Prime thick subcategories and spectra of derived and singularity categories of noetherian schemes
aa r X i v : . [ m a t h . AG ] F e b PRIME THICK SUBCATEGORIES AND SPECTRA OF DERIVED ANDSINGULARITY CATEGORIES OF NOETHERIAN SCHEMES
HIROKI MATSUI
Abstract.
For an essentially small triangulated category T , we introduce the notion of primethick 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 trian-gulated categories that appear in algebraic geometry such as the derived and the singularitycategories 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. Introduction
Classification of thick subcategories is one of the important approaches for understandingthe 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], mod-ular representation theory of finite groups [5, 6, 7], stable homotopy theory [10, 11], and soon. Especially, classifications of thick tensor ideals of tensor triangulated categories are quitesuccessful. For example, the thick tensor ideals of the perfect derived category of a noetherianscheme, 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 givenessentially small tensor triangulated category ( T , ⊗ , ), Balmer defines a topology on the setSpec ⊗ ( T ) of prime thick tensor ideals of T , which is called the Balmer spectrum of T . Balmerestablishes the following monumental work in the theory. Theorem 1.1 (Balmer) . Let T be an essentially small tensor triangulated category. Thenthere 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 tostudy an essentially small tensor triangulated category using its Balmer spectrum.Tensor triangular geometry can be applied to any essentially small tensor triangulated cat-egories, whereas it cannot be directly applied to triangulated categories that are not tensortriangulated. Such triangulated categories include important ones, e.g., the bounded derivedcategory D b ( X ) of coherent sheaves and the singularity category D sg ( X ) of a noetherian scheme X do not have natural tensor triangulated structures, which are used in the study of birational Mathematics Subject Classification.
Key words and phrases. complete intersection, derived category, hypersurface, noetherian scheme, primethick subcategory, singularity category, spectrum, triangulated category.The author was partly supported by JSPS Grant-in-Aid for JSPS Fellows 19J00158. 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 tensortriangular geometry for essentially small triangulated categories without tensor triangulatedstructures.The present paper aims to construct for a given essentially small triangulated category T atopological 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 subcategoriestogether with a topology given in [18]. Moreover, we study the prime thick subcategoriesand spectra of the perfect derived category D perf ( X ), the bounded derived category D b ( X )of coherent sheaves, and the singularity category D sg ( X ) of a noetherian scheme X . One ofthe main theorems of this paper is the following, which means that the spectra contain sometopological 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 △ (D perf ( X )) of topological spaces, which is a homeomorphism if X is quasi-affine. (2) There is an immersion
CI( X ) ֒ → Spec △ (D b ( X )) of topological spaces, which is a homeomorphism if X is quasi-affine and regular. (3) Assume that X is a separated Gorenstein scheme. Then there is an immersion HS( X ) ֒ → Spec △ (D sg ( X )) of topological spaces, which is a homeomorphism if X is quasi-affine and locally a hyper-surface. Here, CI( X ) is the complete intersection locus of X , which is the set of points x of X such that O X,x are complete intersection. Similarly, HS( X ) is the hypersurface locus , which is the set ofsingular points x such that O X,x are hypersurface. If X is excellent and Cohen-Macaulay, thenCI( 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 derivedcategories D b ( X ) and D b ( Y ) are equivalent as triangulated categories. This concept is alsoknown 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 △ (D b ( 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 idealsand 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 trian-gulated 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 = D perf ( X ) of a noetherian scheme X . PECTRA OF TRIANGULATED CATEGORIES 3
The organization of this paper is as follows. In Section 2, we introduce the notion of a primethick 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 D perf ( X ), D b ( X ), and D sg ( X ). The proof of Theorem 1.2 will be givenhere. In Section 4, we investigate our spectrum for an essentially small tensor triangulatedcategory. 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 beessentially 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 assumedto be full and additive. Denote by the zero subcategory of T , that is, the subcategoryconsisting of objects isomorphic to the zero object.(2) For a noetherian scheme X , the bounded derived category D b ( X ) of X is the derived categoryof bounded complexes of coherent sheaves on X . An object F of D b ( X ) is said to be perfect if, for any x ∈ X , there is an open neighborhood U ⊆ X of x such that F | U is isomorphicin D b ( U ) to a bounded complex of free O U -modules of finite rank. Denote by D perf ( X )the thick subcategory of D b ( X ) consisting of all perfect complexes. We call it the perfectderived category of X . Denote by D sg ( X ) the Verdier quotient D sg ( X ) := D b ( X ) / D perf ( X )which we call the singularity category of X .If we consider an affine scheme X = Spec R , then we simply write D perf ( R ), D b ( R ), andD sg ( 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 rolethroughout the paper. Definition 2.2.
We say that a thick subcategory P of T is prime if there is a unique thicksubcategory that is minimal among all thick subcategories X of T satisfying P ( X . Denoteby 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 triangulatedcategory are characterized by a similar condition; see Proposition 4.7. This fact justifiesthe above definition of prime thick subcategories.(2) Recently, Takahashi [30] introduced the notion of core of the singularity category D sg ( R ) of acommutative noetherian local ring R as the intersection of all non-trivial thick subcategoriesof D sg ( R ). Our definition of a prime thick subcategory and the concept of a core are closelyrelated. Indeed, one can easily see that is a prime thick subcategory of D sg ( R ) if andonly if the core of D sg ( R ) is not zero.We then recall the definition of a topology on Spec △ ( T ). HIROKI MATSUI
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 ). • T i ∈ I Z ( E i ) = Z ( S i ∈ I E i ). • 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 onSpec △ ( T ) whose family of closed subsets are { Z ( E ) | E ⊆ T } . We call this topological spacethe spectrum of T .For an object M ∈ T , define the support of M bySupp △ ( M ) := Z ( { M } ) = {P ∈ Spec △ T | M
6∈ P} . Then the family of supports { Supp △ ( M ) } M ∈T forms a closed basis of Spec △ ( T ). More generally,we set Supp △ ( X ) := [ M ∈X Supp △ ( M )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 T -space. Definition 2.6 ([18, Definition 2.7]) . For a thick subcategory X of T , we define its radical by √X := \ X ⊆P∈
Spec △ ( T ) P . We say that X is radical if the equality √X = X holds. Denote by Rad △ ( T ) the set of radicalthick subcategories of T . 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 asfollows. Theorem 2.8 ([18, Theorem 2.9]) . There is a mutually inverse lattice isomorphisms
Rad △ ( T ) Supp △ / / Param (Spec △ T ) . Supp − △ o o Here, we define
Supp − △ ( W ) := { M ∈ T | Supp △ ( M ) ⊆ W } . PECTRA OF TRIANGULATED CATEGORIES 5
We have explained so far basic definitions and general properties along with [18]. The re-mainder 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 − ( Y ) ⊆ T by F ( X ) := { M ′ ∈ T ′ | M ′ ∼ = F ( M ) for some M ∈ X } ,F − ( Y ) := { M ∈ T | F ( M ) ∈ Y } . One can easily check that F − ( Y ) is always a thick subcategory of T . Therefore, we have aposet homomorphism Th ( T ′ ) → Th ( T ) , Y 7→ F − ( 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 7→ F − ( Q ) . (2) For a thick subcategory
Q ∈ Th ( T / K ) , Q is a prime thick subcategory of T / K if and onlyif F − ( Q ) is a prime thick subcategory of T . (3) F induces an immersion a F : Spec △ ( T / K ) → Spec △ ( T ) , Q 7→ F − ( 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 − ( 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 − ( Q ) is a prime thicksubcategory of T .(3) Using (1) and (2), we have a well-defined injective map a F : Spec △ ( T / K ) → Spec △ ( T ) , Q 7→ F − ( Q )with image {P ∈ Spec △ ( T ) | K ⊆ P} . It is not hard to check that the equal-ity ( a F ) − (Supp △ ( M )) = Supp △ ( F ( M )) holds for an object M ∈ T . Since the fam-ily { 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. (cid:4) We say that an exact functor F : T → T ′ of triangulated categories is triangle equivalenceup 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. HIROKI MATSUI (1)
There is a lattice isomorphism Th ( T ′ ) ∼ = −→ Th ( T ) , Y 7→ F − ( Y ) . (2) For a thick subcategory
Q ∈ Th ( T ′ ) , Q is a prime thick subcategory of T ′ if and only if F − ( Q ) is a prime thick subcategory of T . (3) F induces a homeomorphism a F : Spec △ ( T ′ ) ∼ = −→ Spec △ ( T ) , P 7→ F − ( 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 7→ F − ( Y )is a lattice isomorphism. This follows from the same argument as in [4, Proposition 3.13]. (cid:4) Next, we discuss the determination of the spectrum of a given triangulated category using classifying support data . Definition 2.12.
Let T be a triangulated category. A support data for T is a pair ( X, σ ) of atopological 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 variousfields 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 X ( F ) := { x ∈ X | F x = 0 in D b ( O X,x ) } of F ∈ D b ( X ) defines a support data ( X, Supp X ) for D b ( X ).(3) Let X be a noetherian scheme and denote by Sing( X ) the singular locus of X . Then the singular support Supp X ( F ) := { x ∈ Sing( X ) | F x = 0 in D sg ( O X,x ) } of F ∈ D sg ( X ) defines a support data (Sing( X ) , Supp X ) for D sg ( 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 forany 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 followingconditions;(i) X is a noetherian sober space. Here, we say that a topological space X is sober if everyirreducible closed subset has a unique generic point. PECTRA OF TRIANGULATED CATEGORIES 7 (ii) There is a mutually inverse lattice isomorphisms Th ( T ) σ / / Spcl ( X ) , σ − o o where σ ( X ) := S M ∈X σ ( M ) and σ − ( W ) := { M ∈ T | σ ( M ) ⊆ W } .Using a classifying support data, we can translate a problem concerning thick subcategoriesto a problem about specialization-closed subsets. The following lemma characterizes the topo-logical counterpart of prime thick subcategories. Lemma 2.15.
Let X be a sober space and W be a specialization-closed subset of X . Thefollowing conditions are equivalent. (i) There is a unique minimal specialization-closed subset T of X such that W ( T . (ii) There is a unique element x ∈ X such that W = { x ′ ∈ X | x
6∈ { 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 . Fromthe assumption on T , one has the equalities W = W ∪ ( { x } \ { x } ) and T = W ∪ { x } . Thesetwo equalities yield T = W ∪ { x } = W ∪ { x } . Moreover, such x is uniquely determined bythe 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
6∈ { 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
6∈ { x ′ } and hence x ′ ∈ W . Here, we used theassumption that X is sober. Next, we shall check that T is a unique minimal specialization-closed 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 . (cid:4) 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 corre-spond 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 − ( { x ′ ∈ X | x
6∈ { x ′ }} )is a well-defined bijection. Thus it remains to check that this map is a homeomorphism. Tothis 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 intothe union σ ( M ) = { x } ∪ · · · ∪ { x r } of irreducible components. We then obtain the following HIROKI MATSUI equivalences, which mean ϕ ( x ) = { M ∈ T | x σ ( M ) } : σ ( M ) ⊆ { x ′ ∈ X | x
6∈ { x ′ }} ⇔ x i ∈ { x ′ ∈ X | x
6∈ { x ′ }} for all i ⇔ x
6∈ { x i } 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 − (Supp △ ( M )) = σ ( M ) . Hence ϕ : X → Spec △ ( T ) is continuous. On the other hand, for anytwo element x, x ′ ∈ X , x ∈ { x ′ } if and only if ϕ ( x ) ⊆ ϕ ( x ′ ). Indeed, there are the followingequivalences: x ∈ { x ′ } ⇔ x ′ σ ( ϕ ( x ))(= { x ′ ∈ X | x
6∈ { 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 − is also continuous. (cid:4) Recall that a commutative noetherian local ring ( R, m ) is a hypersurface if the m -adic com-pletion of R is a regular local ring modulo a non-zero element. Using Theorem 2.16, we candetermine the spectra of D perf ( X ) and D sg ( 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 △ (D perf ( X )) . (2) Assume further that X is locally a hypersurface (i.e., O X,x is a hypersurface for all x ∈ X ).Then there is a homeomorphism Sing( X ) ∼ = Spec △ (D sg ( X )) .Proof. Since X is quasi-affine, the structure sheaf O X is an ample line bundle. According to[31, Lemma 3.12], every thick subcategory of D perf ( X ) containing O X is D perf ( X ).(1) Let X be a thick subcategory of D perf ( X ). Then we can easily verify that the subcategory Y := {F ∈ D perf ( X ) | F ⊗ L O X G ∈ X for all
G ∈ X } is a thick subcategory of D perf ( X ) containing O X . Therefore Y = D perf ( X ) and this means that X is a thick tensor ideal. Then the classification [31, Theorem 3.15] shows that ( X, Supp X ) isa classifying support data for D perf ( X ). Thus Theorem 2.16 finishes the proof.(2) First note that there is an action ∗ : D perf ( X ) × D sg ( X ) → D sg ( X ) on D sg ( X ) whichmakes D sg ( X ) a D perf ( X )-module; see [27, Section 3]. Let X be a thick subcategory of D sg ( X ).Then the subcategory Y := {F ∈ D perf ( X ) | F ∗ G ∈ X for all G ∈ X } is a thick subcategory of D perf ( X ) containing O X . Again this implies that Y = D perf ( X ) andtherefore X is a D perf ( X )-submodule of D sg ( X ). Then the classification [27, Theorem 7.7] givesus a classifying support data (Sing( X ) , Supp X ) for D sg ( X ). The homeomorphism follows byTheorem 2.16. (cid:4) PECTRA OF TRIANGULATED CATEGORIES 9 Prime thick subcategories of derived and singularity categories ofnoetherian schemes
In this section, we study prime thick subcategories of the derived and the singularity cate-gories 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 S T ( p ) the subcategory of T consisting ofobjects M which satisfy 1 M = 0 in Hom T ( M, M ) p , i.e., a M = 0 in T for some a ∈ R \ p . Lemma 3.1. (a) S T ( p ) is a thick subcategory of T . (b) For any element a ∈ R \ p , the mapping cone C ( a M ) of a M : M → M is in S T ( p ) . (c) The canonical functor Q : T → T / S T ( p ) induces an isomorphism Hom T ( M, N ) p ∼ = −→ Hom T / S T ( p ) ( M, N ) , f /a Q ( a N ) − Q ( f ) of R -modules for any M, N ∈ T .Proof. (a) Obviously, S T ( p ) is closed under direct summands and shifts. It remains to showthat S T ( p ) is closed under extensions. Take a triangle L f −→ M g −→ N → L [1] in T with L, N ∈ S T ( p ). Then there are elements a, b ∈ R \ p such that a L = 0 and b N = 0 in T .Since g ( b M ) = ( b N ) g = 0, there is a morphism u : M → L such that f u = b M . Therefore( ab )1 M = a ( f u ) = f ( a L ) u = 0 and hence M ∈ S T ( p ).(b) Set C := C ( a M ) and prove C ∈ S T ( p ). For any object N ∈ T , the triangle M a M −−→ M → C → M [1] induces an exact sequenceHom T ( N, M ) p a ∼ = −→ Hom T ( N, M ) p → Hom T ( N, C ) p → Hom T ( N, M [1]) p a ∼ = −→ Hom T ( N, M [1]) p of R p -modules. Thus we get Hom T ( N, C ) p = 0 for any N ∈ T . Taking N = C , we concludethat 1 C = 0 in Hom T ( C, C ) p .(c) Thanks to (b), the canonical functor Q : T → T / S T ( p ) induces a homomorphismHom T ( M, N ) p → Hom T / S T ( p ) ( M, N ) , f /a Q ( a N ) − Q ( f )of R -modules.Let f ∈ Hom T ( M, N ) with Q ( f ) = 0 in T / S T ( p ). Then there is a morphism s : N → L in T such that sf = 0 in T and C ( s ) ∈ S T ( p ). Pick an element a ∈ R \ p with a C ( s ) = 0. Then a N factors as a N : N s −→ L t −→ N . This shows that af = ( a N ) f = tsf = 0 and hence theabove homomorphism is injective.Let α : M → N be a morphism in T / S T ( p ). This morphism is given by α = Q ( s ) − Q ( f ) forsome morphisms f : M → L and s : N → L in T with C ( s ) ∈ S T ( p ). Take an element a ∈ R \ p such that a C ( s ) = 0. Then a N factors as a N : N s −→ L t −→ N . Using this factorization, weget α = Q ( s ) − Q ( f ) = Q ( ts ) − Q ( tf ) = Q ( a N ) − Q ( tf ), which belongs to the image of thehomomorphism. This shows the surjectivity of the homomorphism. (cid:4) Concerning our R -linear triangulated categories D perf ( R ), D b ( R ), and D sg ( R ) for a commu-tative 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 D perf ( R ) / S perf ( p ) ∼ = −→ D perf ( R p ) , where S perf ( p ) := { M ∈ D perf ( R ) | M p ∼ = 0 in D perf ( R p ) } . (2) There is a triangle equivalence D b ( R ) / S b ( p ) ∼ = −→ D b ( R p ) , where S b ( p ) := { M ∈ D b ( R ) | M p ∼ = 0 in D b ( R p ) } . (3) Assume further that R is a Gorenstein ring of finite Krull dimension. Then there is atriangle equivalence D sg ( R ) / S sg ( p ) ∼ = −→ D sg ( R p ) , where S sg ( p ) := { M ∈ D sg ( R ) | M p ∼ = 0 in D sg ( R p ) } .Proof. Let
D ∈ { D perf , D b , D sg } , and S ( p ) ∈ {S perf ( p ) , S b ( p ) , S sg ( p ) } the corresponding thicksubcategory of D ( R ). Then there is an isomorphismHom D ( R ) ( M, N ) p ∼ = −→ Hom D ( R p ) ( M p , N p ) , f /a a − f p ( ∗ )of R p -modules for any M, N ∈ D ( R ). Indeed, for D ( R ) = D perf ( R ) or D ( R ) = D b ( R ), see[1, Lemma 5.2(b)]. For D ( R ) = D sg ( R ), since R is Gorenstein, D ( R ) is equivalent to thestable category of maximal Cohen-Macaulay R -modules by [8, Theorem 4.4.1(2)]. Then ( ∗ )follows from [32, Lemma 3.9] for example. From the isomorphism ( ∗ ), one has S D ( R ) ( p ) = S ( p ).Furthermore, the isomorphism ( ∗ ) is equal to the compositionHom D ( R ) ( M, N ) p ∼ = −→ Hom D ( R ) / S ( p ) ( M, N ) → Hom D ( R p ) ( M p , N p ) , 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 localizationfunctor D ( R ) → D ( R p ) induces a fully faithful functor D ( R ) / S ( p ) → D ( R p ) , M M p As a result, it is enough to check that the localization functor D ( R ) → D ( R p ) is essentiallysurjective in each case.(1) Note that D perf ( R ) ∼ = K b ( R ), where K b ( R ) is the homotopy category of bounded com-plexes of finitely generated projective R -modules. Let P ∈ K b ( R p ). Then we shall con-struct e P ∈ K b ( R ) such that e P p ∼ = P in K b ( R p ). By taking shifts, we may assume that P := (0 → P r d r −→ P r − d r − −−→ · · · d −→ P → P i of P is a free R p -module, each differential d i : P i → P i − is given by a matrix with entries in R p . Take s ∈ R \ p so that sα belongs to the image of R in R p for all entries α of differentials of P . Then for each i , sd i comes from a homomorphism e d i : e P i → e P i − between free R -modules, i.e., ( e P i ) p = P i PECTRA OF TRIANGULATED CATEGORIES 11 and ( e d i ) p = sd i . Since ( e d i − ) p ( e d i ) p = s d i − d i = 0, one can find an element t ∈ R \ p such that t e d i e d i − = 0 for all i . In particular, we obtain a bounded complex e P := (0 → e P r t e d r −→ e P r − t e d r − −−−→ · · · → e P t e d −→ e P → R -modules such that e P p = (0 → P r std r −−→ P r − std r − −−−→ · · · → P std −−→ P → . This e P p is isomorphic to P as complexes of R p -modules. In fact, there is a commutative diagram0 −−−→ P r std r −−−→ P r − std r − −−−−→ · · · std −−−→ P std −−−→ P std −−−→ P −−−→ y ( st ) r y ( st ) r − y ( st ) y st (cid:13)(cid:13)(cid:13) −−−→ P r d r −−−→ P r − d r − −−−→ · · · d −−−→ P d −−−→ P d −−−→ P −−−→ R p -modules.(2) Let M ∈ D b ( R p ). Taking a truncation of a projective resolution of M gives rise to atriangle N [ − f −→ P → M → N in D b ( R p ) with N finitely generated R p -module and P ∈ K b ( R p ). Pick a finitely generated R -module e N and e P ∈ K b ( R ) together with isomorphisms ϕ : e N p ∼ = −→ N and ψ : e P p ∼ = −→ P . Theexistence of such an e N is easy and that of a e P is by (1). As we have remarked as ( ∗ ), there isan isomorphismHom D b ( R ) ( e N [ − , e P ) p ∼ = −→ Hom D b ( R p ) ( e N p [ − , e P p ) , g/a a − g p . Therefore, the composition ψ − f ϕ [ −
1] : e N p [ − → e P p is equal to a − g p for some g ∈ Hom D b ( R ) ( e N [ − , e P ) and a ∈ R \ p . Embedding g into a triangle e N [ − g −→ e P → f M → e N in D b ( R ), we obtain an object f M ∈ D b ( R ) with f M p ∼ = M . Indeed, localizing this triangle at p ,we obtain a commutative diagram e N p [ − g p / / aϕ [ − ∼ = (cid:15) (cid:15) e P p / / ψ ∼ = (cid:15) (cid:15) f M p / / (cid:15) (cid:15) e N p aϕ ∼ = (cid:15) (cid:15) N [ − f / / P / / M / / N in D b ( R p ) and hence there is a dotted arrow which is an isomorphism in D b ( R p ).(3) We note that every object of D sg ( R p ) is isomorphic in D sg ( R p ) to a finitely generated R p -module. It is isomorphic to a localization of some finitely generated R -module. (cid:4) Recall that a commutative noetherian local ring ( R, m ) is a complete intersection (of codimen-sion c ) if its m -adic completion is isomorphic to a regular local ring modulo a regular sequence(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 D perf ( X ) andthe bounded derived category D b ( 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 is a prime thicksubcategory of D b ( R ) if and only if R is a complete intersection.Proof. Assume that is a prime thick subcategory of D b ( R ) and let X be a unique minimalnon-zero thick subcategory of D b ( R ). The minimality of X shows that X ⊆ D perf ( R ) and inparticular X contains a non-zero perfect complex. For any non-zero object M ∈ D b ( R ), denoteby thick( M ) the smallest thick subcategory of D b ( 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 anon-zero perfect complex for any non-zero object M ∈ D b ( 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 D b ( R )contains the non-zero thick subcategory Supp − perf ( R ) ( { m } ) := { M ∈ D perf ( R ) | Supp R ( M ) ⊆{ m }} . Hence, Supp − perf ( R ) ( { m } ) is a unique minimal non-zero thick subcategory of D b ( R ). (cid:4) Now we are ready to prove one of the main theorems in this paper, which provides examplesof prime thick subcategories.
Theorem 3.4.
Let X be a noetherian scheme and x ∈ X . (1) The thick subcategory S perf X ( x ) := {F ∈ D perf ( X ) | F x ∼ = 0 in D perf ( O X,x ) } is a prime thick subcategory of D perf ( X ) . (2) Set S b X ( x ) := {F ∈ D b ( X ) | F x ∼ = 0 in D b ( O X,x ) } . Then S b X ( x ) is a prime thick subcategory of D b ( X ) if and only if O X,x is a complete inter-section.Proof. (1) Let U ⊆ X be an affine open neighborhood of x with complement Z := X \ U . Itfollows from [3, Theorem 2.13] that the restriction functor ( − ) | U : D perf ( X ) → D perf ( U ) inducesa triangle equivalence D perf ( X ) / D perfZ ( X ) → D perf ( U )up to direct summands, where D perf Z ( X ) := {F ∈ D perf ( X ) | F | U ∼ = 0 in D perf ( U ) } . On theother hand, D perf Z ( X ) ⊆ S perf X ( x ) follows from their definitions. By Proposition 2.11(1), we havethe following lattice isomorphism Th (D perf ( X ) / D perfZ ( X )) ∼ = Th (D perf ( U ))under which S perf X ( x ) / D perfZ ( X ) corresponds to S perf U ( x ). As a result, we may assume that X isaffine by Proposition 2.9(2). PECTRA OF TRIANGULATED CATEGORIES 13
Let X be an affine noetherian scheme. From Proposition 2.9(2) and the triangle equivalenceD perf ( X ) / S perf ( x ) ∼ = D perf ( O X,x ) which is obtained in Lemma 3.2(1), it is enough to check that is a prime thick subcategory of D perf ( O X,x ). We use the poset isomorphism {X ∈ Th (D perf ( O X,x )) | X 6 = } ∼ = { W ∈ Spcl (Spec O X,x ) | W = ∅} which follows from [20, Theorem 1.5]. Since the right-hand side has a unique minimal element { m X,x } ( m X,x is the unique maximal ideal of O X,x ), so does the left-hand side. This means that is a prime thick subcategory of D perf ( O X,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 : D b ( X ) → D b ( U ) induces a triangle equivalenceD b ( X ) / D b Z ( X ) ∼ = −→ D b ( U ) , where D b Z ( X ) := {F ∈ D b ( X ) | F | U = 0 in D b ( 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). (cid:4) Next, we discuss the singularity category of a noetherian scheme and prove a similar resultto 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.
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 : D sg ( X ) → D sg ( U ) inducesa triangle equivalence D sg ( X ) / D sg Z ( X ) ∼ = −→ D sg ( U ) , where D sg Z ( X ) := {F ∈ D sg ( X ) | F | U ∼ = 0 in D sg ( U ) } . This result is essentially shown by Krause in [15, Proposition 6.9] using the unbounded stablederived category S( X ). Let X be a separated noetherian scheme. Denote by K(Inj X ) thehomotopy category of complexes of injective quasi-coherent O X -modules, and by S( X ) thesubcategory 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 : D sg ( X ) → S( X ) c up to direct summands, where ( − ) c stands for the thick subcategory consisting of all compactobjects. 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], S Z ( 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 equivalence j ∗ : S( X ) c / S Z ( X ) c → S( U ) c up to direct summands. Note that by [15, Theorem 6.6], there is a commutative diagramD sg Z ( X ) (cid:31) (cid:127) / F (cid:15) (cid:15) D sg ( X ) j ∗ / / F (cid:15) (cid:15) D sg ( U ) F (cid:15) (cid:15) S Z ( X ) c (cid:31) (cid:127) / S( X ) c j ∗ / / S( U ) c ( ∗ )of exact functors and hence we get F − (S Z ( X ) c ) = D sg Z ( X ).Consider a morphism f : M ′ → F ( M ) in S( X ) c with M ′ ∈ S Z ( X ) c and M ∈ D sg ( X ). ByLemma 2.10, there is an isomorphism ϕ : M ′ ⊕ M ′ [1] ∼ = −→ F ( L ) for some L ∈ D sg ( X ). Since F ( j ∗ ( L )) ∼ = j ∗ ( F ( L )) ∼ = j ∗ ( M ′ ⊕ M ′ [1]) = 0, one has j ∗ ( L ) ∼ = 0 i.e., L ∈ D sg Z ( X ). Therefore, f factors as M ′ ϕ t (1 , −−−−→ F ( L ) ( f, ϕ − −−−−−→ F ( M ) with L ∈ D sg Z ( X ). Using [13, Proposition 10.2.6(ii)], F induces a triangle equivalence F : D sg ( X ) / D sg Z ( X ) → S( X ) c / S Z ( X ) c up to direct summands. From the commutative diagram ( ∗ ), we have a commutative diagramD sg ( X ) / D sg Z ( X ) j ∗ / / F (cid:15) (cid:15) D sg ( U ) F (cid:15) (cid:15) S( X ) c / S Z ( X ) c j ∗ / / S( U ) c of exact functors, where the vertical functors and the bottom j ∗ are triangle equivalencesup to direct summands. Therefore, the top j ∗ is also fully faithful. On the other hand, j ∗ : D sg ( X ) / D sg Z ( X ) → D sg ( U ) is essentially surjective as the restriction functor j ∗ :D b ( X ) → D b ( U ) is essentially surjective by [25, Theorem 3.3.2]. Hence we conclude that j ∗ : D sg ( X ) / D sg Z ( X ) → D sg ( U ) is a triangle equivalence. (cid:4) The next proposition, which is similar to Proposition 3.3, characterizes hypersurface localrings in terms of prime thick subcategories of the singularity category of a commutative noe-therian local ring.
Proposition 3.6.
Let R be a non-regular commutative noetherian local ring. If R is a hyper-surface, then is a prime thick subcategory of D sg ( R ) . The converse holds if R is a completeintersection.Proof. Assume that R is hypersurface. From [28, Theorem 6.6], there is a poset isomorphism {X ∈ Th (D sg ( R )) | X 6 = } ∼ = { W ∈ Spcl (Sing( R )) | W = ∅} , which proves the first statement. The second one follows from [30, Theorem 3.7]. Indeed, if is prime, then there is a unique minimal non-zero thick subcategory X of D sg ( R ). This X mustbe equal to the intersection of all non-zero thick subcategories of D sg ( R ). (cid:4) 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.
PECTRA OF TRIANGULATED CATEGORIES 15
Theorem 3.7.
Let X be a separated Gorenstein scheme and x ∈ Sing( X ) . If O X,x is ahypersurface, then S sg X ( x ) := {F ∈ D sg ( X ) | F x ∼ = 0 in D sg ( O X,x ) } is a prime thick subcategory of D sg ( X ) . Conversely, if O X,x is a complete intersection and S sg X ( x ) is a prime thick subcategory of D sg ( X ) , then O X,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 triangleequivalence D sg ( X ) / S sg X ( x ) ∼ = D sg ( O X,x ) . It follows from Proposition 2.9(2) that S sg X ( x ) is prime if and only if is a prime thick subcat-egory of D sg ( O X,x ). Then the result follows by Proposition 3.6. (cid:4)
For a noetherian scheme X , we define the complete intersection locus CI( X ) and the hyper-surface locus HS( X ) of X byCI( X ) := { x ∈ X | O X,x is a complete intersection } , HS( X ) := { x ∈ Sing( X ) | O X,x is a hypersurface } . Applying Theorems 3.4 and 3.7, we obtain the following result which has been stated in theintroduction.
Corollary 3.8.
Let X be a noetherian scheme. (1) There is an immersion
X ֒ → Spec △ (D perf ( X )) of topological spaces. (2) There is an immersion
CI( X ) ֒ → Spec △ (D b ( X )) of topological spaces. (3) If X is a separated Gorenstein scheme, then there is an immersion HS( X ) ֒ → Spec △ (D sg ( X )) of topological spaces.Proof. (1) From the equality Supp X ( S perf X ( x )) = { x ′ ∈ X | x
6∈ { x ′ }} and Theorem 3.4(1), thereis an injective map X ֒ → Spec △ (D perf ( X )) , x
7→ S perf X ( x ) . For an object
F ∈ D perf ( X ), one can easily see that the inverse image of Supp △ ( F ) by thisinjection is Supp X ( F ). Since the family { Supp X ( F ) | F ∈ D perf ( X ) } (resp. { Supp △ ( F ) | F ∈ D perf ( X ) } ) forms a closed basis of X (resp. Spec △ (D perf ( X ))), the topology on X is nothingbut the one induced from Spec △ (D perf ( X )).The remained statements (2) and (3) are shown similarly as above. (cid:4) Remark 3.9. If X is an excellent Cohen-Macaulay scheme, then the complete intersectionlocus 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-Macaulayring 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 H ( R ) := { p ∈ Spec R | codim R p ≤ } . Since HS( R ) = H ( R ) ∩ Sing R , it suffices toshow that H ( R ) is open in Spec R . Recall that for a prime ideal p of R , codim R p coincideswith the 1st deviation ǫ ( R p ) because R p is a complete intersection; see [19, Section 21]. Thenthe openness of H ( R ) follows from [24, Proposition 3.6].We say that two noetherian schemes X and Y are • perfectly derived equivalent if D perf ( X ) ∼ = D perf ( Y ) as triangulated categories. • derived equivalent if D b ( X ) ∼ = D b ( Y ) as triangulated categories. • singularly equivalent if D sg ( X ) ∼ = D sg ( Y ) as triangulated categories.If X and Y are derived equivalent, then Y is said to be a Fourier-Mukai partner of X . The fol-lowing result is a direct consequence of Corollary 3.8 because equivalent triangulated categorieshave homeomorphic spectra. Corollary 3.10.
Let X be a noetherian scheme. (1) There is an immersion
Y ֒ → Spec △ (D perf ( X )) of topological spaces for any noetherian scheme Y which is perfectly derived equivalent to X . (2) There is an immersion
CI( Y ) ֒ → Spec △ (D b ( X )) of topological spaces for any noetherian scheme Y which is derived equivalent to X . (3) Let X be a separated Gorenstein scheme. There is an immersion HS( Y ) ֒ → Spec △ (D sg ( X )) of topological spaces for any separated Gorenstein scheme Y which is singularly equivalentto X . Remark 3.11.
For two commutative noetherian rings R and S , it is known as the derivedMorita theorem that there are implicationsperfectly derived equivalent ⇔ derived equivalent ⇒ singularly equivalent . If X and Y are projective schemes over a field and assume that X possesses an ample linebundle satisfying a certain vanishing condition on cohomologies, then the same implicationsperfectly derived equivalent ⇔ derived equivalent ⇒ singularly equivalenthold by [2, Theorem 7.13]. PECTRA OF TRIANGULATED CATEGORIES 17 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 , ⊗ , ). Besides, we discuss more the spectrumof the tensor triangulated category (D perf ( X ) , ⊗ L O X , O X ) 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 , ⊗ , ) is a tensor triangulated category if T is a triangulated categoryequipped with a symmetric monoidal structure ( ⊗ , ) which is compatible with the triangulatedstructure; see [12, Appendix A] for the precise definition. Definition 4.1.
Let ( T , ⊗ , ) 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 onehas 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 by Rad ⊗ ( T ).(3) An ideal P of T is called prime provided P 6 = 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 axiomsfor closed subsets. The set Spec ⊗ ( T ) together with this topology is called the Balmer spectrum of T .For an object M ∈ T , define the Balmer support of M bySupp ⊗ ( M ) := Z ( { M } ) = {P ∈ Spec ⊗ ( T ) | M
6∈ 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 followingcouple 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 T -space. Proposition 4.4 ([4, Lemma 4.2]) . Let I be an ideal of T . Then one has the equality √I = \ I⊆P∈
Spec ⊗ ( T ) P . A subset of a topological space X is said to be Thomason if it is a union of closed subsetswhose 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 Rad ⊗ ( T ) Supp ⊗ / / Thom (Spec ⊗ ( T )) , Supp − ⊗ o o where Supp ⊗ ( X ) := S 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
Thom (Spec ⊗ ( T )) = { Supp ⊗ ( X ) |⊆ X } . holds.The following result shows that prime ideals of T are characterized by a similar conditionto the definition of prime thick subcategories. Therefore, it justifies our definition of a primethick subcategory. Proposition 4.7.
Let P be a radical ideal of T . Assume that there is a unique minimal radicalthick subcategory I of T with I ( T . Then P is a prime ideal. The converse holds if Spec ⊗ ( T ) is noetherian (e.g., T = D perf ( 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 6⊆ Q} = {Q ∈ Spec ⊗ ( T ) | P 6∈ {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 latticeisomorphism in Theorem 4.5, I := Supp − ⊗ ( T ) satisfies the condition in the statement. (cid:4) 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 e X the smallest radical ideal of T containing X . For anyradical ideal I of T with P ( I , the minimality of X shows X ⊆ I and hence e X ⊆ I . Thismeans that e X is a unique minimal radical ideal of T with P ( e X . Then Proposition 4.7 showsthat P is a prime ideal. (cid:4) PECTRA OF TRIANGULATED CATEGORIES 19
For a noetherian scheme X , the thick subcategory S perf X ( x ) is a prime ideal, and actually, everyprime ideal of D perf ( X ) is given in this way; see [4, Corollary 5.6]. Therefore, the immersion inTheorem 3.4(1) can be considered as the inclusionSpec ⊗ (D perf ( X )) ⊆ Spec △ (D perf ( 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 D perf ( X ) . Then P is aprime thick subcategory of D perf ( X ) if and only if P is a prime ideal of D perf ( X ) . We close this paper by giving one concrete example of spectra.
Example 4.10.
Let k be a field. First note that since P is regular, there is a triangle equiva-lence D b ( P ) ∼ = D perf ( P ). It is explained in [16, Section 4.1] that there is a lattice isomorphism Th (D perf ( P )) ∼ = Spcl ( P ) ⊔ Z . Here, Z is considered as the discrete lattice. On the right-hand side, an element of Spcl ( P )corresponds to an ideal of D perf ( P ) and those of Z corresponds to a thick subcategory of theform thick( O P ( i )) for some i ∈ Z . As Corollary 4.9 shows, the prime ideals and the idealswhich are prime thick subcategories are the same. Therefore, restricting the above latticeisomorphism to prime thick subcategories, we get a homeomorphismSpec △ (D perf ( P )) ∼ = P ⊔ Z , where Z is considered as the discrete topological space. This gives an example of noetherianscheme which is not quasi-affine and the immersion X ֒ → Spec △ (D perf ( X )) is not a homeomor-phism. Acknowledgments.
The author thanks Ryo Takahashi for giving helpful comments and usefulsuggestions. The author also thanks Tsutomu Nakamura for fruitful discussions on Proposition3.5.
References [1] L. L. Avramov and H.-B. Foxby,
Homological dimensions of unbounded complexes , J. Pure Appl. Algebra (1991), no. 2, 129–155.[2] M. R. Ballard, Equivalences of derived categories of sheaves on quasi-projective schemes , preprint (2009), arXiv:0905.3148 .[3] P. Balmer,
Presheaves of triangulated categories and reconstruction of schemes , Math. Ann. (2002),no. 3, 557–580.[4] ,
The spectrum of prime ideals in tensor triangulated categories , J. Reine Angew. Math. (2005),149–168.[5] D. J. Benson, J. F. Carlson, and J. Rickard,
Thick subcategories of the stable module category , Fund. Math. (1997), no. 1, 59–80.[6] D. J. Benson, S. B. Iyengar, and H. Krause,
Stratifying modular representations of finite groups , Ann. ofMath. (2) (2011), no. 3, 1643–1684.[7] D. J. Benson, S. B. Iyengar, H. Krause, and J. Pevtsova,
Stratification for module categories of finite groupschemes , J. Amer. Math. Soc. (2018), no. 1, 265–302.[8] R.-O. Buchweitz, Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings , unpub-lished manuscript (1987), 155pp.[9] S. Greco and M.G. Marinari,
Nagata’s criterion and openness of loci for Gorenstein and complete inter-section , Math. Z. (1978), no. 3, 207–216. [10] M. J. Hopkins,
Global methods in homotopy theory , Proceedings of the Durham Symposium on HomotopyTheory (J. D. S. Jones and E. Rees, eds.), LMS Lecture Note Series 117, 1987, pp. 73–96.[11] M. J. Hopkins and J. H. Smith,
Nilpotence and stable homotopy theory II , Ann. of Math. (2) (1998),no. 1, 1–49.[12] M. Hovey, J. H. Palmieri, and N. P. Strickland,
Axiomatic stable homotopy theory , Mem. Amer. Math.Soc., vol. 610, American Mathematical Society, Providence, RI, 1997.[13] M. Kashiwara and P. Schapira,
Categories and sheaves , Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences], vol. 332, Springer-Verlag, Berlin, 2006.[14] Y. Kawamata,
Birational geometry and derived categories, Celebrating the 50th anniversary of the Journalof Differential Geometry , Surv. Differ. Geom., vol. 22, pp. 291–317, Int. Press, Somerville, MA, 2018.[15] H. Krause,
The stable derived category of a Noetherian scheme , Compos. Math. (2005), no. 5, 1128–1162.[16] H. Krause and G. Stevenson,
The derived category of the projective line , Spectral Structures and Topolog-ical Spectral Structures and Topological Methods in Mathematics, European Mathematical Society, 2019,pp. 275–298.[17] H. Matsui,
Singular equivalences of commutative noetherian rings and reconstruction of singular loci , J.Algebra (2019), 170–194.[18] H. Matsui and R. Takahashi,
Construction of spectra of triangulated categories and applications to com-mutative rings , J. Math. Soc. Japan (2020), no. 4, 1283–1307.[19] H. Matsumura, Commutative Ring Theory , second ed., Cambridge Studies in Advanced Math., vol. 8,Cambridge University Press, Cambridge, 1989.[20] A. Neeman,
The chromatic tower for D ( R ) , With an appendix by Marcel B¨okstedt , Topology (1992),no. 3, 519–532.[21] , The connection between the K -theory localization theorem of Thomason, Trobaugh and Yao andthe smashing subcategories of Bousfield and Ravenel , Ann. Sci. ´Ecole Nrom. Sup. (4) (1992), no. 5,547–566.[22] D. Orlov, Triangulated categories of singularities and D -branes in Landau-Ginzburg models , Proc. SteklovInst. Math. (2004), 240–262.[23] J. Pollitz, The derived category of a locally complete intersection , Adv. Math. (2019), no. 106752, 18pp.[24] A. Ragusa,
On openness of H n -locus and semicontinuity of n th deviation , Proc. Amer. Math. Soc. (1980), no. 2, 201–209.[25] M. Schlichting, Higher algebraic K -theory, Topics in algebraic and topological K -theory , Lecture Notes inMath., vol. 2008, pp. 167–241, Springer, Berlin, 2011.[26] G. Stevenson, Support theory via actions of tensor triangulated categories , J. Reine Angew. Math. (2013), 219–254.[27] ,
Subcategories of singularity categories via tensor actions , Compos. Math. (2014), no. 2, 229–272.[28] R. Takahashi,
Classifying thick subcategories of the stable category of Cohen-Macaulay modules , Adv. Math. (2010), no. 4, 2076–2116.[29] ,
Thick subcategories over Gorenstein local rings that are locally hypersurfaces on the puncturedspectra , J. Math. Soc. Japan (2013), no. 2, 357–374.[30] , Intersections of resolving subcategories and intersections of thick subcategories , preprint (2020),available at .[31] R. W. Thomason,
The classification of triangulated subcategories , Compos. Math. (1997), no. 1, 1–27.[32] Y. Yoshino,
Cohen-Macaulay modules over Cohen-Macaulay rings , London Mathematical Society LectureNote Series, Cambridge University Press, 1990.
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
Email address ::