mathcal{W}}$ is just the character group
$X(G)$ of $G$. $X(T)^{\mathcal{W}}_{\mathbb{R}}$ given by:

${\mathcal{H}}=\bigcup_{H\in\bar{{\mathcal{H}}}}(-H+X(T))\cap
X(T)^{\mathcal{W}}_{\mathbb{R}}.$ (4.2)

######  Remark 4.4.

The hyperplane arrangement (4.2) may be degenerate in the sense that
$X(T)^{\mathcal{W}}_{\mathbb{R}}\subset-H+\chi$ for some $\chi\in X(T)$.

######  Example 4.5.

We give a simple example where degeneration occurs. Let
$G=\operatorname{SL}(2)$. If $V$ is the standard representation and $W=V^{n}$
with $n$ even, then $(1/2)\Sigma$ is the interval $]-n/2,n/2[$ (identifying
$X(T)\cong{\mathbb{Z}}$). Moreover $\bar{\rho}=1$. Hence the “hyperplanes”
$-H+X(T)$ are given by the integers. Furthermore
$X(T)^{\mathcal{W}}_{\mathbb{R}}=0$. Thus the induced hyperplane arrangement
in $X(T)^{\mathcal{W}}_{\mathbb{R}}$ is indeed degenerate. If $n$ is odd on
the other hand then it is non-degenerate. See [SVdB17, Theorem 1.4.5] for a
complete treatment of the case $G=\operatorname{SL}(2)$.

This hyperplane arrangement is such that if $\delta$ is the complement of
${\mathcal{H}}$ then

$(-\bar{\rho}+\delta+1{/}2\,\partial\bar{\Sigma})\cap X(T)=\emptyset.$

The following result is a slight variation on [SVdB17, Theorem 1.6.4].

######  Theorem 4.6.

Let $(G,W)$ be generic and assume that $W$ is quasi-symmetric. Let $\delta$ be
an element of the complement of ${\mathcal{H}}$. Put

$\displaystyle{\mathcal{L}}_{\delta}$
$\displaystyle=X(T)^{+}\cap(-\bar{\rho}+\delta+(1/2)\bar{\Sigma}),$ (4.3)
$\displaystyle U_{\delta}$
$\displaystyle=\bigoplus_{\chi\in{\mathcal{L}}_{\delta}}V(\chi),$ (4.4)
$\displaystyle\Lambda_{\delta}$
$\displaystyle=\operatorname{End}_{S^{G}}(M(U_{\delta})).$ (4.5)

If ${\mathcal{L}}_{\delta}\neq\emptyset$ then $\Lambda$ is an NCCR for
$\operatorname{Sym}(W)^{G}$.

It is easy to see that ${\mathcal{L}}_{\delta}$ and hence $U_{\delta}$ depend
only on the connected component of the complement of ${\mathcal{H}}$ to which
$\delta$ belongs.

We obtain some evidence for Conjecture 3.7.101010This is stated in loc. cit.
for $W$ generic. However one easily reduces to this case.

######  Corollary 4.7 ([SVdB17, Theorem 1.6.2]).

If $G=T$ is a torus and $W$ is quasi-symmetric then
$\operatorname{Sym}(W)^{T}$ has a (toric) NCCR.

######  Remark 4.8.

For reference we note that there is extension of Theorem 4.6 that may allow
one to construct twisted NCCRs [SVdB17, Theorem 1.6.4].

We now state some consequences of these results for determinantal varieties.

######  Theorem 4.9 ([SVdB17, Theorem 1.4.1]).

For $l<n$ let $X_{l,n}$ be the variety of $n\times n$-matrices of rank $\leq
l$. Then $k[X_{l,n}]$ has an NCCR.

The variety $X_{l,n}$ was already discussed in Example 3.1 and the NCCR
obtained in loc. cit. is the same as the one we obtain in Theorem 4.9. To
prove Theorem 4.9 we use the classical description of $k[X_{l,n}]$ as an
invariant ring [Wey46]. Put $G=\operatorname{GL}(l)$ and let $V$ be the
standard representation of $G$. Put $W=V^{n}\oplus(V^{\ast})^{n}$. Then
$k[X_{l,n}]=\operatorname{Sym}(W)^{G}$ and we show in [SVdB17] that Theorem
4.9 follows from Theorem 4.6. For the benefit of the reader we describe the
actual module of covariants that gives the NCCR. Let $B_{l,n-l}$ be the set of
partitions that fit in a rectangle of size $l\times(n-l)$. In loc. cit. it is
shown that the following module of covariants defines an NCCR for $R$:

$M=\bigoplus_{\lambda\in B_{l,n-l}}M(S^{\lambda}V),$ (4.6)

where $S^{\lambda}V$ denotes the Schur functor indexed by $\lambda$ applied to
$V$.

######  Theorem 4.10.

For $2l<n$ let $X_{2l,n}^{-}$ be the variety of skew-symmetric $n\times n$
matrices of rank $\leq 2l$. If $n$ is odd then $k[X_{2l,n}^{-}]$ has an NCCR.

This time we put $G=\operatorname{Sp}(2l)$ and $W=V^{n}$ where $V$ is the
standard representation of $G$.

######  Theorem 4.11 ([SVdB17, Theorem 1.4.1]).

For $l<n$ let $X^{+}_{l,n}$ be the variety of symmetric $n\times n$ matrices
of rank $\leq l$. If $l$ and $n$ have opposite parity then $k[X^{+}_{l,n}]$
has an NCCR. If $l$ and $n$ have the same parity then $k[X_{l,n}^{+}]$ has a
twisted NCCR.

Here we put $G=\operatorname{O}(l)$ and again $W=V^{n}$ where $V$ is the
standard representation of $G$. A complication arises since
$\operatorname{O}(l)$ is not connected, so we cannot directly apply Theorem
4.6. So we have to perform a more refined analysis which is carried out in
[SVdB17, §6]. Twisted NCCRs appear because $\operatorname{SO}(l)$, the
connected component of $\operatorname{O}(l)$, is not simply connected.

The NCCRs given in Theorems 4.10, 4.11 have been crucial for establishing
homological projective duality [Kuz07] for determinantal varieties of skew-
symmetric matrices by Rennemo and Segal [RS19]. The corresponding results for
symmetric matrices are work in progress by the same authors [Ren19].

Even if $W$ is not quasi-symmetric then it is still possible that
$\operatorname{Sym}(W)^{G}$ has an NCCR given by a module of covariants but we
are unaware of a general rule like Theorem 4.6 for constructing them. Three-
dimensional affine toric varieties (see §3.4) are an example of this, since
they can be written as $\operatorname{Sym}(W)^{G}$ where $W$ is generally not
quasi-symmetric. Another example is given by the recent work of Doyle:

######  Example 4.12 ([Doy, Theorem 3.11]).

Let $0<l<n$ be integers such that $\operatorname{gcd}(l,n)=1$. Let $V$ be the
standard representation of $G=\operatorname{SL}(l)$ and put $W=V^{n}$.

Then $R:=\operatorname{Sym}(W)^{G}$ is the homogeneous coordinate ring of the
Grassmannian $\operatorname{Gr}(l,n)$ for the Plücker embedding. Let
$P_{l,n-l}$ be the set of partitions whose young tableaux are above the
diagonal in a rectangle of size $l\times(n-l)$. In loc. cit. it is shown that
the following module of covariants defines an NCCR for $R$:

$M=\bigoplus_{\lambda\in P_{l,n-l}}M(S^{\lambda}V)$

(compare with (4.6)).

We reiterate that even if an NCCR exists, there doesn’t have to be one given
by a module of covariants. See Example 3.6.

###  4.2 NCCRs via crepant resolutions obtained by GIT

Here we discuss some results from [HLS20] that shows that in certain cases the
NCCRs for the categorical quotients $X/\mskip-5.40001mu/G$ that we constructed
in §4 can be obtained as the endomorphisms of a tilting bundle on a crepant
resolution, i.e. the method of §3.2. This crepant resolution is constructed
using a geometric invariant theory. It turns out that we have to allow crepant
resolutions by Deligne-Mumford stacks. This occurred already before in Lemma
3.5 and Example 3.6.

######  Remark 4.13.

To construct a resolution of $X/\mskip-5.40001mu/G$ using geometric invariant
theory, one needs a linearized line bundle on $X$. Since here $X$ is a
representation, the only $G$-equivariant line bundles on $X$ are those
obtained from characters of $G$. If $G$ is semi-simple then there are no (non-
trivial) characters so we cannot proceed. Thus we can for example not deal
with determinantal varieties of symmetric and skew-symmetric matrices (see
Theorems 4.10 and 4.11). In those cases the relevant groups were respectively
$\operatorname{Sp}(2l)$ and the connected component $\operatorname{SO}(l)$ of
$\operatorname{O}(l)$, both of which are semi-simple. On the other hand,
ordinary determinantal varieties are fine since in that case
$G=\operatorname{GL}(l)$ which has a non-trivial character given by the
determinant, which may be used to construct a crepant resolution.

######  Remark 4.14.

Geometric invariant theory is still helpful for constructing a resolution of
$X/\mskip-5.40001mu/G$ via a procedure invented by Kirwan [Kir85, Rei89,
ER21]. However these resolutions are usually not crepant. Consistent with
expected minimality of NCCRs (see §3.3) we are able to show that some NCCRs
embed inside them [SVdB19b].

We retain the notations and assumptions of the previous section. We assume
that $W$ is a quasi-symmetric representation of $G$ and
$X=\operatorname{Spec}\operatorname{Sym}(W)=W^{\ast}$. Recall that for a
character $\mu\in X(G)=X(T)^{\mathcal{W}}$ we may define a $G$-invariant open
subset of $X$ as

$X^{ss,\mu}=\\{x\in X\mid\exists k>0\text{ and
}s\in\Gamma({\mathcal{O}}_{X}\otimes\mu^{k})^{G}\text{ such that }s(x)\neq
0\\}.$

The variety $X^{ss,\mu}$ admits a good quotient111111A $G$-equivariant map
$Z\rightarrow Y$ is a good quotient if locally on $Y$ it is of the form
$U\rightarrow U/\mskip-5.40001mu/G$ for $U$ affine.
$X^{ss,\mu}/\mskip-5.40001mu/G$ which is proper over $X/\mskip-5.40001mu/G$.
For $U$ a representation of $G$ we write ${\mathcal{M}}(U)$ for the vector
bundle on $X^{ss,\mu}/G$ given by $U\otimes{\mathcal{O}}_{X^{ss,\mu}/G}$. The
global sections of ${\mathcal{M}}(U)$ are equal to $M(U)$.

Below we let ${\mathcal{H}}_{0}$ be the _central_ hyperplane arrangement on
$X(G)_{\mathbb{R}}=X(T)^{\mathcal{W}}_{\mathbb{R}}$ corresponding to the
affine hyperplane arrangement ${\mathcal{H}}$ introduced in (4.2). Thus the
hyperplanes in ${\mathcal{H}}_{0}$ are the hyperplanes which are induced from
central hyperplanes in $X(T)_{\mathbb{R}}$ which are parallel to the facets of
$\bar{\Sigma}$.

######  Proposition 4.15 ([HLS20, Proposition 2.1]).

Assume that the action of $T$ on $X$ has generically finite stabilizers and
let $\mu\in X(G)$ be in the complement of ${\mathcal{H}}_{0}$. Then
$X^{ss,\mu}/G$ is a Deligne-Mumford stack.

######  Lemma 4.16.

Assume that $(G,W)$ is generic. Then the canonical map
$X^{ss,\mu}/G\rightarrow X/\mskip-5.40001mu/G$ is crepant.

###### Proof.

This is proved in [SVdB20a, Lemma 4.5] in the case that $G$ is a torus, but
this assumption is not relevant for the proof. ∎

The following is one of the main results of [HLS20]. It is proved using
similar combinatorics as in [SVdB17].

######  Theorem 4.17 ([HLS20]).

Assume that the action of $T$ on $X$ has generically finite stabilizers and
let $\mu\in X(G)$ be in the complement of ${\mathcal{H}}_{0}$. Let
$U_{\delta}$ be as in the statement of Theorem 4.6. Then
${\mathcal{M}}(U_{\delta})$ is a tilting bundle on $X^{ss,\mu}/G$ such that
$\operatorname{End}_{X^{ss,\mu}/G}({\mathcal{M}}(U_{\delta}))=M(\operatorname{End}(U_{\delta}))$.

###### Proof.

This follows from combining [HLS20, Theorem 1.2] with [HLS20, Lemma 2.9]. ∎

In this way we obtain more evidence for Conjecture 1.2.

######  Corollary 4.18 ([HLS20, Corollary 1.3]).

Under the hypotheses of Theorem 4.17 if $\mu,\mu^{\prime}\in X(G)$ are in the
complement of ${\mathcal{H}}_{0}$ and the complement of ${\mathcal{H}}$ is
non-empty (i.e. ${\mathcal{H}}$ is non-degenerate) then
${\mathcal{D}}(X^{ss,\mu}/G)\cong{\mathcal{D}}(X^{ss,\mu^{\prime}}/G)$.

We also obtained the promised description of NCCRs via resolutions.

######  Corollary 4.19.

Assume that $(G,W)$ is generic and let $\Lambda$ be an NCCR constructed via
Theorem 4.6. Let $\mu$ be in the complement of ${\mathcal{H}}_{0}$. Then
$\Lambda$ is the endomorphism ring of a tilting bundle on the DM stack
$X^{ss,\mu}/G$.

######  Remark 4.20.

Hidden behind what is discussed in §4.1,§4.2 is the idea of _windows_ ,
pioneered in [DS14]. This is based on the fact that we have a restriction map

$\operatorname{Res}:{\mathcal{D}}(X/G)\rightarrow{\mathcal{D}}(X^{ss,\mu}/G),$

It is then natural to try to find a full subcategory
${\mathcal{D}}\subset{\mathcal{D}}(X/G)$ such that the restriction of
$\operatorname{Res}$ to ${\mathcal{D}}$ yields an equivalence
${\mathcal{D}}\cong{\mathcal{D}}(X^{ss,\mu}/G)$. A very general result in this
direction is [HL15, Theorem 1.1], see also [BFK19].

In concrete cases one may hope to define ${\mathcal{D}}$ as the full
subcategory of ${\mathcal{D}}(X/G)$ which is split generated by
$U\otimes{\mathcal{O}}_{X/G}$ for a suitable $G$-representation $U$ whose
highest weights are restricted to a certain subset ${\mathcal{L}}$ of
$X(T)^{+}$ (a “window”). This is precisely what happens in Theorem 4.17, where
we take ${\mathcal{L}}={\mathcal{L}}_{\delta}$. The resulting category
${\mathcal{D}}$ is a concrete realization of [HL15, Theorem 1.1], see [HLS20,
Lemma 3.5].

One does not actually need to have non-trivial $X^{ss,\mu}/G$ to apply the
window principle. The proof of Theorem 4.6, is based on the fact that
$\operatorname{mod}(\Lambda_{\delta})$ embeds in $\mathop{\text{{coh}}}(X/G)$
as the abelian category with a projective generator
$U_{\delta}\otimes{\mathcal{O}}_{X/G}$.

###  4.3 Local systems, the SKMS and schobers

In this slightly informal section we assume that the hypotheses of Theorem
4.17 hold. While Corollary 4.18 implies that two different
${\mathcal{D}}(X^{ss,\mu}/G)$, ${\mathcal{D}}(X^{ss,\mu^{\prime}}/G)$ are
derived equivalent, the actual derived equivalence depends on the choice of
$\delta$ in the complement of ${\mathcal{H}}$. Moreover by considering
compositions
${\mathcal{D}}(X^{ss,\mu}/G)\xrightarrow{\delta}{\mathcal{D}}(X^{ss,\mu^{\prime\prime}}/G)\xrightarrow{\delta^{\prime}}{\mathcal{D}}(X^{ss,\mu^{\prime}}/G)$
we may produce more derived equivalences. This is consistent with the
assertion in Remark 1.3 that there is no “god-given” derived equivalence
between different crepant resolutions. A different way of saying this is that
a crepant resolution may have a large group of auto-equivalences.

If $M$ is a Calabi-Yau variety then homological mirror symmetry predicts the
existence of a space $S$ (the "stringy Kähler moduli space” or SKMS) such that
$\pi_{1}(S)$ acts on ${\mathcal{D}}(M)$. More precisely $S$ is the moduli
space of complex structures on the mirror dual $M^{\vee}$ of $M$. In many
cases there are good heuristic descriptions of $M^{\vee}$ and $S$.

Even without access to the full mirror symmetric context, which may be
technically challenging or even only heuristic, it turns out to be very
illuminating to represent the derived autoequivalences of an algebraic variety
(or stack) as elements of $\pi_{1}(S)$ for a suitable topological space $S$.
Alternatively we may think of such a representation as a _local system of
triangulated categories on $S$_. Understanding this for
${\mathcal{D}}(X^{ss,\mu}/G)$ was, according to the authors, one of the main
motivations for writing [HLS20]. Indeed when $X$ is a quasi-symmetric
representation, under hypotheses of Theorem 4.17 one may take

$S=(X(G)_{{\mathbb{C}}}-{\mathcal{H}}_{\mathbb{C}})/X(G)$

where ${\mathcal{H}}_{\mathbb{C}}$ denotes the complexification of the real
hyperplane arrangement ${\mathcal{H}}$ [HLS20, Proposition 6.6].

In this case there is a nice way to understand that action of $\pi_{1}(S)$ on
${\mathcal{D}}(X^{ss}/G)$ [HLS20, §6][SVdB19a]. Using Theorem 4.17 again we
may just as well describe the action of $\pi_{1}(S)$ on
${\mathcal{D}}(\Lambda_{\delta})$ for $\delta$ contained in the complement of
${\mathcal{H}}$ and $\Lambda_{\delta}=M(\operatorname{End}(U_{\delta}))$.

For $\delta\in X(G)_{\mathbb{R}}$ define $U_{\delta}$ as in (4.4) and put
${\mathcal{D}}_{\delta}={\mathcal{D}}(\Lambda_{\delta})$. Now ${\mathcal{H}}$
defines a cell decomposition of $X(G)_{\mathbb{R}}$ and it is easy to see that
$U_{\delta}$ only depends on the cell to which $\delta$ belongs. Hence for a
cell $C$ let us write $\Lambda_{C}:=\Lambda_{\delta}$,
${\mathcal{D}}_{C}:={\mathcal{D}}_{\delta}$ for $\delta\in C$. We will refer
to the cells of maximal dimension as chambers. These are also the connected
components of the complement of ${\mathcal{H}}$.

If $C^{\prime}$ is a face of $C$ then there is an idempotent
$e_{C,C^{\prime}}\in\Lambda_{C^{\prime}}$ such that
$\Lambda_{C}=e_{C,C^{\prime}}\Lambda_{C^{\prime}}e_{C,C^{\prime}}$. If $C\neq
C^{\prime\prime}$ are distinct adjacent chambers, sharing a codimension one
face $C^{\prime}$ then the functor
$e_{C^{\prime\prime},C^{\prime}}\Lambda_{C^{\prime}}e_{C,C^{\prime}}\overset{L}{\otimes}_{\Lambda_{C}}-$
defines an equivalence of categories
$\phi_{C,C^{\prime\prime}}:{\mathcal{D}}_{C}\rightarrow{\mathcal{D}}_{C^{\prime\prime}}$.

Put $\tilde{S}=X(G)_{\mathbb{C}}-{\mathcal{H}}_{\mathbb{C}}$ and let
$\Pi_{1}(\tilde{S})$ be the groupoid whose objects are the chambers and whose
morphisms are given by the homotopy classes of paths in
$X(G)_{\mathbb{C}}-{\mathcal{H}}_{\mathbb{C}}$ connecting the chambers. Then
$\Pi_{1}(\tilde{S})$ is equivalent to the fundamental groupoid of
$X(G)_{\mathbb{C}}-{\mathcal{H}}_{\mathbb{C}}$. If $C$, $C^{\prime\prime}$ are
adjacent chambers separated by a hyperplane $H\in{\mathcal{H}}$, such that
$H(C^{\prime\prime})>0$ then there is a canonical (up to homotopy) minimal
path $\nu_{C,C^{\prime\prime}}$ in
$X(G)_{\mathbb{C}}-{\mathcal{H}}_{\mathbb{C}}$ going from $C$ to
$C^{\prime\prime}$ and passing through
$\\{\operatorname{Im}H_{\mathbb{C}}>0\\}$. Sending $C$ to ${\mathcal{D}}_{C}$
and $\nu_{C,C^{\prime\prime}}$ to $\phi_{C,C^{\prime\prime}}$ defines a
representation of the groupoid $\Pi_{1}(\tilde{S})$ in triangulated
categories.

If $\chi\in X(G)$ then tensoring by $\chi$ defines an equivalence
${\mathcal{D}}_{C}\rightarrow{\mathcal{D}}_{C+\chi}$ and in this way the
representation of $\Pi_{1}(\tilde{S})$ may be extended to a representation of
$\Pi_{1}(\tilde{S})\rtimes X(G)$ and the latter is equivalent to the
fundamental groupoid $\Pi_{1}(S)$ of $S=\tilde{S}/X(G)$ [Bro06, Chapter
11][HLS20, §6]. Hence fixing a “base chamber” $C$ we get an action of
$\pi_{1,C}(S)$ on ${\mathcal{D}}_{C}$.

######  Remark 4.21.

It is shown in [SVdB19a] that the family of triangulated categories
$({\mathcal{D}}_{C})_{C}$ for _all_ cells $C$ is a so-called
$X(G)$-equivariant _perverse schober_. This is a categorification of a
perverse sheaf on $X(G)_{\mathbb{C}}/X(G)$ [KS15] (see also [Don19, BKS18]).
Note that $X(G)_{\mathbb{C}}/X(G)$ is a torus and $S$ is the complement of a
“toric hyperplane arrangement”. If $G$ is itself a torus $T$ then
$X(T)_{\mathbb{C}}/X(T)$ may be identified with the dual torus $T^{\vee}$.

######  Remark 4.22.

The $X(G)$-equivariant hyperplane arrangement constructed by Halpern-Leistner
and Sam in [HLS20] is very similar to the $\operatorname{Cl}(R)$-equivariant
hyperplane arrangement associated to a 3-dimensional terminal complete
Gorenstein ring $R$ constructed by Iyama and Wemyss (see Example 4.12). One
would expect there to be an associated equivariant schober also in this case.
In the case of a single curve flop this is essentially contained in [DW19,
§3].

######  Remark 4.23.

As explained we have an action of $\pi_{1}(S)$ on ${\mathcal{D}}_{C}$ for a
chamber $C$ and hence also an action of $\pi_{1}(S)$ on
$K_{0}({\mathcal{D}}_{C})_{\mathbb{C}}$. In other words we have a local system
$L$ on $S$. It is then a natural question if this local system occurs as the
solutions of a natural system of differential equations. In the case that $G$
is a torus we show in [SVdB20c] that a generic “equivariant” deformation of
$L$ is obtained as the solution of a well-known system of differential
equations introduced by Gel’fand, Kapranov and Zelevinsky [GKZ89]. This starts
from a computation by Kite [Kit17] which shows that the hyperplane arrangement
constructed in [HLS20] is up to translation defined by the so-called
“principal $A$-determinant”, an important ingredient in the theory developed
Gel’fand, Kapranov and Zelevinsky. For more information see [Spe21].

######  Remark 4.24.

The themes touched upon in this section occur in many different contexts. See
e.g. [BH06, ABM15, BQS20].

##  5 NCCRs and stringy $E$-functions

In this section we discuss some ongoing work of Timothy De Deyn (see [Dey]).
Let $X$ be an algebraic variety over ${\mathbb{C}}$. The cohomology groups
$H^{i}_{c}(X,{\mathbb{C}})$ carry a natural mixed Hodge structure. We denote
by $h^{p,q}(H^{i}_{c}(X,{\mathbb{C}}))$ the dimension of the $(p,q)$-type
component of $H^{i}_{c}(X,{\mathbb{C}})$. The _Hodge polynomial_ of $X$ is
defined by

$E(X,u,v)=\sum_{p,q,i}(-1)^{i}h^{p,q}(H^{i}_{c}(X,{\mathbb{C}}))u^{p}v^{p}.$

The Hodge polynomial defines a ring homomorphism from the Grothendieck ring of
algebraic varieties $K_{0}(\operatorname{Var}/{\mathbb{C}})$ to
${\mathbb{Z}}[u,v]$.

We put $e(X)=E(X,1,1)$; i.e.

$e(X)=\sum_{i}(-1)^{i}\sum_{p,q}h^{p,q}(H^{i}_{c}(X,{\mathbb{C}}))=\sum_{i}(-1)^{i}\dim
H^{i}_{c}(X,{\mathbb{C}}).$

In other words $e(X)$ is the _Euler characteristic_ (with compact
support121212If $X$ is smooth then, by Poincaré duality, the Euler
characteristic with compact support coincides with the usual Euler
characteristic $\sum_{i}(-1)^{i}\dim H^{i}(X,{\mathbb{C}})$.) of $X$. It
defines a ring homomorphism from $K_{0}(\operatorname{Var}/{\mathbb{C}})$ to
${\mathbb{Z}}$.

######  Definition 5.1 ([Bat98, Definition 3.1]).

Assume that $X$ is a normal ${\mathbb{Q}}$-Gorenstein algebraic
variety$/{\mathbb{C}}$ with at most log-terminal singularities and let
$\pi:Y\rightarrow X$ be a resolution of singularities whose exceptional locus
is a normal crossing divisor. Let $D_{1},\ldots,D_{r}$ be the irreducible
components of the exceptional locus and put $I=\\{1,\ldots,r\\}$. For any
subset $J\subset I$ we set $D_{J}=\bigcap_{j\in J}D_{j}$,
$D^{\circ}_{J}:=D_{J}\setminus\bigcup_{j\in I\setminus J}D_{j}$. The _stringy
$E$-function_ of $X$ is defined as

$E_{st}(X,u,v):=\sum_{J\subset I}E(D^{\circ}_{J},u,v)\prod_{j\in
J}\frac{uv-1}{(uv)^{a_{j}+1}-1}$ (5.1)

where the numbers $a_{j}\in{\mathbb{Q}}\,\cap\,]-1,\infty[$ are defined by

$K_{Y}=\pi^{\ast}K_{X}+\sum_{j=1}^{r}a_{j}D_{j}.$

Putting $e_{st}(X)=\lim_{u,v\rightarrow 1}E_{st}(X,u,v)$ defines the _stringy
Euler characteristic_ of $X$, with the formula

$e_{st}(X)=\sum_{J\subset I}e(D^{\circ}_{J})\prod_{j\in J}\frac{1}{a_{j}+1}.$

It follows from the theory of motivic integration (see e.g. [Kon95, DL99,
Bat00, Cra04, LCMMVVS20]) that $E_{st}(X,u,v)$ is independent of the chosen
resolution $Y$ [Bat98, Theorem 3.4]. Indeed $E_{st}(X,u,v)$ may be obtained by
integrating over the arc space associated to $X$ [DL99]. In a similar vein
$E_{st}(X,u,v)=E_{st}(Y,u,v)$ holds for birational maps $\pi:Y\rightarrow X$
satisfying $\pi^{\ast}K_{X}=K_{Y}$ [Bat98, Theorem 3.12].

If $X$ is smooth then the stringy $E$-function coincides with the Hodge
polynomial. Hence one has:

######  Theorem 5.2 ([Bat98, Theorem 3.12]).

If $X$ has a crepant resolution $Y$ then the stringy $E$-function of $X$
coincides with the Hodge polynomial of $Y$. In particular, it is a polynomial.
Similarly, the stringy Euler characteristic of $X$ coincides with the usual
Euler characteristic of $Y$ and hence it is an integer.

The following conjecture seems natural:

######  Conjecture 5.3 ([Dey]).

If $X$ is a normal Gorenstein variety$/{\mathbb{C}}$ with an NCCR then its
stringy $E$-function is a polynomial.

We give some evidence for this conjecture below, but at this point it is
probably safer to regard it as a question. We illustrate below in Remark 5.8
and Example 5.11 that reasonable extensions of this conjecture are false.

######  Example 5.4.

Quotient varieties of the form ${\mathbb{C}}^{n}/\mskip-5.40001mu/G$ for
$G\subset\operatorname{SL}(n)$ finite always have a stringy $E$-function which
is a polynomial by [BD96][DL02, Theorem 3.6]. They also have an NCCR by §3.1.
So in this case Conjecture 5.3 is true.

######  Example 5.5.

Batyrev proves in [Bat00, Proposition 4.4] that the stringy $E$-function of
any toric variety with Gorenstein singularities is a polynomial. Hence
Conjecture 5.3 is compatible with Conjecture 3.7.

A good test for Conjecture 5.3 is given by cones over Fano varieties.

######  Proposition 5.6 ([Dey]).

Let $Z$ be a smooth projective variety$/{\mathbb{C}}$ with ample line bundle
${\mathcal{O}}_{Z}(1)$ and let $X=\operatorname{Spec}R$ be the corresponding
cone. Assume $\omega_{Z}={\mathcal{O}}_{Z}(-n)$ for $n\geq 1$. Then $R$ is
Gorenstein and

$E_{st}(X,u,v)=E(Z,u,v)\frac{(q-1)q^{n}}{q^{n}-1}$ (5.2)

with $q=uv$. In particular:

$e_{st}(X)=\frac{e(Z)}{n}.$ (5.3)

######  Example 5.7.

Consider the Grassmannian $Z:=\operatorname{Gr}(d,n)$. Then (e.g. [BL19,
Proposition A.4])

$E_{st}(Z,u,v)={n\choose d}_{q}$ (5.4)

where

${n\choose
d}_{q}=\frac{(q^{n}-1)(q^{n-1}-1)\cdots(q^{n-d+1}-1)}{(q-1)(q^{2}-1)\cdots(q^{d}-1)}.$

Hence

$e_{st}(Z)={n\choose d}.$ (5.5)

Let $X$ be the cone over $Z$ with respect to the Plucker embedding and let $R$
be the coordinate ring of $X$. Using (5.2)(5.4) it is shown in [Dey] that in
this case $E_{st}(X,u,v)$ is a polynomial precisely when
$\operatorname{gcd}(d,n)=1$. On the other hand by [Doy, Theorem 3.11] (see
Example 4.12 above) $R$ has an NCCR when $\gcd(d,n)=1$. So Conjecture 5.3 is
true in this case.

######  Remark 5.8.

It is shown in in [Dey, §4.2] that the cone over an arbitrary Grassmannian
$\operatorname{Gr}(d,n)$ always has a weakly crepant categorical resolution
(see §2.2). Hence it follows that Conjecture 5.3 is false for weakly crepant
categorical resolutions. On the other hand it seems reasonable to extend
Conjecture 5.3 to strongly crepant categorical resolutions.

######  Remark 5.9.

In view of Theorem 5.2 one may naively ask if it is true that $E_{st}(X,u,v)$
being a polynomial implies that $X$ has a crepant resolution. Not unexpectedly
this fails drastically. Finite group quotients and affine toric varieties have
a polynomial stringy $E$-function, as we have seen above, but they don’t need
to have a crepant resolution. In fact, the example
${\mathbb{C}}^{4}/\mskip-5.40001mu/{\mathbb{Z}}_{2}$ given in §3.1 of a
Gorenstein singularity with an NCCR but without a crepant resolution, lives in
both classes. On the other hand these classes of counterexamples are not very
convincing since they admit crepant resolutions by smooth Deligne Mumford
stacks, which is just as good. In the case of finite quotient singularities
this is clear, and for toric varieties it follows from Lemma 3.5.

In contrast one may show that for $X$ as in Example 5.7 there is no crepant
resolution by a smooth DM-stack (this is mainly because $R$ is factorial). So
in some sense it is a “better” counterexample (when $\gcd(d,n)\neq 1$).

GIT quotients form an important class of toric varieties (see [CLS11,
Corollary 14.2.16]). So in view of Example 5.5 as well as Example 5.4 the
following question by Batyrev suggests itself.

######  Question 5.10 ([Bat98, Question 5.5]).

Does a GIT quotient of ${\mathbb{C}}^{n}$ for a linear action of
$G\subset\operatorname{SL}(n)$ always have a stringy $E$-function that is a
polynomial?

Alas the answer is negative. Indeed Example 5.7 for
$\operatorname{gcd}(d,n)\neq 1$ gives a simple counterexample since the cone
over a Grassmannian is a GIT quotient for $\operatorname{SL}(d)$ acting on $n$
copies of its standard representation.

The first counterexample however was constructed much earlier in [Kie03].

######  Example 5.11.

Let $W$ be given by three copies of the adjoint representation of
$G=\operatorname{SL}(2)$. Then by a quite involved computation it is shown in
[Kie03, Corollary 1.2] that the stringy $E$-function of $\operatorname{Spec}R$
for $R=\operatorname{Sym}(W)^{G}$ is not a polynomial. This example is
interesting since by [SVdB17, Theorem 1.4.5] $R$ has a twisted NCCR. In other
words, Conjecture 5.3 is also false for twisted NCCRs.

As a side remark we note that this twisted NCCR is a rather classical object.
It is the trace ring generated by 3 generic traceless $2\times 2$ matrices
[Art69, Raz74, Pro76, Pro87, LB87, LBVdB88].

In the setting of Theorem 5.2 the Euler characteristic of $Y$ can be computed
using periodic cyclic homology, thanks to the Hochschild-Kostant-Rosenberg
theorem. Below we define the Euler characteristic $e(\Lambda)$ of an algebra
$\Lambda$ or a sheaf of algebras as
$\dim\operatorname{HP}^{\text{even}}(\Lambda)-\dim\operatorname{HP}^{\text{odd}}(\Lambda)$.
If $\Lambda$ is a quasi-coherent sheaf of algebras then we use
$\operatorname{HP}^{*}(\Lambda):=\operatorname{HP}^{*}(\operatorname{Perf}_{dg}{\Lambda})$
(where $\operatorname{Perf}_{dg}(\Lambda)$ is a standard dg-enhancement of
$\operatorname{Perf}(\Lambda)$). The following conjecture appears plausible.

######  Conjecture 5.12.

The stringy Euler characteristic of a normal Gorenstein variety can be
computed as the Euler characteristic of an NCCR, computed via periodic cyclic
homology.

######  Remark 5.13.

One can again not expect this conjecture to hold for twisted NCCRs. An
interesting example is given in [BW19]. It was shown by [BO02, Kuz08a] that
for a generic complete intersection $Y$ of $n$ quadrics in
${\mathbb{P}}^{2n-1}$ one has a derived equivalence between $Y$ and
$({\mathbb{P}}^{n-1},{\mathcal{B}}_{0})$ where ${\mathcal{B}}_{0}$ is the even
part of the universal Clifford algebra corresponding to the quadrics defining
$Y$. Because of the derived equivalence we then have
$e(Y)=e({\mathcal{B}}_{0})$ [Kel99]. One may show that ${\mathcal{B}}_{0}$ is
a twisted NCCR of its center which is a double cover $Z$ of
${\mathbb{P}}^{n-1}$ [BW19, §1]. It is shown in [BW19] that in general
$e(Y)\neq e_{st}(Z)$ and hence $e({\mathcal{B}}_{0})\neq e_{st}(Z)$. So
Conjecture 5.12 does not extend to twisted NCCRs.

In suitable “local” contexts (e.g. [VdB15, Theorem 9.1]) Conjecture 5.12 leads
to a more concrete conjecture:131313To handle the complete case one has to use
a “completed” version of periodic cyclic homology. See [VdB15]

######  Conjecture 5.14 ([Dey]).

Let $R$ a normal Gorenstein ring which is either a complete local ring, or
else connected ${\mathbb{N}}$-graded (i.e. $R_{0}={\mathbb{C}}$). Assume that
$R$ has an NCCR $\operatorname{End}_{R}(M)$. Then the number of non-isomorphic
indecomposable summands of $M$ is equal to $e_{st}(X)$ for
$X=\operatorname{Spec}R$.

######  Example 5.15.

If $X=\operatorname{Spec}R$ is an affine toric variety as in §3.4, with
Gorenstein singularities, then Batyrev [Bat98, Proposition 4.10] proves that
$e_{st}(X)$ is equal to the volume of the associated polytope $P$ (see §3.4).
So Conjecture 5.14 is compatible with Conjecture 3.8.

######  Example 5.16.

For varieties of the form
$X=W/\mskip-5.40001mu/G=\operatorname{Spec}{\mathbb{C}}[W]^{G}$ for
$G\subset\operatorname{SL}(n)$ finite and $W$ a finite dimensional
representation of $G$ it follows from [Bat99, Theorem 8.4] that $e_{st}(X)$ is
equal to the number of conjugacy classes in $G$. This number is in turn equal
to the number of irreducible representations of $G$ and hence equal to the
number of non-isomorphic indecomposable summands of the reflective
${\mathbb{C}}[W]^{G}$-module ${\mathbb{C}}[W]$ which defines an NCCR for
${\mathbb{C}}[W]^{G}$ by §3.1. So Conjecture 5.14 is true in this specific
example.

######  Example 5.17.

Let $X$ be the cone over $\operatorname{Gr}(d,n)$ as in Example 5.7. Then by
(5.3)(5.5) we have

$e_{st}(X)=\frac{1}{n}{n\choose d}.$

We check that Conjecture 5.14 is compatible with the NCCR constructed by Doyle
in [Doy, Theorem 3.11] (see Example 4.12 above). Conjecture 5.14 amounts to

$|P_{d,n-d}|=\frac{1}{n}{n\choose d}.$

which is indeed true by [Loe11, §12.1].

Example 5.17 can be put in a more general context. Let us first state a lemma.

######  Lemma 5.18.

Let $Z$ be a smooth projective variety with a tilting complex. Then one has
$e(Z)=\operatorname{rk}K_{0}(Z)$.

###### Proof.

Let ${\mathcal{T}}$ be the tilting complex and put
$A=\operatorname{End}_{Z}({\mathcal{T}})$. We have:

  1. 1.

Euler characteristics may be computed with periodic cyclic homology.

  2. 2.

Periodic cyclic homology is invariant under derived equivalence [Kel99], and
so are Grothendieck groups;

  3. 3.

$\operatorname{HP}^{\ast}(A)=\operatorname{HP}^{\ast}(A/\operatorname{rad}A)$
by Goodwillie’s theorem [Goo85, Theorem II.5.1], and the standard fact that
$K_{0}(A)=K_{0}(A/\operatorname{rad}A)$.

So we conclude

$e(Z)\overset{(1,2)}{=}e(A)\overset{(3)}{=}\operatorname{rk}K_{0}(A)\overset{(2)}{=}\operatorname{rk}K_{0}(Z).\qed$

Let us go back to the setting of Proposition 5.6 but assume now in addition
that $Z$ has a tilting complex. Then by (5.3) combined with Lemma 5.18 we get

$e_{st}(X)=\frac{\operatorname{rk}K_{0}(Z)}{n}.$

Assuming that $R$ has a graded NCCR $\Lambda$ then Conjecture 5.14 implies

$\operatorname{rk}K_{0}(\Lambda)=\frac{\operatorname{rk}K_{0}(Z)}{n}.$ (5.6)

######  Example 5.19.

This formula holds for the NCCRs constructed via Proposition 3.3. Indeed
$\operatorname{rk}K_{0}(\Lambda)$ is given by the number $u$ of non-isomorphic
indecomposable summands of ${\mathcal{E}}$. On the other hand
${\mathcal{D}}(Z)$ has a semi-orthogonal decomposition consisting of $n$ parts
whose $K_{0}$ also has rank $u$. So (5.6) does indeed hold and we obtain again
some evidence for Conjecture 5.14.

######  Remark 5.20.

One way to think of this example as the realization of the (conjectured)
“motivic” identity (5.6) via semi-orthogonal decompositions of derived
categories. See [PVdB19] for another (deeper) instance of this principle.

First and foremost I am grateful to my coauthor and friend Špela Špenko for
contributing much of the mathematics of our joint work. Without her input this
survey would have been a lot shorter.

Furthermore I thank Shinnosuke Okawa for readily answering all my questions
about the minimal model program. Likewise I thank Michael Wemyss for input on
the non-commutative geometry of cDV singularities.

This work was partially supported by the Advanced ERC grant 885203 “Schobers,
Mutations and Stability”.

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