SINGULAR EQUIVALENCES OF COMMUTATIVE NOETHERIANRINGS AND RECONSTRUCTION OF SINGULAR LOCIHIROKI MATSUI
Abstract. Two left noetherian rings R and S are said to be singularly equivalent if their
singularity categories are equivalent as triangulated categories. The aim of this paper
is to give a necessary condition for two commutative noetherian rings to be singularly
equivalent. To do this, we develop a support theory for triangulated categories without
tensor structure.

1. Introduction
Let R be a left noetherian ring. The singularity category of R is by definition the
Verdier quotientDsg (R) := Db (modR)/Kb (projR),
which has been introduced by Buchweitz [Buc]. Here, mod R stands for the categoryof finitely generated left R-modules and Db (modR) its bounded derived category, andKb (projR) the homotopy category of finitely generated projective R-modules. The singularity categories have been deeply investigated from algebro-geometric and representationtheoretic motivations [Che, IW, Kra, Ste, ZZ] and recently connected to the Homological
Mirror Symmetry Conjecture by Orlov [Orl04].
One of the important subjects in representation theory of rings is to classify rings up
to certain category equivalences. For example, left noetherian rings R and S are said tobe:• Morita equivalent if mod R ∼= mod S as abelian categories,b• derived equivalent if D (mod R) ∼= Db (mod S) as triangulated categories,• singularly equivalent if Dsg (R) ∼= Dsg (S) as triangulated categories.
It is well-known that the following implications hold.
Morita equivalence ⇒ derived equivalence ⇒ singular equivalence.
Complete characterizations of Morita equivalence and derived equivalence have alreadybeen obtained in [Mor, Ric], while singular equivalence is quite diﬃcult to characterizeeven in the case of commutative rings. Indeed, only a few examples of singular equivalencesof commutative noetherian rings are known. For all of such known examples, the singularloci of rings are homeomorphic. Thus, it is natural to ask the following question.
2010 Mathematics Subject Classification. 13C14, 13D09, 18E30.
Key words and phrases. triangulated category, triangle equivalence, (classifying) support data, quasiaﬃne scheme, finite p-group, Gorenstein ring.
The author was partly supported by Grant-in-Aid for JSPS Fellows 16J01067.
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2

HIROKI MATSUIQuestion 1.1. Let R and S be commutative noetherian rings. Are their singular locihomeomorphic if R and S are singularly equivalent?
The main purpose of this paper is to show that this question is aﬃrmative for certainclasses of commutative noetherian rings. To be precise, we shall prove the following
theorem.
Theorem 1.2 (Theorem 4.4). Let R and S be excellent commutative noetherian local
rings that are locally hypersurfaces on the punctured spectra. Assume that R and S are
either(a) Gorenstein rings, or(b) Cohen-Macaulay rings with quasi-decomposable maximal ideal.
If R and S are singularly equivalent, then their singular loci Sing R and Sing S are homeomorphic.
Here, we say that an ideal I of a commutative noetherian ring R is quasi-decomposable
if there is an R-regular sequence x in I such that I/(x) is decomposable as an R-module.
The key role to prove this theorem is played by a support theory for triangulated categories. A support theory has been developed by Balmer [Bal02, Bal05] for tensor triangulated categories and is a powerful tool to show such a reconstruction theorem. However,singularity categories do not have tensor triangulated structure in general. For this reason, we develop a support theory without tensor structure which is motivated by Balmer’s
work [Bal05].
We have considered only topological structures of singular loci so far. Of course, thetopological structure gives us much less information than the scheme structure. For
instance, we can not distinguish isolated singularities via the topological structures of
those singular loci. Therefore, reconstructing scheme structures of singular loci fromsingularity categories is a more interesting problem. For a Henselian Gorenstein local ring
R with an isolated singularity (i.e., dim Sing R = 0), its singular locus can be reconstructedfrom its singularity category. For details, see Remark 5.1. Thus, the next step we haveto consider is the case of commutative noetherian rings R with dim Sing R = 1. In thispaper, we consider the following particular singularities.
Let k be an algebraically closed field of characteristic 0. We say that a hypersurface
singularity R = k[[x0 , x1 , . . . , xd ]]/(f ) is of type (A∞ ) or (D∞ ) if f is(A∞ ) x20 + x22 + · · · + x2d , or(D∞ ) x20 x1 + x22 + · · · + x2d .
For these singularities, we obtain the following result.
Theorem 1.3 (Theorem 5.3). Let R be a hypersurface singularity of type (A∞ ) or (D∞ )and M ∈ Dsg (R) an indecomposable object with Sing R = SSuppR (M ). Then EndDsg (R) (M )is a commutative ring and there is a scheme isomorphism
Sing R ∼= Spec EndDsg (R) (M ).
Here, we think of Sing R as the reduced closed subscheme of Spec R.

SINGULAR EQUIVALENCES AND RECONSTRUCTION OF SINGULAR LOCI3

The symbol SSuppR (M ) stands for the singular support of M , which is by definition
the set of prime ideals p of R such that Mp ∼
̸= 0 in Dsg (Rp ). As it will be proved in Theorem 4.7, the condition Sing R = SSuppR (M ) is characterized by using the triangulated
structure on Dsg (R). Thus, this theorem states that the scheme structure on Sing R canbe reconstructed from Dsg (R).
The organization of this paper is as follows. In section 2, we introduce the notionsof a support data and a classifying support data for a given triangulated category anddevelop our support theory without tensor structure. In section 3, we connect the results
obtained in section 2 with the support theory for tensor triangulated categories and study
reconstructing the topologies of the Balmer spectra without tensor structure. Moreover,
we give some applications to algebraic geometry and modular representation theory. In
section 4, we prove Theorem 1.2 and give examples of commutative rings which are notsingularly equivalent. In section 5, we consider reconstructing scheme structures of singular loci from singularity categories. Actually, we prove that the singular locus of a
hypersurface local ring of countable Cohen-Macaulay representation type appears as the
Zariski spectrum of the endomorphism ring of some object of the singularity category.
Throughout this paper, all categories are assumed to be essentially small. For two
triangulated category T , T ′ (resp. topological spaces X, X ′ ), the notation T ∼= T ′ (resp.
′′∼
X = X ) means that T and T are equivalent as triangulated categories (resp. X and X ′are homeomorphic) unless otherwise specified.
2. The support theory without tensor structure
In this section, we discuss the support theory for triangulated categories without tensor
structure. Throughout this section, T denotes a triangulated category with shift functor
Σ.
First of all, let us recall some basic definitions which are used in this section.
Definition 2.1. Let X be a topological space and T a triangulated category.
(1) We say that X is sober if every irreducible closed subset of X is the closure of exactly
one point.
(2) We say that X is noetherian if every descending chain of closed subspaces stabilizes.
(3) We say that a subset W of X is specialization-closed if it is closed under specialization,
namely if an element x of X belongs to W , then the closure {x} is contained in W .
Note that W is specialization-closed if and only if it is a union of closed subspaces ofX.
(4) We say that a non-empty additive full subcategory X of T is thick if it satisfies thefollowing conditions:(i) closed under taking shifts: ΣX = X .
(ii) closed under taking extensions: for a triangle L → M → N → ΣL in T , if L
and N belong to X , then so does M .
(iii) closed under taking direct summands: for two objects L, M of T , if the direct
sum L ⊕ M belongs to X , then so do L and M .
For a subcategory X of T , denote by thickT X the smallest thick subcategory of T
containing X .

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We introduce the notion of a support data for a triangulated category.
Definition 2.2. Let T be a triangulated category. A support data for T is a pair (X, σ)
where X is a topological space and σ is an assignment which assigns to an object M of
T a closed subset σ(M ) of X satisfying the following conditions:(1) σ(0) = ∅.
(2) σ(Σn M ) = σ(M ) for any M ∈ T and n ∈ Z.
(3) σ(M ⊕ N ) = σ(M ) ∪ σ(N ) for any M, N ∈ T .
(4) σ(M ) ⊆ σ(L) ∪ σ(N ) for any triangle L → M → N → ΣL in T .
Support data naturally appear in various areas of algebras. Before giving examples of
support data, let us recall the notion of the stable category of maximal Cohen-Macaulay
modules.
Definition 2.3. Let R be a Gorenstein ring. Denote by CM(R) the category of maximal
Cohen-Macaulay R-modules. (i.e., modules M satisfying ExtiR (M, R) = 0 for all integers
i > 0). The stable category CM(R) of CM(R) is the category whose objects are the sameas CM(R) and the set of morphisms from M to N is given by
HomR (M, N ) := HomR (M, N )/PR (M, N ),where PR (M, N ) consists of all R-linear maps from M to N factoring through some free
R-module.
Example 2.4. Let R be a commutative noetherian ring. For M ∈ Dsg (R), we define thesingular support of M by
SSuppR (M ) := {p ∈ Sing R | Mp ∼
̸= 0 in Dsg (Rp )}.
Then (Sing R, SSuppR ) is a support data for Dsg (R). Indeed, SSuppR clearly satisfies thecondition (1) in Definition 2.2 and it follows from [AIL, Theorem 1.1] and [BM, Lemma
4.5] that SSuppR (M ) is a closed subset of Sing R. The remained conditions (2)-(4) also
hold true because the localization functor Dsg (R) → Dsg (Rp ) is exact.
Assume that R is Gorenstein. Then the stable category CM(R) has the structure of atriangulated category; see [Hap]. Moreover, the natural inclusion induces a triangle equiv∼=→ Dsg (R) by [Buc]. Thus we obtain the support data (Sing R, SuppR )alence F : CM(R) −for CM(R) by using this equivalence. Here,SuppR (M ) := SSuppR (F (M )) = {p ∈ Sing R | Mp ̸∼= 0 in CM(Rp )}for M ∈ CM(R).
Example 2.5. Let X be a noetherian scheme. For F ∈ Dperf (X), we define the cohomological support of F by
SuppX (F) := {x ∈ X | Fx ∼
̸= 0 in Dperf (OX,x )}.
∪Then, SuppX (F) = n∈Z SuppX (Hn (F)) is a finite union of supports of coherent OX modules and hence is a closed subspace of X. Moreover, (X, SuppX ) is a support data for
Dperf (X) because the localization is exact. For details, please see [Tho].

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Example 2.6. Let k be a field of characteristic p > 0 and G a finite group such that p
divides the order of G. Then as in the case of Gorenstein rings, we can define the stable
category mod kG of mod kG and it is also a triangulated category.
We denote by{⊕i∈Z Hi (G; k) p = 2
H∗ (G; k) =⊕i∈2Z Hi (G; k) p : odd
the direct sum of cohomologies of G with coeﬃcient k. Then H∗ (G; k) has the structureof a graded-commutative noetherian ring by using the cup product and we can consider
its homogeneous prime spectrum Proj H∗ (G; k). Denote by VG (M ) the support variety fora finitely generated kG-module M which is a closed space of Proj H∗ (G; k). Then the pair
(Proj H∗ (G; k), VG ) becomes a support data for mod kG. For details, please refer to [Ben,
Chapter 5].
Remark 2.7. Actually, the above examples of support data satisfy the following stronger
condition:(1′ ) σ(M ) = ∅ if and only if M ∼= 0.
Let us fix the following notations:Notation 2.8. Let T be a triangulated category and X a topological space. Then weset:• Th(T ) := {thick subcategories of T },• Spcl(X) := {specialization closed subsets of X},• Cl(X) := {closed subsets of X},• Irr(X) := {irreducible closed subsets of X}.
Let (X, σ) be a support data for T , X a thick subcategory of T , and W ∪a specializationclosed subset of X. Then one can easily check that fσ (X ) := σ(X ) := M ∈X σ(M ) is a
specialization-closed subset of X and gσ (W ) := σ −1 (W ) := {M ∈ T | σ(M ) ⊆ W } is athick subcategory of T . Therefore, we obtain two order-preserving maps
Th(T ) o

fσ

/

gσ

Spcl(X)

with respect to the inclusion relations.
Definition 2.9. Let (X, σ) be a support data for T . Then we say that (X, σ) is aclassifying support data for T if(i) X is a noetherian sober space, and(ii) the above maps fσ and gσ are mutually inverse bijections:Th(T ) ofσgσ/

Spcl(X).

One can easily check that the classifying support data (X, σ) for T satisfies the condition(1 ) in Remark 2.7.
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Here, we have to mention that our definition of a (classifying) support data is motivatedby the (classifying) support data for a tensor triangulated category which was introduced
and discussed by Balmer [Bal05]. In this paper, Balmer’s (classifying) support data will
be called tensorial (classifying) support data and the relationship between them and our(classifying) support data will be discussed in the next section.
Every classifying support data automatically satisfies the following realization property.
Lemma 2.10. Let (X, σ) be a classifying support data for T . Then for any closed subset
Z of X, there is an object M of T such that Z = σ(M ).
Proof. Since X is a noetherian sober space and σ(M ) ∪ σ(N ) = σ(M ⊕ N ), we may
assume that Z = {x} for some x ∈ X. From the assumption, one has Z = fσ gσ (Z) =∪M ∈gσ (Z) σ(M ). Hence, there is an element x of σ(M ) for some M ∈ gσ (Z). Then we■
obtain x ∈ σ(M ) ⊆ Z = {x} and this implies that σ(M ) = {x} = Z.
Next, let us introduce the notion of the spectrum of a triangulated category.
Definition 2.11. (1) We say that a thick subcategory X of T is principal if there is anobject M of T such that X = thickT M . Denote by PTh(T ) the set of all principal
thick subcategories of T .
(2) We say that a non-zero principal thick subcategory X of T is irreducible if X =thickT (X1 ∪ X2 ) (X1 , X2 ∈ PTh(T )) implies that X1 = X or X2 = X . Denote bySpec T the set of all irreducible thick subcategories of T .
(3) For M ∈ T , set
SuppT (M ) := {X ∈ Spec T | X ⊆ thick M } ⊆ Spec T .
We consider a topology on Spec T with closed subbasis {SuppT (M ) | M ∈ T }. We
call this topological space the spectrum of T .
The following lemma shows that by using classifying support data, we can also classify
principal thick subcategories and irreducible thick subcategories.
Lemma 2.12. Let (X, σ) be a classifying support data for T , then the one-to-one correspondence
Th(T ) o

fσ

/

gσ

Spcl(X),

which restricts to one-to-one correspondences
PTh(T ) o
Spec T o

fσ

/

gσ
fσ
gσ

/

Cl(X),

Irr(X).

Proof. Note that fσ (thickT M ) = σ(M ) for any M ∈ T . Therefore, the injective map
fσ : Th(T ) → Spcl(X) induces a well defined injective map fσ : PTh(T ) → Cl(X). The
surjectivity has been already proved in Lemma 2.10.

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Next, we show the second one-to-one correspondence. For X1 , X2 ∈ Th(T ), one has∪(1)
fσ (thickT (X1 ∪ X2 )) =σ(M )M ∈thickT (X1 ∪X2 )

∪

=

σ(M )

M ∈X1 ∪X2=(∪

σ(M )) ∪ (

M ∈X1

∪

σ(M ))M ∈X2= fσ (X1 ) ∪ fσ (X2 ).
On the other hand, for Z1 , Z2 ∈ Spcl(X), one has
fσ (thickT (gσ (Z1 ) ∪ gσ (Z2 ))) = fσ (gσ (Z1 )) ∪ fσ (σ(Z2 ))= Z1 ∪ Z2 .
Applying gσ to this equality, we get(2)

thickT (gσ (Z1 ) ∪ gσ (Z2 )) = gσ (Z1 ∪ Z2 ).

Let W be an irreducible closed subset of X. Assume gσ (W ) = thickT (X1 ∪ X2 ) for some
X1 , X2 ∈ PTh(T ). Then, from the above equality (1), we obtain
W = fσ (gσ (W )) = fσ (thickT (X1 ∪ X2 )) = fσ (X1 ) ∪ fσ (X2 ).
Since W is irreducible, fσ (X1 ) = W or fσ (X2 ) = W and hence X1 = gσ (fσ (X1 )) = gσ (W )or X2 = gσ (fσ (X2 )) = gσ (W ). This shows that gσ (W ) is irreducible.
Conversely, take a irreducible thick subcategory X of T and assume fσ (X ) = Z1 ∪ Z2
for some closed subsets Z1 , Z2 of X. From the above equality (2), we get
X = gσ (fσ (X )) = gσ (Z1 ∪ Z2 ) = thickT (gσ (Z1 ) ∪ gσ (Z2 )).
Since X is irreducible, X = gσ (Z1 ) or X = gσ (Z2 ) and therefore, Z1 = fσ (gσ (Z1 )) = fσ (X )or Z2 = fσ (gσ (Z2 )) = fσ (X ). Thus, fσ (X ) is irreducible.
These observations show the second one-to-one correspondence.
■
The following theorem is the main result of this section which is an analogous result to[Bal05, Theorem 5.2].
Theorem 2.13. Let T be an essentially small triangulated category which admits a clas∼=
sifying support data (X, σ). Then there is a homeomorphism φ : X −→ Spec T which∼=
restricts to a homeomorphism φ : σ(M ) −→ SuppT (M ) for each M ∈ T .
Proof. First note that for a topological space X, the natural map ιX : X → Irr(X), x 7→{x} is bijective if and only if X is sober.
From Lemma 2.12, we have a bijective mapgσιXφ : X −→
Irr(X) −→ Spec T .

As X ⊆ thick M ⇔ fσ (X ) ⊆ fσ (thick M ) = σ(M ), this bijection restricts to∼=φ : σ(M ) −→ SuppT M.

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Since the topology on Spec T is given by the closed subbasis {SuppT M | M ∈ T }, weconclude that φ : X → Spec T is a homeomorphism.
■
Note that Spec T is determined by the triangulated structure of T . Therefore, if two
triangulated categories T and T ′ are equivalent as triangulated categories, then Spec T
and Spec T ′ are homeomorphic. From this observation, we obtain the following corollary,which shows that a classifying support data is unique up to homeomorphism.
Corollary 2.14. Consider the following settings:• T and T ′ are triangulated categories.
• (X, σ) and (Y, τ ) are classifying support data for T and T ′ .
∼=Suppose that there is a triangle equivalence F : T −→ T ′ . Then there is a homeomorphism∼∼==→ τ (F (M )) for each M ∈ T .
→ Y which restricts to a homeomorphism φ : σ(M ) −φ:X−
Proof. From the assumption, F induces a one-to-one correspondence∼=F̃ : Th(T ) −→ Th(T ′ ), X 7→ F̃ (X ),where F̃ (X ) := {N ∈ T ′ | ∃M ∈ X such that N ∼= F (M )}. For an object M of T , setFτ (M ) := τ (F (M )). Then we can easily verify that the pair (Y, τ F ) is a support data for
T . Furthermore, it becomes a classifying support data for T . Indeed, for X ∈ Th(T ) andW ∈ Cl(Y ), we obtain∪∪∪fτ F (X ) =τ F (M ) =τ (F (M )) =
τ (N ) = fτ (F̃ (X )),
M ∈X

M ∈X

N ∈F̃ (X )F̃ (gτ F (W )) = F̃ ({M ∈ T | τ F (M ) ⊆ W })= {N ∈ T ′ | τ (N ) ⊆ W } = gτ (W ).
From these equalities, we get equalities fτ F = fτ ◦ F̃ and F̃ ◦ gτ F = gτ and thus fτ F
and gτ F give mutually inverse bijections between Th(T ) and Cl(Y ). Consequently, we
obtain two classifying support data (X, σ) and (Y, τ F ) for T , and hence both X and Y
are homeomorphic to Spec T by Theorem 2.13.
■
3. Comparison with tensor triangulated structure
In this section, we discuss relation between the support theory we discussed in section
2 and the support theory for tensor triangulated categories.
Recall that a tensor triangulated category (T , ⊗, 1) consists of a triangulated category
T together with a symmetric monoidal tensor product ⊗ with unit object 1 which iscompatible with the triangulated structure of T . For the precise definition, refer to [HPS,Appendix A].
Example 3.1. (1) Let X be a noetherian scheme. Then (Dperf (X), ⊗LOX , OX ) is a tensor
triangulated category. Here, ⊗LOX denotes the derived tensor product.
(2) Let k be a field and G a finite group. Then (mod kG, ⊗k , k) is a tensor triangulated
category.

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Throughout this section, fix a tensor triangulated category (T , ⊗, 1). We begin withrecalling some basic definitions which are used in the support theory of tensor triangulated
categories.
Definition 3.2. (1) A full subcategory X of T is called a thick tensor ideal if it is a thicksubcategory of T and is closed under the action of T by ⊗: M ⊗ N ∈ X for any
M ∈ X and N ∈ T . For a subcategory X of T , denote by ⟨X ⟩ the smallest thicktensor ideal of T containing X .
(2) For a thick subcategory X of T , define its radical by√
X := {M ∈ T | ∃n > 0 such that M ⊗n ∈ X }.
Here, M ⊗n denotes the n-fold tensor product of M . By [Bal05, Lemma 4.2], theradical of a thick subcategory is always a thick tensor ideal.
√
A thick tensor ideal X of T is called radical if it satisfies X = X .
(3) A thick tensor ideal X of T is called prime if it satisfies
M ⊗ N ∈ X ⇒ M ∈ X or N ∈ X .
Denote by Spc T the set of all prime thick tensor ideals of T .
(4) For M ∈ T , the Balmer support of M is defined as SppT M := {P ∈ Spc(T , ⊗) | M ∈/P}. The set Spc(T , ⊗) is a topological space with closed basis {SppT M | M ∈ T }and call it the Balmer spectrum of T .
(5) Let X be a topological space. We say that a subset W of X is a Thomason subset
if it is a union of closed subsets whose complements are quasi-compact. Denote byThom(X) the set of all Thomason subsets of X. Note that Thom(X) ⊆ Spcl(X).
We say that a support data (X, σ) for T is tensorial if it satisfies:σ(M ⊗ N ) = σ(M ) ∩ σ(N )for any M, N ∈ T . In [Bal05], tensorial support data are called simply support data.
Then gσ (W ) is a radical thick tensor ideal of T for every specialization-closed subset W
of X. We say that a tensorial support data (X, σ) is classifying if X is a noetherian sober
space and there is a one-to-one correspondence:{radical thick tensor ideals of T } o

fσ
gσ

/

Spcl(X).

Here we recall the celebrated result due to Balmer.
Theorem 3.3. [Bal05, Lemma 2.6, Theorem 4.10](1) The pair (Spc(T , ⊗), SppT ) is a tensorial support data for T .
(2) There is a one-to-one correspondence:{radical thick tensor ideals of T } o

fSpp
gSpp

/

Thom(Spc(T , ⊗)).

Remark 3.4. If a topological space X is noetherian, then every specialization-closed
subset of X is Thomason. Therefore, the above theorem shows that (Spc(T , ⊗), Spp) is aclassifying tensorial support data for T provided Spc(T , ⊗) is noetherian.

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Recall that a tensor triangulated category T is rigid if(1) T is closed i.e., the functor M ⊗ − : T → T has a right adjoint F (M, −) : T → T
for each M ∈ T and(2) every object M is strongly dualizable (i.e., the natural map F (M, 1) ⊗ N →F (M, N ) is an isomorphism for each N ∈ T ).
If T is rigid, then (Spc(T , ⊗), SppT ) satisfies the stronger condition.
Lemma 3.5. Assume that T is rigid. Then the support data (Spc(T , ⊗), SppT ) satisfies
the condition (1′ ) in Remark 2.7.
Proof. Take an object M ∈ T with Spp(M ) = ∅. By [Bal05, Corollary 2.4], there is apositive integer n such that M ⊗n ∼= 0 where M ⊗n stands for the n-fold tensor product of⊗2i
n-copies of M . On the other hand, by [HPS, Lemma A 2.6], M ⊗i belongs to thick⊗)T (Mfor any positive integer since every object is strongly dualizable. Therefore, by using
induction, we conclude that M ∼■= 0.
Note that a tensorial classifying support data for T is a classifying tensorial support
data for T . Indeed, for a tensorial classifying support data (X, σ) for T and X ∈ Th(T ),we obtain an equalities√√
X = gσ (fσ (X )) = gσ (fσ ( thick⊗ X )) = thick⊗ X .
The following lemma gives a criterion for the converse implication of this fact.
Lemma 3.6. Let (X, σ) be a classifying tensorial support data for T . Suppose that T is
closed. Then the following are equivalent:(1) (X, σ) is a classifying support data for T .
(2) Every thick subcategory of T is a thick ⊗-ideal.
(3) T = thickT 1.
Proof. (1) ⇒ (2): From the assumption, every thick subcategory X of T is of the form
X = gσ (W ) for some specialization-closed subset W of X. On the other hand, gσ (W ) isa radical thick ⊗-ideal as (X, σ) is a tensorial support data.
(2) ⇒ (3): By assumption, the thick subcategory thickT 1 is a thick tensor ideal. Thus,
for any M ∈ T , M ∼= M ⊗ 1 belongs to thickT 1.
(3) ⇒ (1): Note that 1 is strongly dualizable and the family of all strongly dualizable
objects forms a thick subcategory of T by [HPS, Theorem A.2.5 (a)]. Therefore, every
object of T = thickT 1 is strongly dualizable. Thus, for any object M ∈ T , M belongs tothick⊗T (M ⊗ M ) by [HPS, Lemma A.2.6]. Then [Bal05, Proposition 4.4] shows that every
thick tensor ideal of T is radical.
On the other hand, for any thick subcategory X of Y, one can easily verify that the
subcategory Y := {M ∈ T | M ⊗ X ⊆ X } is a thick ⊗-ideal of T containing 1. Thus, we
obtain Y = thickT 1 = T and hence X is a thick ⊗-ideal.
From these discussion, we conclude that every thick subcategory of T is a radical thick⊗-ideal and this shows the implication (3) ⇒ (1).
■The following corollaries are direct consequences of this lemma, Theorem 2.13 andCorollary 2.14.

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Corollary 3.7. Let T be a closed tensor triangulated category. Assume that the Balmer
spectrum Spc(T , ⊗) of T is noetherian and T = thickT 1. Then, there is a homeomor∼∼==
phism φ : Spc(T , ⊗) −→ Spec T which restricts to a homeomorphism φ : SppT M −→SuppT M for each M ∈ T .
Corollary 3.8. Let (T , ⊗, 1) and (T ′ , ⊗′ , 1′ ) be closed tensor triangulated categories suchthat
• Spc(T , ⊗) and Spc(T ′ , ⊗′ ) are noetherian, and• T and T ′ are generated by their units 1 and 1′ , respectively.
If T and T ′ are equivalent just as triangulated categories, then Spc(T , ⊗) and Spc(T ′ , ⊗′ )are homeomorphic.
Next, we consider several applications of results we discussed in this section to tensor
triangulated categories appeared in Example 3.1.
Thomason showed the following classification theorem of thick tensor ideas of Dperf (X).
Theorem 3.9. [Tho, Theorem 3.15] Let X be a noetherian scheme. Then (X, SuppX ) isa classifying tensorial support data for Dperf (X).
As an application of Theorem 2.13, we can reconstruct underlying topological spaces ofa certain class of schemes from their perfect derived categories without tensor structure.
Theorem 3.10. Let X be a noetherian quasi-aﬃne scheme. Then there is a homeomorphism∼=
φ:X−→ Spec Dperf (X)∼=which restricts to a homeomorphism φ : SuppX (F) −→ SuppDperf (X) (F) for each F ∈perf
D (X).
Proof. First, let me remark that the functor F ⊗LOX − : Dperf (X) → Dperf (X) has a rightadjoint RHomOX (F, −) : Dperf (X) → Dperf (X) for each F ∈ Dperf (X) and moreover
Dperf (X) is rigid.
Note that a scheme X is quasi-aﬃne if and only if its structure sheaf OX is ample.
Thus, every thick subcategory of Dperf (X) is thick tensor ideal by [Tho, Proposition
3.11.1]. Applying Theorem 2.13, we obtain the result.
■
As a direct consequence of the theorem, we have a necessary condition for derived
equivalences for noetherian quasi-aﬃne schemes.
Corollary 3.11. Let X and Y be noetherian quasi-aﬃne schemes (i.e., open subschemesof aﬃne schemes). If X and Y are derived equivalent (i.e., Dperf (X) ∼= Dperf (Y ) as
triangulated categories), then X and Y are homeomorphic. In particular, topologicallydetermined properties, such as the dimensions and the numbers of irreducible componentsof quasi-aﬃne noetherian schemes are preserved by derived equivalences.
Remark 3.12. Let X and Y be noetherian schemes.
(1) By [Ric, Proposition 9.2], if X and Y are aﬃne, then a derived equivalence Dperf (X) ∼=Dperf (Y ) implies that X and Y are isomorphic as schemes.

12

HIROKI MATSUI(2) By [Bal02, Theorem 9.7], if Dperf (X) and Dperf (Y ) are equivalent as tensor triangulated
categories, then X and Y are isomorphic as schemes.
Next consider stable module categories over group rings of finite groups. In this case,the following classification theorem is given by Benson-Carlson-Rickard for algebraicallyclosed field k and by Benson-Iyengar-Krause for general k.
Theorem 3.13. [BCR, BIK] Let k be a field of characteristic p > 0 and G a finitegroup such that p divides the order of G. Then the support data (Proj H∗ (G; k), VG ) is aclassifying tensorial support data for mod kG.
Applying Theorem 2.13 to this classifying tensorial support data, we obtain the following result:
Theorem 3.14. Let k be a field of characteristic p and G a finite p-group. Then there isa homeomorphism∼=
φ : Proj H∗ (G; k) −→ Spec mod kG∼=

which restricts to a homeomorphism φ : VG (M ) −→ Suppmod kG (M ) for each M ∈ mod kG.
Proof. For each M ∈ mod kG, the functor M ⊗k − : mod kG → mod kG has a right adjoint
Homk (M, −) : mod kG → mod kG and in addition mod kG is rigid. Moreover, for a pgroup G, kG has only one simple module k. Therefore, we have mod kG = thickmod kG k.
Applying Theorem 2.13, we are done.
■
As in the case of perfect derived categories, we have a necessary condition for stableequivalences for group algebras of p-groups.
Corollary 3.15. Let k (resp. l) be field of characteristic p (resp. q), G (resp. H) be a
finite p-group (resp. q-group). If kG and lH are stably equivalent (i.e., mod kG ∼= mod lH
as triangulated categories), then Proj H∗ (G; k) and Proj H∗ (H; l) are homeomorphic.
Recall that the p-rank of a finite group G is by definition the maximal rank of elementary
abelian p-subgroup:rp (G) := sup{r | (Z/p)r ⊆ G}.
Quillen [Qui] showed that the dimension of the cohomology ring H ∗ (G; k) is equal to thep-rank of G. Thus, the p-rank is an invariant of stable equivalences:
Corollary 3.16. Let k, l, G, H be as in Corollary 3.15. Assume that there is a stableequivalence between kG and lH, then rp (G) = rq (H).
Other invariants of stable equivalences are studied by Linckelmann [Lin].
Remark 3.17. Let G and H be a p-group and k a field of characteristic p.
(1) By [Lin, Corollary 3.6], if there exists a stable equivalence between kG and kH, then
|G| = |H|.
(2) By [Lin, Corollary 3.2], if there exists a stable equivalence of Morita type between kG
and kH, then G ∼= H.
We will end this section by proving the following corollary which is a combination of
the previous corollary and remark.

SINGULAR EQUIVALENCES AND RECONSTRUCTION OF SINGULAR LOCI

13

Corollary 3.18. Let k be a field of characteristic p, G and H p-groups. Suppose that
there is a stable equivalence between kG and lH. Then G is an elementary abelian p-group
if and only if H is an elementary abelian p-group. When this is the case, G and H are
isomorphic.
Proof. Note that for a p-group G, it is elementary abelian p-group if and only if |G| =p · rp (G). By Corollary 3.16, one has rp (G) = rp (H) and |G| = |H| by [Lin, Corollary3.6]. Thus the statement holds.
■
4. A necessary condition for singular equivalences
Recall that commutative noetherian rings R and S are said to be singularly equivalent
if their singularity categories are equivalent as triangulated categories. The only known
examples of singular equivalences are the following:
Example 4.1. (1) If R ∼= S, then Dsg (R) ∼= Dsg (S).
∼(2) If R and S are regular, then Dsg (R) = 0 ∼= Dsg (S).
(3) (Knörrer’s periodicity [Yos, Chapter 12]) Let k be an algebraically closed field of
characteristic 0. Set R := k[[x0 , x1 , ..., xd ]]/(f ) and S := k[[x0 , x1 , ..., xd , u, v]]/(f +uv).
Then Dsg (R) ∼= Dsg (S).
Remark 4.2. All of these singular equivalences, the singular loci Sing R and Sing S are
homeomorphic. In fact, the cases (1) and (2) are clear. Consider the case of R :=k[[x0 , x1 , ..., xd ]]/(f ) and S := k[[x0 , x1 , ..., xd , u, v]]/(f + uv). ThenSing S = V(∂f /∂x0 , . . . ∂f /∂xd , u, v)∼= Spec(S/(∂f /∂x0 , . . . , ∂f /∂xd , u, v))∼= Spec(k[[x0 , x1 , ..., xd , u, v]]/(f + uv, ∂f /∂x0 , . . . , ∂f /∂xd , u, v))∼= Spec(k[[x0 , x1 , ..., xd ]]/(f, ∂f /∂x0 , . . . , ∂f ∂xd )∼= V(∂f /∂x0 , . . . ∂f /∂xd ) = Sing R.
Here, the first and the last equalities follow from the Jacobian criterion.
Let me give some definitions appearing in the statement of the main theorem of thissection.
Definition 4.3. Let (R, m, k) be a commutative noetherian local ring.
(1) We say that an ideal I of R is quasi-decomposable if there is an R-regular sequence
x of I such that I/(x) is decomposable as an R-module.
(2) A local ring R is said to be complete intersection if there is a regular local ring
S and an S-regular sequence x such that the completion R̂ of R is isomorphic
to S/(x). We say that R is a hypersurface if we can take x to be an S-regularsequence of length 1.
(3) A local ring R is said to be locally a hypersurface on the punctured spectrum if Rp
is a hypersurface for every non-maximal prime ideal p.
The following theorem is the main result of this section.

14

HIROKI MATSUI

Theorem 4.4. Let R and S be excellent commutative noetherian local rings that are
locally hypersurfaces on the punctured spectra. Assume that R and S are either(a) Gorenstein rings, or(b) Cohen-Macaulay rings with quasi-decomposable maximal ideal.
If R and S are singularly equivalent, then Sing R and Sing S are homeomorphic.
For a ring R satisfying the condition (b) in Theorem 4.4, Nasseh-Takahashi [NT, Theorem B] shows that (Sing R, SSuppR ) is a classifying support data for Dsg (R). Therefore,
the statement of Theorem 4.4 follows from Corollary 2.14. Hence the problem is the caseof (a).
For a ring R satisfying the condition (a) in Theorem 4.4, Takahashi [Tak] classified
thick subcategories of Dsg (R) containing the residue field k of R by using the singularlocus Sing R and the singular support SSuppR . We would like to apply Corollary 2.14 also
for this case. The problem is that whether the condition “containing the residue field k”
is preserved by singular equivalences. As we will show later, this condition is actuallypreserved by singular equivalences for Gorenstein local rings.
For a while, we consider Gorenstein local rings and the stable category CM(R) insteadof Dsg (R). The following lemma gives a categorical characterization of thick subcategory
generated by the d-th syzygy of the residue field.
Lemma 4.5. Let (R, m, k) be a d-dimensional Gorenstein local ring. Then the following
are equivalent for M ∈ CM(R).
(1) EndR (M ) is a right artinian.
(2) M is locally free on the punctured spectrum (i.e., Mp ∼= 0 for any non-maximal prime
ideal p).
(3) M ∈ thickCM(R) Ωd k.
Proof. The implications (3) ⇒ (2) ⇒ (1) is clear and the implication (2) ⇒ (3) is shownin [Tak, Corollary 2.6].
The only thing remained to prove is the implication (1) ⇒ (2). Assume that EndR (M )is a right artinian ring. Then the right artinianity of EndR (M ) and Nakayama’s lemma
imply mi EndR (M ) = 0 for some integer i ≥ 0 and hence EndR (M ) is of finite length asan R-module. Let a be the kernel of the natural map R → EndR (M ). Since EndR (M ) isof finite length as an R-module, so is R/a. This means that a is m-primary. Therefore,
for any non-maximal prime ideal p, the natural map Rp → EndRp (Mp ) is zero and hence
Mp ∼
■
= 0.
Denote by CM0 (R) the thick subcategory of CM(R) consisting of maximal CohenMacaulay modules which are locally free on the punctured spectrum. As we have shownin the previous lemma, CM0 (R) = thickCM(R) Ωd k and the objects are categorically determined objects. Therefore, we have the following result.
Proposition 4.6. Let R and S be Gorenstein local rings. If there is a triangle equivalence
CM(R) ∼= CM(S), then it induces triangle equivalences
CM (R) ∼= CM (S), and
00CM(R)/CM0 (R) ∼= CM(S)/CM0 (S).

SINGULAR EQUIVALENCES AND RECONSTRUCTION OF SINGULAR LOCI

15

∼
=

Proof. Let F : CM(R) −→ CM(S) be a triangle equivalence. Then for any object M ∈CM(R), M belongs to CM0 (R) if and only if EndR (M ) ∼= EndS (F (M )) is right artinian if
and only if F (M ) belongs to CM0 (S). Thus, F restricts to the first equivalence and also
induces the second one.
■
Gathering [Tak, Theorem 6.7], [NT, Theorem B] and Theorem 2.13, we obtain thefollowing proposition.
Theorem 4.7. Let R be an excellent noetherian local ring.
(1) If R satisfies the condition (a) in Theorem 4.4, then there is a homeomorphism∼=

φ : Sing R \ {m} −→ Spec CM(R)/CM0 (R).
(2) If R satisfies the condition (b) in Theorem 4.4 or is a hypersurface, then there is ahomeomorphism∼=→ Spec Dsg (R)φ : Sing R −∼=which restricts to a homeomorphism φ : SSuppR (M ) −→ SuppDsg (R) (M ) for each M ∈Dsg (R).
Proof. The statement (2) directly follows from [Tak, Theorem 6.7], [NT, Theorem B] andTheorem 2.13.
By [Tak, Theorem 6.7] again, the assignment
CM(R)/CM0 (R) ∋ M 7→ SSuppR (M ) \ {m} ⊆ Sing R \ {m}defines a classifying support data for CM(R)/CM0 (R). Thus the statement (1) follows
from Theorem 2.13 again.
■
Now, the combination of Proposition 4.6 and Theorem 4.7 completes the proof of Theorem 4.4.
Proof of Theorem 4.4. Assume that R and S satisfy the condition (a). Using Proposition
4.6 and Theorem 4.7, if there is a triangle equivalence CM(R) ∼= CM(S), we get a homeomorphism Sing R \ {m} ∼= Sing S \ {n}. Then adding one closed point on both sides showsthat there is a homeomorphism Sing R ∼= Sing S. The proof for the case (b) is similar. ■Remark 4.8. For a hypersurface ring R, the triangulated category Dsg (R) becomes a
pseudo tensor triangulated category (i.e., tensor triangulated category without unit). It
is shown by Yu implicitly in the paper [Yu] that for two hypersurfaces R and S, if asingular equivalence between R and S preserves tensor products, then Sing R and Sing S
are homeomorphic. Indeed, Sing R is reconstructed from Dsg (R) by using its pseudo tensor
triangulated structure.
Since Theorem 4.4 gives a necessary condition for singular equivalences, we can generatemany pairs of rings which are not singularly equivalent. Let us start with the following
lemma.
Lemma 4.9. Let R be a Gorenstein local ring with only an isolated singularity and r >1 an integer. Then the ring R[[u]]/(ur ) is a Gorenstein local ring which is locally ahypersurface on the punctured spectrum, and Sing(R[[u]]/(ur )) is homeomorphic to Spec R.

16

HIROKI MATSUIProof. Set T := R[[u]]/(ur ). The natural inclusion R → T induces a homeomorphism f :∼=Spec T −→ Spec R. Then one can easily check that P = (f (P ), u)T for any P ∈ Spec T andTP ∼= Rf (P ) [[u]]/(ur ). Therefore, T is locally a hypersurface on the punctured spectrum
and Sing T = Spec T .
■Corollary 4.10. Let R and S be excellent Gorenstein local rings which have only isolated
singularities. Assume that Spec R and Spec S are not homeomorphic. Then for any
integers r, s > 1, one has
Dsg (R[[u]]/(ur )) ≁= Dsg (S[[v]]/(v s )).
In particular, Dsg (R ∗ R) ∼
̸= Dsg (S ∗ S). Here R ∗ R denotes the trivial extension ring ofa commutative ring R.
Proof. From the above lemma, we obtain(1) R[[u]]/(ur ) and S[[v]]/(v s ) satisfies the condition (a) in Theorem 4.4,
(2) Sing R[[u]]/(ur ) ∼= Spec R and Sing S[[u]]/(v r ) ∼= Spec S are not homeomorphic.
r∼Thus, we conclude Dsg (R[[u]]/(u )) ̸= Dsg (S[[v]]/(v s )) by Theorem 4.4.
The second statement follows from an isomorphism R ∗ R ∼= R[[u]]/(u2 ).

■The following corollary says that a Knörrer-type equivalence fails over a non-regularring.
Corollary 4.11. Let S be an excellent Gorenstein local ring and f an S-regular element.
Assume that S/(f ) has an isolated singularity. Then one has
Dsg (S[[u]]/(f, u2 )) ̸∼= Dsg (S[[u, v, w]]/(f + vw, u2 )).
Proof. Sing S[[u]]/(f, u2 ) ∼= Spec S/(f ) and Sing S[[u, v, w]]/(f +vw, u2 ) ∼= Spec S[[v, w]]/(f +vw) have diﬀerent dimensions and hence are not homeomorphic.
■
The last purpose of this section is to show that singular equivalence localizes. We needa lemma for this.
Lemma 4.12. Let R be a d-dimensional Gorenstein local ring and p a prime ideal of R.
Then a full subcategory Xp := {M ∈ Dsg (R) | Mp ∼= 0 in Dsg (Rp )} is thick and there is atriangle equivalence
Dsg (R)/Xp ∼= Dsg (Rp ).
Proof. By using the triangle equivalence Dsg (R) ∼= CM(R), we may show the triangle
equivalence
CM(R)/Xp ∼= CM(Rp ),∼where Xp := {M ∈ CM(R) | Mp = 0 in CM(Rp )}.
Note that the localization functor Lp : CM(R) → CM(Rp ), M 7→ Mp is triangulated.
Since Xp = Ker Lp , Xp is a thick subcategory of CM(R) and Lp induces a triangulated
functor Lp : CM(R)/Xp → CM(Rp ). Thus, we have only to verify that Lp is dense andfully faithful.
(i): Lp is dense.
δ
Let U be an Rp -module. Take a finite free presentation Rpn −→ Rpm → U → 0 of U .
Then δ can be viewed as an m × n-matrix (αij ) with entries in Rp . Write αij = aij /s forSINGULAR EQUIVALENCES AND RECONSTRUCTION OF SINGULAR LOCI

17

some aij ∈ R and s ∈ R \ p. Then the cokernel M := Coker((aij ) : Rn → Rm ) is a finitely
generated R-module and Mp ∼= U . Since Mp is a maximal Cohen-Macaulay Rp -module,
we obtain isomorphismsd∼∼∼ −d d(Ω−dR ΩR M )p = ΩRp ΩRp Mp = Mp = U

in CM(Rp ). This shows that the functor Lp is dense.
(ii): Lp is faithful.
Let α : M → N be a morphism in CM(R)/Xp . Then α is given by a fraction f /sof morphisms f : M → Z and s : N → Z in CM(R) such that the mapping coneC(s) of s belongs to Xp . Assume Lp (α) = Lp (s)−1 Lp (f ) = (sp )−1 fp = 0. Then fp = 0 in
HomRp (Mp , Zp ). From the isomorphism HomR (M, Z)p ∼= HomRp (Mp , Zp ), there is a ∈ R\psuch that af = 0 in HomR (M, Z). Since a : Zp → Zp is isomorphism, the mapping coneof the morphism a : Z → Z in CM(R) belongs to Xp . Thus, α = f /s = (af )/(as) = 0 in
CM(R)/Xp . This shows that Lp is faithful.
(iii): Lp is full.
Let g : Mp → Np be a morphism in CM(Rp ) where M, N ∈ CM(R). By the isomorphism
HomR (M, N )p ∼= HomRp (Mp , Np ), there is a morphism f : M → N in CM(R) and a ∈ R\psuch that g = fp /a. Since the mapping cone of a : N → N is in Xp , we obtain a morphism
f /a : M → N in CM(R)/Xp and Lp (f /a) = fp /a = g. This shows that Lp is full.
■
Corollary 4.13. Let R and S be excellent Gorenstein local rings which are locally hypersurfaces on the punctured spectra. If R and S are singularly equivalent, then there is a∼=homeomorphism φ : Sing R −→ Sing S such that Rp and Sφ(p) are singularly equivalent forany p ∈ Sing R.
Proof. As in Lemma 4.12, we may consider the category CM(R).
Let F : CM(R) → CM(S) be a triangle equivalence. Take a homeomorphism φ :Sing R → Sing S given in Proposition 2.13 and Theorem 2.14. Then by construction, it
satisfies∪{φ(p)} =SuppS F (M )M ∈CM(R), SuppR (M )⊆V(p)for each p ∈ Sing R. Moreover, the following diagram is commutative:F̃ThT(CM(R)) (CM(R)) −−−→ ThT(CM(S)) (CM(S))
fSuppfSupp yy S
R
Nesc(Sing R)

−−−→
φ̃

Nesc(Sing S),

where the map F̃ and φ̃ are defined by F̃ (X ) := {N ∈ T ′ | ∃M ∈ X such that N ∼=F (M )} and φ̃(W ) := φ(W ), respectively.
Let p be an element of Sing R. Set Wp := {q ∈ Sing R | q ̸⊆ p} which is a specializationclosed subset of Sing R. We establish two claims.
Claim 1. gSuppR (Wp ) = Xp .

18

HIROKI MATSUIProof of Claim 1. Let M ∈ Xp . Since Mp = 0 in CM(Rp ), one has p ̸∈ SuppR (M ). Thus,
SuppR (M ) ⊆ Wp and hence M ∈ gSuppR (Wp ).
Next, take M ∈ gSuppR (Wp ). Then SuppR (M ) ⊆ Wp means that p does not belong to
SuppR (M ). Therefore, Mp = 0 in CM(Rp ) and hence M ∈ Xp .
■Claim 2. φ(Wp ) = Wφ(p) := {q ∈ Sing S | q ̸⊆ φ(p)}.
Proof of Claim 2. One can easily check that φ is order isomorphism with respect to theinclusion relations. Since Sing R \ Wp has a unique maximal element p, φ(Sing R \ Wp ) =Sing S \ φ(Wp ) also has a unique maximal element φ(p). This shows φ(Wp ) = Wφ(p) . ■From the above two claims, we obtain
F̃ (Xp ) = F̃ (gSuppR (Wp )) = gSuppS (φ̃(Wp )) = gSuppS (Wφ(p) ) = Xφ(p) ,where the second equality comes from the above commutative diagram and the last equality is shown by the same proof as Claim 1. Consequently, the triangle equivalence F
induces triangle equivalences:
CM(Rp ) ∼= CM(R)/Xp ∼= CM(S)/Xφ(p) ∼= CM(Sφ(p) ).
■5. Reconstruction of scheme scheme structure
Finally, we consider reconstructing the scheme structure of a singular locus from thestable category of maximal Cohen-Macaulay modules over a hypersurface.
In this section, we always consider a hypersurface R := S/(f ), where S is a regular
local ring with algebraically closed residue field k and f be a non-zero element of S. We
always consider reduced closed subscheme structure of Spec R on Sing R. The following
remarks are an easy observation and a known result.
Remark 5.1 (The case of isolated singularities : dim Sing R = 0). (1) Assume that S is
Henselian. Then EndR (M ) is a local finite dimensional algebra over the residue
field k of R for any non-zero indecomposable object M of CM(R).
Thus,
EndR (M )/J(EndR (M )) is a finite dimensional division algebra over k, where
J(EndR (M )) denotes the Jacobson radical of EndR (M ). As k is algebraically closed,
it is isomorphic to k. Therefore,
Sing R ∼= Spec End (M )/J(End (M )).
R

R(2) ([Dyc, Corollary 6.5]) Assume that S is essentially of finite type. Note that for a
hypersurface singularity S/(f ), there is a Z/2Z-graded DG category MF(S, f ) suchthat H0 (MF(S, f )) ∼= CM(R). Then there is an isomorphism of schemes:
Sing R ∼= Spec HH∗ (MF(S, f )).
Here, HH∗ (MF(S, f )) is the Hochschild cohomology ring of MF(S, f ).
Thus, the next step we should consider is the case of hypersurfaces R with dim Sing R =1. Here, we consider the following singularity, which satisfies this condition.

SINGULAR EQUIVALENCES AND RECONSTRUCTION OF SINGULAR LOCI19

Let k be an algebraically closed field of characteristic 0. We say that a hypersurface
singularity R = k[[x0 , x1 , . . . , xd ]]/(f ) is of type (A∞ ) or (D∞ ) if f is(A∞ ) x20 + x22 + · · · + x2d , or(D∞ ) x20 x1 + x22 + · · · + x2d .
Remark 5.2. (1) If R is of type (A∞ ) or (D∞ ), then dim Sing R = 1.
(2) For a hypersurface R, CM(R) is of countable representation type (i.e., there are only
countably many indecomposable objects) if and only if R is isomorphic to the singularity of type (A∞ ) or (D∞ ). Moreover, this is the case, all indecomposable objects ofCM(R) have been classified completely; see [Sch].
We say that an object M ∈ CM(R) has full-support if SuppCM(R) (M ) = Spec CM(R).
By Theorem 4.7, M is full-support if and only if
SuppR (M ) = Sing R.
Therefore, the condition SuppR (M ) = Sing R is categorically determined.
For a hypersurface singularity of type (A∞ ) or (D∞ ), we can reconstruct the scheme
structure of the singular locus by using a maximal Cohen-Macaulay module with full
support.
Theorem 5.3. Let R be a hypersurface of type (A∞ ) or (D∞ ) and M an indecomposable
object of CM(R) with full-support. Then EndR (M ) is a commutative ring and one has ascheme isomorphism
Sing R ∼= Spec EndR (M ).
Proof. Thanks to Knörrer’s periodicity, we have only to prove in dimension one and two.
Case : dim R = 1 & (A∞ )In this case, R = k[[x, y]]/(x2 ) and it has only one indecomposable maximal CohenMacaulay R-module M = R/(x) with full-support. Note that the natural map
R/(x) → EndR (M )is an isomorphism. Since HomR (R/(x), R) ∼= (0 :R x) = (x), one has PR (M, M ) = 0.
∼
Thus R/(x) = EndR (M ), and there is a scheme isomorphism Sing R ∼= Spec EndR (M ).
The proofs for the cases dim R = 1 & (D∞ ) and dim R = 2 & (A∞ ) are similar since theindecomposable maximal Cohen-Macaulay modules with full-supports are cyclic. Therefore, we have only to prove the following case.
Case : dim R = 2 & (D∞ )In this case, R = k[[x, y, z]]/(x2 y + z 2 ) and p := (x, z) ⊆ R is the only one indecomposable maximal Cohen-Macaulay R-module with full-support. Notice that p is a prime
ideal of R. Then EndR (p) ∼= p : p and which sends PR (p, p) to (R : p)p = p. Therefore,∼
EndR (p) = (p : p)/p. Clearly, R + (z/x)R is contained in p : p. Conversely, for an elementa/s ∈ p : p, one has ax ∈ ap ⊆ sp = sxR + szR and hence a/s ∈ R + (z/x)R. As
a result, we get p : p = R + (z/x)R. Since (p : p)/p = R/p + (z/x)(R/p), there is asurjective R-algebra homomorphism ϕ : (R/p)[t] → (p : p)/p, t → z/x. As (z/x)2 = −yand a + b(z/x) ∈ p with a, b ∈ R implies a, b ∈ p, the kernel of ϕ is (t2 + y) and we
conclude that
EndR (p) ∼= (p : p)/p ∼= (R/p)[t]/(t2 + y) ∼= k[[y]].

20

HIROKI MATSUIThus, we get an isomorphism Sing R ∼= Spec EndR (M ) of schemes.

■

Remark 5.4. As we mentioned in Remark 5.2(2), the singular locus of hypersurfaceof types (A∞ ) and (D∞ ) are characterized in terms of the stable categories of maximal
Cohen-Macaulay modules. The advantage of this theorem is that it gives more explicit
description of their singular loci.
Acknowledgments. The author is grateful to his supervisor Ryo Takahashi for many supports and helpful comments. Also, the author very much thank the anonymous referee
for his/her careful reading, and for many helpful comments, corrections and suggestions
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Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya,
Aichi 464-8602, Japan
Email address: m14037f@math.nagoya-u.ac.jp