PRIME THICK SUBCATEGORIES AND SPECTRA OF DERIVED AND
SINGULARITY CATEGORIES OF NOETHERIAN SCHEMESHIROKI MATSUI
Abstract. For an essentially small triangulated category T , we introduce the notion of prime
thick subcategories and define the spectrum of T , which shares the basic properties with thespectrum of a tensor triangulated category introduced by Balmer. We mainly focus on triangulated categories that appear in algebraic geometry such as the derived and the singularity
categories of a noetherian scheme X. We prove that certain classes of thick subcategories areprime thick subcategories of these triangulated categories. Furthermore, we use this result toshow that certain subspaces of X are embedded into their spectra as topological spaces.

1. Introduction
Classification of thick subcategories is one of the important approaches for understanding
the structure of a given triangulated category and is a common problem in various fields ofmathematics, such as commutative algebra [10, 20, 27, 28], algebraic geometry [27, 31], modular representation theory of finite groups [6, 7, 8], stable homotopy theory [10, 11], and soon. Especially, classifications of thick tensor ideals of tensor triangulated categories are quite
successful. For example, the thick tensor ideals of the perfect derived category of a noetherian
scheme, those of the stable module category, and those of the stable homotopy category havebeen classified; see the references above. Recently, Balmer [4] has established a theory called
tensor triangular geometry, which gives a unified perspective on such classifications. For a given
essentially small tensor triangulated category (T , ⊗, 1), Balmer defines a topology on the setSpec⊗ (T ) of prime thick tensor ideals of T , which is called the Balmer spectrum of T . Balmer
establishes the following monumental work in the theory.
Theorem 1.1 (Balmer). Let T be an essentially small tensor triangulated category. Then
there is a lattice isomorphism between the set of radical thick tensor ideals of T and the set ofThomason subsets of Spec⊗ (T ).
By virtue of this result, tensor triangular geometry provides us an algebro-geometric way to
study an essentially small tensor triangulated category using its Balmer spectrum.
Tensor triangular geometry can be applied to any essentially small tensor triangulated categories, whereas it cannot be directly applied to triangulated categories that are not tensor
triangulated. Such triangulated categories include important ones, e.g., the bounded derived
category Db (X) of coherent sheaves and the singularity category Dsg (X) of a noetherian scheme
X do not have natural tensor triangulated structures, which are used in the study of birational
2020 Mathematics Subject Classification. 13D09, 13H10, 14F08, 18G80.
Key words and phrases. complete intersection, derived category, hypersurface, noetherian scheme, prime
thick subcategory, singularity category, spectrum, triangulated category.
The author was partly supported by JSPS Grant-in-Aid for JSPS Fellows 19J00158.
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geometry and homological mirror symmetry conjecture; see [14, 22] and references therein.
Therefore it is a natural and important problem to develop an analogous theory of tensor
triangular geometry for essentially small triangulated categories without tensor triangulated
structures.
The present paper aims to construct for a given essentially small triangulated category T a
topological space Spec△ (T ) which we call the spectrum of T . To do this, we define the notionof prime thick subcategories of T and the spectrum of T as the set of prime thick subcategories
together with a topology given in [18]. Moreover, we study the prime thick subcategories
and spectra of the perfect derived category Dperf (X), the bounded derived category Db (X)of coherent sheaves, and the singularity category Dsg (X) of a noetherian scheme X. One of
the main theorems of this paper is the following, which means that the spectra contain some
topological information on X.
Theorem 1.2 (Corollaries 2.17 and 3.8). Let X be a noetherian scheme.
(1) There is an immersion
X ↩→ Spec△ (Dperf (X))
of topological spaces, which is a homeomorphism if X is quasi-aﬃne.
(2) There is an immersion
CI(X) ↩→ Spec△ (Db (X))
of topological spaces, which is a homeomorphism if X is quasi-aﬃne and regular.
(3) Assume further that X is separated. Then there is an immersion
HS(X) ↩→ Spec△ (Dsg (X))

of topological spaces, which is a homeomorphism if X is quasi-aﬃne and locally a hypersurface.
Here, CI(X) is the complete intersection locus of X, which is the set of points x of X such that
OX,x are complete intersection. Similarly, HS(X) is the hypersurface locus, which is the set of
singular points x such that OX,x are hypersurface. If X is excellent and Cohen-Macaulay, then
CI(X) and HS(X) are open subsets of X and Sing(X), respectively; see Remark 3.9.
We say that two noetherian schemes X and Y are derived equivalent if their bounded derived
categories Db (X) and Db (Y ) are equivalent as triangulated categories. This concept is also
known as Fourier-Mukai partners. Since by definition equivalent triangulated categories havehomeomorphic spectra, Theorem 1.2 immediately implies the following result.
Corollary 1.3 (Corollary 3.10). Let X be a noetherian scheme. Then there is an immersion
CI(Y ) ↩→ Spec△ (Db (X))

of topological spaces for any Fourier-Mukai partner Y of X.
If T is a tensor triangulated category, then we have two topological spaces, Spec⊗ (T ) andSpec△ (T ). Although these are not equal in general, there is a relation between prime ideals
and prime thick subcategories, which justifies the definition of prime thick subcategories.
Theorem 1.4 (Proposition 4.8 and Corollary 4.9). Let T be an essentially small tensor triangulated category and P be a radical thick tensor ideal of T . If P is a prime thick subcategory of
T , then it is a prime thick tensor ideal of T . Moreover, the converse holds for T = Dperf (X)of a noetherian scheme X.

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The organization of this paper is as follows. In Section 2, we introduce the notion of a prime
thick subcategory, give the definition of the spectrum, and study them for a given essentiallysmall triangulated category T . Most arguments therein are along the same line as in [18,Section 2]. In Section 3, we study prime thick subcategories and the spectra of derived andsingularity categories Dperf (X), Db (X), and Dsg (X). The proof of Theorem 1.2 will be given
here. In Section 4, we investigate our spectrum for an essentially small tensor triangulated
category.
2. Spectra of triangulated categories
In this section, fix an essentially small triangulated category T and let us define the spectrum
Spec△ (T ) of T . A general construction of spectra has been given in [18] for a given set of thicksubcategories of T . We adopt the prime thick subcategories, which will be defined soon, as theunderlying set of the spectrum. We begin with our convention.
Convention 2.1. (1) Throughout this paper, all triangulated categories T are assumed to be
essentially small so that the set Th(T ) of all thick subcategories forms a set. Always weconsider Th(T ) as a lattice via the inclusion relation. Also, all subcategories are assumed
to be full and additive. Denote by 0 the zero subcategory of T , that is, the subcategory
consisting of objects isomorphic to the zero object.
(2) For a noetherian scheme X, the bounded derived category Db (X) of X is the derived categoryof bounded complexes of coherent sheaves on X. An object F of Db (X) is said to be perfectif, for any x ∈ X, there is an open neighborhood U ⊆ X of x such that F|U is isomorphic
in Db (U ) to a bounded complex of free OU -modules of finite rank. Denote by Dperf (X)the thick subcategory of Db (X) consisting of all perfect complexes. We call it the perfectderived category of X. Denote by Dsg (X) the Verdier quotient Dsg (X) := Db (X)/Dperf (X)which we call the singularity category of X.
If we consider an aﬃne scheme X = Spec R, then we simply write Dperf (R), Db (R), andsg
D (R) for the perfect derived, the bounded derived, and the singularity categories of X.
Now, we introduce the definition of a prime thick subcategory which plays a central role
throughout the paper.
Definition 2.2. We say that a thick subcategory P of T is prime if the set {X ∈ Th(T ) | P ⊊X } has a unique minimal element. Denote by Spec△ (T ) the set of prime thick subcategories
of T .
Remark 2.3. (1) We will see in Section 4 that prime thick tensor ideals of a tensor triangulated
category are characterized by a similar condition; see Proposition 4.7. This fact justifies
the above definition of prime thick subcategories.
(2) Recently, Takahashi [30] introduced the notion of core of the singularity category Dsg (R) of acommutative noetherian local ring R as the intersection of all non-trivial thick subcategories
of Dsg (R). Our definition of a prime thick subcategory and the concept of a core are closely
related. Indeed, one can easily see that 0 is a prime thick subcategory of Dsg (R) if andonly if the core of Dsg (R) is not zero.
We then recall the definition of a topology on Spec△ (T ).

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Definition 2.4 ([18, Definition 2.1]). For a family E of objects of T , we set
Z(E) := {P ∈ Spec△ (T ) | P ∩ E = ∅}.

We can easily check the following conditions hold:• Z(T! ) = ∅ and Z(∅)
" = Spec△ (T ).
• i∈I Z(Ei ) = Z( i∈I Ei ).
• Z(E) ∪ Z(E ′ ) = Z(E ⊕ E ′ ), where E ⊕ E ′ := {M ⊕ M ′ | M ∈ E, M ′ ∈ E ′ }.
Here the first and the second conditions hold trivially, and the third one follows since thicksubcategories are closed under taking direct sums. Therefore we can define a topology on
Spec△ (T ) whose family of closed subsets are {Z(E) | E ⊆ T }. We call this topological space
the spectrum of T .
For an object M ∈ T , define the support of M by
Supp△ (M ) := Z({M }) = {P ∈ Spec△ T | M ∕∈ P}.

Then the family of supports {Supp△ (M )}M ∈T forms a closed basis of Spec△ (T ). More generally,
we set#Supp△ (X ) :=Supp△ (M )M ∈X

for a thick subcategory X of T .

Here, we list some definitions and general properties which are given in [18].
Proposition 2.5 ([18, Proposition 2.3]). For any prime thick subcategory P of T , one has{P} = {Q ∈ Spec△ (T ) | Q ⊆ P}.
In particular, Spec△ (T ) is a T0 -space.
Definition 2.6 ([18, Definition 2.7]). For a thick subcategory X of T , we define its radical by
$
√
X :=
P.
We say that X is radical if the equality
thick subcategories of T .

√X ⊆P∈Spec△ (T )X = X holds. Denote by Rad△ (T ) the set of radical

Definition 2.7. We define the parameter set of T by

Param(Spec△ (T )) := {Supp△ (X ) | X ∈ Th(T )}.

The reason why we call this so is that it parametrizes the radical thick subcategories of T as
follows.
Theorem 2.8 ([18, Theorem 2.9]). There is a mutually inverse lattice isomorphisms
Supp△

Rad△ (T ) "!

Supp−1△

Param(Spec△ T ).

Here, we define Supp−1△ (W ) := {M ∈ T | Supp△ (M ) ⊆ W }.

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5

We have explained so far basic definitions and general properties along with [18]. The remainder of this section is devoted to the things that are specific to our definition.
First, we study the functorial properties of spectra. Let F : T → T ′ be an exact functor oftriangulated categories. Let X ⊆ T and Y ⊆ T ′ be thick subcategories. Then we define thesubcategories F (X ) ⊆ T ′ and F −1 (Y) ⊆ T by
F (X ) := {M ′ ∈ T ′ | M ′ ∼= F (M ) for some M ∈ X },F −1 (Y) := {M ∈ T | F (M ) ∈ Y}.

One can easily check that F −1 (Y) is always a thick subcategory of T . Therefore, we have aposet homomorphism
Th(T ′ ) → Th(T ), Y ,→ F −1 (Y).
Proposition 2.9. Let T be a triangulated category and K be its thick subcategory. Denote by
F : T → T /K the canonical functor.
(1) There is a lattice isomorphism∼=

Th(T /K) −→ {X ∈ Th(T ) | K ⊆ X },Q ,→ F −1 (Q).

(2) For a thick subcategory Q ∈ Th(T /K), Q is a prime thick subcategory of T /K if and only
if F −1 (Q) is a prime thick subcategory of T .
(3) F induces an immersiona

F : Spec△ (T /K) → Spec△ (T ),Q ,→ F −1 (Q)of topological spaces with image {P ∈ Spec△ (T ) | K ⊆ P}.

Proof. (1) The first statement is by [29, Lemma 3.1].
(2) The lattice isomorphism in (1) restricts to a poset isomorphism{Y ∈ Th(T /K) | Q ⊊ Y} ∼= {X ∈ Th(T ) | F −1 (Q) ⊊ X }.

The left-hand side admits a unique minimal element if and only if so does the right-handside, that is, Q is a prime thick subcategory of T /K if and only if F −1 (Q) is a prime thicksubcategory of T .
(3) Using (1) and (2), we have a well-defined injective mapa

F : Spec△ (T /K) → Spec△ (T ),Q ,→ F −1 (Q)with image {P ∈ Spec△ (T ) | K ⊆ P}. It is not hard to check that the equality (a F )−1 (Supp△ (M )) = Supp△ (F (M )) holds for an object M ∈ T . Since the family {Supp△ (M )}M ∈T (resp. {Supp△ (F (M ))}M ∈T ) forms a closed basis of Spec△ (T ) (resp.
Spec△ (T /K)), we conclude that a F : Spec△ (T /K) → Spec△ (T ) is an immersion of topologicalspaces.
!
We say that an exact functor F : T → T ′ of triangulated categories is triangle equivalence
up to direct summands if it is fully faithful and any object M ′ ∈ T ′ is a direct summand of
F (M ) for some M ∈ T . The following lemma is useful.

Lemma 2.10 ([4, (3.2)]). Let F : T → T ′ be a triangle equivalence up to direct summands.
Then one has M ′ ⊕ M ′ [1] ∈ F (T ) for any M ′ ∈ T ′ .
Proposition 2.11. Let F : T → T ′ be a triangle equivalence up to direct summands.

6

HIROKI MATSUI(1) There is a lattice isomorphism∼=

Th(T ′ ) −→ Th(T ),Y ,→ F −1 (Y).

(2) For a thick subcategory Q ∈ Th(T ′ ), Q is a prime thick subcategory of T ′ if and only if
F −1 (Q) is a prime thick subcategory of T .
(3) F induces a homeomorphisma∼=

F : Spec△ (T ′ ) −→ Spec△ (T ),P ,→ F −1 (P).

Proof. By the same reason as in the proof of Proposition 2.9, it is enough to show that the map
Th(T ′ ) → Th(T ),Y ,→ F −1 (Y)is a lattice isomorphism. This follows from the same argument as in [4, Proposition 3.13].

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Next, we discuss the determination of the spectrum of a given triangulated category usingclassifying support data.
Definition 2.12. Let T be a triangulated category. A support data for T is a pair (X, σ) of a
topological space X and an assignment σ which assigns to an object M of T a closed subset
σ(M ) of X satisfying the following conditions;(1) σ(0) = ∅.
(2) σ(M [n]) = σ(M ) holds for each object M of T and integer n ∈ Z.
(3) σ(M ) ⊆ σ(L) ∪ σ(N ) holds for each triangle L → M → N → L[1] in T .
(4) σ(M ⊕ N ) = σ(M ) ∪ σ(N ) holds for each pair of objects M, N of T .
A lot of examples of support data for triangulated categories naturally appear in various
fields of mathematics. Here we present the ones that will appear in this paper.
Example 2.13. (1) ([18, Remark 2.2]) The assignment T ∋ M ,→ Supp△ (M ) ⊆ Spec△ (T )defines a support data (Spec△ (T ), Supp△ ) for T .
(2) Let X be a noetherian scheme. One can easily verify that the cohomological support
Supp (F) := {x ∈ X | Fx ∼∕= 0 in Db (OX,x )}X

bof F ∈ D (X) defines a support data (X, SuppX ) for Db (X).
(3) Let X be a noetherian scheme and denote by Sing(X) the singular locus of X. Then the
singular support
Supp (F) := {x ∈ Sing(X) | Fx ∼∕= 0 in Dsg (OX,x )}X

sgof F ∈ D (X) defines a support data (Sing(X), SuppX ) for Dsg (X), see [17, Example 2.4].

We say that a subset W of a topological space X is specialization-closed if it is a union ofclosed subsets of X. We easily see that W is specialization-closed if and only if {x} ⊆ W for
any x ∈ W . Denote by Spcl(X) the set of specialization-closed subsets of X.
Definition 2.14. A support data (X, σ) for T is said to be classifying if it satisfies the following
conditions;(i) X is a noetherian sober space. Here, we say that a topological space X is sober if every
irreducible closed subset has a unique generic point.

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7(ii) There is a mutually inverse lattice isomorphisms
Th(T ) "where σ(X ) :="M ∈X σ(M ) and σ

−1

σ
σ −1

!

Spcl(X),(W ) := {M ∈ T | σ(M ) ⊆ W }.

Using a classifying support data, we can translate a problem concerning thick subcategories
to a problem about specialization-closed subsets. The following lemma characterizes the topological counterpart of prime thick subcategories.
Lemma 2.15. Let X be a sober space and W be a specialization-closed subset of X. The
following conditions are equivalent.
(i) There is a unique minimal element T of {Y ∈ Spcl(X) | W ⊊ Y }.
(ii) There is a unique element x ∈ X such that W = {x′ ∈ X | x ∕∈ {x′ }}.
Proof. (i) ⇒ (ii). Take an element x ∈ T \ W . Since T is specialization-closed, there areinclusions W ⊆ W ∪ ({x} \ {x}) ⊊ W ∪ {x} ⊆ T of specialization-closed subsets of X. From
the assumption on T , one has the equalities W = W ∪ ({x} \ {x}) and T = W ∪ {x}. These
two equalities yield T = W ∪ {x} = W ∪ {x}. Moreover, such x is uniquely determined by
the equality {x} = T \ W . In summary, there is a unique element x ∈ T \ W which has thefollowing property: each specialization-closed subset V of X with W ⊊ V satisfies x ∈ V .
Set W ′ := {x′ ∈ X | x ∕∈ {x′ }} and prove the equality W = W ′ . The inclusion W ⊆ W ′ istrivial since W is specialization-closed and x ∕∈ W . On the other hand, one can easily checkthat W ′ is a specialization-closed subset of X which does not contain x. Thus we conclude theequality W = W ′ from the above argument.
(ii) ⇒ (i). First we prove that T := W ∪ {x} is specialization-closed, i.e., {x} ⊆ W ∪ {x}.
For any element x′ ∈ {x} \ {x}, it satisfies x ∕∈ {x′ } and hence x′ ∈ W . Here, we used theassumption that X is sober. Next, we shall check that T is a unique minimal specializationclosed subset that properly contains W . Let V be a specialization-closed subset of X suchthat W ⊊ V . Take an element x′ ∈ V \ W . Then we have x ∈ {x′ } ⊆ V and this implies
T = W ∪ {x} ⊆ V .
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Now, we are ready to prove the main theorem in this section.
Theorem 2.16. Let T be a triangulated category and (X, σ) be a classifying support data for
T . Then there is a homeomorphism X ∼= Spec△ (T ).
Proof. By the lattice isomorphism in Definition 2.14(ii), the prime thick subcategories correspond to the specialization-closed subsets W of X which satisfy the equivalent conditions inLemma 2.15. Therefore the map
ϕ : X → Spec△ (T ),x ,→ σ −1 ({x′ ∈ X | x ∕∈ {x′ }})is a well-defined bijection. Thus it remains to check that this map is a homeomorphism. To
this end, let us check the equality ϕ(x) = {M ∈ T | x ∕∈ σ(M )} first. Let M be an objectof T . Since σ(M ) is a closed subset of a noetherian sober space X, we can decompose it into
the union σ(M ) = {x1 } ∪ · · · ∪ {xr } of irreducible components. We then obtain the following

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equivalences, which mean ϕ(x) = {M ∈ T | x ∕∈ σ(M )}:σ(M ) ⊆ {x′ ∈ X | x ∕∈ {x′ }} ⇔ xi ∈ {x′ ∈ X | x ∕∈ {x′ }} for all i⇔ x ∕∈ {xi } for all i⇔ x ∕∈ σ(M ).

For any x ∈ X and M ∈ T , the equality ϕ(x) = {M ∈ T | x ∕∈ σ(M )} immediately implies
ϕ−1 (Supp△ (M )) = σ(M ). Hence ϕ : X → Spec△ (T ) is continuous. On the other hand, for any
two element x, x′ ∈ X, x ∈ {x′ } if and only if ϕ(x) ⊆ ϕ(x′ ). Indeed, there are the following
equivalences:x ∈ {x′ } ⇔ x′ ∕∈ σ(ϕ(x))(= {x′ ∈ X | x ∕∈ {x′ }})⇔ x′ ∕∈ σ(M ) for any M ∈ ϕ(x)⇔ M ∈ ϕ(x′ ) for any M ∈ ϕ(x)⇔ ϕ(x) ⊆ ϕ(x′ ).

Then Proposition 2.5 yields that ϕ({x}) = {ϕ(x)} and therefore ϕ−1 is also continuous.

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Recall that a commutative noetherian local ring (R, m) is a hypersurface if the m-adic completion of R is a regular local ring modulo a non-zero element. Using Theorem 2.16, we candetermine the spectra of Dperf (X) and Dsg (X) for some special cases.
Corollary 2.17 (cf. [17, Theorems 3.10 and 4.7]). Let X be a noetherian quasi-aﬃne scheme.
(1) There is a homeomorphism X ∼= Spec△ (Dperf (X)).
(2) Assume further that X is locally a hypersurface (i.e., OX,x is a hypersurface for all x ∈ X).
Then there is a homeomorphism Sing(X) ∼= Spec△ (Dsg (X)).
Proof. Since X is quasi-aﬃne, the structure sheaf OX is an ample line bundle. According to[31, Lemma 3.12], every thick subcategory of Dperf (X) containing OX is Dperf (X).
(1) Let X be a thick subcategory of Dperf (X). Then we can easily verify that the subcategory
Y := {F ∈ Dperf (X) | F ⊗LOX G ∈ X for all G ∈ X }is a thick subcategory of Dperf (X) containing OX . Therefore Y = Dperf (X) and this means that
X is a thick tensor ideal. Then the classification [31, Theorem 3.15] shows that (X, SuppX ) isa classifying support data for Dperf (X). Thus Theorem 2.16 finishes the proof.
(2) First note that there is an action ∗ : Dperf (X) × Dsg (X) → Dsg (X) on Dsg (X) whichmakes Dsg (X) a Dperf (X)-module; see [27, Section 3]. Let X be a thick subcategory of Dsg (X).
Then the subcategory
Y := {F ∈ Dperf (X) | F ∗ G ∈ X for all G ∈ X }is a thick subcategory of Dperf (X) containing OX . Again this implies that Y = Dperf (X) andtherefore X is a Dperf (X)-submodule of Dsg (X). Then the classification [27, Theorem 7.7] gives
us a classifying support data (Sing(X), SuppX ) for Dsg (X). The homeomorphism follows by
Theorem 2.16.
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9

3. Prime thick subcategories of derived and singularity categories ofnoetherian schemes
In this section, we study prime thick subcategories of the derived and the singularity categories of a noetherian scheme X and investigate their spectra.
We start with several lemmas, which allow us to study prime thick subcategories locally.
First, let us discuss a comparison between the Verdier localization and the localization by aprime ideal of a commutative noetherian ring. Let R be a commutative noetherian ring and T
be an R-linear triangulated category. We denote by ST (p) the subcategory of T consisting ofobjects M which satisfy 1M = 0 in HomT (M, M )p , i.e., a1M = 0 in T for some a ∈ R \ p.
Lemma 3.1. (a) ST (p) is a thick subcategory of T .
(b) For any element a ∈ R \ p, the mapping cone C(a1M ) of a1M : M → M is in ST (p).
(c) The canonical functor Q : T → T /ST (p) induces an isomorphism∼=

HomT (M, N )p −→ HomT /ST (p) (M, N ),f /a ,→ Q(a1N )−1 Q(f )of R-modules for any M, N ∈ T .
Proof. (a) Obviously, ST (p) is closed under direct summands and shifts. It remains to showfgthat ST (p) is closed under extensions. Take a triangle L −→ M −→ N → L[1] in T with
L, N ∈ ST (p). Then there are elements a, b ∈ R \ p such that a1L = 0 and b1N = 0 in T .
Since g(b1M ) = (b1N )g = 0, there is a morphism u : M → L such that f u = b1M . Therefore(ab)1M = a(f u) = f (a1L )u = 0 and hence M ∈ ST (p).
a1(b) Set C := C(a1M ) and prove C ∈ ST (p). For any object N ∈ T , the triangle M −−M→
M → C → M [1] induces an exact sequencea∼=

HomT (N, M )p −→ HomT (N, M )p → HomT (N, C)pa∼=→ HomT (N, M [1])p −→ HomT (N, M [1])pof Rp -modules. Thus we get HomT (N, C)p = 0 for any N ∈ T . Taking N = C, we conclude
that 1C = 0 in HomT (C, C)p .
(c) Thanks to (b), the canonical functor Q : T → T /ST (p) induces a homomorphism
HomT (M, N )p → HomT /ST (p) (M, N ),f /a ,→ Q(a1N )−1 Q(f )of R-modules.
Let f ∈ HomT (M, N ) with Q(f ) = 0 in T /ST (p). Then there is a morphism s : N → L in
T such that sf = 0 in T and C(s) ∈ ST (p). Pick an element a ∈ R \ p with a1C(s) = 0. Thenst
a1N factors as a1N : N −→L−→ N . This shows that af = (a1N )f = tsf = 0 and hence theabove homomorphism is injective.
Let α : M → N be a morphism in T /ST (p). This morphism is given by α = Q(s)−1 Q(f ) forsome morphisms f : M → L and s : N → L in T with C(s) ∈ ST (p). Take an element a ∈ R \pstsuch that a1C(s) = 0. Then a1N factors as a1N : N −→L−→ N . Using this factorization, we
get α = Q(s)−1 Q(f ) = Q(ts)−1 Q(tf ) = Q(a1N )−1 Q(tf ), which belongs to the image of thehomomorphism. This shows the surjectivity of the homomorphism.
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Concerning our R-linear triangulated categories Dperf (R), Db (R), and Dsg (R) for a commutative noetherian ring R, the above lemma is translated into the following statement.
Lemma 3.2. Let R be a commutative noetherian ring and p be a prime ideal of R.
(1) There is a triangle equivalence∼=

Dperf (R)/S perf (p) −→ Dperf (Rp ),where S perf (p) := {M ∈ Dperf (R) | Mp ∼= 0 in Dperf (Rp )}.
(2) There is a triangle equivalence∼=

Db (R)/S b (p) −→ Db (Rp ),where S b (p) := {M ∈ Db (R) | Mp ∼= 0 in Db (Rp )}.
(3) There is a triangle equivalence∼=

Dsg (R)/S sg (p) −→ Dsg (Rp ),where S sg (p) := {M ∈ Dsg (R) | Mp ∼= 0 in Dsg (Rp )}.

Proof. Let D ∈ {Dperf , Db , Dsg }, and S(p) ∈ {S perf (p), S b (p), S sg (p)} the corresponding thicksubcategory of D(R). Then there is an isomorphism∼=

HomD(R) (M, N )p −→ HomD(Rp ) (Mp , Np ),(∗)

f /a ,→ a−1 fp

of Rp -modules for any M, N ∈ D(R). Indeed, for D(R) = Dperf (R) or D(R) = Db (R), see[1, Lemma 5.2(b)]. For D(R) = Dsg (R), it follows from [5, Theorem 3.8] that every object in
Dsg (R) is isomorphic to M [m] for some finitely generated R-module M and m ∈ Z, and there
is a natural isomorphism for any finitely generated R-modules M, N :colim Hom (Ωi−m M, Ωi−n N ) ∼= HomDsg (R) (M [m], N [n]).
Ri≫0Since the left-hand side commutes with localizations by [32, Lemma 3.9], we are done.
From the isomorphism (∗), one has SD(R) (p) = S(p). Furthermore, the isomorphism (∗) isequal to the composition∼=

HomD(R) (M, N )p −→ HomD(R)/S(p) (M, N ) → HomD(Rp ) (Mp , Np ),where the first map is the one given in Lemma 3.1(c) and the second one is induced from thelocalization at p. Therefore, the second map is an isomorphism and hence the localization
functor D(R) → D(Rp ) induces a fully faithful functor
D(R)/S(p) → D(Rp ),M ,→ Mp

As a result, it is enough to check that the localization functor D(R) → D(Rp ) is essentiallysurjective in each case.
(1) Note that Dperf (R) ∼= Kb (R), where Kb (R) is the homotopy category of bounded complexes of finitely generated projective R-modules. Let P ∈ Kb (Rp ). Then we shall construct P% ∈ Kb (R) such that P%p ∼= P in Kb (Rp ). By taking shifts, we may assume that
d

dr−1

d

r
1
P := (0 → Pr −→Pr−1 −−→ · · · −→P0 → 0). Because each component Pi of P is a free Rp module, each diﬀerential di : Pi → Pi−1 is given by a matrix with entries in Rp . Take s ∈ R \ p
so that sα belongs to the image of R in Rp for all entries α of diﬀerentials of P . Then for eachSPECTRA OF TRIANGULATED CATEGORIES

11

i, sdi comes from a homomorphism d%i : P%i → P%i−1 between free R-modules, i.e., (P%i )p = Pi
and (d%i )p = sdi . Since (d%i−1 )p (d%i )p = s2 di−1 di = 0, one can find an element t ∈ R \ p such that
td%i d%i−1 = 0 for all i. In particular, we obtain a bounded complex!

tdr−1
td!1 %
td!r %
P% := (0 → P%r −→Pr−1 −−−→ · · · → P%1 −→P0 → 0)of finitely generated free R-modules such thatstdr−1
std1
stdr
P%p = (0 → Pr −−→
Pr−1 −−−→ · · · → P1 −−→
P0 → 0).

This P%p is isomorphic to P as complexes of Rp -modules. In fact, there is a commutative diagram
std

stdr−1

std

std

std

d

dr−1

d

d

d

r
3
2
1
0 −−−→ Pr −−−→
Pr−1 −−−−→ · · · −−−→
P2 −−−→
P1 −−−→
P0 −−−→ 0&&&&(&(st)r&(st)r−1&(st)2
&
(
'
'
'
'st
(
r
3
2
1
0 −−−→ Pr −−−→ Pr−1 −−−→ · · · −−−→ P2 −−−→ P1 −−−→ P0 −−−→ 0

whose vertical arrows are all isomorphisms of Rp -modules.
(2) Let M ∈ Db (Rp ). Taking a truncation of a projective resolution of M gives rise to atrianglef

N [−1] −→P →M →Nbin D (Rp ) with N finitely generated Rp -module and P ∈ Kb (Rp ). Pick a finitely generated∼∼==% and P% ∈ Kb (R) together with isomorphisms ϕ : N%p −R-module N→ N and ψ : P%p −→ P . The% is easy and that of a P% is by (1). As we have remarked as (∗), there isexistence of such an N
an isomorphism∼=% [−1], P%)p −%p [−1], P%p ),HomDb (R) (N→ HomDb (Rp ) (Ng/a ,→ a−1 gp .

%p [−1] → P%p is equal to a−1 gp for some g ∈Therefore, the composition ψ −1 f ϕ[−1] : N% [−1], P%) and a ∈ R \ p. Embedding g into a triangle
HomDb (R) (Ng% [−1] −)→N%
N→ P% → M) ∈ Db (R) with M)p ∼in Db (R), we obtain an object M= M . Indeed, localizing this triangle at p,
we obtain a commutative diagram%p [−1] gp
N

aϕ[−1] ∼=#N [−1]

f

%p
!P

ψ ∼
=

#
!P

)p
!M
#
!M

%p
!N

aϕ ∼
=

#
!N

in Db (Rp ) and hence there is a dotted arrow which is an isomorphism in Db (Rp ).
(3) We note that every object of Dsg (Rp ) is isomorphic in Dsg (Rp ) to a finitely generated
Rp -module. It is isomorphic to a localization of some finitely generated R-module.
!
Recall that a commutative noetherian local ring (R, m) is a complete intersection (of codimension c) if its m-adic completion is isomorphic to a regular local ring modulo a regular sequence

12

HIROKI MATSUI(of length c). We note that complete intersection local rings of codimension 1 coincide withhypersurface local rings.
Now, let us first study prime thick subcategories of the perfect derived category Dperf (X) andthe bounded derived category Db (X) of a noetherian scheme X. The next proposition givesa characterization of complete intersection local rings in terms of prime thick subcategories ofthe bounded derived category.
Proposition 3.3. Let (R, m) be a commutative noetherian local ring. Then 0 is a prime thicksubcategory of Db (R) if and only if R is a complete intersection.
Proof. Assume that 0 is a prime thick subcategory of Db (R) and let X be a unique minimal
non-zero thick subcategory of Db (R). The minimality of X shows that X ⊆ Dperf (R) and inparticular X contains a non-zero perfect complex. For any non-zero object M ∈ Db (R), denoteby thick(M ) the smallest thick subcategory of Db (R) containing M . Since M is non-zero,again using the minimality of X , we get X ⊆ thick(M ). This shows that thick(M ) contains a
non-zero perfect complex for any non-zero object M ∈ Db (R). Consequently, the ‘only if’ partfollows by the result [23, Theorem 5.2].
To show the ‘if part’, assume that R is a complete intersection. The combination of [20,Theorem 1.5] and [23, Theorem 5.2] yields that every non-zero thick subcategory X of Db (R)contains the non-zero thick subcategory Supp−1({m}) := {M ∈ Dperf (R) | SuppR (M ) ⊆Dperf (R){m}}. Hence, Supp−1({m}) is a unique minimal non-zero thick subcategory of Db (R). !
Dperf (R)Now we are ready to prove one of the main theorems in this paper, which provides examples
of prime thick subcategories.
Theorem 3.4. Let X be a noetherian scheme and x ∈ X.
(1) The thick subcategory
S perf (x) := {F ∈ Dperf (X) | Fx ∼= 0 in Dperf (OX,x )}X

is a prime thick subcategory of Dperf (X).
(2) SetbSX(x) := {F ∈ Db (X) | Fx ∼= 0 in Db (OX,x )}.
b
Then SX(x) is a prime thick subcategory of Db (X) if and only if OX,x is a complete intersection.
Proof. (1) Let U ⊆ X be an aﬃne open neighborhood of x with complement Z := X \ U . It
follows from [3, Theorem 2.13] that the restriction functor (−)|U : Dperf (X) → Dperf (U ) inducesa triangle equivalence
perf
Dperf (X)/Dperf(U )Z (X) → D
perf
up to direct summands, where D (X) := {F ∈ Dperf (X) | F|U ∼= 0 in Dperf (U )}. On theZ

perf
other hand, Dperf
Z (X) ⊆ SX (x) follows from their definitions. By Proposition 2.11(1), we havethe following lattice isomorphism
Th(Dperf (X)/Dperf (X)) ∼= Th(Dperf (U ))
Z

perf
perf
under which SX(x)/Dperf
Z (X) corresponds to SU (x). As a result, we may assume that X isaﬃne by Proposition 2.9(2).

SPECTRA OF TRIANGULATED CATEGORIES

13

Let X be an aﬃne noetherian scheme. From Proposition 2.9(2) and the triangle equivalence
D (X)/S perf (x) ∼= Dperf (OX,x ) which is obtained in Lemma 3.2(1), it is enough to check that0 is a prime thick subcategory of Dperf (OX,x ). We use the poset isomorphism
perf{X ∈ Th(Dperf (OX,x )) | X ∕= 0} ∼= {W ∈ Spcl(Spec OX,x ) | W ∕= ∅}which follows from [20, Theorem 1.5]. Since the right-hand side has a unique minimal element{mX,x } (mX,x is the unique maximal ideal of OX,x ), so does the left-hand side. This means that0 is a prime thick subcategory of Dperf (OX,x ).
(2) Let U ⊆ X be an aﬃne open neighborhood of x with complement Z := X \ U . By [25,
Theorem 3.3.2], the restriction functor (−)|U : Db (X) → Db (U ) induces a triangle equivalence∼=

Db (X)/DbZ (X) −→ Db (U ),where DbZ (X) := {F ∈ Db (X) | F|U = 0 in Db (U )}. With the same argument as above, thistriangle equivalence and Proposition 2.9(2) allow us to assume that X is aﬃne. From thisobservation, the result follows by Propositions 2.9(2), 3.3 and Lemma 3.2(2).
!
Next, we discuss the singularity category of a noetherian scheme and prove a similar result
to Theorem 3.4. To reduce to the aﬃne case, we need the following result. Although it may bewell known, we give a proof since we have not been able to locate a specific reference, see [22,Proposition 1.14] for the case of Sing(X) ⊆ U .
Proposition 3.5. Let X be a separated noetherian scheme and U an aﬃne open subset of X
with complement Z := X \ U . Then the restriction functor (−)|U : Dsg (X) → Dsg (U ) inducesa triangle equivalence∼=

Dsg (X)/Dsg→ Dsg (U ),Z (X) −
where Dsg (X) := {F ∈ Dsg (X) | F|U ∼= 0 in Dsg (U )}.
ZThis result is essentially shown by Krause in [15, Proposition 6.9] using the unbounded stable
derived category S(X). Let X be a separated noetherian scheme. Denote by K(Inj X) the
homotopy category of complexes of injective quasi-coherent OX -modules, and by S(X) the
subcategory of K(Inj X) consisting of all acyclic complexes. By the result of [15, Corollary 5.4],
S(X) is compactly generated triangulated category and there is a triangle equivalence
F : Dsg (X) → S(X)cup to direct summands, where (−)c stands for the thick subcategory consisting of all compact
objects.
Proof of Proposition 3.5. Let j : U ↩→ X be the open immersion. Since j ∗ : K(Inj X) →K(Inj U ) preserves acyclicity, it restricts to an exact functor j ∗ : S(X) → S(U ). By virtueof [26, Corollary 4.11] and [27, Lemma 7.2], SZ (X) is generated by compact objects of S(X).
Then it follows from [15, Corollary 6.9] and [21, Theorem 2.1] that j ∗ : S(X) → S(U ) inducesa triangle equivalencej ∗ : S(X)c /SZ (X)c → S(U )c14

HIROKI MATSUI

up to direct summands. Note that by [15, Theorem 6.6], there is a commutative diagram
!
Dsg
Z (X)

(∗)

"

! Dsg (X)Fj∗

! Dsg (U )F#

SZ (X)

F

#

c! "

j∗

! S(X)c#! S(U )cof exact functors and hence we get F −1 (SZ (X)c ) = Dsg
Z (X).
Consider a morphism f : M ′ → F (M ) in S(X)c with M ′ ∈ SZ (X)c and M ∈ Dsg (X). By∼=
Lemma 2.10, there is an isomorphism ϕ : M ′ ⊕ M ′ [1] −→ F (L) for some L ∈ Dsg (X). Sincesg
F (j ∗ (L)) ∼= j ∗ (F (L)) ∼= j ∗ (M ′ ⊕ M ′ [1]) = 0, one has j ∗ (L) ∼= 0 i.e., L ∈ DZ (X). Therefore, f
ϕt (1,0)(f,0)ϕ−1

factors as M ′ −−−−→ F (L) −−−−−→ F (M ) with L ∈ Dsg
Z (X). Using [13, Proposition 10.2.6(ii)],
F induces a triangle equivalencecc
F : Dsg (X)/Dsg
Z (X) → S(X) /SZ (X)up to direct summands. From the commutative diagram (∗), we have a commutative diagram
Dsg (X)/Dsg
Z (X)

j∗

! Dsg (U )FF#

S(X)c /SZ (X)cj∗#
! S(U )cof exact functors, where the vertical functors and the bottom j ∗ are triangle equivalences
up to direct summands. Therefore, the top j ∗ is also fully faithful. On the other hand,sg∗j ∗ : Dsg (X)/Dsg:
Z (X) → D (U ) is essentially surjective as the restriction functor jbb
D (X) → D (U ) is essentially surjective by [25, Theorem 3.3.2]. Hence we conclude thatsgj ∗ : Dsg (X)/Dsg!
Z (X) → D (U ) is a triangle equivalence.
The next proposition, which is similar to Proposition 3.3, characterizes hypersurface local
rings in terms of prime thick subcategories of the singularity category of a commutative noetherian local ring.
Proposition 3.6. Let R be a non-regular commutative noetherian local ring. If R is a hypersurface, then 0 is a prime thick subcategory of Dsg (R). The converse holds if R is a complete
intersection.
Proof. Assume that R is hypersurface. From [28, Theorem 6.6], there is a poset isomorphism{X ∈ Th(Dsg (R)) | X ∕= 0} ∼= {W ∈ Spcl(Sing(R)) | W ∕= ∅},which proves the first statement. The second one follows from [30, Theorem 3.7]. Indeed, if 0
is prime, then there is a unique minimal non-zero thick subcategory X of Dsg (R). This X must
be equal to the intersection of all non-zero thick subcategories of Dsg (R).
!
In view of the proof of Theorem 3.4, we can deduce the following result from Lemma 3.2(3)and Propositions 3.5, 3.6.

SPECTRA OF TRIANGULATED CATEGORIES

15

Theorem 3.7. Let X be a separated noetherian scheme and x ∈ Sing(X). If OX,x is a hypersurface, thensgSX(x) := {F ∈ Dsg (X) | Fx ∼= 0 in Dsg (OX,x )}is a prime thick subcategory of Dsg (X). Conversely, if OX,x is a complete intersection andsgSX(x) is a prime thick subcategory of Dsg (X), then OX,x is a hypersurface.
Proof. Using Proposition 3.5, the same argument as in the proof of Theorem 3.4 allows us toreduce to the aﬃne case. Assume X is an aﬃne scheme. Then Lemma 3.2(3) gives us a triangle
equivalencesg
Dsg (X)/SX(x) ∼= Dsg (OX,x ).
sg
It follows from Proposition 2.9(2) that SX(x) is prime if and only if 0 is a prime thick subcatsg
egory of D (OX,x ). Then the result follows by Proposition 3.6.
!

For a noetherian scheme X, we define the complete intersection locus CI(X) and the hypersurface locus HS(X) of X by
CI(X) := {x ∈ X | OX,x is a complete intersection},

HS(X) := {x ∈ Sing(X) | OX,x is a hypersurface}.

Applying Theorems 3.4 and 3.7, we obtain the following result which has been stated in the
introduction.
Corollary 3.8. Let X be a noetherian scheme.
(1) There is an immersion
X ↩→ Spec△ (Dperf (X))
of topological spaces.
(2) There is an immersion
CI(X) ↩→ Spec△ (Db (X))
of topological spaces.
(3) If X is furthermore separated, then there is an immersion
HS(X) ↩→ Spec△ (Dsg (X))
of topological spaces.
perf
Proof. (1) From the equality SuppX (SX(x)) = {x′ ∈ X | x ∕∈ {x′ }} and Theorem 3.4(1), there
is an injective map

X ↩→ Spec△ (Dperf (X)),

perf
x ,→ SX(x).

For an object F ∈ Dperf (X), one can easily see that the inverse image of Supp△ (F) by thisinjection is SuppX (F). Since the family {SuppX (F) | F ∈ Dperf (X)} (resp. {Supp△ (F) | F ∈Dperf (X)}) forms a closed basis of X (resp. Spec△ (Dperf (X))), the topology on X is nothing
but the one induced from Spec△ (Dperf (X)).
The remained statements (2) and (3) are shown similarly as above.
!

16

HIROKI MATSUI

Remark 3.9. If X is an excellent Cohen-Macaulay scheme, then the complete intersection
locus CI(X) and the hypersurface locus HS(X) of X are open subsets of X and Sing(X),respectively.
Indeed, because the problem is local, we may assume that R is an excellent Cohen-Macaulay
ring and prove that the complete intersection locus CI(R) and the hypersurface locus HS(R) areopen in Spec R and Sing R, respectively. The openness of the complete intersection locus CI(R)has been already known in [9, Corollary 3.3]. Therefore, assume that R is an excellent locallycomplete intersection ring and show that the hypersurface locus HS(R) is open in Sing R.
Set H1 (R) := {p ∈ Spec R | codim Rp ≤ 1}. Since HS(R) = H1 (R) ∩ Sing R, it suﬃces toshow that H1 (R) is open in Spec R. Recall that for a prime ideal p of R, codim Rp coincides
with the 1st deviation &1 (Rp ) because Rp is a complete intersection; see [19, Section 21]. Thenthe openness of H1 (R) follows from [24, Proposition 3.6].
We say that two noetherian schemes X and Y are• perfectly derived equivalent if Dperf (X) ∼= Dperf (Y ) as triangulated categories.
bb• derived equivalent if D (X) ∼= D (Y ) as triangulated categories.
sg• singularly equivalent if D (X) ∼= Dsg (Y ) as triangulated categories.
If X and Y are derived equivalent, then Y is said to be a Fourier-Mukai partner of X. The following result is a direct consequence of Corollary 3.8 because equivalent triangulated categories
have homeomorphic spectra.
Corollary 3.10. Let X be a noetherian scheme.
(1) There is an immersion
Y ↩→ Spec△ (Dperf (X))

of topological spaces for any noetherian scheme Y which is perfectly derived equivalent toX.
(2) There is an immersion
CI(Y ) ↩→ Spec△ (Db (X))
of topological spaces for any noetherian scheme Y which is derived equivalent to X.
(3) Assume further that X is separated. Then there is an immersion
HS(Y ) ↩→ Spec△ (Dsg (X))
of topological spaces for any separated noetherian scheme Y which is singularly equivalent
to X.
Remark 3.11. For two commutative noetherian rings R and S, it is known as the derived
Morita theorem that there are implications
perfectly derived equivalent ⇔ derived equivalent ⇒ singularly equivalent.
If X and Y are projective schemes over a field and assume that X possesses an ample line
bundle satisfying a certain vanishing condition on cohomologies, then the same implications
perfectly derived equivalent ⇔ derived equivalent ⇒ singularly equivalenthold by [2, Theorem 7.13].

SPECTRA OF TRIANGULATED CATEGORIES

17

4. Comparison with Balmer spectra
In this section, we compare the spectrum Spec△ (T ) discussed so far to the Balmer spectrumSpec⊗ (T ) for a tensor triangulated category (T , ⊗, 1). Besides, we discuss more the spectrumof the tensor triangulated category (Dperf (X), ⊗LOX , OX ) for a noetherian scheme X. To thisend, let us start with a brief survey on Balmer’s tensor triangular geometry.
Recall that a triple (T , ⊗, 1) is a tensor triangulated category if T is a triangulated category
equipped with a symmetric monoidal structure (⊗, 1) which is compatible with the triangulated
structure; see [12, Appendix A] for the precise definition.
Definition 4.1. Let (T , ⊗, 1) be a tensor triangulated category.
(1) A thick subcategory I of T is called a (thick tensor) ideal if for M ∈ T and N ∈ X one
has M ⊗ N ∈ X .
(2) For an ideal I of T , define its radical by√
I := {M ∈ T | M ⊗n ∈ T for some non-negative integer n}.
√
We say that I is radical if I = I holds. The set of radical ideals of T is denoted byRad⊗ (T ).
(3) An ideal P of T is called prime provided P ∕= T and if M ⊗ N is in P, then so is either M
or N . Denote by Spec⊗ (T ) the set of prime ideals of T .
Now, let us define a topology on Spec⊗ (T ) whose definition is the same as those of Spec△ (T ),except it uses prime ideals instead of prime thick subcategories.
Definition 4.2 ([4, Definition 2.1]). For a family E of objects of T , we set
Z(E) := {P ∈ Spec⊗ (T ) | P ∩ E = ∅}.
We can easily check that the family {Z(E) | E ⊆ T } of subsets of Spec⊗ (T ) satisfies the axioms
for closed subsets. The set Spec⊗ (T ) together with this topology is called the Balmer spectrumof T .
For an object M ∈ T , define the Balmer support of M by
Supp⊗ (M ) := Z({M }) = {P ∈ Spec⊗ (T ) | M ∕∈ P}.
Then the family of the Balmer supports forms a closed basis of Spec⊗ (T ).
Proposition 2.5, Definition 2.6, and Theorem 2.8 should be compared with the following
couple of results.
Proposition 4.3 ([4, Proposition 2.9]). For a prime ideal P of T , one has{P} = {Q ∈ Spec⊗ (T ) | Q ⊆ P}.
In particular, Spec△ (T ) is a T0 -space.
Proposition 4.4 ([4, Lemma 4.2]). Let I be an ideal of T . Then one has the equality
$
√
I=
P.
I⊆P∈Spec⊗ (T )

18

HIROKI MATSUIA subset of a topological space X is said to be Thomason if it is a union of closed subsets
whose complements are quasi-compact. We denote by Thom(X) the set of Thomason subsetsof X. It is immediately from the definition that Thomason subsets are specialization-closed. If
X is noetherian, then the converse also holds true i.e., Thom(X) = Spcl(X).
Theorem 4.5 ([4, Theorem 4.10]). Let T be a tensor triangulated category. Then there is amutually inverse lattice isomorphisms
Supp⊗Rad⊗ (T ) "where Supp⊗ (X ) :="!

Supp−1⊗

Thom(Spec⊗ (T )),−1
M ∈X Spec⊗ (M ) and Supp⊗ (W ) := {M ∈ T| Supp⊗ (M ) ⊆ W }.

Remark 4.6 (cf. Definition 2.7). We remark that by [4, Proposition 2.14] the equality
holds.

Thom(Spec⊗ (T )) = {Supp⊗ (X ) |⊆ X }.

The following result shows that prime ideals of T are characterized by a similar condition
to the definition of prime thick subcategories. Therefore, it justifies our definition of a prime
thick subcategory.
Proposition 4.7. Let P be a radical ideal of T . Assume that the set {I ∈ Rad⊗ (T ) | P ⊊ I}
has a unique minimal element I. Then P is a prime ideal. The converse holds if Spec⊗ (T ) isnoetherian (e.g., T = Dperf (X) for a noetherian scheme).
Proof. Since P is a radical ideal, by Proposition 4.4, P is the intersection of all prime ideals Q
which contain P. If P is not a prime ideal, these Q properly√ contain P and hence also contain
I by assumption. Therefore, we conclude that P ⊊ I ⊆ P = P, a contradiction. As a result,
P is a prime ideal.
Assume that Spec⊗ (T ) is noetherian and P is a prime ideal. By the lattice isomorphism inTheorem 4.5, P corresponds to the specialization-closed subset
W := {Q ∈ Spec⊗ (T ) | P ∕⊆ Q} = {Q ∈ Spec⊗ (T ) | P ∕∈ {Q}}.

Here, the second equality follows from Proposition 4.3. Then Lemma 2.15 shows that there is aunique minimal specialization-closed subset T of Spec⊗ (T ) with W ⊊ T . Again using the lattice
isomorphism in Theorem 4.5, I := Supp−1!
⊗ (T ) satisfies the condition in the statement.
Now we establish the following result which gives a connection between prime ideals andprime thick subcategories.
Proposition 4.8. Let T be a tensor triangulated category and P be a radical ideal of T . If P
is a prime thick subcategory of T , then it is a prime ideal of T .
Proof. As P is a prime thick subcategory of T , there is a unique minimal thick subcategory X
of T such that X ⊊ T . Denote by X% the smallest radical ideal of T containing X . For any
radical ideal I of T with P ⊊ I, the minimality of X shows X ⊆ I and hence X% ⊆ I. Thismeans that X% is a unique minimal radical ideal of T with P ⊊ X%. Then Proposition 4.7 showsthat P is a prime ideal.
!

SPECTRA OF TRIANGULATED CATEGORIES

19

perf
For a noetherian scheme X, the thick subcategory SX(x) is a prime ideal, and actually, every
prime ideal of Dperf (X) is given in this way; see [4, Corollary 5.6]. Therefore, the immersion in
Theorem 3.4(1) can be considered as the inclusion

Spec⊗ (Dperf (X)) ⊆ Spec△ (Dperf (X)).

From this observation and Proposition 4.8, we obtain the following corollary.
Corollary 4.9. Let X be a noetherian scheme and P be an ideal of Dperf (X). Then P is aprime thick subcategory of Dperf (X) if and only if P is a prime ideal of Dperf (X).
We close this paper by giving one concrete example of spectra.

Example 4.10. Consider the projective line P1 over a field k. First note that since P1 isregular, there is a triangle equivalence Db (P1 ) ∼= Dperf (P1 ). It is explained in [16, Section 4.1]
that there is a lattice isomorphism
Th(Dperf (P1 )) ∼= Spcl(P1 ) ⊔ Z.
Here, the above Z is considered as the discrete poset Z together with a maximum and aminimum elements. On the right-hand side, an element of Spcl(P1 ) corresponds to an idealof Dperf (P1 ) and those of Z corresponds to a thick subcategory of the form thick(OP1 (i)) forsome i ∈ Z. As Corollary 4.9 shows, the prime ideals and the ideals which are prime thicksubcategories are the same. Therefore, restricting the above lattice isomorphism to prime thicksubcategories, we get a homeomorphism
Spec (Dperf (P1 )) ∼= P1 ⊔ Z,△

where Z is considered as the discrete topological space. This gives an example of noetherian
scheme which is not quasi-aﬃne and the immersion X ↩→ Spec△ (Dperf (X)) is not a homeomorphism.
Acknowledgments. The author thanks Ryo Takahashi for giving helpful comments and usefulsuggestions, and Tsutomu Nakamura for fruitful discussions on Proposition 3.5. Also, theauthor is grateful to the anonymous referee for his/her careful reading and valuable comments.
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71 (1991), no. 2, 129–155.
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arXiv:0905.3148.
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no. 3, 557–580.
[4], The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588 (2005),
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Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguroku, Tokyo 153-8914, Japan
Email address: mhiroki@ms.u-tokyo.ac.jp