MAXIMAL MODIFICATIONS AND AUSLANDER–REITEN DUALITY
FOR NON-ISOLATED SINGULARITIES.
OSAMU IYAMA AND MICHAEL WEMYSS

arXiv:1007.1296v4 [math.AG] 14 Nov 2013

In memory of Kentaro Nagao
Abstract. We first generalize classical Auslander–Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the
notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated
singularities, CT modules recover the classical notion of cluster tilting modules but
in general the two concepts differ. Then, wanting to generalize the notion of NCCRs
to cover partial resolutions of Spec R, in the main body of this paper we introduce a
theory of modifying and maximal modifying modules. Under mild assumptions all the
corresponding endomorphism algebras of the maximal modifying modules for threedimensional Gorenstein rings are shown to be derived equivalent. We then develop a
theory of mutation for modifying modules which is similar but different to mutations
arising in cluster tilting theory. Our mutation works in arbitrary dimension, and in
dimension three the behavior of our mutation strongly depends on whether a certain
factor algebra is artinian.

Contents
1. Introduction
1.1. Motivation and History
1.2. Auslander–Reiten Duality for Non-Isolated Singularities
1.3. Maximal Modifications and NCCRs
1.4. Derived Equivalences
1.5. Mutation of Modifications
1.6. Conventions
2. Preliminaries
2.1. Depth and CM Modules
2.2. Reflexive Equivalence and Symmetric Algebras
2.3. Non-Singular and Gorenstein Orders
2.4. d-sCY Algebras
2.5. Global–local properties
3. Auslander–Reiten Duality for Non-Isolated Singularities
4. Modifying and Maximal Modifying Modules
4.1. Modifications in Dimension d
4.2. Derived Equivalence in Dimension 3
5. Relationship Between CT modules, NCCRs and MM modules
6. Mutations of Modifying Modules
6.1. Mutations and Derived Equivalences in Dimension d
6.2. Mutations and Derived Equivalences in Dimension 3
6.3. Complete Local Case
References

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The first author was partially supported by JSPS Grant-in-Aid for Scientific Research 24340004,
23540045, 20244001 and 22224001. The second author was partially supported by a JSPS Postdoctoral
Fellowship and by the EPSRC.
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OSAMU IYAMA AND MICHAEL WEMYSS

1. Introduction
1.1. Motivation and History. One of the basic results in representation theory of commutative algebras is Auslander–Reiten (=AR) duality [Aus78, AR, Y90] for isolated singularities, which gives us many important consequences, e.g. the existence of almost split
sequences and the Calabi-Yau property of the stable categories of Cohen–Macaulay (=CM)
modules over Gorenstein isolated singularities. One of the aims of this paper is to establish a version of AR duality for singularities with one dimensional singular loci. As an
application, the stable categories of CM modules over Gorenstein singularities with one
dimensional singular loci enjoy a generalized Calabi-Yau property. This is a starting point
of our research to apply the methods of cluster tilting in representation theory to study
singularities.
One of the highlights of representation theory of commutative algebras is AR theory
of simple surface singularities [Aus86]. They have only finitely many indecomposable CM
modules, and the Auslander algebras (i.e. the endomorphism algebras of the direct sums of
all indecomposable CM modules) enjoy many nice properties. If we consider singularities
of dimension greater than two, then there are very few representation-finite singularities,
and their Auslander algebras do not satisfy such nice properties. The reason is that the
categories of CM modules do not behave nicely in the sense that the homological properties
of simple functors corresponding to free modules are very hard to control. Motivated to
obtain the correct category on which higher AR theory should be performed, in [Iya07] the
first author introduced the notion of a maximal n-orthogonal subcategory and maximal
n-orthogonal module for the category mod Λ, later renamed cluster tilting subcategories
and cluster tilting modules respectively. Just as classical AR theory on mod Λ was moved
to AR theory on CM Λ following the work of Auslander on the McKay correspondence for
surfaces Λ [Aus86], this suggests that in the case of a higher dimensional CM singularity
R we should apply the definition of a maximal n-orthogonal subcategory/modules to
CM R and hope that this provides a suitable framework for tackling higher-dimensional
geometric problems. Strong evidence for this is provided when R is a three dimensional
normal isolated Gorenstein singularity, since in this case it is known [IR08, 8.13] that such
objects have an intimate relationship with Van den Bergh’s noncommutative crepant
resolutions (NCCRs) [V04b]. Requiring R to be isolated is absolutely crucial to this
relationship (by normality the singularities are automatically isolated in the surfaces case);
from an algebraic perspective this perhaps should not be surprising since AR theory only
works well for isolated singularities. It turns out that the study of maximal n-orthogonal
modules in CM R is not well-suited to non-isolated singularities since the Ext vanishing
condition is far too strong; the question arises as to what subcategories of CM R should
play the role of the maximal n-orthogonal subcategories above.
Although in this paper we answer this question, in fact we say much more. Philosophically, the point is that we are asking ourselves the wrong question. The restriction
to studying maximal orthogonal modules is unsatisfactory since crepant resolutions need
not exist (even for 3-folds) and so we develop a theory which can deal with singularities
in the crepant partial resolutions. Since the endomorphism rings of maximal orthogonal
modules have finite global dimension, these will not do the job for us.
We introduce the notion of maximal modifying modules (see 1.12 below) which intuitively we think of as corresponding to shadows of maximal crepant partial resolutions.
Geometrically this level always exists, but only sometimes will it be smooth. With regards
to this viewpoint maximal modifying modules are a more natural class of objects to work
with compared to noncommutative crepant resolutions; we should thus always work in
this level of generality and simply view the case when the geometry is smooth as being
a happy coincidence. Pushing this philosophy further, everything that we are currently
able to do with NCCRs we should be able to do with maximal modifying modules, and
this motivates much of the work in this paper.
In fact in many regards restricting our attention to only studying maximal crepant
partial resolutions misses much of the picture and so we should (and do) work even more
generally. When one wants to flop curves between varieties with canonical singularities
which are not terminal this does not take place on the maximal level but we should still
be able to understand this homologically. This motivates our definition and the study of

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

3

modifying modules (see 1.12 below). Viewing our modifying modules M as conjectural
shadows of partial resolutions we should thus be able to track the birational transformations between the geometrical spaces by using some kind of homological transformation
between the corresponding modifying modules. This leads us to develop a theory of
mutation for modifying modules, which we do in Section 6.
We note that some parts of the theory of (maximal) modifying modules developed
in this paper are analogues of cluster tilting theory [GLS, IR08], especially when the ring
has Krull dimension three. One main difference is that we do not assume that the ring is
an isolated singularity, so we need to introduce (maximal) modifying modules which are
much more general than (maximal) rigid modules in cluster tilting theory. Some of the
main properties in cluster tilting theory are still true in our setting, for example, mutation
is an involution in dimension three (see 1.25 below) and gives a derived equivalence in any
dimension (see 1.23 below). On the other hand, new features also appear in our setting.
For example, mutation sometimes does not change the given modifying modules (see 1.25
below). This feature is necessary, since it exactly reflects the geometry of partial crepant
resolutions.
Although in this paper we are principally interested in the geometrical and commutative algebraic statements, the proofs of our theorems require a slightly more general
noncommutative setting. For this reason throughout this paper we use the language of
singular Calabi–Yau algebras:
Definition 1.1. Let Λ be a module finite R-algebra, then for d ∈ Z we call Λ d-Calabi–Yau
(=d-CY) if there is a functorial isomorphism
HomD(Mod Λ) (X, Y [d]) ∼
= D0 HomD(Mod Λ) (Y, X)
for all X ∈ Db (fl Λ), Y ∈ Db (mod Λ), where D0 is the Matlis dual (see §2.4 for more
details). Similarly we call Λ singular d-Calabi–Yau (=d-sCY) if the above functorial
isomorphism holds for all X ∈ Db (fl Λ) and Y ∈ Kb (proj Λ).
Clearly d-sCY (respectively d-CY) algebras are closed under derived equivalence
[IR08, 3.1(1)]. When Λ = R, it is known (see 2.20) that R is d-sCY if and only if R
is Gorenstein and equi-codimensional with dim R = d. Thus throughout this paper, we
use the phrase ‘R is d-sCY’ as a convenient shorthand for this important property.
We remark that by passing to mildly noncommutative d-sCY algebras we increase the
technical difficulty, but we emphasize that we are forced to do this since we are unable to
prove the purely commutative statements without passing to the noncommutative setting.
We now describe our results rigorously, and in more detail.
1.2. Auslander–Reiten Duality for Non-Isolated Singularities. Throughout this
subsection let R be an equi-codimensional (see 1.3 below) CM ring of dimension d with a
canonical module ωR . Recall that for a non-local CM ring R, a finitely generated R-module
ωR is called a canonical module if (ωR )m is a canonical Rm -module for all m ∈ Max R [BH,
3.3.16]. In this case (ωR )p is a canonical Rp -module for all p ∈ Spec R since canonical
modules localize for local CM rings [BH, 3.3.5].
We denote CM R to be the category of CM R-modules (see §2.1), CMR to be the
stable category and CMR to be the costable category. The AR translation is defined to be
τ := HomR (Ωd Tr(−), ωR ) : CMR → CMR.
When R is an isolated singularity one of the fundamental properties of the category CM R
is the existence of Auslander–Reiten duality [Aus78, I.8.8] [AR, 1.1(b)], namely
HomR (X, Y ) ∼
= D0 Ext1R (Y, τ X)
for all X, Y ∈ CM R where D0 is the Matlis dual (see §3). Denoting D1 := Extd−1
R (−, ωR )
to be the duality on the category of Cohen–Macaulay modules of dimension 1, we show
that AR duality generalizes to mildly non-isolated singularities as follows:
Theorem 1.2 (=3.1). Let R be a d-dimensional, equi-codimensional CM ring with a
canonical module ωR and singular locus of Krull dimension less than or equal to one.

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Then there exist functorial isomorphisms
fl HomR (X, Y ) ∼
= D0 (fl Ext1R (Y, τ X)),


Ext1R (Y, τ X)
HomR (X, ΩY ) ∼
= D1
fl HomR (X, ΩY )
fl Ext1R (Y, τ X)
for all X, Y ∈ CM R, where for an R-module M we denote fl M to be the largest finite
length R-submodule of M .
In fact we prove 3.1 in the setting of noncommutative R-orders (see §3 for precise
details). In the above and throughout this paper, for many of the global-local arguments
to work we often have to add the following mild technical assumption.
Definition 1.3. A commutative ring R is equi-codimensional if all its maximal ideals
have the same height.
Although technical, such rings are very common; for example all domains finitely
generated over a field are equi-codimensional [Ei95, 13.4]. Since our main applications
are three-dimensional normal domains finitely generated over C, in practice this adds no
restrictions to what we want to do. We will use the following well-known property [Mat,
17.3(i), 17.4(i)(ii)]:
Lemma 1.4. Let R be an equi-codimensional CM ring, and let p ∈ Spec R. Then
ht p + dim R/p = dim R.
The above generalized Auslander–Reiten duality implies the following generalized
(d − 1)-Calabi-Yau property of the triangulated category CMR.
Corollary 1.5 (=3.7). Let R be a d-sCY ring with dim Sing R ≤ 1. Then
(1) There exist functorial isomorphisms
fl HomR (X, Y ) ∼
= D0 (fl HomR (Y, X[d − 1])),


HomR (Y, X[d − 2])
HomR (X, Y ) ∼
D
= 1
fl HomR (X, Y )
fl HomR (Y, X[d − 2])
for all X, Y ∈ CM R.
(2) (=4.4) For all X, Y ∈ CM R, HomR (X, Y ) ∈ CM R if and only if HomR (Y, X) ∈
CM R.
Note that 1.5(2) also holds (with no assumptions on the singular locus) provided that
R is normal (see 2.9). This symmetry in the Hom groups gives us the technical tool we
require to move successfully from the cluster tilting level to the maximal modification
level below, and is entirely analogous to the symmetry given by [CB, Lemma 1] as used
in cluster theory (e.g. [GLS]).
1.3. Maximal Modifications and NCCRs. Here we introduce our main definitions,
namely modifying, maximal modifying and CT modules, and then survey our main results.
Throughout, an R-algebra is called module finite if it is a finitely generated R-module.
As usual, we denote (−)p := − ⊗R Rp to be the localization functor. For an R-algebra
Λ, clearly Λp is an Rp -algebra and we have a functor (−)p : mod Λ → mod Λp . Recall
[Aus78, Aus84, CR90]:
Definition 1.6. Let R be a CM ring and let Λ be a module finite R-algebra. We say
(1) Λ is an R-order if Λ ∈ CM R.
(2) An R-order Λ is non-singular if gl.dim Λp = dim Rp for all primes p of R.
(3) An R-order Λ has isolated singularities if Λp is a non-singular Rp -order for all nonmaximal primes p of R.
In the definition of non-singular R-order above, gl.dim Λp = dim Rp means that
gl.dim Λp takes the smallest possible value. In fact for an R-order Λ we always have
that gl.dim Λp ≥ dim Rp := tp for all primes p of R since proj.dimΛp Λp /(x1 , ..., xtp )Λp =
dim Rp for a Λp -regular sequence x1 , . . . , xtp . We also remark that since the localization
functor is exact and dense, we always have gl.dim Λp ≤ gl.dim Λ for all p ∈ Spec R.

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

5

Throughout this paper we denote
(−)∗ := HomR (−, R) : mod R → mod R
and we say that X ∈ mod R is reflexive if the natural map X → X ∗∗ is an isomorphism.
We denote ref R to be the category of reflexive R-modules. By using Serre’s (S2 )-condition
(see for example [EG85, 3.6], [BH, 1.4.1(b)]), when R is a normal domain the category
ref R is closed under both kernels and extensions.
Definition 1.7. Let R be CM, then by a noncommutative crepant resolution (NCCR) of
R we mean Γ := EndR (M ) where M ∈ ref R is non-zero such that Γ is a non-singular
R-order.
We show in 2.17 that under very mild assumptions the condition in 1.6(2) can in
fact be checked at only maximal ideals, and we show in 2.23 that 1.7 is equivalent to the
definition of NCCR due to Van den Bergh [V04b] when R is a Gorenstein normal domain.
Recall the following:
Definition 1.8. Let A be a ring. We say that an A-module M is a generator if A ∈
add M . A projective A-module M which is a generator is called a progenerator.
Motivated by wanting a characterization of the reflexive generators which give NCCRs, we define:
Definition 1.9. Let R be a d-dimensional CM ring with a canonical module ωR . We call
M ∈ CM R a CT module if
add M = {X ∈ CM R : HomR (M, X) ∈ CM R} = {X ∈ CM R : HomR (X, M ) ∈ CM R}.
Clearly a CT module M is always a generator and cogenerator (i.e. add M contains
both R and ωR ). We show in 5.12 that this recovers the established notion of maximal
(d − 2)-orthogonal modules when R is d-dimensional and has isolated singularities. The
following result in the complete case is shown in [Iya07, 2.5] under the assumption that
G is a small subgroup of GL(d, k), and S G is an isolated singularity. We can drop all
assumptions under our definition of CT modules:
Theorem 1.10 (=5.7). Let k be a field of characteristic zero, and let S be the polynomial ring k[x1 , . . . , xd ] (respectively formal power series ring k[[x1 , . . . , xd ]]). For a finite
subgroup G of GL(d, k), let R = S G . Then S is a CT R-module.
One of our main results involving CT modules is the following, where part (2) answers
a question of Van den Bergh posed in [V04b, 4.4].
Theorem 1.11 (=5.9). Let R be a normal 3-sCY ring. Then
(1) CT modules are precisely those reflexive generators which give NCCRs.
(2) R has a NCCR ⇐⇒ R has a NCCR given by a CM generator M ⇐⇒ R has a CT
module.
However in many cases R need not have a NCCR so we must weaken the notion of
CT module and allow for our endomorphism rings to have infinite global dimension. We
do so as follows:
Definition 1.12. Let R be a d-dimensional CM ring. We call N ∈ ref R a modifying
module if EndR (N ) ∈ CM R, whereas we call N a maximal modifying (MM) module if
N is modifying and furthermore it is maximal with respect to this property, that is to say
if there exists X ∈ ref R with N ⊕ X modifying, necessarily X ∈ add N . Equivalently, we
say N is maximal modifying if
add N = {X ∈ ref R : HomR (N ⊕ X, N ⊕ X) ∈ CM R}.
If N is an MM module (respectively modifying module), we call EndR (N ) a maximal
modification algebra (=MMA) (respectively modification algebra).
In this paper we will mainly be interested in the theoretical aspects of MMAs, but
there are many natural examples. In fact NCCRs are always MMAs (see 4.5) and so this
gives one rich source of examples. However MMAs need not be NCCRs, and for examples

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OSAMU IYAMA AND MICHAEL WEMYSS

of this type of behaviour, together with the links to the geometry, we refer the reader to
[IW11].
When R is d-dimensional with isolated singularities we show in 5.12 that modifying
modules recover the established notion of (d − 2)-rigid modules, whereas MM modules
recover the notion of maximal (d − 2)-rigid modules. However, other than pointing out
this relationship, throughout we never assume that R has isolated singularities.
When an NCCR exists, we show that MMAs are exactly the same as NCCRs:
Proposition 1.13 (=5.11). Let R be a normal 3-sCY ring, and assume that R has a
NCCR (equivalently, by 1.11, a CT module). Then
(1) MM modules are precisely the reflexive modules which give NCCRs.
(2) MM modules which are CM (equivalently, by 4.2, the MM generators) are precisely
the CT modules.
(3) CT modules are precisely those CM modules which give NCCRs.
The point is that R need not have a NCCR, and our definition of maximal modification algebra is strictly more general.
1.4. Derived Equivalences. We now explain some of our results involving derived equivalences of modifying modules. We say that two module finite R-algebras A and B are
derived equivalent if D(Mod A) ≃ D(Mod B) as triangulated categories, or equivalently
Db (mod A) ≃ Db (mod B) by adapting [R89, 8.1, 8.2]. First, we show that any algebra
derived equivalent to a modifying algebra also has the form EndR (M ).
Theorem 1.14 (=4.6). Let R be a normal d-sCY ring, then
(1) Modifying algebras of R are closed under derived equivalences, i.e. any ring derived
equivalent to a modifying algebra is isomorphic to a modifying algebra.
(2) NCCRs of R are closed under derived equivalences.
The corresponding statement for MMAs is slightly more subtle, but we show it is
true in dimension three (1.17), and also slightly more generally in 4.8(2).
Throughout this paper we freely use the notion of a tilting module which we always
assume has projective dimension less than or equal to one:
Definition 1.15. Let Λ be a ring. Then T ∈ mod Λ is called a partial tilting module if
proj.dimΛ T ≤ 1 and Ext1Λ (T, T ) = 0. If further there exists an exact sequence
0 → Λ → T0 → T1 → 0
with each Ti ∈ add T , we say that T is a tilting module.
Our next result details the relationship between modifying and maximal modifying
modules on the level of derived categories.
Theorem 1.16 (=4.15). Let R be a normal 3-sCY ring with MM module M . Then
(1) If N is any modifying module, then T := HomR (M, N ) is a partial tilting EndR (M )module that induces a recollement [BBD, §1.4]
K

D(Mod EndR (M ))

F

D(Mod EndR (N ))

where F = RHom(T, −) and K is a certain triangulated subcategory of D(Mod EndR (M )).
(2) If further N is maximal modifying then the above functor F is an equivalence.
Theorems 1.14 and 1.16 now give the following, which we view as the noncommutative
analogue of a result of Chen [C02, 1.1].
Corollary 1.17 (=4.16). Let R be a normal 3-sCY ring with an MM module. Then all
MMAs are derived equivalent, and further any algebra derived equivalent to an MMA is
also an MMA.
In our study of modifying modules, reflexive modules over noncommutative R-algebras
play a crucial role.
Definition 1.18. Let R be any commutative ring. If A is any R-algebra then we say that
M ∈ mod A is a reflexive A-module if it is reflexive as an R-module.

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

7

Note that we do not require that the natural map M → HomAop (HomA (M, A), A) is
an isomorphism. However when A is 3-sCY and M is a reflexive A-module in the sense of
1.18, automatically it is. Our main theorem regarding maximal modifying modules is the
following remarkable relationship between modifying modules and tilting modules. Note
that (3) below says that R has a maximal modification algebra if and only if it has a
maximal modification algebra EndR (N ) where N is a CM generator, a generalization of
1.11(2).
Theorem 1.19 (=4.17, 4.18). Let R be a normal 3-sCY ring with an MM module M .
Then
(1) The functor HomR (M, −) : mod R → mod EndR (M ) induces bijections
1:1

{maximal modifying R-modules}
{modifying R-modules}

←→ {reflexive tilting EndR (M )-modules}.
1:1
←→ {reflexive partial tilting EndR (M )-modules}.

(2) N is modifying ⇐⇒ N is a direct summand of a maximal modifying module.
(3) R has an MM module which is a CM generator.
1.5. Mutation of Modifications. Recall:
Definition 1.20. Let Λ be a ring. For Λ-modules M and N , we say that a morphism
f : N0 → M is a right (add N )-approximation if N0 ∈ add N and further
·f

HomΛ (N, N0 ) → HomΛ (N, M )
is surjective. Dually we define a left (add N )-approximation.
Now let R be a normal d-sCY ring. We introduce categorical mutations as a method
of producing modifying modules (together with a derived equivalence) from a given one.
For a given modifying R-module M , and N such that 0 6= N ∈ add M we consider
a
(1) a right (add N )-approximation of M , denoted N0 → M .
b

(2) a right (add N ∗ )-approximation of M ∗ , denoted N1∗ → M ∗ .
Note that the above a and b are surjective if N is a generator. In what follows we denote
the kernels by
c

a

0 → K 0 → N0 → M

d

b

and 0 → K1 → N1∗ → M ∗

and call these exchange sequences.
Definition 1.21. With notation as above, we define the right mutation of M at N to be
−
∗
µ+
N (M ) := N ⊕ K0 and we define the left mutation of M at N to be µN (M ) := N ⊕ K1 .
+
∗ ∗
Note that by definition µ−
N (M ) = (µN ∗ (M )) .
−
Theorem 1.22 (=6.10, 6.5). (1) Both µ+
N (M ) and µN (M ) are modifying R-modules.
+
−
+
(2) µN and µN are mutually inverse operations, i.e. we have that µ−
N (µN (M )) = M and
−
µ+
N (µN (M )) = M , up to additive closure.

What is remarkable is that this process always produces derived equivalences, even
in dimension d:
Theorem 1.23 (=6.8, 6.10). Let R be a normal d-sCY ring with modifying module M .
Suppose that 0 6= N ∈ add M . Then
+
(1) EndR (M ), EndR (µ−
N (M )) and EndR (µN (M )) are all derived equivalent.
+
(2) If M gives an NCCR, so do µN (M ) and µ−
N (M ).
−
(3) Whenever N is a generator, if M is a CT module so are µ+
N (M ) and µN (M ).
(4) Whenever dim Sing R ≤ 1 (e.g. if d = 3), if M is a MM module so are µ+
N (M ) and
µ−
(M
).
N
In particular the above allows us to mutate any NCCR, in any dimension, at any
direct summand, and will give another NCCR together with a derived equivalence. In
particular, we can do this when the ring R is not complete local, and also we can do
this when the NCCR may be given by a quiver with relations where the quiver has both
loops and 2-cycles, in contrast to cluster theory. This situation happens very frequently in

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OSAMU IYAMA AND MICHAEL WEMYSS

the study of one-dimensional fibres, where this form of mutation seems to have geometric
consequences.
One further corollary in full generality is the following surprising result on syzygies
Ω and cosyzygies Ω−1 , since they are a special case of left and right mutation. Note that
we have to be careful when defining our syzygies and cosyzygies so that they have free
summands; see §6.1 for more details.
Corollary 1.24 (=6.11). Suppose that R is a normal d-sCY ring and M ∈ ref R is a
modifying generator. Then
(1) Ωi M ∈ CM R is a modifying generator for all i ∈ Z, and further all EndR (Ωi M ) are
derived equivalent.
(2) If M is CT (i.e. gives an NCCR), then all Ωi M ∈ CM R are CT, and further all
EndR (Ωi M ) are derived equivalent.
We remark that when dim R = 3, in nice situations we can calculate the mutated algebra from the original algebra by using various combinatorial procedures (see e.g. [BIRS,
5.1], [KY11, 3.2] and [V09, 3.5]), but we note that our mutation is categorical and much
more general, and expecting a combinatorial rule is unfortunately too optimistic a hope.
We also remark that since we are dealing with algebras that have infinite global dimension,
there is no reason to expect that they possess superpotentials and so explicitly describing
their relations is in general a very difficult problem.
When dim R = 3 and R is complete local, we can improve the above results. In this
setting, under fairly weak assumptions it turns out that left mutation is the same as right
mutation, as in the case of cluster theory [IY08]. If 0 6= N ∈ add M then we define [N ]
to be the two-sided ideal of EndR (M ) consisting of morphisms M → M which factor
through a member of add N , and denote ΛN := Λ/[N ]. The behaviour of mutation at N
is controlled by ΛN , in particular whether or not ΛN is artinian. Note that when R is
finitely generated over a field k, ΛN is artinian if and only if it is finite dimensional over
k (see 6.15).
For maximal modifying modules the mutation picture is remarkably clear, provided
that we mutate at only one indecomposable summand at a time:
Theorem 1.25 (=6.25). Suppose R is complete normal 3-sCY with MM module M .
Denote Λ = EndR (M ), let Mi be an indecomposable summand of M and consider Λi :=
Λ/Λ(1 − ei )Λ where ei is the idempotent in Λ corresponding to Mi . To ease notation
+
−
−
denote µ+
i = µ M and µi = µ M . Then
Mi

Mi

−
(1) If Λi is not artinian then µ+
i (M ) = M = µi (M ).
−
+
(2) If Λi is artinian then µi (M ) = µi (M ) and this is not equal to M .
−
In either case denote µi := µ+
i = µi then it is also true that
(3) µi µi (M ) = M .
(4) µi (M ) is a MM module.
(5) EndR (M ) and EndR (µi (M )) are derived equivalent, via the tilting EndR (M )-module
HomR (M, µi (M )).

Some of the above proof works in greater generality, but we suppress the details here.
1.6. Conventions. We now state our conventions. All modules will be left modules, so
for a ring A we denote mod A to be the category of finitely generated left A-modules.
Throughout when composing maps f g will mean f then g, similarly for quivers ab will
mean a then b. Note that with this convention HomR (M, X) is a EndR (M )-module and
HomR (X, M ) is a EndR (M )op -module. For M ∈ mod A we denote add M to be the full
subcategory consisting of summands of finite direct sums of copies of M and we denote
proj A := add A to be the category of finitely generated projective A-modules. Throughout
we will always use the letter R to denote some kind of commutative noetherian ring. We
always strive to work in the global setting, so we write (R, m) if R is local. We use the
cp to denote the completion of the localization Rp at its unique maximal ideal.
notation R
2. Preliminaries

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

9

2.1. Depth and CM Modules. Here we record the preliminaries we shall need in subsequent sections, especially some global-local arguments that will be used extensively. For
a commutative noetherian local ring (R, m) and M ∈ mod R recall that the depth of M
is defined to be
depthR M := inf{i ≥ 0 : ExtiR (R/m, M ) 6= 0},
which coincides with the maximal length of a M -regular sequence. Keeping the assumption that (R, m) is local we say that M ∈ mod R is maximal Cohen-Macaulay (or simply,
CM ) if depthR M = dim R. This definition generalizes to the non-local case as follows: if
R is an arbitrary commutative noetherian ring we say that M ∈ mod R is CM if Mp is
CM for all prime ideals p in R, and we say that R is a CM ring if R is a CM R-module.
It is often convenient to lift the CM property to noncommutative rings, which we do
as follows:
Definition 2.1. Let Λ be a module finite R-algebra, then we call M ∈ mod Λ a CM Λmodule if it is CM when viewed as an R-module. We denote the category of CM Λ-modules
by CM Λ.
To enable us to bring the concept of positive depth to non-local rings, the following
is convenient:
Definition 2.2. Let R be a commutative noetherian ring and M ∈ mod R. We denote
fl M to be the unique maximal finite length R-submodule of M .
It is clear that fl M exists because of the noetherian property of M ; when (R, m) is
local fl M = {x ∈ M : ∃ r ∈ N with mr x = 0}. The following is well-known.
Lemma 2.3. Let (R, m) be a local ring of dimension d ≥ 2 and let Λ be a module finite Ralgebra. Then for all M, N ∈ mod Λ with depthR N ≥ 2 we have depthR HomΛ (M, N ) ≥
2.
Proof. A free presentation Λa → Λb → M → 0 gives 0 → HomΛ (M, N ) → N b → N a so
the result follows from the depth lemma.

In particular if depth R ≥ 2 then reflexive R-modules always have depth at least two.
Lemma 2.4. Suppose R is a d-dimensional CM ring with d ≥ 2 and let Λ be a module
finite R-algebra. For any X ∈ ref Λ we have Xp ∈ CM Λp for all p ∈ Spec R with ht p ≤ 2.
Proof. Since X is reflexive as an R-module we can find an exact sequence 0 → X → P → Q
with P, Q ∈ add R and so on localizing we see that Xp is a second syzygy for all primes p.
Consequently if p has height ≤ 2 then Xp is a second syzygy for the CM ring Rp which
has dim Rp ≤ 2 and so Xp ∈ CM Rp .

2.2. Reflexive Equivalence and Symmetric Algebras. Here we introduce and fix
notation for reflexive modules and symmetric algebras. All the material in this subsection
can be found in [IR08]. Recall from the introduction (1.18) our convention on the definition
of reflexive modules. Recall also that if Λ is a module finite R-algebra, we say M ∈
ref Λ is called a height one progenerator (respectively, height one projective) if Mp is a
progenerator (respectively, projective) over Λp for all p ∈ Spec R with ht p ≤ 1.
In this paper, when the underlying commutative ring R is a normal domain the
following reflexive equivalence is crucial:
Lemma 2.5. If Λ is a module finite R-algebra, then
(1) If M ∈ mod Λ is a generator then
HomΛ (M, −) : mod Λ → mod EndΛ (M )
≃

is fully faithful, restricting to an equivalence add M → proj EndΛ (M ).
If further R is a normal domain, then the following assertions hold.
(2) HomΛ (X, Y ) ∈ ref R for any X ∈ mod Λ and any Y ∈ ref Λ.
(3) Every non-zero M ∈ ref R is a height one progenerator.
(4) Suppose Λ is a reflexive R-module and let M ∈ ref Λ be a height one progenerator.
Then
HomΛ (M, −) : ref Λ → ref EndΛ (M )

10

OSAMU IYAMA AND MICHAEL WEMYSS

is an equivalence. In particular HomR (N, −) : ref R → ref EndR (N ) is an equivalence for
all non-zero N ∈ ref R.
Proof. (1) is standard.
(2) follows easily from the fact that reflexives are closed under kernels; see [IR08, 2.4(1)].
(3) If p is a height one prime then by 2.4 Mp ∈ CM Rp . But R is normal so Rp is regular;
thus Mp is free.

(4) follows by (3) and [RV89, 1.2] (see also [IR08, 2.4(2)(i)]).
Throughout this paper, we will often use the following observation.
Lemma 2.6. Let R be a CM ring, X, Y ∈ CM R. Then SuppR ExtiR (X, Y ) ⊆ Sing R for
all i > 0. In particular, if R has isolated singularities then ExtiR (X, Y ) is a finite length
R-module for all X, Y ∈ CM R and i > 0.
Proof. This is well-known [Aus78], [Y90, 3.3].



The following lemma is convenient and will be used extensively.
Lemma 2.7. Let R be a 3-dimensional, equi-codimensional CM ring and let Λ be a module
finite R-algebra. If X ∈ mod Λ and Y ∈ ref Λ then
HomΛ (X, Y ) ∈ CM R ⇒ fl Ext1Λ (X, Y ) = 0.
If further Y ∈ CM Λ then the converse holds.
Proof. (⇒) For each m ∈ Max R there is an exact sequence
fm

0 → HomΛ (X, Y )m → HomΛ (P, Y )m → HomΛ (ΩX, Y )m → Ext1Λ (X, Y )m → 0

(2.A)

obtained by localizing the exact sequence obtained from 0 → ΩX → P → X → 0 with P ∈
add Λ. Now depthRm HomΛm (Xm , Ym ) = 3 and further by 2.3 depthRm HomΛm (Pm , Ym ) ≥
2, thus depthRm Cok fm ≥ 2. Since again by 2.3 depthRm HomΛm (ΩXm , Ym ) ≥ 2, we
conclude that depthRm Ext1Λm (Xm , Ym ) > 0 for all m ∈ Max R and so fl Ext1Λ (X, Y ) = 0.
(⇐) Suppose now that Y ∈ CM Λ. Then in (2.A) depthRm HomΛm (Pm , Ym ) = 3 and so by
a similar argument depthRm Ext1Λm (Xm , Ym ) > 0 implies that depthRm HomΛm (Xm , Ym ) =
3.

Definition 2.8. Let Λ be a module finite R-algebra where R is an arbitrary commutative
ring. We call Λ a symmetric R-algebra if HomR (Λ, R) ∼
= Λ as Λ-bimodules. We call Λ a
locally symmetric R-algebra if Λp is a symmetric Rp -algebra for all p ∈ Spec R.
Note that if Λ is a symmetric R-algebra then as functors mod Λ → mod Λop
HomΛ (−, Λ) ∼
= HomΛ (−, HomR (Λ, R)) ∼
= HomR (Λ ⊗Λ −, R) = HomR (−, R).
We have the following well-known observation. Recall that throughout our paper, (−)∗ :=
HomR (−, R).
Lemma 2.9. Let R be a normal domain and Λ be a symmetric R-algebra. Then there is
a functorial isomorphism HomΛ (X, Y ) ∼
= HomΛ (Y, X)∗ for all X, Y ∈ ref Λ such that Y
is height one projective.
Proof. For the convenience of the reader, we give a detailed proof here. We have a natural
map f : HomΛ (Y, Λ) ⊗Λ X → HomΛ (Y, X) sending a ⊗ x to (y 7→ a(y)x). Consider the
map f ∗ : HomΛ (Y, X)∗ → (HomΛ (Y, Λ) ⊗Λ X)∗ between reflexive R-modules. Since Y is
height one projective, fp and (f ∗ )p are isomorphisms for any prime p of height at most
one. Thus f ∗ is an isomorphism since R is normal. Thus we have
f∗

HomΛ (Y, X)∗ ∼
= HomΛ (X, Y ∗∗ ) ∼
= HomΛ (X, Y )
= (Y ∗ ⊗Λ X)∗ ∼
= (HomΛ (Y, Λ) ⊗Λ X)∗ ∼
as required.



This immediately gives the following result, which implies that symmetric algebras
are closed under reflexive equivalence.

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

11

Lemma 2.10. [IR08, 2.4(3)] If Λ is a symmetric R-algebra then so is EndΛ (M ) for any
height one projective M ∈ ref Λ. In particular if R is a normal domain and N ∈ ref R
then EndR (N ) is a symmetric R-algebra.
When discussing derived equivalence of modifying algebras later, we will require the
following result due to Auslander–Goldman.
Proposition 2.11 (cf. [AG60, 4.2]). Let R be a normal domain with dim R ≥ 1, and let
Λ be a module finite R-algebra. Then the following conditions are equivalent:
(1) There exists M ∈ ref R such that Λ ∼
= EndR (M ) as R-algebras.
(2) Λ ∈ ref R and further Λp is Morita equivalent to Rp for all p ∈ Spec R with ht p = 1.
Proof. (1)⇒(2) is trivial.
(2)⇒(1) For the convenience of the reader, we give a simple direct proof. Let K be the
quotient field of R. As we have assumed that Λp is Morita equivalent to Rp , K ⊗R Λ ∼
=
Mn (K) as K-algebras for some n > 0. Thus for any M ∈ mod Λ, we can regard K ⊗R M
as an Mn (K)-module. We denote by V the simple Mn (K)-module.
First, we show that there exists M ∈ ref Λ such that K ⊗R M ≃ V as Mn (K)modules. For example, take a split epimorphism f : Mn (K) → V of Mn (K)-modules and
let M := (f (Λ))∗∗ . Then clearly M satisfies the desired properties.
Next, for M above, we show that the natural map g : Λ → EndR (M ) given by the
action of Λ on M is an isomorphism. Since both Λ and EndR (M ) are reflexive R-modules
by our assumption, it is enough to show that gp is an isomorphism for all p ∈ Spec R with
ht p = 1. Since K ⊗R EndR (M ) = EndK (K ⊗R M ) = EndK (V ) = Mn (K), we have that
K ⊗g : K ⊗R Λ → K ⊗R EndR (M ) is an isomorphism. In particular, gp : Λp → EndR (M )p
is injective. Since Rp is local and Λp is Morita equivalent to Rp , we have that Λp is a full
matrix ring over Rp , which is well-known to be a maximal order over Rp (see e.g. [AG60,
3.6], [CR90, §37]). Thus we have that gp is an isomorphism.

2.3. Non-Singular and Gorenstein Orders. Recall from 1.6 the definition of a nonsingular R-order. By definition the localization of a non-singular R-order is again a
non-singular R-order — we shall see in 2.17 that in most situations we may check whether
an algebra is a non-singular R-order by checking only at the maximal ideals.
For some examples of non-singular R-orders, recall that for a ring Λ and a finite group
G together with a group homomorphism G → Autk−alg (Λ), we define the skew group ring
Λ#G as follows [Aus86, Y90]: As a set, it is a free Λ-module with the basis G. The
multiplication is given by
(sg)(s′ g ′ ) := (s · g(s′ ))(gg ′ )
for any s, s′ ∈ S and g, g ′ ∈ G.
Lemma 2.12. Let R be a CM ring containing a field k. Let Λ be a non-singular R-order,
let G be a finite group together with a group homomorphism G → AutR-alg (Λ) and suppose
|G| 6= 0 in k. Then Λ#G is a non-singular R-order.
Proof. Since Λ#G is a direct sum of copies of Λ as an R-module and Λ ∈ CM R, we have
Λ#G ∈ CM R. Now if X, Y ∈ mod Λ#G then G acts on HomΛ (X, Y ) by
(gf )(x) := g · f (g −1 x)
for all g ∈ G, f ∈ HomΛ (X, Y ) and x ∈ X. Clearly we have a functorial isomorphism
HomΛ#G (X, Y ) = HomΛ (X, Y )G
for all X, Y ∈ mod Λ#G. Taking G-invariants (−)G is an exact functor since kG is
semisimple. Thus we have a functorial isomorphism
ExtiΛ#G (X, Y ) = ExtiΛ (X, Y )G
for all X, Y ∈ mod Λ#G and i ≥ 0. In particular, gl.dim Λ#G ≤ gl.dim Λ holds, and we
have the assertion.

In the remainder of this subsection we give basic properties of non-singular orders.
Lemma 2.13. Non-singular R-orders are closed under Morita equivalence.

12

OSAMU IYAMA AND MICHAEL WEMYSS

Proof. Suppose that Λ is a non-singular R-order and Γ is Morita equivalent to Λ. Then
Λp is Morita equivalent to Γp for all primes p. Thus since global dimension is an invariant
of the abelian category, we have gl.dim Γp = gl.dim Λp = dim Rp for all primes p. To see
why the CM property passes across the Morita equivalence let P denote the progenerator
in mod Λ such that Γ ∼
= EndΛ (P ). Since P is a summand of Λn for some n we know that
Γ is a summand of EndΛ (Λn ) = Mn (Λ) as an R-module. Since Λ is CM, so is Γ.

Recall from the introduction (§1.2) the definition of a canonical module ωR for a
non-local CM ring R. If Λ is an R-order we have an exact duality
HomR (−, ωR ) : CM Λ ↔ CM Λop
and so the Λ-module ωΛ := HomR (Λ, ωR ) is an injective cogenerator in the category
CM Λ.
Definition 2.14. [CR90, GN02] Assume R has a canonical module ωR . An R-order Λ
is called Gorenstein if ωΛ is a projective Λ-module.
It is clear that if Λ is a Gorenstein R-order then Λp is a Gorenstein Rp -order for all
p ∈ Spec R. If R is Gorenstein and Λ is a symmetric R-order (i.e. an R-order which is a
symmetric R-algebra), then Λ is clearly a Gorenstein R-order. Moreover if both R and Λ
are d-sCY (see §2.4), we shall see in 2.21 that Λ is a Gorenstein order. Also, we have the
following.
Lemma 2.15. Let Λ be an R-order, then the following are equivalent.
(1) Λ is Gorenstein.
(2) addΛ (Λ) = addΛ (ωΛ ).
(3) addΛop (Λ) = addΛop (ωΛ ).
(4) Λop is Gorenstein.
Proof. Due to 2.26, we can assume that R is complete local, and so mod Λ is Krull–
Schmidt [CR90, 6.12].
(1)⇒(2) If Λ is Gorenstein then by definition addΛ (ωΛ ) ⊆ addΛ (Λ). The number of
non-isomorphic indecomposable projective Λ-modules is equal to that of Λop -modules.
Moreover, the latter is equal to the number of non-isomorphic indecomposable summands
of ωΛ by the duality HomR (−, ωR ). By the Krull–Schmidt property, addΛ (Λ) ⊆ addΛ (ωΛ ).
(2)⇒(1) is trivial.
(2)⇔(3) follows by applying the duality HomR (−, ωR ).
(3)⇔(4) is identical to (1)⇔(2).

When R is local, Gorenstein R-orders Λ are especially important since we have the following Auslander–Buchsbaum type equality, which in particular says that the Λ-modules
which have finite projective dimension and are CM as R-modules are precisely the projective Λ-modules.
Lemma 2.16. Let (R, m) be a local CM ring with a canonical module ωR and let Λ be a
Gorenstein R-order. Then for any X ∈ mod Λ with proj.dimΛ X < ∞,
depthR X + proj.dimΛ X = dim R.
Proof. Let X be a Λ-module with proj.dimΛ X < ∞.
(i) We will show that if X ∈ CM Λ then X is projective. We know that ExtiΛ (−, ωΛ ) = 0
on CM Λ for all i > 0 since ωΛ is injective in CM Λ. Since Λ is Gorenstein add Λ = add ωΛ
by 2.15 and so we have ExtiΛ (−, Λ) = 0 on CM Λ for all i > 0. Since ExtnΛ (X, Λ) 6= 0 for
n = proj.dim X, we have that X is projective.
(ii) Let n = proj.dim X and t = depth X. Take a minimal projective resolution
0 → Pn → . . . → P0 → X → 0.
By the depth lemma necessarily t ≥ d−n. On the other hand by the depth lemma we have
Ωd−t X ∈ CM Λ. By (i) we know Ωd−t X is projective so n ≤ d − t. Thus d = n + t.

The following result is well-known to experts (e.g. [Aus84, 1.5]).

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

13

Proposition 2.17. Let Λ be an R-order where R is a CM ring with a canonical module
ωR . Then the following are equivalent:
(1) Λ is non-singular.
(2) gl.dim Λm = dim Rm for all m ∈ Max R.
(3) CM Λ = proj Λ.
(4) Λ is Gorenstein and gl.dim Λ < ∞.
Proof. (1)⇒(2) This is immediate.
(2)⇒(3) Let X ∈ CM Λ. Then Xm ∈ CM Λm . Let x1 , . . . , xd be a Xm -regular sequence
with d = dim Rm . Since we have an exact sequence
x

1
Xm → Xm /x1 Xm → 0
0 → Xm −→

which induces an exact sequence
x

1
i+1
ExtiΛm (Xm , −) → Exti+1
ExtiΛm (Xm , −) −→
Λm (Xm /x1 Xm , −) → ExtΛm (Xm , −),

we have proj.dimΛm (Xm /x1 Xm ) = proj.dimΛm Xm + 1 by Nakayama’s Lemma. Using
this repeatedly, we have proj.dimΛm (Xm /(x1 , . . . , xd )Xm ) = proj.dimΛm Xm + d. Since
gl.dim Λm = d, this implies that Xm is a projective Λm -module. Since this holds for all
m ∈ Max R, X is a projective Λ-module (see e.g. 2.26).
(3)⇒(4) We have ωΛ ∈ CM Λ = proj Λ. Since Ωdim R X ∈ CM Λ = proj Λ for any X ∈
mod Λ, we have gl.dim Λ ≤ dim R.
(4)⇒(1) Pick p ∈ Spec R and suppose Y ∈ mod Λp . Since gl.dim Λp < ∞, by Auslander–
Buchsbaum 2.16 proj.dimΛp Y ≤ dim Rp and so gl.dim Λp ≤ dim Rp . Since Λp is an
Rp -order, the Λp -regular sequence x1 , . . . , xd with d = dim Rp gives an Λp -module X :=

Λp /(x1 , . . . , xd )Λp with proj.dimΛp X = d. Thus we have gl.dim Λp ≥ dim Rp .
2.4. d-sCY Algebras. Throughout this paper we shall freely use the notion of d-CY and
d-sCY as in [IR08, §3]: let R be a commutative noetherian ring with dim R = d and let
Λ be a module finite
L R-algebra. For any X ∈ mod Λ denote by E(X) the injective hull of
X, and set E := m∈Max R E(R/m). This gives rise to Matlis duality D0 := HomR (−, E)
(see for example [O76, §1]). Matlis duality always gives a duality from the category of
finite length R-modules to itself. This is true without assuming that R is (semi-)local
because any finite length R-module is the finite direct sum of finite length Rm -modules
for maximal ideals m, so the statement follows from that for the local setting [BH, 3.2.12].
Recall from the introduction:
Definition 2.18. For n ∈ Z we call Λ n-CY if there is a functorial isomorphism
HomD(Mod Λ) (X, Y [n]) ∼
= D0 HomD(Mod Λ) (Y, X)
for all X ∈ Db (fl Λ), Y ∈ Db (mod Λ). Similarly we call Λ n-sCY if the above functorial
isomorphism holds for all X ∈ Db (fl Λ) and Y ∈ Kb (proj Λ).
The next three results can be found in [IR08]; we include them here since we will use
and refer to them extensively.
Proposition 2.19. (1) Λ is d-CY if and only if it is d-sCY and gl.dim Λ < ∞.
(2) d-sCY (respectively d-CY) algebras are closed under derived equivalences.
Proof. (1) is [IR08, 3.1(7)] whilst (2) is [IR08, 3.1(1)].



Proposition 2.20. [IR08, 3.10] Let R be a commutative noetherian ring, d ∈ N. Then
R is d-sCY if and only if R is Gorenstein and equi-codimensional with dim R = d.
From this, for brevity we often say ‘R is d-sCY’ instead of saying ‘R is Gorenstein
and equi-codimensional with dim R = d’.
Proposition 2.21. Let R be d-sCY and let Λ be a module finite R-algebra. Then
Λ is d-sCY ⇐⇒ Λ is an R-order which is a locally symmetric R-algebra.
Thus if Λ is d-sCY then Λ is a Gorenstein R-order.
Proof. The first statement is [IR08, 3.3(1)]. For the second, suppose Λ is d-sCY then since
it is locally symmetric we have Λm ∼
= HomRm (Λm , Rm ) = HomR (Λ, R)m is a projective Λm module for all m ∈ Max R. Hence HomR (Λ, R) is a projective Λ-module, as required. 

14

OSAMU IYAMA AND MICHAEL WEMYSS

The following picture for d-sCY rings R may help the reader navigate the terminology
introduced above.
d-CY R-algebra

non-singular R-order

if gl.dim<∞

d-sCY R-algebra

2.21

2.17

locally symmetric R-order

2.21

if gl.dim<∞

Gorenstein R-order

symmetric R-order
The following non-local result is also useful.
Lemma 2.22. Suppose that R is a d-sCY normal domain.
(1) If Λ is a module finite R-algebra which is d-sCY and M ∈ ref Λ is a height one
projective, then EndΛ (M ) is d-sCY ⇐⇒ EndΛ (M ) ∈ CM R.
(2) If N ∈ ref R then EndR (N ) is d-sCY ⇐⇒ EndR (N ) ∈ CM R.
Proof. (1) Let Γ := EndR (M ). By 2.21 Λm is a symmetric Rm -algebra for all m ∈ Max R,
thus Γm is a symmetric Rm -algebra by 2.10. By 2.21, Γ is d-sCY if and only if Γm ∈ CM Rm
for all m ∈ Max R if and only if Γ ∈ CM R. Thus the assertion follows.
(2) This follows immediately from (1) since any N ∈ ref R is a height one progenerator

by 2.5(3).
Throughout we shall use the definition of NCCR in the introduction (1.7) due to its
suitability for global-local arguments. However, we have the following:
Lemma 2.23. Let R be a d-sCY normal domain, then M ∈ ref R gives a NCCR if and
only if gl.dim EndR (M ) < ∞ and EndR (M ) ∈ CM R.
Proof. (⇒) obvious.
(⇐) Set Λ := EndR (M ), d := dim R. By 2.22(2) Λ is d-sCY hence by 2.21 Λ is a

Gorenstein order, with gl.dim Λ < ∞. By 2.17 Λ is non-singular.
2.5. Global–local properties. In this paper we work in the global setting of non-local
rings so that we can apply our work to algebraic geometry [IW11]. To do this requires
the following technical lemmas.
Lemma 2.24. Derived equivalences between module finite R-algebras are preserved under
localization and completion.
Proof. Let A and B be module finite R-algebras with A derived equivalent to B via a
cp
tilting complex T [R89, 6.5]. Since Ext groups localize (respectively, complete), Tp and T
both have no self-extensions. Further A can be constructed from T using cones, shifts and
summands of T , so using the localizations (respectively, completions) of these triangles
cp can be reached from T
cp .
we conclude that Ap can be constructed from Tp and also A
cp is a tilting A
cp complex.
Thus Tp is a tilting Ap complex and T

The following ensure that membership of add M can be shown locally or even complete locally, and we will use this often.

Lemma 2.25. Let Λ be a module finite R-algebra, where R is a commutative noetherian
g
ring, and let M, N ∈ mod Λ. Denote by N0 → M a right (add N )-approximation of M .
(·g)

Then add M ⊆ add N if and only if the induced map HomΛ (M, N0 ) −−→ EndΛ (M ) is
surjective.
(·g)

Proof. (⇐) If HomΛ (M, N0 ) −−→ EndΛ (M ) is surjective we may lift idM to obtain a
splitting for g and hence M is a summand of N0 .
a
b
(⇒) If M ∈ add N then there exists M → N n → M with ab = idM . Since g is an

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

15

approximation, for every ϕ ∈ EndΛ (M ) there is a commutative diagram
Nn

b

ϕ

ψ

N0

M

g

M

Consequently ϕ = abϕ = aψg and so ϕ is the image of aψ under the map (·g).



Proposition 2.26. Let Λ be a module finite R-algebra, where R is a commutative noetherian ring, and let M, N ∈ mod Λ. Then the following are equivalent:
1. add M ⊆ add N .
2. add Mp ⊆ add Np for all p ∈ Spec R.
3. add Mm ⊆ add Nm for all m ∈ Max R.
cp ⊆ add N
bp for all p ∈ Spec R.
4. add M
cm ⊆ add N
bm for all m ∈ Max R.
5. add M
Furthermore we can replace ⊆ by equality throughout and the result is still true.
Proof. Let g be as in 2.25. Then gp : (N0 )p → Mp is a right (add Np )-approximation
\
c
c
and gbp : (N
0 )p → Mp is a right (add Np )-approximation for any p ∈ Spec R. Since
(·g)

the vanishing of Cok(HomΛ (M, N0 ) −−→ EndΛ (M )) can be checked locally or complete
locally, all conditions are equivalent. The last statement holds by symmetry.

3. Auslander–Reiten Duality for Non-Isolated Singularities

Let R be a d-dimensional, equi-codimensional CM ring with a canonical module
ωR , and let Λ be an R-order. If R is d-sCY, we always choose ωR := R. We denote CMΛ to be the stable category of maximal CM Λ-modules and CMΛ to be the
costable category. By definition these have the same objects as CM Λ, but morphism
spaces are defined as HomΛ (X, Y ) := HomΛ (X, Y )/P(X, Y ) (respectively HomΛ (X, Y ) :=
HomΛ (X, Y )/I(X, Y )) where P(X, Y ) (respectively I(X, Y )) is the subspace of morphisms factoring through add Λ (respectively add ωΛ ).
We denote Tr := TrΛ : modΛ → modΛop the Auslander–Bridger transpose duality,
and ΩΛop : modΛop → modΛop the syzygy functor. Then we have AR translation
τ := HomR (ΩdΛop TrΛ (−), ωR ) : CMΛ → CMΛ.
We denote Di := Extd−i
R (−, ωR ) to be the duality of the category of Cohen–Macaulay
modules of dimension i, so D0 is the Matlis dual (as in §2.4).
If Λ is an R-order as above we define SingR Λ := {p ∈ Spec R : gl.dim Λp > dim Rp }
to be the singular locus of Λ (see 1.6(2)). Our main theorem is the following:
Theorem 3.1. Let R be a d-dimensional, equi-codimensional CM ring with a canonical
module ωR . Let Λ be an R-order with dim SingR Λ ≤ 1. Then there exist functorial
isomorphisms
fl HomΛ (X, Y ) ∼
= D0 (fl Ext1Λ (Y, τ X)),


HomΛ (X, ΩY ) ∼
Ext1Λ (Y, τ X)
= D1
fl HomΛ (X, ΩY )
fl Ext1Λ (Y, τ X)
for all X, Y ∈ CM Λ.
In fact 3.1 immediately follows from the more general 3.2 below. Recall for X ∈ mod Λ
that NP(X) := {p ∈ Spec R : Xp ∈
/ proj Λp } and CM1 Λ := {X ∈ CM Λ : dim NP(X) ≤
1}.
Theorem 3.2. Let R be a d-dimensional, equi-codimensional CM ring with a canonical
module ωR . Let Λ be an R-order. Then there exist functorial isomorphisms
fl HomΛ (X, Y ) ∼
= D0 (fl Ext1Λ (Y, τ X)),


Ext1Λ (Y, τ X)
HomΛ (X, ΩY ) ∼
= D1
fl HomΛ (X, ΩY )
fl Ext1Λ (Y, τ X)
for all X ∈ CM1 Λ and Y ∈ CM Λ

16

OSAMU IYAMA AND MICHAEL WEMYSS

The proof of 3.2 requires the next three easy lemmas. For a finitely generated Rmodule M , denote ER (M ) to be the injective hull of M .
Lemma 3.3. If X ∈ mod R and Y ∈ Mod R satisfies Supp X ∩ Ass Y = ∅, then
HomR (X, Y ) = 0.
Proof. Let f : X → Y be any map and X ′ := Im f . Then X ′ ⊂ Y is a finitely generated
submodule such that Ass X ′ ⊂ Supp X ∩ Ass Y . Thus Ass X ′ = ∅ and so since X ′ is
finitely generated, X ′ = 0.

Lemma 3.4. [BH, 3.2.7(a)] We have Ass ER (R/p) = {p}.
Now recall that if R is a d-dimensional equi-codimensional CM ring with canonical
ωR then the minimal R-injective resolution of ωR , denoted
0 → ωR → I0 → I1 → . . . → Id−1 → Id → 0,
satisfies
Ii

[BH, 3.2.9, 3.3.10(b)]

=

M

M

1.4

E(R/p) =

(3.A)

E(R/p).

(3.B)

p:dim R/p=d−i

p:htp=i

In particular the Matlis dual is D0 = HomR (−, Id ).
Lemma 3.5. Let R be a d-dimensional equi-codimensional CM ring with canonical module
ωR . If N ∈ mod R with dimR N ≤ 1, then
d−1 N
∼
(1) Extd−1
R (N, ωR ) = ExtR ( fl N , ωR ).
d
d
(2) ExtR (N, ωR ) ∼
= ExtR (fl N, ωR ).
(3) There is an exact sequence
d
N
0 → Extd−1
R ( fl N , ωR ) → HomR (N, Id−1 ) → HomR (N, Id ) → ExtR (fl N, ωR ) → 0.

Proof. There is an exact sequence 0 → fl N → N →
HomR (−, ωR ) gives

N
fl N

→ 0 from which applying

d−1
d−1
d−1 N
Extd−2
R (fl N, ωR ) → ExtR ( fl N , ωR ) → ExtR (N, ωR ) → ExtR (fl N , ωR ).

Since dimR (fl N ) = 0, it is well-known that ExtiR (fl N, ωR ) = 0 for all i 6= d [BH, 3.5.11],
hence the outer two ext groups vanish, establishing (1). But we also have an exact sequence
N
ExtdR ( flNN , ωR ) → ExtdR (N, ωR ) → ExtdR (fl N , ωR ) → Extd+1
R ( fl N , ωR )

and so since flNN has positive depth (or is zero) at all maximal ideals, ExtiR ( flNN , ωR ) = 0
for all i > d − 1, again by [BH, 3.5.11]. This establishes (2). For (3), note first that
HomR (N, Id−2 ) = 0 by 3.3, since by 3.4 and the assumption that dim N ≤ 1 we have
that Supp N ∩ Ass Id−2 = ∅. Consequently simply applying HomR (N, −) to (3.A) gives
an exact sequence
d
0 → Extd−1
R (N, ωR ) → HomR (N, Id−1 ) → HomR (N, Id ) → ExtR (N, ωR ) → 0,

and so (3) follows from (1) and (2).



We are now ready to prove 3.2. To ease notation, we often drop Tor and Ext, and
R
for example write 1R (X, Y ) for Ext1R (X, Y ), and R
1 (X, Y ) for Tor1 (X, Y ).
Proof. Denote T := Tr X. Now since Y ∈ CM Λ we have ExtiR (Y, ωR ) = 0 for all i > 0
and so applying HomR (Y, −) to (3.A) gives an exact sequence
0 → R (Y, ωR ) → R (Y, I0 ) → R (Y, I1 ) → . . . → R (Y, Id−1 ) → R (Y, Id ) → 0
of left Λop -modules, which we split into short exact sequences as
0

R (Y, ωR )

R (Y, I0 )

R (Y, I1 )

C1

R (Y, I2 )

C2

...

R (Y, Id−2 )

R (Y, Id−1 )

Cd−1

R (Y, Id )

0.

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

17

Applying HomΛop (T, −) gives exact sequences
1
1
2
2
2
Λop (T, R (Y, Id−1 )) Λop (T, R (Y, Id )) Λop (T, Cd−1 ) Λop (T, R (Y, Id−1 )) Λop (T, R (Y, Id ))
2
2
3
3
Λop (T, R (Y, Id−2 ))
Λop (T, Cd−1 )
Λop (T, Cd−2 )
Λop (T, R (Y, Id−2 ))

..
.
d−1
Λop (T, R (Y, I1 ))

d−1
Λop (T, C2 )

d
Λop (T, C1 )

d
Λop (T, R (Y, I1 ))

d
Λop (T, R (Y, I0 ))

d
Λop (T, C1 )

d+1
Λop (T, R (Y, ωR ))

d+1
Λop (T, R (Y, I0 ))

.

By the functorial isomorphism [CE99, VI.5.1]
∼ HomR (TorΛ (A, B), I)
Extj (A, R (B, I)) =
j

Λ

where I is an injective R-module, we have exact sequences
Λ
R (1 (T, Y

), Id−1 )

Λ
R (2 (T, Y

Λ
R (1 (T, Y

Λ
R (d (T, Y

2
Λop (T, Cd−1 )

2
Λop (T, Cd−1 )

), Id−2 )

Λ
R (d−1 (T, Y

), Id )

d−1
Λop (T, C2 )

), I1 )

d
Λop (T, C1 )

), I0 )

Λ
R (2 (T, Y

Λ
R (2 (T, Y

), Id ) (3.C)

3
Λ

(T,
C
)
(
(T,
Y
),
I
)
op

d−2
R 3
d−2
Λ




..

.
(3.D)
Λ
d


R (d (T, Y ), I1 )
Λop (T, C1 )




Λ
d+1

.
(
(T,
Y
),
I
)
(T,
(Y,
ω
))
R
0
op
R
R
d+1
Λ
), Id−1 )

By the assumption that X ∈ CM1 Λ, for all primes p such that dim R/p > 1, we
have Xp ∈ proj Λp and so Tp ∈ proj Λop
p . Thus for all such primes and any j > 0,
Λp
∼
(T
,
Y
)
=
0. Hence for all i = 0, 1, . . . , d − 2 and all
Tor
we have TorΛ
(T,
Y
)
=
p
p
p
j
j
j > 0, by 3.4 and (3.B) it follows that Supp TorΛ
j (T, Y ) ∩ Ass Ii = ∅ and so consequently
(T,
Y
),
I
)
=
0
for
all
j
>
0
and
all
i = 0, 1, . . . , d − 2 by 3.3. Thus (3.D)
HomR (TorΛ
i
j
reduces to
∼ Ext3 op (T, Cd−2 ) =
∼ ... =
∼ Extd op (T, C1 ) =
∼ Extd+1
Ext2 op (T, Cd−1 ) =
op (T, R (Y, ωR ))
Λ

Λ

Λ

Λ

and so it follows that
Ext2Λop (T, Cd−1 ) ∼
= Ext1Λ (Y, R (ΩdΛop Tr X, ωR )) = Ext1Λ (Y, τ X).
= Ext1Λop (ΩdΛop Tr X, R (Y, ωR )) ∼

(3.E)
Using the well-known functorial isomorphism [Aus78, 3.2],[Y90, 3.9]
TorΛ (Tr X, Y ) ∼
= Hom (X, Y ),

(3.F)

Λ

1

(3.C), (3.E) and (3.F) combine to give the following commutative diagram of exact sequences:
Λ
R (Tor1 (T, Y
∼
=

), Id−1 )

∼
=

(3.F)

R (HomΛ (X, Y

Λ
R (Tor1 (T, Y

), Id−1 )

), Id )

Ext2Λop (T, Cd−1 )
∼
=

(3.F)

R (HomΛ (X, Y

), Id )

ψ

Λ
R (Tor2 (T, Y
∼
=

(3.E)

Ext1Λ (Y, τ X)

), Id−1 )

Λ
R (Tor2 (T, Y
∼
=

(3.F)

R (HomΛ (X, ΩY

), Id−1 )

), Id )

(3.F)

R (HomΛ (X, ΩY

), Id )

which we splice as
R (HomΛ (X, Y

), Id−1 ) → R (HomΛ (X, Y ), Id ) → Im ψ → 0

(3.G)

0 → Im ψ → Ext1Λ (Y, τ X) → Cok ψ → 0

(3.H)

0 → Cok ψ → R (HomΛ (X, ΩY ), Id−1 ) → R (HomΛ (X, ΩY ), Id ).

(3.I)

By applying 3.5(3) to N := HomΛ (X, Y ) and comparing to (3.G) we see that
Im ψ ∼
= Extd (fl Hom (X, Y ), ωR ) = D0 (fl Hom (X, Y )).
R

Λ

Λ

(3.J)

Similarly, applying 3.5(3) to N := HomΛ (X, ΩY ) and comparing to (3.I) we see that




HomΛ (X,ΩY )
HomΛ (X,ΩY )
(3.K)
Cok ψ ∼
= Extd−1
R
fl Hom (X,ΩY ) , ωR = D1 fl Hom (X,ΩY ) .
Λ

Λ

fl Ext1Λ (Y, τ X),

Now (3.J) and (3.K) show that Im ψ =
and together with (3.J) this
establishes the first required isomorphism, and together with (3.H) and (3.K) this establishes the second required isomorphism.


18

OSAMU IYAMA AND MICHAEL WEMYSS

When R has only isolated singularities the above reduces to classical Auslander–
Reiten duality. If moreover R is a d-sCY ring with isolated singularities (i.e. R is a
Gorenstein d-dimensional equi-codimensional ring with isolated singularities), AR duality
implies that the category CMR is (d − 1)-CY. We now apply 3.1 to possibly non-isolated
d-sCY rings to obtain some analogue of this (d − 1)-CY property (see 3.7(1) below). The
following lemma is well-known [Aus78, III.1.3].
Lemma 3.6. Suppose R is d-sCY and let Λ be a symmetric R-order. Then τ ∼
= Ω2−d
Λ .
Proof. We have Ω2 Tr(−) ∼
= HomΛ (−, Λ). Since R is d-sCY (and so ωR := R), and Λ is
symmetric, we have Ω2 Tr(−) ∼
= HomΛ (−, Λ) ∼
= HomR (−, R). Thus
d
∼
τ = HomR (Ω op Tr(−), R) = HomR (Ωd−2
op HomR (−, R), R)
Λ

Λ

∼
HomR (HomR (−, R), R)
= Ω2−d
Λ
∼
= Ω2−d .
Λ


Corollary 3.7. Let R be a d-sCY ring and let Λ be a symmetric R-order with dim SingR Λ ≤
1. Then
(1) There exist functorial isomorphisms
fl HomΛ (X, Y ) ∼
= D0 (fl HomΛ (Y, X[d − 1])),


HomΛ (X, Y ) ∼
HomΛ (Y, X[d − 2])
= D1
fl HomΛ (X, Y )
fl HomΛ (Y, X[d − 2])
for all X, Y ∈ CM Λ.
(2) If d = 3 then for all X, Y ∈ CM Λ, HomΛ (X, Y ) ∈ CM R if and only if HomΛ (Y, X) ∈
CM R.
Proof. (1) It is well-known that in CMΛ the shift functor [1] = Ω−1 so by 3.6 τ = [d − 2].
Thus the result follows directly from 3.1, using the fact that HomΛ (A, B[1]) ∼
= Ext1Λ (A, B)
for all A, B ∈ CM Λ.

(2) Immediate from (1) and 2.7.
Note that by 2.9, 3.7(2) also holds for arbitrary d (with no assumptions on the singular
locus) provided that R is normal. When R is not necessarily normal, we improve 3.7(2)
in 4.4 below.
4. Modifying and Maximal Modifying Modules
Motivated by the fact that Spec R need not have a crepant resolution, we want to
be able to control algebras of infinite global dimension and hence partial resolutions of
singularities.
4.1. Modifications in Dimension d. We begin with our main definition.
Definition 4.1. Let R be a d-dimensional CM ring, Λ a module finite R-algebra. We
call N ∈ ref Λ a modifying module if EndΛ (N ) ∈ CM R, whereas we call N a maximal
modifying (MM) module if N is modifying and furthermore it is maximal with respect to
this property, that is to say if there exists X ∈ ref Λ with N ⊕ X modifying, necessarily
X ∈ add N .
The following is immediate from the definition.
Lemma 4.2. Suppose R is a d-dimensional CM ring, Λ a module finite R-algebra. Then
(1) The modifying Λ-modules which are generators are always CM.
(2) If further Λ is a Gorenstein R-order then the MM generators are precisely the MM
modules which are CM.
Proof. (1) Since M is a modifying EndΛ (M ) ∈ CM R, and since M is a generator, Λ ∈
add M . Hence M ⊕n ∼
= HomΛ (Λ, M ⊕n ) ∈ CM R is a summand of HomΛ (M ⊕n , M ⊕n ) ∼
=
⊕n2
EndΛ (M )
for some n ∈ N, thus M ⊕n and so consequently M itself are CM.
(2) Conversely suppose that M is an MM module which is CM. Then certainly we have

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

19

HomΛ (Λ, M ) ∼
= M ∈ CM R and also HomΛ (M, ωΛ ) ∼
= HomR (M, ωR ) ∈ CM R. Since Λ is
a Gorenstein R-order, add Λ = add ωΛ by 2.15, thus EndΛ (M ⊕ Λ) ∈ CM R. Since M is
maximal necessarily Λ ∈ add M .

Under assumptions on the singular locus, we can check whether a CM module is
modifying by examining Ext groups. The following is a generalization of 2.7 for d = 3,
and [Iya07, 2.5.1] for isolated singularities.
Theorem 4.3. Suppose that R is d-sCY with d = dim R ≥ 2 and dim Sing R ≤ 1,
let Λ be an R-order and let X, Y ∈ CM Λ. Then HomΛ (X, Y ) ∈ CM R if and only if
ExtiΛ (X, Y ) = 0 for all i = 1, . . . , d − 3 and fl Extd−2
Λ (X, Y ) = 0.
Proof. Without loss of generality, we can assume that R is local. Consider a projective
resolution . . . → P1 → P0 → X → 0. Applying HomΛ (−, Y ), we have a complex
0 → HomΛ (X, Y ) → HomΛ (P0 , Y ) → . . .
. . . → HomΛ (Pd−3 , Y ) → HomΛ (Ωd−2 X, Y ) → Extd−2
Λ (X, Y ) → 0 (4.A)
with homologies ExtiΛ (X, Y ) at HomΛ (Pi , Y ) for i = 1, . . . , d − 3.
(⇐) By assumption the sequence (4.A) is exact. Since depth Extd−2
Λ (X, Y ) ≥ 1,
depth HomΛ (Ωd−2 X, Y ) ≥ 2 by 2.3, and HomΛ (Pi , Y ) ∈ CM R, we have HomΛ (X, Y ) ∈
CM R by the depth lemma.
(⇒) By 2.6 and our assumption dim Sing R ≤ 1, we have dim ExtiΛ (X, Y ) ≤ 1 for
any i > 0. Assume ExtiΛ (X, Y ) 6= 0 for some i = 1, . . . , d − 3. Take minimal i such that
ExtiΛ (X, Y ) 6= 0. We have an exact sequence
0 → HomΛ (X, Y ) → HomΛ (P0 , Y ) → . . .
. . . → HomΛ (Pi−1 , Y ) → HomΛ (Ωi X, Y ) → ExtiΛ (X, Y ) → 0.
Localizing at prime ideal p of R with height at least d−1 and using depthRp HomΛp (Ωi Xp , Yp ) ≥
2 by 2.3, HomΛp (Pip , Yp ) ∈ CM Rp and HomΛp (Xp , Yp ) ∈ CM Rp by our assumption,
we have depthRp ExtiΛ (X, Y )p ≥ 1 by the depth lemma. If p has height d − 1, then
dimRp ExtiΛ (X, Y )p = 0 and we have ExtiΛ (X, Y )p = 0. Thus dim ExtiΛ (X, Y ) = 0 holds,
and we have ExtiΛ (X, Y ) = 0, a contradiction.
Thus we have ExtiΛ (X, Y ) = 0 for all i = 1, . . . , d−3 and so the sequence (4.A) is exact.
Since depth HomΛ (Ωd−2 X, Y ) ≥ 2, HomΛ (Pi , Y ) ∈ CM R and HomΛ (X, Y ) ∈ CM R, we
have depth Extd−2

Λ (X, Y ) ≥ 1 by the depth lemma.
The following improves 3.7(2).
Corollary 4.4. Let R be a d-sCY ring and let Λ be a symmetric R-order with dim SingR Λ ≤
1. Then for all X, Y ∈ CM Λ, HomΛ (X, Y ) ∈ CM R if and only if HomΛ (Y, X) ∈ CM R.
Proof. By the statement and proof of 2.3, when d ≤ 2, if M and N are CM then so are
both HomΛ (M, N ) and HomΛ (N, M ). Thus we can assume that d ≥ 3. By symmetry, we
need only show (⇒). Assume that HomΛ (X, Y ) ∈ CM R. Then by 4.3 ExtiΛ (X, Y ) = 0
ExtiΛ (X, Y )
for any i = 1, . . . , d − 3 and fl Extd−2
= 0 for any
Λ (X, Y ) = 0. Since
fl ExtiΛ (X, Y )
ExtiΛ (Y, X)
i = 1, . . . , d − 3, the D1 duality in 3.7(1) implies
= 0 for any i = 1, . . . , d − 3.
fl ExtiΛ (Y, X)
i
Thus ExtΛ (Y, X) has finite length for any i = 1, . . . , d − 3. Since fl ExtiΛ (X, Y ) = 0 for any
i = 1, . . . , d − 2, the D0 duality in 3.7(1) implies fl ExtiΛ (Y, X) = 0 for any i = 1, . . . , d − 2.
Consequently we have ExtiΛ (Y, X) = 0 for any i = 1, . . . , d − 3 and fl Extd−2
Λ (Y, X) = 0.
Again by 4.3 we have HomΛ (Y, X) ∈ CM R.

Recall from 1.7 the definition of an NCCR. The following asserts that, in arbitrary
dimension, NCCRs are a special case of MMAs:
Proposition 4.5. Let R be a d-dimensional, normal, equi-codimensional CM ring with
a canonical module ωR (e.g. if R is a normal d-sCY ring). Then reflexive R-modules M
giving NCCRs are MM modules.

20

OSAMU IYAMA AND MICHAEL WEMYSS

Proof. Assume that X ∈ ref R satisfies EndR (M ⊕ X) ∈ CM R. Then HomR (M, X) ∈
CM Γ for Γ := EndR (M ). By 2.17 we have HomR (M, X) ∈ proj Γ. By 2.5(4) X ∈ add M
as required.

We now investigate the derived equivalence classes of modifying algebras, maximal
modifying algebras, and NCCRs.
Theorem 4.6. Let R be a normal d-sCY ring, then
(1) Modifying algebras of Λ are closed under derived equivalences.
(2) NCCRs of Λ are closed under derived equivalences.
Proof. (1) Let Λ = EndR (M ) be a modifying algebra of R, and let Γ be a ring that is
derived equivalent to Λ. Then Γ is a module finite R-algebra since it is the endomorphism
ring of a tilting complex of Λ. Since Λ is a modifying algebra of R, it is d-sCY by 2.22(2).
But d-sCY algebras are closed under derived equivalences, hence Γ is also d-sCY and so
Γ ∈ CM R by 2.21. In particular, Γ is reflexive as an R-module.
Now we fix a height one prime ideal p of R. Since Mp is a free Rp -module of finite
rank, Λp = EndRp (Mp ) is a full matrix algebra of Rp . Since Rp is local, the Morita
equivalence class of Rp coincides with the derived equivalence class of Rp [RZ03, 2.12],
so we have that Γp is Morita equivalent to Rp . Thus Γ satisfies the conditions in 2.11,
so there exists a reflexive R-module N such that Γ ∼
= EndR (N ) as R-algebras. We have
already observed that Γ ∈ CM R, hence it is a modifying algebra of R.
(2) Since R is normal d-sCY, by 2.23 NCCRs of R are nothing but modifying algebras of
R which have finite global dimension. We know modifying algebras are d-sCY by 2.22, so
the result follows by combining (1) and 2.19.

Question 4.7. Let R be a normal d-sCY ring. Are the maximal modifying algebras of
R closed under derived equivalences?
We now show that the question has a positive answer when d ≥ 2 provided that
dim Sing R ≤ 1. In particular, this means that 4.7 is true when d ≤ 3.
Theorem 4.8. Suppose R is a normal d-sCY ring with dim R = d ≥ 2 and dim Sing R ≤
1. Let N be a modifying module and set Γ := EndR (N ). Then
(1) Then N is MM if and only if CMΓ has no non-zero objects Y satisfying HomΓ (Y, Y [i]) =
0 for all i = 1, . . . , d − 3 and fl HomΓ (Y, Y [d − 2]) = 0.
(2) MMAs are closed under derived equivalences.
Proof. (1) By reflexive equivalence 2.5(4) it is easy to show that there exists X ∈ ref R
with X ∈
/ add N such that EndR (N ⊕ X) ∈ CM R if and only if there exists Y ∈ CM Γ
with Y ∈
/ add Γ such that EndΓ (Y ) ∈ CM R. Since for Y ∈ CM Γ we have ExtiΓ (Y, Y ) =
HomΓ (Y, Y [i]), by 4.3 we have the assertion.
(2) Suppose that Λ is derived equivalent to Γ = EndR (N ) where Γ is an MMA. By 4.6(1)
we know that Λ ∼
= EndR (M ) for some modifying M . Since the equivalence D(Mod Λ) ≃
D(Mod Γ) induces equivalences Db (mod Λ) ≃ Db (mod Γ) and Kb (proj Λ) ≃ Kb (proj Γ)
by [R89, 8.1, 8.2], we have CMΛ ≃ CMΓ by [Bu86, 4.4.1]. By (1), the property of being
an MMA can be characterized on the level of this stable category, hence Λ is also an
MMA.

4.2. Derived Equivalence in Dimension 3. We now restrict to dimension three. In
this case, we can strengthen 4.6 to obtain one of our main results (4.16). Leading up to our
next proposition (4.12) we require three technical lemmas. Recall from the introduction
(1.20) the notion of an approximation.
Lemma 4.9. Let R be a normal 3-sCY ring and let Λ be a module finite R-algebra which
is 3-sCY. Let B ∈ ref Λ be a modifying height one progenerator and let C ∈ ref Λ. If
g
f
0 → A → B0 → C → 0 is an exact sequence where g is a right (add B)-approximation,
then the cokernel of the natural map
f·

HomΛ (B0 , B) → HomΛ (A, B)
has finite length.

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

21

Proof. Set Γ := EndΛ (B). Since B is a height one progenerator we have a reflexive
equivalence F := HomΛ (B, −) : ref Λ → ref Γ by 2.5(4). Moreover Γ is 3-sCY by 2.22.
Since g is a right (add B)-approximation, we have an exact sequence
0 → FA → FB0 → FC → 0
of Γ-modules. Then since FB0 ∈ proj Γ, applying HomΓ (−, Γ) = HomΓ (−, FB) gives an
exact sequence
HomΓ (FB0 , FB)
∼
=

HomΛ (B0 , B)

HomΓ (FA, FB)
f·

Ext1Γ (FC, Γ)

0

∼
=

HomΛ (A, B)

and thus Cok(f ·) = Ext1Γ (FC, Γ). Hence we only have to show that Ext1Γ (FC, Γ)p = 0
for any non-maximal prime ideal p of R. By 2.3 and 2.4 we have (FC)p ∈ CM Γp .
Since Γ is 3-sCY, Γp is a Gorenstein Rp -order by 2.21. Consequently Ext1Γ (FC, Γ)p =

Ext1Γp ((FC)p , Γp ) = 0, as required.
Lemma 4.10. Let R be a normal 3-sCY ring and let Λ be a module finite R-algebra which
is 3-sCY. Suppose N ∈ ref Λ and M ∈ CM Λ with both M and N modifying such that M
h
is a height one progenerator. If 0 → L → M0 → N → 0 is an exact sequence where h is
a right (add M )-approximation, then EndΛ (L ⊕ M ) ∈ CM R.
Proof. Note first that since N is reflexive and M ∈ CM Λ we have L ∈ CM Λ by the depth
lemma. From the exact sequence
0 → HomΛ (M, L) → HomΛ (M, M0 ) → HomΛ (M, N ) → 0
with HomΛ (M, M0 ) ∈ CM R we see, using 2.3 and the depth lemma, that HomΛ (M, L) ∈
CM R. By 2.9 HomΛ (L, M ) ∈ CM R. Since EndΛ (M ) ∈ CM R by assumption, it suffices
to show that EndΛ (L) ∈ CM R. By 2.7 we only need to show that fl Ext1Λ (L, L) = 0.
Consider now the following exact commutative diagram
Cok f
HomΛ (L, M0 )

f

c

b

HomΛ (M0 , M0 )

Ext1Λ (L, L)

HomΛ (L, N )

t

Ext1Λ (L, M0 ) .
K

HomΛ (M0 , N )

Since HomΛ (L, M ) ∈ CM R we know by 2.7 that fl Ext1Λ (L, M0 ) = 0 and so fl K = 0.
Hence to show that fl Ext1Λ (L, L) = 0 we just need to show that fl Cok f = 0. To do this
consider the exact sequence
Cok b → Cok bf → Cok f → 0.

(4.B)

By 4.9 applied with B = M0 , Cok b has finite length and thus the image of the first map
in (4.B) has finite length. Second, note that Cok bf = Cok tc = Cok c and fl Cok c = 0
since Cok c embeds inside Ext1Λ (N, N ) and furthermore fl Ext1Λ (N, N ) = 0 by 2.7. This
means that the image of the first map is zero, hence Cok f ∼
= Cok c and so in particular
fl Cok f = 0.

In fact, using reflexive equivalence we have the following improvement of 4.10 which
does not assume that M is CM, which is the analogue of [GLS, 5.1].
Lemma 4.11. Let R be a normal 3-sCY ring and let M and N be modifying modules. If
h
0 → L → M0 → N is an exact sequence where h is a right (add M )-approximation, then
L ⊕ M is modifying.
Proof. Note that L is reflexive since R is normal. Denote Λ := EndR (M ) and F :=
HomR (M, −) : ref R → ref Λ the reflexive equivalence in 2.5(4). Then Λ is 3-sCY by 2.22,
FN ∈ ref Λ, FM ∈ CM Λ and both FN and FM are modifying Λ-modules. Further
Fh

0 → FL → FM0 −→ FN → 0

22

OSAMU IYAMA AND MICHAEL WEMYSS

is exact and FM = Λ so trivially Fh is a right (add FM )-approximation. It is also clear
that FM = Λ is a height one progenerator. By 4.10 we see that EndΛ (FL ⊕ FM ) ∈ CM R,

hence EndR (L ⊕ M ) ∼
= EndΛ (FL ⊕ FM ) ∈ CM R as required.
Now we are ready to prove the following crucial result (c.f. 5.10 later), which is the
analogue of [GLS, 5.2].
Theorem 4.12. Let R be a normal 3-sCY ring and let M be a non-zero modifying module.
Then the following are equivalent
(1) M is an MM module.
f

(2) If N is any modifying module then there exists an exact sequence 0 → M1 → M0 → N
with each Mi ∈ add M such that f is a right (add M )-approximation.
Proof. Set Λ := EndR (M ). Since M is a height one progenerator, we have a reflexive
equivalence F := HomR (M, −) : ref R → ref Λ by 2.5(4). Moreover Λ is 3-sCY by 2.22
and so a Gorenstein R-order by 2.21.
f
(1)⇒(2) We have an exact sequence 0 → L → M0 → N where f is a right (add M )approximation of N . By 4.11 EndR (L ⊕ M ) ∈ CM R thus since M is an MM module,
L ∈ add M .
(2)⇒(1) Suppose N is reflexive with EndR (M ⊕ N ) ∈ CM R. Then FN ∈ CM R. We
have proj.dimΛ FN ≤ 1 since N is a modifying module and so there is an exact sequence
0 → FM1 → FM0 → FN → 0 by assumption. Since Λ is a Gorenstein R-order it follows
that FN is a projective Λ-module by using localization and Auslander–Buchsbaum 2.16.
Hence N ∈ add M .

The following version of the Bongartz completion [B80][ASS, VI.2.4] is convenient for
us. Recall from the introduction that throughout this paper when we say tilting module
we mean a tilting module of projective dimension ≤ 1 (see 1.15).
Lemma 4.13. Suppose R is normal, M ∈ ref R and denote Λ := EndR (M ). If N ∈ ref R
is such that HomR (M, N ) is a partial tilting Λ-module then there exists L ∈ ref R such
that HomR (M, N ⊕ L) is a tilting Λ-module.
Proof. By 2.5 T := HomR (M, N ) and Λ are both reflexive. Thus since R is normal we
can invoke [IR08, 2.8] to deduce that there exists an X ∈ ref Λ such that T ⊕ X is tilting.

Again by 2.5 X = HomR (M, L) for some L ∈ ref R.
We have the following analogue of [IR08, 8.7].
Proposition 4.14. Let R be a normal 3-sCY ring and assume M is an MM module.
Then
(1) HomR (M, −) sends modifying R-modules to partial tilting EndR (M )-modules.
(2) HomR (M, −) sends MM R-modules to tilting EndR (M )-modules.
Proof. (1) Denote Λ := EndR (M ), let N be a modifying module and denote T :=
HomR (M, N ). Note first that proj.dimΛ T ≤ 1 by 4.12 and also Λ is a Gorenstein Rorder by 2.21 and 2.22.
Since projective dimension localizes proj.dimΛp Tp ≤ 1 for all primes p, and further
if ht p = 2 then Tp ∈ CM Rp by 2.3. Since Λp is a Gorenstein Rp -order, Auslander–
Buchsbaum (2.16) implies that Tp is a projective Λp -module and so Ext1Λp (Tp , Tp ) = 0
for all primes p with ht p = 2. Consequently Ext1Λm (Tm , Tm ) has finite length for all
m ∈ Max R. On the other hand Λ is 3-sCY by 2.22 and EndΛ (T ) ∼
= EndR (N ) ∈ CM R
by 2.5. Thus fl Ext1Λm (Tm , Tm ) = 0 for all m ∈ Max R by 2.7 and so Ext1Λ (T, T ) = 0, as
required.
(2) Now suppose that N is also MM. By Bongartz completion 4.13 we may find L ∈ ref R
such that HomR (M, N ⊕L) is a tilting EndR (M )-module, thus EndR (M ) and EndR (N ⊕L)
are derived equivalent. Since EndR (M ) is 3-sCY so is EndR (N ⊕ L) by 2.19 and thus by
2.22 EndR (N ⊕ L) ∈ CM R. Consequently L ∈ add N and so HomR (M, N ) is a tilting
module.
Now for the convenience of the reader we give a second proof, which shows us more
explicitly how our tilting module generates the derived category. If N is also MM then

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

23

since (−)∗ : ref R → ref R is a duality, certainly N ∗ (and M ∗ ) is MM. By 4.12 we can find
0 → N1∗ → N0∗ → M ∗
such that
0 → FN1∗ → FN0∗ → FM ∗ → 0
∗

(4.C)
∗

Ext1Γ (FM ∗ , FM ∗ )

is exact, where F = HomR (N , −). Denote Γ := EndR (N ), then
=
0 by (1). Thus applying HomΓ (−, FM ∗ ) to (4.C) gives us the following commutative
diagram
0

HomΓ (FM ∗ , FM ∗ )

HomΓ (FN0∗ , FM ∗ )

HomΓ (FN1∗ , FM ∗ )

0

0

HomR (M ∗ , M ∗ )

HomR (N0∗ , M ∗ )

HomR (N1∗ , M ∗ )

0

where the top row is exact and the vertical maps are isomorphisms by 2.5. Hence the
bottom row is exact. Since (−)∗ : ref R → ref R is a duality, this means that
0 → HomR (M, M ) → HomR (M, N0 ) → HomR (M, N1 ) → 0
is exact. But denoting Λ := EndR (M ) and T := HomR (M, N ), this means we have an
exact sequence
0 → Λ → T0 → T1 → 0
with each Ti ∈ add T . Hence T is a tilting Λ-module.

The following is now immediate:
Corollary 4.15. Let R be a normal 3-sCY ring and assume M is an MM module. Then
(1) If N is any modifying module then the partial tilting EndR (M )-module T := HomR (M, N )
induces a recollement
K

D(Mod EndR (M ))

F

D(Mod EndR (N ))

where F = RHom(T, −) and K is a certain triangulated subcategory of D(Mod EndR (M )).
(2) If further N is maximal modifying then the above functor F is an equivalence.
Proof. (1) Set Λ := EndR (M ) then T := HomR (M, N ) is a partial tilting Λ-module
by 4.14. The fact that EndΛ (T ) ∼
= EndR (N ) follows since HomR (M, −) is a reflexive
equivalence by 2.5(4). By Bongartz completion T is a summand of a tilting Λ-module U .
We have a derived equivalence D(Mod EndR (M )) ≃ D(Mod EndΛ (U )). Moreover there
exists an idempotent e of EndΛ (U ) such that e EndΛ (U )e ∼
= EndΛ (T ) ∼
= EndR (N ). Thus
we have the desired recollement (e.g. [K10, 4.16], see also [M03]).
(2) is an immediate consequence by taking U := T in the argument above.

We can now improve 4.8.
Theorem 4.16. Let R be a normal 3-sCY ring. Then MMAs of R form a complete
derived equivalence class.
Proof. By 4.8(2), MMAs of R are closed under derived equivalence. On the other hand,

all MMAs are derived equivalent by 4.15(2).
Moreover, we have the following bijections (cf. [IR08, 8.9]).
Theorem 4.17. Let R be a normal 3-sCY ring and assume M is an MM module. Then
the functor HomR (M, −) : mod R → mod EndR (M ) induces bijections
(1) {maximal modifying R-modules}
(2)
{modifying R-modules}

1:1

←→ {reflexive tilting EndR (M )-modules}.
1:1
←→ {reflexive partial tilting EndR (M )-modules}.

Proof. (1) In light of 4.14(2) we only need to show that every reflexive tilting module is
the image of some MM module. Thus let X be a reflexive tilting EndR (M )-module; by
reflexive equivalence 2.5(4) there exists some N ∈ ref R such that HomR (M, N ) ∼
= X. We
claim that N is MM. Since HomR (M, N ) is a tilting EndR (M )-module certainly EndR (M )
and EndR (N ) are derived equivalent; the fact that N is MM follows from 4.8(2) above.
(2) By 4.14(1) we only need to show that every reflexive partial tilting EndR (M )-module

24

OSAMU IYAMA AND MICHAEL WEMYSS

is the image of some modifying module. Suppose X is a reflexive partial tilting EndR (M )module, say X ∼
= HomR (M, N ). Then by Bongartz completion 4.13 there exists N1 ∈
ref R such that HomR (M, N ⊕ N1 ) is a tilting EndR (M )-module. By (1) N ⊕ N1 is MM,
thus EndR (N ) is a summand of the CM R-module EndR (N ⊕ N1 ) and so is itself CM. 
Corollary 4.18. Let R be a normal 3-sCY ring and assume M is an MM module. Then
(1) N is a modifying module ⇐⇒ N is the direct summand of an MM module.
(2) R has an MM module which is a CM generator.
Proof. (1) ‘if’ is clear. For the ‘only if’ let N be a modifying module, then by 4.14(1)
HomR (M, N ) is a partial tilting EndR (M )-module, so the proof of 4.17(2) shows that
there exists N1 ∈ ref R such that N ⊕ N1 is MM.

(2) Apply (1) to R. The corresponding MM module is necessarily CM by 4.2.
Recall from 1.6(3) we say that an R-order Λ has isolated singularities if gl.dim Λp =
dim Rp for all non-maximal primes p of R.
Remark 4.19. It is unclear in what generality every maximal modification algebra
EndR (M ) has isolated singularities. In many cases this is true — for example if R is
itself an isolated singularity this holds, as it does whenever M is CT by 5.4. Also, if
R is Gorenstein, X → Spec R projective birational with M ∈ ref R modifying such that
Db (coh X) ∼
= Db (mod EndR (M )), then provided X has at worst isolated singularities (e.g.
if X is a 3-fold with terminal singularities) then EndR (M ) has isolated singularities too.
This is a direct consequence of the fact that in this case the singular derived category has
finite dimensional Hom-spaces. Also note that if R is normal 3-sCY then the existence of
an MM algebra EndR (M ) with isolated singularities implies that Rp has finite CM type
for all primes p of height 2 by a result of Auslander (see [IW08, 2.13]). Finally note that
it follows immediately from 4.15(2) (after combining 2.19 and 2.24) that if R is normal
3-sCY and there is one MM algebra EndR (M ) with isolated singularities then necessary
all MM algebras EndR (N ) have isolated singularities.
The above remark suggests the following conjecture.
Conjecture 4.20. Let R be a normal 3-sCY ring with rational singularities. Then
(1) R always has an MM module M (which may be R).
(2) For all such M , EndR (M ) has isolated singularities.
This is closely related to a conjecture of Van den Bergh regarding the equivalence
of the existence of crepant and noncommutative crepant resolutions when R is a rational
normal Gorenstein 3-fold [V04b, 4.6]. We remark that given the assumption on rational
singularities, any proof is likely to be geometric. We also remark that the restriction
to rational singularities is strictly necessary, since we can consider any normal surface
singularity S of infinite CM type. Since S is a surface EndS (M ) ∈ CM S for all M ∈ CM S,
so since S has infinite CM type it cannot admit an MMA. Now consider R := S ⊗C C[t],
then R is a 3-fold that does not admit an MMA. A concrete example is given by R =
C[x, y, z, t]/x3 + y 3 + z 3 .
5. Relationship Between CT modules, NCCRs and MM modules
In this section we define CT modules for singularities that are not necessarily isolated,
and we show that they are a special case of the MM modules introduced in §4. We also
show (in 5.12) that all these notions recover the established ones when R is an isolated
singularity.
When R is a normal 3-sCY domain, below we prove the implications in the following picture which summarizes the relationship between CT modules, NCCRs and MM
modules:
5.4

CT modules

4.5

modules giving NCCRs

if generator
5.4

MM modules

modifying modules

if ∃NCCR
5.11

We remark that the relationship given by 4.5 and 5.4 holds in arbitrary dimension d,
whereas 5.11 requires d = 3.

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

25

Definition 5.1. Let R be a d-dimensional CM ring with a canonical module ωR . We call
M ∈ CM R a CT module if
add M = {X ∈ CM R : HomR (M, X) ∈ CM R} = {X ∈ CM R : HomR (X, M ) ∈ CM R}.
The name CT is inspired by (but is normally different than) the notion of ‘cluster
tilting’ modules. We explain the connection in 5.12.
Lemma 5.2. (1) Any CT module is a generator-cogenerator.
(2) Any CT module is maximal modifying.
Proof. Let M be a CT module.
(1) This is clear from HomR (M, ωR ) ∈ CM R and HomR (R, M ) ∈ CM R.
(2) Suppose N ∈ ref R with EndR (M ⊕N ) ∈ CM R, then certainly HomR (M, N ) ∈ CM R.
Since R ∈ add M by (1), we have N ∈ CM R. Hence since M is a CT module, necessarily
N ∈ add M .

Not every MM module is CT, however in the situation when R has a CT module
(equivalently, by 5.9(2) below, R has a NCCR) we give a rather remarkable relationship
between CT modules, MM modules and NCCRs in 5.11 at the end of this subsection.
If R is a CM ring with a canonical module ωR we denote the duality (−)∨ :=
HomR (−, ωR ). We shall see shortly that if M or M ∨ is a generator then we may test
the above CT condition on one side (see 5.4 below), but before we do this we need the
following easy observation.
Lemma 5.3. Let R be a CM ring with a canonical module ωR , M ∈ CM R. If EndR (M )
is a non-singular R-order, then R ∈ add M ⇐⇒ ωR ∈ add M .
∼ EndR (M )op
Proof. Since (−)∨ : CM R → CM R is a duality we know that EndR (M ∨ ) =
∨
∨
∨
and so EndR (M ) is also a non-singular R-order. Moreover R = ωR and ωR
= R so by
the symmetry of this situation we need only prove the ‘only if’ part. Thus assume that
R ∈ add M . In this case since HomR (M, ωR ) = M ∨ ∈ CM R, by 2.17 HomR (M, ωR ) is a
projective EndR (M )-module and thus ωR ∈ add M by 2.5(1).

We reach one of our main characterizations of CT modules. Note that if R is a normal
d-sCY ring, then by 2.9 the definition of CT modules is equivalent to
add M = {X ∈ CM R | HomR (M, X) ∈ CM R},
however the following argument works in greater generality:
Theorem 5.4. Let R be a d-dimensional, equi-codimensional CM ring (e.g. if R is dsCY) with a canonical module ωR . Then for any M ∈ CM R the following are equivalent
(1) M is a CT module.
(2) R ∈ add M and add M = {X ∈ CM R : HomR (M, X) ∈ CM R}.
′
(2) ωR ∈ add M and add M = {X ∈ CM R : HomR (X, M ) ∈ CM R}.
(3) R ∈ add M and EndR (M ) is a non-singular R-order.
′
(3) ωR ∈ add M and EndR (M ) is a non-singular R-order.
In particular CT modules are precisely the CM generators which give NCCRs.
Proof. (2)⇒(3) By assumption we have R ∈ add M and EndR (M ) ∈ CM R. Now let Y ∈
mod EndR (M ) and consider a projective resolution Pd−1 → Pd−2 → ... → P0 → Y → 0.
f

By 2.5(1) there is an exact sequence Md−1 → Md−2 → ... → M0 with each Mi ∈ add M
such that the projective resolution above is precisely
·f

HomR (M, Md−1 ) → HomR (M, Md−2 ) → ... → HomR (M, M0 ) → Y → 0.
Denote Kd = Ker f . Then we have an exact sequence
0 → HomR (M, Kd ) → Pd−1 → Pd−2 → ... → P0 → Y → 0.
Localizing the above and counting depths we see that HomR (M, Kd )m ∈ CM Rm for all
m ∈ Max R, thus HomR (M, Kd ) ∈ CM R and so by definition Kd ∈ add M . Hence
proj.dimEndR (M) Y ≤ d and so gl.dim EndR (M ) ≤ d.
(3)⇒(2) Since EndR (M ) ∈ CM R, automatically add M ⊆ {X ∈ CM R : HomR (M, X) ∈
CM R}. To obtain the reverse inclusion assume that X ∈ CM R with HomR (M, X) ∈

26

OSAMU IYAMA AND MICHAEL WEMYSS

CM R, then since EndR (M ) is a non-singular R-order HomR (M, X) is a projective EndR (M )module by 2.17. This implies that X ∈ add M by 2.5(1).
′
′
(2) ⇐⇒ (3) We have a duality (−)∨ : CM R → CM R thus apply (2) ⇐⇒ (3) to M ∨
and use the fact that EndR (M ∨ ) = EndR (M )op has finite global dimension if and only if
EndR (M ) does.
(3) ⇐⇒ (3)′ This is immediate from 5.3.
In particular by the above we have (2) ⇐⇒ (2)′ . Since we clearly have (1) ⇐⇒ (2)+(2)′ ,
the proof is completed.

Note that the last assertion in 5.4 is improved when R is a 3-sCY normal domain in
5.11(3). From the definition it is not entirely clear that CT is a local property:
Corollary 5.5. Let R be a d-dimensional, equi-codimensional CM ring (e.g. if R is dsCY) with a canonical module ωR . Then the following are equivalent
(1) M is a CT R-module
(2) Mp is a CT Rp -module for p ∈ Spec R.
(3) Mm is a CT Rm -module for m ∈ Max R.
cp is a CT R
cp -module for p ∈ Spec R.
(4) M
d
c
(5) M
m is a CT Rm -module for m ∈ Max R.
Thus CT can be checked locally, or even complete locally.
Proof. By 5.4(3) M is a CT R-module if and only if R ∈ add M and EndR (M ) is a nonsingular R-order. Non-singular R-orders can be checked either locally or complete locally

(2.17), and R ∈ add M can be checked locally or complete locally by 2.26.
Theorem 5.4 also gives an easy method to find examples of CT modules. Recall that
an element g ∈ GL(d, k) is called pseudo-reflection if the rank of g − 1 is at most one.
A finite subgroup G of GL(d, k) is called small if it does not contain pseudo-reflections
except the identity. The following is well-known:
Proposition 5.6. Let k be a field of characteristic zero, let S be the polynomial ring
k[x1 , . . . , xd ] (respectively formal power series ring k[[x1 , . . . , xd ]]) and let G be a finite
subgroup of GL(d, k).
(1) If G is generated by pseudo-reflections, then S G is a polynomial ring (respectively a
formal power series ring) in d variables.
(2) If G is small, then the natural map S#G → EndR (S) given by sg 7→ (t 7→ s · g(t))
(s, t ∈ S, g ∈ G) is an isomorphism.
Proof. (1) See [Bou68, §5 no. 5] for example.
(2) This is due to Auslander [Aus86, §4], [Y90, 10.8]. See also [IT10, 3.2] for a detailed
proof.

This immediately gives us a rich source of CT modules. The following result is shown
in [Iya07, 2.5] under the assumption that G is a small subgroup of GL(d, k) and S G is an
isolated singularity. We can drop both assumptions under our definition of CT modules.
Theorem 5.7. Let k be a field of characteristic zero, and let S be the polynomial ring
k[x1 , . . . , xd ] (respectively formal power series ring k[[x1 , . . . , xd ]]). For a finite subgroup
G of GL(d, k), let R = S G . Then S is a CT R-module.
Proof. We prove the case when S is a polynomial ring, since the case when S is a formal
power series ring then follows by 5.5. We proceed by induction on |G|, the case |G| = 1
being trivial. If G is small, then EndR (S) is isomorphic to S#G by 5.6(2). This shows,
by 2.12, that EndR (S) is a non-singular R-order and so S is a CT R-module by 5.4.
Hence we can assume that G is not small, so if N denotes the subgroup of G generated
by pseudo-reflections, we have |G/N | < |G|. Now G/N acts on S N , which by 5.6(1) is
a polynomial ring. In fact the graded subring S N of S has a free generating set of
homogeneous polynomials [C55, Thm. A]. Let V (d) be the vector space of N -invariant
polynomials of degree d (with respect to the original grading of S). We prove, by induction
on d, that generators of S N can be chosen so that G/N acts linearly on these generators.
Clearly the action of G/N is linear on U (d1 ) := V (d1 ), where d1 > 0 is the smallest
such that V (d1 ) is non-empty. Consider now V (d). It has a subspace, say U (d), of linear

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

27

combinations of products of N -invariant polynomials of smaller degree. By induction,
G/N acts on U (d), so U (d) is a G/N -submodule. By Maschke’s theorem we can take
the G/N -complement to U (d) in V (d). Then G/N acts linearly on this piece, so it acts
linearly on V (d). Hence a k-basis of U (d) for each d > 0 gives a free generating set on
which G/N acts linearly.
Hence, with these new generators, we have S G = (S N )G/N ∼
= k[X1 , . . . , Xd ]G/N where
G/N a subgroup of GL(d, k). Thus
CM S G ≃ CM k[X1 , . . . , Xd ]G/N
and further under this correspondence
addS G S ≃ addk[X1 ,...,Xd ]G/N S N = addk[X1 ,...,Xd ]G/N k[X1 , . . . , Xd ].
Hence EndS G (S) is Morita equivalent to Endk[X1 ,...,Xd ]G/N (k[X1 , . . . , Xd ]) := Λ. By induction Λ is a non-singular R-order, so it follows that EndS G (S) is a non-singular R-order
by 2.13. Consequently S is a CT S G -module by 5.4 since R is a direct summand of the
R-module S by the Reynolds operator.

As another source of CT modules, we have:
f

Example 5.8. Let Y → X = Spec R be a projective birational morphism such that
Rf∗ OY = OX and every fibre has dimension at most one, where R is a d-dimensional
normal Gorenstein ring R finitely generated over a field. Then provided Y is smooth and
crepant there exists a NCCR EndR (M ) [V04a, 3.2.10] in which M is CM containing R as
a summand. By 5.4 M is a CT module.
We now show that for R normal 3-sCY, the existence of a CT module is equivalent
to the existence of a NCCR. Note that (2) below answers a question of Van den Bergh
[V04b, 4.4].
Corollary 5.9. Let R be a 3-sCY normal domain. Then
(1) CT modules are precisely those reflexive generators which give NCCRs.
(2) R has a NCCR ⇐⇒ R has a NCCR given by a CM generator M ⇐⇒ CM R
contains a CT module.
Proof. Notice that any reflexive generator M which gives a NCCR is CM since R is a
summand of M and further M ∼
= HomR (R, M ) is a summand of EndR (M ) ∈ CM R as an
R-module.
(1) By 5.4 CT modules are precisely the CM generators which give NCCRs. The assertion
follows from the above remark.
(2) The latter equivalence was shown in 5.4. We only have to show (⇒) of the former
assertion. If R has a NCCR Λ, then Λ is an MMA (by 4.5) and so by 4.18(2) R has an
MMA Γ = EndR (M ) where M is a CM generator. But by 4.15(2) Γ and Λ are derived
equivalent, so since Λ is an NCCR, so too is Γ (4.6(2)).

Below is another characterization of CT modules, which is analogous to [Iya07, 2.2.3].
Compare this to the previous 4.12.
Proposition 5.10. Assume R is a 3-sCY normal domain and let M ∈ CM R with R ∈
add M . Then the following are equivalent
(1) M is a CT module.
(2) EndR (M ) ∈ CM R and further for all X ∈ CM R there exists an exact sequence
f

0 → M1 → M0 → X → 0 with M1 , M0 ∈ add M and a right (add M )-approximation f .
Proof. (1)⇒(2). Fix X ∈ CM R. Since R is 3-sCY, we have an exact sequence 0 → X →
P0 → P1 with each Pi ∈ add R. Applying HomR (M, −) gives an exact sequence
g

0 → HomR (M, X) → HomR (M, P0 ) → HomR (M, P1 ) → Cok g → 0.
Since both HomR (M, Pi ) are projective EndR (M )-modules by 2.5(1), and gl.dim EndR (M ) =
3 by 5.4, it follows that proj.dimEndR (M) HomR (M, X) ≤ 1. Consequently we may take
a projective resolution 0 → HomR (M, M1 ) → HomR (M, M0 ) → HomR (M, X) → 0 which
necessarily comes from a complex 0 → M1 → M0 → X → 0, again using 2.5(1). This

28

OSAMU IYAMA AND MICHAEL WEMYSS

complex is itself exact since M is a generator.
(2)⇒(1). Denote Γ = EndR (M ). By 2.22 and 2.21 Γ is a Gorenstein R-order. By
5.4 we only have to show that add M = {X ∈ CM R : HomR (M, X) ∈ CM R}. The
assumption EndR (M ) ∈ CM R shows that the inclusion ⊆ holds so let X ∈ CM R
be such that HomR (M, X) ∈ CM R. By assumption we may find M1 , M0 ∈ add M
such that 0 → HomR (M, M1 ) → HomR (M, M0 ) → HomR (M, X) → 0 is exact, hence
proj.dimΓm HomR (M, X)m ≤ 1 for all m ∈ Max R by 2.5(1). Since HomR (M, X)m ∈
CM Rm , Auslander–Buchsbaum (2.16) implies that proj.dimΓm HomR (M, X)m = 0 for all
m ∈ Max R and hence HomR (M, X) is a projective Γ-module. Since M is a generator,
X ∈ add M .

Provided an NCCR exists, the following shows the precise relationship between MM
modules, CT modules and NCCRs. Note that 5.11(2) says that CT modules are really a
special case of MM modules.
Proposition 5.11. Let R be a 3-sCY normal domain, and assume that R has a NCCR
(equivalently, by 5.9, a CT module). Then
(1) MM modules are precisely the reflexive modules which give NCCRs.
(2) MM modules which are CM (equivalently, by 4.2, the MM generators) are precisely
the CT modules.
(3) CT modules are precisely those CM modules which give NCCRs.
Proof. (1) (⇐) This is shown in 4.5 above.
(⇒) Suppose that M is an MM module, and let EndR (N ) be a NCCR. Then EndR (N )
is an MMA by 4.5 and so EndR (M ) and EndR (N ) are derived equivalent by 4.16. This
implies that EndR (M ) is also an NCCR by 4.6(2).
(2) By (1) MM generators are precisely the CM generators which give NCCRs. By 5.4
these are precisely the CT modules.
(3) Follows immediately from (1) and (2).

In the remainder of this section we relate our work to that of the more common
notions of n-rigid, maximal n-rigid and maximal n-orthogonal (=cluster tilting) modules
in the case when R is an isolated singularity.
Recall that M ∈ ref R is called n-rigid if ExtiR (M, M ) = 0 for all 1 ≤ i ≤ n. We call
M ∈ ref R maximal n-rigid if M is n-rigid and furthermore it is maximal with respect
to this property, namely if there exists X ∈ ref R such that M ⊕ X is n-rigid, then
X ∈ add M .
Recall that M ∈ CM R is called a maximal n-orthogonal module if
add M

= {X ∈ CM R | ExtiR (M, X) = 0 for all 1 ≤ i ≤ n}
= {X ∈ CM R | ExtiR (X, M ) = 0 for all 1 ≤ i ≤ n}.

Proposition 5.12. Let R be d-sCY with only isolated singularities, M ∈ CM R. Then
(1) M is a modifying module if and only if it is (d − 2)-rigid.
(2) M is a maximal modifying module if and only if it is maximal (d − 2)-rigid.
(3) M is a CT module if and only if it is maximal (d − 2)–orthogonal.
Proof. Let X, Y ∈ CM R. By 4.3 and 2.6, it follows that HomR (X, Y ) ∈ CM R if and
only if ExtiR (X, Y ) = 0 for all 1 ≤ i ≤ d − 2. Thus the assertions for (1), (2) and (3)
follow.

6. Mutations of Modifying Modules
6.1. Mutations and Derived Equivalences in Dimension d. Mutation is a technique used to obtain new modifying, maximal modifying and CT modules from a given
one. Many of our arguments work in the full generality of modifying modules although
sometimes it is necessary to restrict to the maximal modifying level to apply certain
arguments.
Throughout this section R will be a normal d-sCY ring, d ≥ 2, and M will be a
modifying module with N such that 0 6= N ∈ add M . Note that N may or may not be

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

29

decomposable. Given this, we define left and right mutation as in 1.21 in the introduction:
we have exact sequences
a

c

0 → K 0 → N0 → M
d

0 → K1 →

b
N1∗ →

(6.A)

M∗

(6.B)
∗

where a is a right (add N )-approximation and b is a right (add N )-approximation. We call
−
∗
them exchange sequences. From this we define µ+
N (M ) := N ⊕ K0 and µN (M ) := N ⊕ K1 .
Note that by the definition of approximations, N0 , N1 ∈ add N and we have exact
sequences
·c

·a

0 → HomR (N, K0 ) → HomR (N, N0 ) → HomR (N, M ) → 0
·d

∗

0 → HomR (N , K1 ) → HomR (N

∗

·b
, N1∗ ) →

∗

(6.C)

∗

HomR (N , M ) → 0.

(6.D)

−
Remark 6.1. (1) In general µ+
N (M ) and µN (M ) are not the same. Nevertheless, we will
+
see later in some special cases that µN (M ) = µ−
N (M ) holds (6.25), as in cluster tilting
theory [IY08, 5.3], [GLS, B06].
(2) A new feature of our mutation which is different from cluster tilting theory is that
−
G
µ+
N (M ) = M = µN (M ) can happen. A concrete example is given by taking R = k[x, y, z]
1
with G = 2 (1, 1, 0), M = k[x, y, z] and N = R.
−
Remark 6.2. If d = 3, then both µ+
N (M ) and µN (M ) are modifying R-modules by 4.11.
We will show in 6.10 that this is the case in any dimension.

We note that mutation is unique up to additive closure. This can be improved if R
is complete local.
a′

a

Lemma 6.3. Suppose N0 → M and N0′ → M are two right (add N )-approximations
of M . Then add(N ⊕ Ker a) = add(N ⊕ Ker a′ ). A similar statement holds for left
approximations.
Proof. Let K := Ker a and K ′ := Ker a′ . Then we have a commutative diagram
0

K

c

N0

s

K′

0

a

M

t
c′

N0′

a′

M

of exact sequences, giving an exact sequence
′

(ct )
(−s c)
0 → K −−−−→ K ′ ⊕ N0 −−→ N0′ .

(6.E)

From the commutative diagram
0

HomR (N, K)

·c

HomR (N, N0 )

·s

0

HomR (N, K ′ )

·a

HomR (N, M )

0

HomR (N, M )

0

·t
·c′

HomR (N, N0′ )

·a′

we see that
′

′

·(ct )

HomR (N, K ⊕ N0 ) −−−→ HomR (N, N0′ ) → 0
is exact. Thus (6.E) is a split short exact sequence, so in particular K ∈ add(N ⊕ K ′ ).
Similarly K ′ ∈ add(N ⊕ K).

Proposition 6.4. Let R be a normal d-sCY ring and let M be a modifying module with
0 6= N ∈ add M (i.e. notation as above). Then
(1) Applying HomR (−, N ) to (6.A) induces an exact sequence
a·

c·

0 → HomR (M, N ) → HomR (N0 , N ) → HomR (K0 , N ) → 0.

(6.F)

In particular c is a left (add N )-approximation.
(2) Applying HomR (−, N ∗ ) to (6.B) induces an exact sequence
b·

d·

0 → HomR (M ∗ , N ∗ ) → HomR (N1∗ , N ∗ ) → HomR (K1 , N ∗ ) → 0

(6.G)

30

OSAMU IYAMA AND MICHAEL WEMYSS

In particular d is a left (add N ∗ )-approximation.
(3) We have that
a∗

c∗

0 → M ∗ → N0∗ → K0∗
∗

b

d

(6.H)

∗

0 → M → N1 → K1∗

(6.I)

are exact, inducing exact sequences
·a∗

·c∗

c∗ ·

a∗ ·

0 → HomR (N ∗ , M ∗ ) → HomR (N ∗ , N0∗ ) → HomR (N ∗ , K0∗ ) → 0

(6.J)

0 → HomR (K0∗ , N ∗ ) → HomR (N0∗ , N ∗ ) → HomR (M ∗ , N ∗ ) → 0
·b∗

(6.K)

·d∗

0 → HomR (N, M ) → HomR (N, N1 ) → HomR (N, K1∗ ) → 0
d∗ ·

(6.L)

b∗ ·

0 → HomR (K1∗ , N ) → HomR (N1 , N ) → HomR (M, N ) → 0

(6.M)

Proof. Denote Λ := EndR (N ) and F := HomR (N, −).
(1) We note that (6.C) is
0 → FK0 → FN0 → FM → 0
so applying HomΛ (−, FN ) gives
0 → HomΛ (FM, FN ) → HomΛ (FN0 , FN ) → HomΛ (FK0 , FN ) → Ext1Λ (FM, Λ).
But by 2.22 Λ is d-sCY and thus a Gorenstein R-order by 2.21. Since FM ∈ CM Λ
and add Λ = add ωΛ by 2.15, it follows that Ext1Λ (FM, Λ) = 0 and hence we have a
commutative diagram of complexes
0

HomΛ (FM, FN )

0

HomR (M, N )

HomΛ (FN0 , FN )
a·

HomR (N0 , N )

c·

HomΛ (FK0 , FN )

0

HomR (K0 , N )

0

in which the top row is exact and the vertical maps are isomorphisms by reflexive equivalence 2.5(4). It follows that the bottom row is exact.
(2) is identical to (1) since HomR (N ∗ , M ∗ ) ∈ CM R.
(3) As in (1) applying HomΛ (−, FR) to (6.C) gives an commutative diagram of complexes
0

HomΛ (FM, FR)

0

HomR (M, R)

HomΛ (FN0 , FR)
a∗

HomR (N0 , R)

HomΛ (FK0 , FR)
c∗

HomR (K0 , R)

in which the top row is exact. Hence the bottom row (i.e. (6.H)) is exact. The proof that
(6.I) is exact is identical. Now since (−)∗ : ref R → ref R is a duality, the sequences (6.J),
(6.K), (6.L) and (6.M) are identical with (6.F), (6.C), (6.G) and (6.D) respectively. Thus
they are exact.

−
−
+
Proposition 6.5. µ+
N and µN are mutually inverse operations, i.e. we have µN (µN (M )) =
+
−
M and µN (µN (M )) = M , up to additive closure.
+
Proof. Since (6.H) and (6.J) are exact, we have µ−
N (µN (M )) = M . The other assertion
follows dually.


The following is standard in the theory of tilting mutation [RS91].
Lemma 6.6. Let Λ be a ring, let Q be a projective Λ-module and consider an exact
f
g
sequence Λ → Q′ → Cok f → 0 where f is a left (add Q)-approximation. If f is injective
then Q ⊕ Cok f is a tilting Λ-module of projective dimension at most one.
Proof. For the convenience of the reader we give a complete proof here. It is clear that
proj.dimΛ (Q ⊕ Cok f ) ≤ 1 and it generates the derived category. We need only to check
that Ext1Λ (Q ⊕ Cok f, Q ⊕ Cok f ) = 0. Applying HomΛ (−, Q), we have an exact sequence
f·

HomΛ (Q′ , Q) → HomΛ (Λ, Q) → Ext1Λ (Cok f, Q) → 0.

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

31

Since (f ·) is surjective, we have Ext1Λ (Cok f, Q) = 0. Applying HomΛ (−, Cok f ), we have
an exact sequence
f·

HomΛ (Q′ , Cok f ) → HomΛ (Λ, Cok f ) → Ext1Λ (Cok f, Cok f ) → 0.
f·

·g

Here (f ·) is surjective since HomΛ (Q′ , Q′ ) → HomΛ (Λ, Q′ ) and HomΛ (Λ, Q′ ) → HomΛ (Λ, Cok f )
are surjective. Thus we have Ext1Λ (Cok f, Cok f ) = 0. Consequently we have Ext1Λ (Q ⊕
Cok f, Q ⊕ Cok f ) = 0 since Q is projective.

The proof of 6.8 requires the following technical lemma.
Lemma 6.7. Let R be a normal domain, let Λ ∈ ref R be a module finite R-algebra and let
T ∈ mod Λ be a height one projective (i.e. Tp is a projective Λp -module for all p ∈ Spec R
with ht p ≤ 1) such that EndΛ (T ) ∈ ref R. Then EndΛ (T ) ∼
= EndΛop (T ∗ )op .
Proof. Consider the natural ring homomorphism
ψ:=(−)∗

EndΛ (T ) −−−−→ EndΛop (T ∗ )op
where recall (−)∗ := HomR (−, R). Note that T ∗ ∈ ref R by 2.5(2), i.e. T ∗ ∈ ref Λop . This
implies EndΛop (T ∗ )op ∈ ref R by 2.5(2).
Since T is a height one projective and Λ ∈ ref R, it follows that Tp ∈ ref Λp for all
height one primes p. Hence, via the anti–equivalence
(−)∗
p

ref Λp −−→ ref Λop
p ,
we have that ψ is a height one isomorphism.
By assumption EndΛ (T ) ∈ ref R holds. Since R is normal, ψ, being a height one
isomorphism between reflexive R-modules, is actually an isomorphism.

Theorem 6.8. Let R be a normal d-sCY ring with modifying module M . Suppose 0 6=
N ∈ add M . Then
(1) EndR (M ) and EndR (µ−
N (M )) are derived equivalent.
(2) EndR (M ) and EndR (µ+
N (M )) are derived equivalent.
Proof. (1) Denote Λ := EndR (M ) and F := HomR (M, −) : ref R → ref Λ. Applying F to
(6.I) and denoting V := Cok(·b∗ ), we have an exact sequence
0

FM

(·b∗ )

FK1∗ .

FN1
V

(6.N)

h

We now claim that (·b∗ ) is a left (add Q)-approximation where Q := HomR (M, N ) = FN .
Simply applying HomΛ (−, Q) = HomΛ (−, FN ) to the above we obtain
HomΛ (FN1 , FN )

HomΛ (FM, FN )

HomR (N1 , N )

HomR (M, N )

0

where the bottom is just (6.M) and so is exact, and the vertical maps are isomorphisms
by reflexive equivalence 2.5(4). Hence the top is surjective, showing that (·b∗ ) is a left
(add Q)-approximation. By 6.6 it follows that Q ⊕ V is a tilting Λ-module.
We now show that EndΛ (V ⊕ Q) ∼
= EndR (µ−
N (M )) by using 6.7. To do this, note
first that certainly Λ ∈ ref R since Λ ∈ CM R, and further Λ is d-sCY by 2.22(2). Hence
EndΛ (V ⊕ Q), being derived equivalent to Λ, is also d-sCY and so EndΛ (V ⊕ Q) ∈ ref R
by 2.21. We now claim that V ⊕ Q is a height one projective Λ-module.
Let p ∈ Spec R be a height one prime, then Mp ∈ ref Rp = add Rp . Hence Mp is a
free Rp -module, and so add Np = add Mp . Localizing (6.L) gives an exact sequence
0 → HomRp (Np , Mp ) → HomRp (Np , (N1 )p ) → HomRp (Np , (K1∗ )p ) → 0
and so since add Np = add Mp ,
0 → HomRp (Mp , Mp ) → HomRp (Mp , (N1 )p ) → HomRp (Mp , (K1∗ )p ) → 0

32

OSAMU IYAMA AND MICHAEL WEMYSS

is exact. This is (6.N) localized at p, hence we conclude that h is a height one isomorphism.
In particular Vp = HomRp (Mp , (K1∗ )p ) with both Mp , (K1∗ )p ∈ add Rp . Consequently V ,
thus V ⊕ Q, is a height one projective Λ-module.
Thus by 6.7 we have an isomorphism
EndΛ (V ⊕ Q) ∼
= EndΛop (V ∗ ⊕ Q∗ )op .
Now since h is a height one isomorphism, it follows that h∗ is a height one isomorphism.
But h∗ is a morphism between reflexive modules, so h∗ must be an isomorphism. We thus
have
V ∗ ⊕ Q∗ = (F(K1∗ ))∗ ⊕ Q∗ = (F(K1∗ ))∗ ⊕ (FN )∗ = (F(K1∗ ⊕ N ))∗ .
Consequently
∼ EndΛ (F(K ∗ ⊕ N ))
∼ EndΛop ((F(K ∗ ⊕ N ))∗ )op =
EndΛ (V ⊕ Q) =
1

1

since
(−)∗

(F(K1∗ ⊕ N ))∗ ∈ ref Λop −−−−→ ref Λ
is an anti–equivalence. This then yields
EndΛ (V ⊕ Q) ∼
= EndΛ (F(K ∗ ⊕ N )) ∼
= EndR (K ∗ ⊕ N ) = EndR (µ− (M )),
1

1

N

where the second isomorphism follows from reflexive equivalence 2.5.
∗
(2) Since M ∗ is a modifying R-module, by (1) EndR (M ∗ ) and EndR (µ−
N ∗ (M )) are derived
−
+
+
∗
∗
∗
equivalent. But µN ∗ (M ) = (µN (M )) , so EndR (M ) and EndR ((µN (M ))∗ ) are derived
op
equivalent. Hence EndR (M )op and EndR (µ+
are derived equivalent, which forces
N (M ))
+
EndR (M ) and EndR (µN (M )) to be derived equivalent [R89, 9.1].

Remark 6.9. By 6.8, for every 0 6= N ∈ add M we obtain an equivalence
TN := RHom(V ⊕ Q, −) : Db (mod EndR (M )) → Db (mod EndR (µ−
N (M ))).
Sometimes µ−
N (M ) = M can happen (see next subsection), but the functor TN is never
the identity provided add N 6= add M . This gives a way of generating autoequivalences of
the derived category.
Theorem 6.10. Let R be a normal d-sCY ring with modifying module M . Suppose
0 6= N ∈ add M . Then
−
(1) µ+
N (M ) and µN (M ) are modifying R-modules.
−
(2) If M gives an NCCR, so do µ+
N (M ) and µN (M ).
−
(3) Whenever N is a generator, if M is a CT module so are µ+
N (M ) and µN (M ).
(4) Whenever dim Sing R ≤ 1 (e.g. if d = 3), if M is a MM module so are µ+
N (M ) and
µ−
(M
).
N
+
Proof. Set Λ := EndR (M ). By 6.8, Λ, EndR (µ−
N (M )) and EndR (µN (M )) are all derived
equivalent. Hence (1) follows from 4.6(1), (2) follows from 4.6(2) and (4) follows from
4.8(2).
(3) Since M is CT, by definition M ∈ CM R. But N is a generator, so the a and b in
the exchange sequences (6.A) and (6.B) are surjective. Consequently both µ+
N (M ) and
5.4.
µ−
(M
)
are
CM
R-modules,
so
the
result
follows
from
(2)
and
N


One further corollary to 6.10 is the following application to syzygies and cosyzygies.
Usually syzygies and cosyzygies are only defined up to free summands, so let us first settle
some notation. Suppose that R is a normal d-sCY ring and M is a modifying generator.
Since M and M ∗ are finitely generated we can consider exact sequences
0 → K0 → P0 → M → 0

(6.O)

P1∗

(6.P)

0 → K1 →

∗

→M →0

−1
where P0 , P1 ∈ add R. We define ΩM := R ⊕ K0 = µ+
M := R ⊕ K1∗ =
R (M ) and Ω
−
i
µR (M ). Inductively we define Ω M for all i ∈ Z.
Our next result shows that modifying modules often come in infinite families, and
that in particular NCCRs often come in infinite families:

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

33

Corollary 6.11. Suppose that R is a normal d-sCY ring and M ∈ ref R is a modifying
generator. Then
(1) EndR (Ωi M ) are derived equivalent for all i ∈ Z.
(2) Ωi M ∈ CM R is a modifying generator for all i ∈ Z.
Proof. The assertions are immediate from 6.8 and 6.10.



6.2. Mutations and Derived Equivalences in Dimension 3. In the special case
d = 3, we can extend some of the above results, since we have more control over the
tilting modules produced from the procedure of mutation. Recall from the introduction
that given 0 6= N ∈ add M we define [N ] to be the two-sided ideal of Λ := EndR (M )
consisting of morphisms M → M which factor through a member of add N .
The factors ΛN := Λ/[N ] are, in some sense, replacements for simple modules in the
infinite global dimension setting. For example, we have the following necessary condition
for a module to be MM.
Proposition 6.12. Suppose that R is a normal 3-sCY ring, let M be an MM R-module
and denote Λ = EndR (M ). Then proj.dimΛ ΛN ≤ 3 for all N such that 0 6= N ∈ add M .
Proof. The sequence (6.A) 0 → K0 → N0 → M gives
0 → HomR (M, K0 ) → HomR (M, N0 ) → Λ → ΛN → 0
where HomR (M, N0 ) and Λ are projective Λ-modules. But K0 is a modifying module by 6.10, so by 4.12 we know that proj.dimΛ HomR (M, K0 ) ≤ 1. Hence certainly
proj.dimΛ ΛN ≤ 3.

Remark 6.13. The converse of 6.12 is not true, i.e. there exists non-maximal modifying
modules M such that proj.dimΛ ΛN ≤ 3 for all 0 6= N ∈ add M . An easy example is
given by M := R ⊕ (a, c2 ) for R := C[[a, b, c, d]]/(ab − c4 ). In this case the right (add R)approximation
2
(ca2 )
(− c inc)
0 → (a, c2 ) −−−a−−−→ R ⊕ R −−−→ (a, c2 ) → 0

shows that proj.dimΛ (Λ/[(a, c2 )]) = 2, whilst the right (add(a, c2 ))-approximation


2

(−a c )

inc
c2
a



0 → R −−−−−→ (a, c2 ) ⊕ (a, c2 ) −−−−−→ R
shows that proj.dimΛ (Λ/[R]) = 2. Also Λ/[M ] = 0 and so trivially proj.dimΛ (Λ/[M ]) = 0.
Hence proj.dimΛ ΛN ≤ 3 for all 0 6= N ∈ add M , however EndR (M ⊕ (a, c)) ∈ CM R with
(a, c) ∈
/ add M , so M is not an MM module.
Roughly speaking, mutation in dimension d = 3 is controlled by the factor algebra
ΛN , in particular whether it is artinian or not. When it is artinian, the derived equivalence
in 6.8 is given by a very explicit tilting module.
Theorem 6.14. Let R be a normal 3-sCY ring with modifying module M . Suppose that
0 6= N ∈ add M and denote Λ = EndR (M ). If ΛN = Λ/[N ] is artinian then
−
∼
(1) T1 := HomR (M, µ−
N (M )) is a tilting Λ-module such that EndΛ (T1 ) = EndR (µN (M )).
+
op
∗
∗
op
∼
(2) T2 := HomR (M , µN (M ) ) is a tilting Λ -module such that EndΛop (T2 ) = EndR (µ+
N (M )) .
Remark 6.15. In the setting of 6.14, we have the following.
(1) ΛN is artinian if and only if add Mp = add Np for all p ∈ Spec R with ht p = 2.
(2) If R is finitely generated over a field k then ΛN is artinian if and only if dimk ΛN < ∞.
Thus if the reader is willing to work over C, they may replace the condition ΛN is artinian
by dimC ΛN < ∞ throughout.
Proof of 6.14. (1) Denote G := HomR (N, −) and Γ := EndR (N ). Applying HomR (M, −)
to (6.I) and HomΓ (GM, −) to (6.L) gives a commutative diagram
0

HomΓ (GM, GM )

0

HomR (M, M )

HomΓ (GM, GK1∗ )

HomΓ (GM, GN1 )
·b∗

HomR (M, N1 )

·d∗

HomR (M, K1∗ )

Ext1Γ (GM, GM )
C

0

34

OSAMU IYAMA AND MICHAEL WEMYSS

where the vertical maps are isomorphisms by 2.5(4), hence C ⊆ Ext1Γ (GM, GM ). We
first claim that C = 0. Since EndΓ (GM ) ∼
= Λ by reflexive equivalence 2.5, by 2.7
fl Ext1Γ (GM, GM ) = 0. On the other hand HomR (N, −) applied to (6.I) is exact (by
6.4), so C is annihilated by [N ] and consequently C is a ΛN -module. Since ΛN is artinian
so too is C, thus it has finite length. Hence C = 0 and so
0 → HomR (M, M ) → HomR (M, N1 ) → HomR (M, K1∗ ) → 0

(6.Q)

is exact. Thus the tilting module V ⊕ Q in the proof of 6.8(1) is simply HomR (M, K1∗ ) ⊕
HomR (M, N ) = T1 . The remaining statements are contained in 6.8(1).
(2) Similarly to the above one can show that applying HomR (M ∗ , −) to (6.H) gives an
exact sequence
0 → HomR (M ∗ , M ∗ ) → HomR (M ∗ , N0∗ ) → HomR (M ∗ , K0∗ ) → 0

(6.R)
∗

and so the tilting module V ⊕ Q in the proof of 6.8(2) is simply HomR (M , K0∗ ) ⊕
∗
HomR (M ∗ , N ∗ ) = HomR (M ∗ , µ+

N (M ) ).
Remark 6.16. Note that the statement in 6.14 is quite subtle. There are examples where
−
∗
∗
HomR (M, µ+
N (M )) (respectively, HomR (M , µN (M ) )) is not a tilting EndR (M )-module
op
(respectively, EndR (M ) -module). Note however that these are always tilting modules
if M is an MM module, by combining 4.14(2) and 6.10(4).
If ΛN is artinian, the module M changes under mutation:
Proposition 6.17. Let R be a normal 3-sCY ring with modifying module M . Suppose
0 6= N ∈ add M , denote Λ = EndR (M ) and define ΛN := Λ/[N ]. If ΛN is artinian then
(1) If add N 6= add M then add µ+
N (M ) 6= add M .
(2) If add N 6= add M then add µ−
N (M ) 6= add M .
Proof. (1) Since ΛN is artinian, the sequence (6.R)
0 → HomR (M ∗ , M ∗ ) → HomR (M ∗ , N0∗ ) → HomR (M ∗ , K0∗ ) → 0
is exact. If this splits then by reflexive equivalence (2.5(4)) M ∗ is a summand of N0∗ ,
contradicting add N 6= add M . Thus the above cannot split so HomR (M ∗ , K0∗ ) cannot be
projective, hence certainly K0∗ ∈
/ add M ∗ and so K0 ∈
/ add M . This implies add µ+
N (M ) 6=
add M .
(2) Similarly, the exact sequence (6.Q) cannot split, so K1∗ ∈
/ add M .

Remark 6.18. It is natural to ask under what circumstances the hypothesis ΛN is artinian in 6.14, 6.17 holds. In the situation of 5.8 the answer seems to be related to the
contractibility of the corresponding curves; we will come back to this question in future
work.
One case where ΛN is always artinian is when R has isolated singularities:
Lemma 6.19. Suppose R is a normal 3-sCY ring. Let M be a modifying module with
0 6= N ∈ add M , denote Λ = EndR (M ) and set ΛN = Λ/[N ]. Then
(1) dimR ΛN ≤ 1.
(2) depthRm (ΛN )m ≤ 1 for all m ∈ Max R.
(3) If R is an isolated singularity then ΛN is artinian.
(4) If proj.dimΛ ΛN < ∞ then inj.dimΛN ΛN ≤ 1.
Proof. (1) We have (EndR (M )/[N ])p = EndRp (Mp )/[Np ] for all p ∈ Spec R. Since R
is normal, add Mp = add Rp = add Np for all p ∈ Spec R with ht p = 1. Thus we have
(EndR (M )/[N ])p = EndRp (Mp )/[Np ] = 0 for all these primes, and so the assertion follows.
(2) is immediate from (1).
(3) If R is isolated then by the argument in the proof of (1) we have dimR ΛN = 0 and so
ΛN is supported only at a finite number of maximal ideals. Hence ΛN has finite length
and so ΛN is artinian.
(4) Notice that Λ is 3-sCY by 2.22. Hence the assertion follows from [IR08, 5.5(3)]
for 3-CY algebras, which is also valid for 3-sCY algebras under the assumption that
proj.dimΛ ΛN < ∞.


MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

35

We now show that mutation does not change the factor algebra ΛN . Suppose M is
modifying and N is such that 0 6= N ∈ add M , and consider an exchange sequence (6.A)
a

c

0 → K0 → N0 → M.
We know by definition that a is a right (add N )-approximation, and by (6.F) that c is a
left (add N )-approximation.
Since ΛN is by definition EndR (M ) factored out by the ideal of all morphisms M → M
which factor through a module in add N , in light of the approximation property of the
map a, this ideal is the just the ideal Ia of all morphisms M → M which factor as xa
where x is some morphism M → N0 . Thus ΛN = EndR (M )/Ia .
On the other hand taking the choice µ+
N (M ) = K0 ⊕ N coming from the above
′
exchange sequence, ΛN is by definition EndR (µ+
N (M )) = EndR (K0 ⊕ N ) factored out by
the ideal of all morphisms K0 ⊕ N → K0 ⊕ N which factor through a module in add N .
Clearly this is just EndR (K0 ) factored out by those morphisms which factor through
add N . In light of the approximation property of the map c, Λ′N = EndR (K0 )/Ic where
Ic is the ideal of all morphisms K0 → K0 which factor as cy where y is some morphism
K 0 → N0 .
Theorem 6.20. Let R be a normal d-sCY ring, and let M be a modifying module with
0 6= N ∈ add M . With the notation and choice of exchange sequence as above, we have
ΛN ∼
= Λ′N as R-algebras. In particular
(1) Λ′N is independent of the choice of exchange sequence, up to isomorphism.
(2) ΛN is artinian if and only if Λ′N is artinian.
Proof. We construct a map α : ΛN = EndR (M )/Ia → EndR (K0 )/Ic = Λ′N as follows:
given f ∈ EndR (M ) we have
0

c

K0

N0

∃ hf

0

a

M

∃ gf
c

K0

N0

f
a

M

where the gf exists (non-uniquely) since a is an approximation. Define α by α(f + Ia ) =
hf + Ic . We will show that α : ΛN → Λ′N is a well-defined map, which is independent of
the choice of gf . Take f ′ ∈ Λ satisfying f − f ′ ∈ Ia . We have a commutative diagram
0

K0

c

N0

∃ hf ′

0

K0

a

M

∃gf ′

c

N0

f′
a

M

There exists x : M → N0 such that xa = f − f ′ . Thus (gf − gf ′ − ax)a = 0 so there exists
y : N0 → K0 such that yc = gf −gf ′ −ax. This implies cyc = c(gf −gf ′ −ax) = (hf −hf ′ )c,
so since c is a monomorphism we have cy = hf − hf ′ . Thus hf + Ic = hf ′ + Ic holds, and
we have the assertion.
It is easy to check that α is an R-algebra homomorphism since α is independent of
the choice of gf .
We now show that α is bijective by constructing the inverse map β : Λ′N → ΛN . Let
t : K0 → K0 be any morphism then on dualizing we have
0

M∗

a∗

∃r

0

M∗

N0∗

c∗

t∗

∃s
a

∗

N0∗

K0∗

c

∗

K0∗

where the rows are exact by (6.H), s exists (non-uniquely) by (6.J) and r exists since a∗
is the kernel of c∗ . Let β(t + Ic ) := r∗ + Ia . By the same argument as above, we have
that β : Λ′N → ΛN is a well-defined map.

36

OSAMU IYAMA AND MICHAEL WEMYSS

Dualizing back gives a commutative diagram
c

K0

0

M

∗

t

r∗

s
c

K0

0

a

N0
N0

a

M

which shows that β is the inverse of α.



6.3. Complete Local Case. In this subsection we assume that R is a complete local
normal Gorenstein d-dimensional ring, then since we have Krull–Schmidt decompositions
we can say more than in the previous section. Note that with these assumptions R is
automatically d-sCY by 2.20. For a modifying module M we write
M
M = M1 ⊕ . . . ⊕ Mn =
Mi
i∈I

as its Krull–Schmidt decomposition into indecomposable submodules, where I = {1, . . . , n}.
Throughout we assume that M is basic, i.e. the Mi ’s are mutually non-isomorphic. With
the new assumption on R we may take minimal approximations
and so the setupLin the preL
M
vious section can be simplified: for ∅ 6= J ⊆ I set MJ := j∈J Mj and M
:= i∈I\J Mi .
J
Then
a
M
(a) Denote L0 → MJ to be a right (add M
)-approximation of MJ which is right
J
M
minimal. If MJ contains R as a summand then necessarily a is surjective.
b

∗

∗
(b) Similarly denote L∗1 → MJ∗ to be a right (add M
M ∗ )-approximation of MJ which is

right minimal. Again if

M
MJ

J

contains R as a summand then b is surjective.

Recall that a morphism a : X → Y is called right minimal if any f ∈ EndR (X) satisfying
a = f a is an automorphism. In what follows we denote the kernels of the above right
minimal approximations by
a

c

0 → C0 → L0 → MJ

d

b

and 0 → C1 → L∗1 → MJ∗ .

This recovers the mutations from the previous subsection:
Lemma 6.21. With notation as above, µ+M (M ) =
MJ

M
MJ

⊕ C0 and µ−M (M ) =
MJ

M
MJ

⊕ C1∗ .

Proof. There is an exact sequence
(c 0)

0 → C0 → L0 ⊕

M
MJ

(a0 10)
→ MJ ⊕

M
MJ

=M →0

M
)-approximation ( a0 10 ). Thus the assertion follows.
with a right (add M
J



Since we have minimal approximations from now on we define our mutations in terms
−
M
M
∗
of them. We thus define µ+
J (M ) := C0 ⊕ MJ and µJ (M ) := C1 ⊕ MJ . When J = {i}
−
+
−
+
we often write µi and µi instead of µ{i} and µ{i} respectively. Note that using this
−
new definition of mutation involving minimal approximations, µ+
J and µJ are now inverse
operations up to isomorphism, not just additive closure. This strengthens 6.5.
We now investigate, in dimension three, the mutation of an MM module at an indecomposable summand Mi . Let ei denote the idempotent in Λ := EndR (M ) corresponding
to the summand Mi , then the theory depends on whether or not Λi := Λ/Λ(1 − ei )Λ is
artinian.
Theorem 6.22. Suppose R is complete local normal 3-sCY and let M be an MM module
with indecomposable summand Mi . Denote Λ = EndR (M ), let ei be the idempotent corre−
sponding to Mi and denote Λi = Λ/Λ(1 − ei )Λ. If Λi is artinian, then µ+
i (M ) = µi (M )
and this is not equal to M .
−
Proof. We know that M , µ+
i (M ) and µi (M ) are all MM modules by 6.10, thus by 4.17(1)
+
it follows that HomR (M, µi (M )) and HomR (M, µ−
i (M )) are both tilting EndR (M )−
modules. But since µ+
i (M ) 6= M and µi (M ) 6= M by 6.17, neither of these tilting modules
equal HomR (M, M ). Further by construction, as EndR (M )-modules HomR (M, µ+
i (M ))

MM MODULES AND AR DUALITY FOR NON-ISOLATED SINGULARITIES.

37

and HomR (M, µ−
i (M )) share all summands except possibly one, thus by a Riedtmann–
∼
Schofield type theorem [IR08, 4.2] [RS91, 1.3], they must coincide, i.e. HomR (M, µ+
i (M )) =
+
−
−
∼
HomR (M, µi (M )). By reflexive equivalence 2.5(4) we deduce that µi (M ) = µi (M ). 
The case when Λi is not artinian is very different:
Theorem 6.23. Suppose R is complete local normal 3-sCY and let M be a modifying
module with indecomposable summand Mi . Denote Λ = EndR (M ), let ei be the idempotent
corresponding to Mi and denote Λi = Λ/Λ(1 − ei )Λ. If Λi is not artinian, then
−
(1) If proj.dimΛ Λi < ∞, then µ+
i (M ) = µi (M ) = M .
+
(2) If M is an MM module, then always µi (M ) = µ−
i (M ) = M .
Proof. (1) It is always true that depthR Λi ≤ dimR Λi ≤ inj.dimΛi Λi by [GN02, 3.5]
(see [IR08, 2.1]). Since proj.dimΛ Λi < ∞, by 6.19(4) we know that inj.dimΛi Λi ≤ 1.
Since Λi is local and inj.dimΛi Λi ≤ 1, depthR Λi = inj.dimΛi Λi by Ramras [Ram, 2.15].
If dimR Λi = 0 then Λi has finite length, contradicting the assumption that Λi is not
artinian. Thus depthR Λi = dimR Λi = inj.dimΛi Λi = 1. In particular Λi is a CM
R-module of dimension 1.
Now Λ is a Gorenstein R-order by 2.21 and 2.22 so by Auslander–Buchsbaum 2.16,
since proj.dimΛ Λi < ∞ necessarily proj.dimΛ Λi = 3 − depthR Λi = 2. Thus we have a
minimal projective resolution
f

0 → P2 → P1 → Λei → Λi → 0.

(6.S)

where f is a minimal right (add Λ(1 − ei ))-approximation since it is a projective cover of
Λ(1 − ei )Λei . By [IR08, 3.4(5)] we have
Ext2Λ (Λi , Λ) ∼
= Ext2R (Λi , R)
op
and this is a projective Λop
i -module by [GN02, 1.1(3)]. It is a free Λi -module since Λi is a
2
local ring. Since Λi is a CM R-module of dimension 1, we have ExtR (Ext2R (Λi , R), R) ∼
= Λi
op
as Λi -modules. Thus the rank has to be one and we have Ext2R (Λi , R) ∼
= Λi as Λi modules. Applying HomΛ (−, Λ) to (6.S) gives an exact sequence

HomΛ (P1 , Λ) → HomΛ (P2 , Λ) → Λi → 0
which gives a minimal projective presentation of the Λop -module Λi . Thus we have
HomΛ (P2 , Λ) ∼
= Λei .
= ei Λ and P2 ∼
Under the equivalence HomR (M, −) : add M → proj Λ, the sequence (6.S) corresponds to a complex
h

g

0 → Mi → L0 → Mi
M
)-approximation. Since the induced morphism Mi → Ker g
with g a minimal right (add M
i
is sent to an isomorphism under the reflexive equivalence HomR (M, −) : ref R → ref Λ
M
(2.5(4)), it is an isomorphism and so h = ker g. Consequently we have µ+
i (M ) = Mi ⊕Mi =
−
M . This implies that µi (M ) = M by 6.5.
(2) This follows from (1) since proj.dimΛ Λi < ∞ by 6.12.


Remark 6.24. The above theorem needs the assumption that Mi is indecomposable. If
we assume that |J| ≥ 2 and ΛJ is still not artinian, then both examples with µ−
J (M ) 6= M
and those with µ−
(M
)
=
M
exist.
See
for
example
[IW12,
§5]
for
more
details.
J
In dimension three when the base R is complete local, we have the following summary,
which completely characterizes mutation at an indecomposable summand.
Summary 6.25. Suppose R is complete normal 3-sCY with MM module M . Denote Λ =
EndR (M ), let Mi be an indecomposable summand of M and consider Λi := Λ/Λ(1 − ei )Λ
where ei is the idempotent in Λ corresponding to Mi . Then
−
(1) If Λi is not artinian then µ+
i (M ) = M = µi (M ).
+
−
(2) If Λi is artinian then µi (M ) = µi (M ) and this is not equal to M .
−
In either case denote µi := µ+
i = µi then it is also true that
(3) µi µi (M ) = M .
(4) µi (M ) is a MM module.

38

OSAMU IYAMA AND MICHAEL WEMYSS

(5) EndR (M ) and EndR (µi (M )) are derived equivalent, via the tilting EndR (M )-module
HomR (M, µi (M )).
Proof. (1) is 6.23 and (2) is 6.22. The remainder is trivially true in the case when Λi is not
−
artinian (by 6.23), thus we may assume that Λi is artinian. Now µi µi (M ) = µ+
i (µi M ) =

M by 6.3, proving (3). (4) is contained in 6.10 and (5) is 6.14(1).
Example 6.26. Consider the subgroup G = 12 (1, 1, 0) ⊕ 12 (0, 1, 1) of SL(3, k) and let
R = k[[x, y, z]]G . We know by 5.7 that M = k[[x, y, z]] is a CT R-module, and in this
example it decomposes into 4 summands R ⊕ M1 ⊕ M2 ⊕ M3 with respect to the characters
of G. The quiver of EndR (M ) is the McKay quiver
y
y

M1
z
xx
R

M2
z

z

xx

z
y
y

M3

and so to mutate at M2 it is clear that the relevant approximation is


z
y
x



R ⊕ M1 ⊕ M3 → M2 → 0
Thus the mutation at vertex M2 changes M = R ⊕ M1 ⊕ M2 ⊕ M3 into R ⊕ M1 ⊕ K2 ⊕ M3
where K2 is the kernel of the above map which (by counting ranks) has rank 2. On the
level of quivers of the endomorphism rings, this induces the mutation

M1

M1

M2

M3
K2

µ2

R

M3
R

Due to the relations in the algebra EndR (µ2 (M )) (which we suppress), the mutation at
R, M1 and M3 in the new quiver are trivial, thus in EndR (µ2 (M )) the only vertex we can
non-trivially mutate at is K2 , which gives us back our original M . By the symmetry of
the situation we obtain the beginning of the mutation graph:
R⊕M1 ⊕M2 ⊕K3
µ3
R⊕K1 ⊕M2 ⊕M3

µ1

R⊕M1 ⊕M2 ⊕M3

µ2

R⊕M1 ⊕K2 ⊕M3

We remark that mutating at any of the decomposable modules M1 ⊕ M2 , M1 ⊕ M3 ,
M2 ⊕ M3 or M1 ⊕ M2 ⊕ M3 gives a trivial mutation. Note that the mutation µ+
M/R (M )
at the vertex R is not a CM R-module, and so we suppress the details.
Acknowledgements. We would like to thank Michel Van den Bergh, Vanya Cheltsov,
Constantin Shramov, Ryo Takahashi and Yuji Yoshino for stimulating discussions and
valuable suggestions. We also thank the anonymous referee for carefully reading the
paper, and offering many useful insights and suggested improvements.
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Osamu Iyama, Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya,
464-8602, Japan
E-mail address: iyama@math.nagoya-u.ac.jp
Michael Wemyss, The Maxwell Institute, School of Mathematics, James Clerk Maxwell
Building, The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK.
E-mail address: wemyss.m@googlemail.com

