TRIANGULATED CATEGORIES OF SINGULARITIES AND D-BRANES IN LANDAU-GINSBURG MODELS

DMITRI ORLOV Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend

Introduction

In spite of physics terms in the title, this paper is pure mathematical and its purpose is to fill some gaps in our knowledges about singularities of algebraic varieties and about categories related to them. It seems that two different types of categories can be associated with singularities (or singularities of maps). The first type categories are connected with vanishing cycles and closely related to the categories which were introduced in [18] for symplectic Picard-Lefschetz pencil. The second type categories are pure algebraic and are coming from the derived categories of coherent sheaves. This type categories will be the center of this work. An important notion here is a concept of a perfect complex, which was introduced in [1]. A perfect complex is a complex of sheaves which locally quasi-isomorphic to a complex of locally free sheaves of finite type ( for good reference see also [19]).

We can attach to any algebraic variety X the bounded derived category of coherent sheaves D^{b}(coh(X)) . This category admits a triangulated structure. The derived category of coherent sheaves has a triangulated subcategory 𝔓𝔢𝔯𝔣(X) formed by the perfect complexes. If the variety X is smooth then any coherent sheaf has a finite resolution of locally frees of finite type and the subcategory of perfect complexes coincides with whole D^{b}(coh(X)) . But for singular varieties this property does not fulfill. We introduce a notion of triangulated category of singularities D_{Sg}(X) as the quotient of the triangulated category D^{b}(coh(X)) by the full triangulated subcategory of perfect complexes 𝔓𝔢𝔯𝔣(X) . The category D_{Sg}(X) reflects the properties of the singularities of X and does not depend on whole X . For example we prove that it is invariant with respect to a localization in the Zarisky topology (Proposition 1.14). The category D_{Sg}(X) has good properties when the variety X is Gorenstein. In this case if the locus of singularities is proper then all Hom’s between any objects are finite-dimensional vector spaces (Corollary 1.24).

The investigation of such type categories is inspired by the Homological Mirror Symmetry Conjecture ([15]).

Algebra Section, Steklov Mathematical Institute RAN, Gubkin str. 8, GSP-1, Moscow 117966, RUSSIA, E-mail address: orlov@mi.ras.ru.

Mainly works on topological string theory consider the case of topological N=2 superconformal sigma-models with a Calabi-Yau target space. In this case the field theory has two topologically twisted versions: A-models and B-models. The corresponding D-branes are called A-branes and B-branes. The mirror symmetry should exchange these two classe of D-branes. From the mathematical point of view the category of B-branes on Calabi-Yau is the derived category of coherent sheaves ([15],[3]). For A-branes on Calabi-Yau manifolds it has been proposed that the relevant category is the so-called Fukaya category, whose objects are, roughly speaking, Lagrangian submanifolds equipped with flat vector bundles ([15]). The Homological Mirror Symmetry Conjecture asserts that if two Calabi-Yaus X and Y are mirror to each other, then the derived category of coherent sheaves on X is equivalent to the Fukaya category of Y , and vice versa.

On the other hand, phisicists also consider more general topological twisted N=2 field theories and corresponding D-branes. One class of such theories is given by the sigma-model with a Fano variety as a target space. Another set of examples is provided by N=2 Landau-Ginsburg models. In many cases these two classes of N=2 theories are related by mirror symmetry ([10]). For example, the sigma model with target ℙ^{n} is mirror to a Landau-Ginsburg model which is given by superpotential

W = x_{1} + ⋯ + x_{n} + \frac{1}{x_{1} ⋯ x_{n}}

on ℂ^{∗n} . A general definition of Landau-Ginsburg model involves, besides a choice of a target space, a choice of a holomorphic function W on this manifold. In particular, non-trivial Landau-Ginsburg models require non-compact target space.

For Fano varieties we have the derived categories of coherent sheaves (B-branes) and also if we have a symplectic form we can propose the relevant Fukaya category (A-branes). Thus if one wants to extend the Homological Mirror Symmetry Conjecture to the non-Calabi-Yau case, one should understand D-branes in Landau-Ginsburg models.

Categories of A-branes in Landau-Ginsburg models are studied in [9] and in [18] from the mathematical point of view. Mirror symmetry relates B-branes on a Fano variety (coherent sheaves) with A-branes in a LG model. In the case when the Fano variety is ℙ^{n} , this prediction has been tested in [9] (and for ℙ^{2} in [18] from the mathematical viewpoint).

One can also consider the Fukaya category (A-branes) on a Fano variety. One can expect that in this case the Fukaya category is equivalent to the category of B-branes in the mirror Landau-Ginsburg model. If we assume the mirror symmetry, we can predict a description of the Fukaya category on a Fano variety by studying B-branes in the mirror LG model. As a rule it is easy to understand B-branes, rather that A-branes.

The mathematical definition for the category of B-branes in Landau-Ginsburg models was proposed by M.Kontsevich. Roughly speaking, he suggests that the superpotential W deforms complexes of coherent sheaves to ”twisted” complexes, i.e the composition of differentials now is not zero, but it is equal to the multiplication with W . This ”twisting” also breaks ℤgrading down to ℤ/2-grading. The equivalence of this definition with the physics notion of B-branes in LG models was tested in paper [11] in the case of usual quadratic superpotential W = x_{1}^{2} + ⋯ + x_{n}^{2} .

We establish a connection between categories of B-branes in Landau-Ginsburg models and triangulated categories of singularities. We consider the singular fibres of the map W and show that the triangulated categories of singularities of these fibres are equivalent to the categories of B-branes (Theorem 3.8). In particular this gives us that, in spite of the fact that the category of B-branes is defined using the total space X , it depends only on the singular fibres of the superpotential. It seems that this result can be useful for precise calculations of the categories of B-branes in Landau-Ginsburg models.

Another result which is also useful for calculations connects the categories of B-branes in different dimensions (Theorem 2.1). It says the following: Let W : X → ℂ be a superpotential. Consider the manifold Y = X × ℂ^{2} and another superpotential W′ = W + xy on Y , where x, y are coordinates on ℂ^{2} . Then the category of B-branes in the Landau-Ginsburg model on Y is equivalent to the category of B-branes in the Landau-Ginsburg model on X . Actually, we prove that the triangulated category of singularities of the fiber X_{0} over point 0 is equivalent to the triangulated category of singularities of the fiber Y_{0} (Theorem 2.1). But taking in mind that the categories of singularities are equivalent to the categories of B-branes (Theorem 3.8) we obtain the connection between B-branes mentioned above.

In the end we give a calculation of the category of B-branes in the Landau-Ginsburg model with the superpotential W = z_{0}^{n}+z_{1}^{2}+⋯+z_{2k}^{2} on ℂ^{2k+1} . The singularity of this superpotential corresponds to the Dynkin diagram A_{n−1} . This category has n − 1 indecomposible objects. We describe morphisms between them, the translation functor and the triangulated structure on this category.

1. Singularities and triangulated categories

1.1. Triangulated categories and localization. In this section we remind definitions of a triangulated category and its localization which were introduced in [20] (see also [5],[12],[13]). Let D be an additive category. The structure of a triangulated category on D is defined by giving of the following data:

a) an additive autoequivalence [1] : D ⟶ D (it is called a translation functor),

b) a class of exact (or distinguished) triangles:

X ⟶^{u} Y ⟶^{v} Z ⟶^{w} X[1],

which must satisfy the set of axioms T1–T4.

T1. a) For each object X the triangle X ⟶^{id} X ⟶ 0 ⟶ X[1] is exact. b) Each triangle isomorphic to an exact triangle is exact.

c) Any morphism X ⟶^{u} Y can be included in an exact triangle

X ⟶^{u} Y ⟶^{v} Z ⟶^{w} X[1] .

T2. A triangle X ⟶^{u} Y ⟶^{v} Z ⟶^{w} X[1] is exact if and only if the triangle

Y ⟶^{v} Z ⟶^{w} X[1] ⟶^{−u[1]} Y [1] is exact.

T3. For any two exact triangles and two morphisms f, g the diagram below

[figure: X; u; f; □; Y; v; g; Z; w; h; X[1]; f[1]; X′; u′; Y′; v′; Z′; w′; X′[1].]

can be completed to a morphism of triangles by a morphism h : Z → Z′ . T4. For each pair of morphisms X ⟶^{u} Y ⟶^{v} Z there is a commutative diagram

X ⟶^{u} Y ⟶^{x} Z′ ⟶ X[1]
∥ ^{v}↓ ↓^{w} ∥
X ⟶ Z ⟶^{y} Y′ ⟶^{w′} X[1]
↓ ↓^{t} ↓^{u[1]}
X′  X′ ⟶^{r} Y [1]
↓^{r}  ↓
Y [1] ⟶^{x[1]} Z′[1]

where the first two rows and the two central columns are exact triangles. A functor F : D ⟶ D′ between two triangulated categories D and D′ is called exact if

it commutes with the translation functors, i.e there fixed a natural isomorphism t : F ∘ [1]_{D} →^{∼} [1]_{D′} ∘ F and it transforms all exact triangles into exact triangles, i.e for each exact triangle X →^{u} Y →^{v} Z →^{w} X[1] in D the triangle

F X ⟶^{F u} F Y ⟶^{F v} F Z ⟶^{t_{X}F w} F X[1]

is exact.

A full additive subcategory N ⊂ D is called full triangulated subcategory, if the following conditions hold: it is closed with respect to the translation functor in D and if it contains any two objects of an exact triangle in D then it contains the third object of this triangle as well.

Now we remind the definition of a localization of categories. Let C be a category and let Σ be a class of morphisms in C . It is well-known [4] that there is a category C[Σ^{−1}] and a functor Q : C → C[Σ^{−1}] which is universal among the functors making the elements of Σ invertible. The category C[Σ^{−1}] has a good description if Σ is a multiplicative system.

A family of morphisms Σ in a category C is called a multiplicative system if it satisfies the following conditions:

M1 all identical morphisms id_{X} belongs to Σ ;

M2 the composition of two elements of Σ belong to Σ ;

M3 any diagram X′ ⟵^{s} X ⟶^{u} Y , with s ∈ Σ can be completed to a commutative square

\xymatrix{X \ar[r]^{u} \ar[d]_{s} & Y \ar@{-->}[d]^{t} \\ X′ \ar@{-->}[r]^{u′} & Y′}

with t ∈ Σ (the same when all arrows reversed); M4 for any two morphisms f, g the existence of s ∈ Σ with fs = gs is equivalent to the

existence of t ∈ Σ with tf = tg . If Σ is a multiplicative system then C[Σ^{−1}] has the following description. The objects of

C[Σ^{−1}] are the objects of C . The morphisms from X to Y in C[Σ^{−1}] are pairs (s, f) in C of the form

X ⟶^{f} Y′ ⟵^{s} Y, s ∈ Σ

modulo the following equivalence relation: (f, s) ∼ (g, t) iff there is a commutative diagram

\xymatrix{ & Y′ \ar[d] \\ X \ar[ur]^{f} \ar[r]^{h} \ar[dr]_{g} & Y′′′ & Y \ar[ul]_{s} \ar[l]_{r} \ar[dl]^{t} \\  & Y′′ \ar[u]}

with r ∈ Σ .

The composition of the morphisms (f, s) and (g, t) is a morphism (g′f, s′t) defined from the following diagram, which exists by M3:

\xymatrix{ & & Z′′ \\ & Y′ \ar@{-->}[ur]^{g′} & & Z′ \ar@{-->}[ul]_{s′} \\ X \ar[ur]^{f} & & Y \ar[ul]_{s} \ar[ur]^{g} & & Z \ar@{-->}[ul]_{t}} .

There is a quotient functor

Q : C ⟶ C[Σ^{−1}], X ↦ X, f ↦ (f, 1)

which inverts all elements of Σ and it is universal in this sense (see [4]).

Let D be a triangulated category and N ⊂ D be a full triangulated subcategory. Denote by Σ(N) a class of morphisms s in D embedding into an exact triangle

X ⟶^{s} Y ⟶ N ⟶ X[1]

with N ∈ N . It can be checked that Σ(N) is a multiplicative system. We define

D/N := D[Σ(N)^{−1}].

We endow the category D/N with a translation functor induced by the translation functor in the category D . Lemma 1.1. The category D/N becomes a triangulated category by taking for exact triangles such that are isomorphic to the images of exact triangles in D . The quotient functor Q : D ⟶ D/N anihilates N . Moreover, any exact functor F : D ⟶ D′ of triangulated categories for which F (X) ≃ 0 when X ∈ N factors uniquely through Q .

The following lemma, which will be necessary in the future, is evident. Lemma 1.2. Let M and N be full triangulated subcategories in triangulated categories C and D respectively. Let F : C → D and G : D → C be adjoint pair of exact functors such that F (M) ⊂ N and G(N) ⊂ M . Then they induce functors

\overline{F} : C/M ⟶ D/N , \overline{G} : D/N ⟶ C/M

which are adjoint. Proposition 1.3. ([20], [12]). Let D be a triangulated category and D′, N be full triangulated subcategories. Let N′ = D′ ∩ N . Assume that any morphism N → X′ with N ∈ N and X′ ∈ D′ admits a factorization N → N′ → X′ with N′ ∈ N′ . Then the natural functor

D′/N′ ⟶ D/N

is fully faithful.

1.2. Triangulated categories of singularities. Let X be a separated noetherian scheme of finite Krull dimension over a field k such that the category of coherent sheaves coh(X) has enough locally free sheaves. The last condition means that for any coherent sheaf F there is a vector bundle E and an epimorphism E ↠ F. For example any quasi-projective scheme satisfies these conditions. Note that any closed and any open subscheme of X is also noetherian, finite dimension and has enough locally free sheaves. It is clear for a closed subscheme while for an open subscheme U it follows from the fact that any coherent sheaf on U can be obtained as the restriction of a coherent sheaf on X (see [7], ex.5.15).

Denote by D^{b}(coh(X)) (resp. D^{b}(Qcoh(X)) ) the bounded derived categories of coherent (resp. quasi-coherent) sheaves on X . These categories have canonical triangulated structures.

Since X is noetherian the natural functor D^{b}(coh(X)) ⟶ D^{b}(Qcoh(X)) is fully faithful and realizes an equivalence of D^{b}(coh(X)) with the full subcategory D^{b}(Qcoh(X))_{coh} ⊂ D^{b}(Qcoh(X)) consisting of all complexes with coherent cohomologies (see [1] II, 2.2.2). Definition 1.4. A bounded complex of coherent sheaves will be called a perfect complex if it is quasi-isomorphic to a bounded complex of locally free sheaves of finite type. Lemma 1.5. Let X be as above. Then for any bounded above complex of coherent sheaves C^{·} on X there is a bounded above complex of locally frees of finite type P^{·} and an epimorphism of the complexes P^{·} ↠ C^{·} which is a quasi-isomorphism. Lemma 1.6. Any complex C^{·}, which is isomorphic to a bounded complex of locally frees in D^{b}(coh(X)), is perfect. Proof. We can represent the isomorphism in the derived category via calculus of fractions as P^{·} ←^{s} E^{·} →^{t} C^{·} , where P^{·} is a bounded complex of locally frees and s, t are quasi-isomorphisms. By Lemma 1.5, there is a bounded above complex Q^{·} of locally frees and quasi-isomorphism Q^{·} → E^{·} . Consider a good truncation τ^{≥−k}Q^{·} for sufficiently large k . As E^{·} is bounded there is a morphism r : τ^{≥−k}Q^{·} → E^{·} that is also a quasi-isomorphism. To prove the lemma it is sufficient to show that τ^{≥−k}Q^{·} is a complex of locally frees. Consider the composition sr : τ^{≥−k}Q^{·} → P^{·} which is a quasi-isomorphism. The cone of sr is a bounded acyclic complex all terms of which, excepted maybe the leftmost term, are locally free. This implies that the leftmost term is locally free as well, because the kernel of an epimorphism of locally frees is locally free. Thus τ^{≥−k}Q^{·} is a bounded complex of locally free sheaves and hence C^{·} is perfect. □

The perfect complexes form a full triangulated subcategory 𝔓𝔢𝔯𝔣(X) ⊂ D^{b}(coh(X)) , which is thick. Remark 1.7. Actually, a perfect complex is defined as a complex of O_{X} -modules locally quasiisomorphic to a bounded complex of locally frees of finite type. But under our assumption on the scheme any such complex is quasi-isomorphic to a bounded complex of locally frees of finite type(see [1] II, or [19] §2). Definition 1.8. We define a triangulated category D_{Sg}(X) as the quotient of the triangulated category D^{b}(coh(X)) by the full triangulated subcategory 𝔓𝔢𝔯𝔣(X) and call it as a triangulated category of singularities of X .

Remark 1.9. It is known that if a scheme X is as above and is regular in addition then the subcategory of perfect complexes coincides with the whole bounded derived category of coherent sheaves. Hence, the triangulated category of singularities is trivial in this case.

We also denote as 𝔏𝔣𝔯(X) ⊂ D^{b}(Qcoh(X)) the full triangulated subcategory consisting of objects which are isomorphic to bounded complexes of locally free sheaves in D^{b}(Qcoh(X)) . By the same argument as in Lemma 1.6 we can show that for any complex C^{·} ∈ 𝔏𝔣𝔯(X) there is a bounded complex of locally frees P^{·} and a quasi-isomorphism P^{·} → C^{·} . It can be easily checked that the subcategory D^{b}(coh(X)) ∩ 𝔏𝔣𝔯(X) coincides with the subcategory of perfect complexes 𝔓𝔢𝔯𝔣(X) .

Define a triangulated category DSg^{′}(X) as the quotient D^{b}(Qcoh(X))/𝔏𝔣𝔯(X) . The full embedding D^{b}(coh(X)) ⟶ D^{b}(Qcoh(X)) induces a functor D_{Sg}(X) ⟶ DSg^{′}(X) . We will show that this functor is also full embedding.

Lemma 1.10. Let X be as above and let F be a quasicoherent sheaf on X such that for any point x ∈ Sing(X) it is locally free in some neighbourhood of x . Then it belongs to the subcategory 𝔏𝔣𝔯(X) . If F is coherent in addition then it is perfect as a complex. Proof. By Lemma 1.5, there is a bounded above complex Q^{·} of locally frees and quasiisomorphism Q^{·} → F .Consider a good truncation τ^{≥−k}Q^{·} for sufficiently large k . There is a morphism r : τ^{≥−k}Q^{·} → F that is also a quasi-isomorphism. To prove the lemma it is sufficient to show that τ^{≥−k}Q^{·} is a complex of locally free sheaves. All terms of this complex, except maybe for the leftmost term, are locally free. But for any point x ∈ Sing(X) the leftmost term is also locally free, because F is locally free. If now x ∉ Sing(X) then there is a neighbourhood U of x which is regular. Hence, the leftmost term is locally free on U under assumption that k > dim X .

If now F is coherent then we can take the resolution Q^{·} such that all terms are of finite type. Hence, F is perfect. □

In particular, we get from this lemma that if the support of an object E^{·} ∈ D^{b}(coh(X)) does not meet Sing(X) then E^{·} is perfect.

Lemma 1.11. Let X be as above. Then any object A ∈ D_{Sg}(X) is isomorphic to an object F[k] where F is a coherent sheaf. Proof. The object A is a bounded complex of coherent sheaves. Let us take locally free bounded above resolution P^{·} →^{∼} A which exists by Lemma 1.5. Consider a brutal truncation σ^{≥−k}P^{·} for sufficiently large k ≫ 0 . Denote by F the cohomology H^{−k}(σ^{≥−k}P^{·}) . It is clear that A ≅ F[k + 1] in D_{Sg}(X) . □

Lemma 1.12. Let X be as above. Then for any locally free sheaf E and for any quasi-coherent sheaf F

Ext^{i}(E, F) = 0, for i > n.

Proof. Let U_{1} ∪ ... ∪ U_{n} be an affine cover of X . For all subsequence of indices I = (i_{1}, ..., i_{k}) , let U_{I} = U_{i_{1}} ∩ ... ∩ U_{i_{k}}, and let j_{I} : U_{I} ↪ X be the open immersion. As X is separated, each U_{I} is affine and each j_{I} is an affine map. Hence, j_{I∗} is exact and preserves quasi-coherence.

We consider the C̆ech hypercover complex of quasi-coherent sheaves

(1)  0 → F → ⨁_{i=1}^{n} j_{i∗}j_{i}^{∗}F → ⨁_{I=(i_{1},i_{2})} j_{I∗}j_{I}^{∗}F → ⨁_{I=(i_{1},i_{2},i_{3})} j_{I∗}j_{I}^{∗}F → ⋯

This is an exact sequence of sheaves.

Since U_{I} = Spec(A_{I}) is affine the category of quasi-coherent sheaves on U_{I} is equivalent to the category of A_{I} -modules. Therefore, we have

Ext^{i}(E, j_{I∗}j_{I}^{∗}F) = Ext^{i}(j_{I}^{∗}E, j_{I}^{∗}F) = 0  for all  i > 0,

because j_{I}^{∗}E corresponds to a projective module over A_{I} . Thus the nontrivial Ext’s from E to F are bounded by the length of the complex (1) which is equal to n . □ Proposition 1.13. For a scheme X as above the natural functor D_{Sg}(X) ⟶ DSg^{′}(X) is fully faithful. Proof. Consider the full embedding D^{b}(coh(X)) ↪ D^{b}(Qcoh(X)) . We have 𝔓𝔢𝔯𝔣(X) = D^{b}(coh(X)) ∩ 𝔏𝔣𝔯(X) . To prove the proposition we check that the conditions of Proposition 1.3 are fulfilled. Let Q^{·} →^{t} F^{·} be a morphism in D^{b}(Qcoh(X)) such that Q^{·} ∈ 𝔏𝔣𝔯(X) and F^{·} ∈ D^{b}(coh(X)) . Let P^{·} →^{s} F^{·} be a quasi-isomorphism where P^{·} is a bounded above complex of locally frees of finite type. It exists by Lemma 1.5. The brutal truncation σ^{≥−k}P^{·} is a perfect complex. Consider the map r : σ^{≥−k}P^{·} → F^{·} induced by s . A cone of the map r is isomorphic to G[k + 1] where G is a coherent sheaf. Since Q^{·} is a bounded complex of locally frees by Lemma 1.12, we obtain that

Hom(Q^{·}, G[k + 1]) = 0

for sufficiently large k . Hence, the map t can be lifted to some map Q^{·} → σ^{≥−k}P^{·} . □

Consider a morphism f : X → Y of finite Tor-dimension (for example flat). It defines a functor Lf^{∗} : D^{b}(coh(Y)) ⟶ D^{b}(coh(X)) . It is clear that the functor Lf^{∗} sends a perfect complex on Y to a perfect complex on X . Therefore, we get a functor Lf̄^{∗} : D_{Sg}(Y) ⟶ D_{Sg}(X) . By the same reason there is a functor Lf̄^{∗} : DSg^{′}(Y) ⟶ DSg^{′}(X) .

Let f : X → Y be a morphism of finite Tor-dimension. Suppose that it is proper of locally finite type. Then there is the functor of direct image Rf_{∗} : D^{b}(coh(X)) → D^{b}(coh(Y)) and, moreover, this functor takes perfect complexes to perfect complexes (see [1] III, or [19]). In this case we get a functor Rf̄_{∗} : D_{Sg}(X) → D_{Sg}(Y) , which is the right adjoint to Lf̄^{∗} .

Now we prove a local property for triangulated categories of singularities. Proposition 1.14. Let X be as above and let j : U ↪ X be an embedding of an open subscheme such that Sing(X) ⊂ U . Then the functor j̄^{∗} : D_{Sg}(X) ⟶ D_{Sg}(U) is an equivalence of triangulated categories. Proof. Since X is separated the map j is affine and, hence, it is exact and it preserves quasicoherence. This implies that there is the functor j_{∗} : D^{b}(Qcoh(U)) → D^{b}(Qcoh(X)) which is right adjoint to j^{∗} . Take an object B ∈ 𝔏𝔣𝔯(U) and consider j_{∗}(B) . It is easy to see that the object j_{∗}(B) belongs to 𝔏𝔣𝔯(X) . Actually, this condition is local, for U it is fulfilled and for X \ Sing(X) as for smooth scheme it is evident. Thus the functor j_{∗} induces the functor

j̄_{∗} : DSg^{′}(U) ⟶ DSg^{′}(X).

Moreover, the functor j̄_{∗} is right adjoint to j̄^{∗}

For any object A ∈ D^{b}(Qcoh(X)) we have a canonical map μ_{A} : A ⟶ j_{∗}j^{∗}A . A cone C(μ_{A}) of this map is an object whose support belongs to X\U and does not intersect Sing(X) . Hence, by Lemma 1.10 the object C(μ_{A}) belongs to the subcategory 𝔏𝔣𝔯(X) . This gives that μ_{A} becomes an isomorphism in DSg^{′}(X) . Therefore, the functor

j̄^{∗} : DSg^{′}(X) ⟶ DSg^{′}(U)

is fully faithful. On the other hand, we know that j^{∗}j_{∗}(B) ≅ B for any B ∈ D^{b}(Qcoh(U)) . Hence, j̄^{∗} is an equivalence.

The functor j^{∗} preserves coherence. Thus, using Proposition 1.13, we obtain that the functor

j̄^{∗} : D_{Sg}(X) ⟶ D_{Sg}(U)

is fully faithful. Now note that by Lemma 1.11 any object B ∈ D_{Sg}(U) is isomorphic to F[k] where F is a coherent sheaf on U and any coherent sheaf on U can be obtained as the restriction of a coherent sheaf on X ([7], Ex.5.15). This implies that j̄^{∗} : D_{Sg}(X) ⟶ D_{Sg}(U) is an equivalence. □

1.3. Triangulated categories of singularities for Gorenstein schemes. Remind the definition of a Gorenstein local ring and a Gorenstein scheme. Definition 1.15. A local noetherian ring A is called Gorenstein if A as a module over itself has a finite injective resolution.

It can be shown that if A is Gorenstein than A is a dualizing complex for itself (see [8]). This means that A has finite injective dimension and the natural map

M ⟶ R Hom^{·}(R Hom^{·}(M, A), A)

is an isomorphism for any coherent A-module M and as consequence for any object from D^{b}(coh(Spec(A))) . Definition 1.16. A scheme X is Gorenstein if all of its local rings are Gorenstein local rings. Remark 1.17. If X is Gorenstein and has finite Krull dimension, then O_{X} is a dualizing complex for X , i.e. it has finite injective dimension as quasi-coherent sheaf and the natural map

F ⟶ R\underline{Hom}^{·}(R\underline{Hom}^{·}(F, O_{X}), O_{X})

is an isomorphism for any coherent sheaf F . In particular, we have that there is an integer n_{0} such that \underline{Ext}^{i}(F, O_{X}) = 0 for each quasi-coherent sheaf F and all i > n_{0} . Lemma 1.18. Let X be as above and Gorenstein. Then for any coherent sheaf F and P^{·} ∈ 𝔓𝔢𝔯𝔣(X) there is an integer m depending only on P^{·} such that

Hom^{i}(F, P^{·}) = 0

for all i > m . Proof. For any coherent sheaf F and for any locally free sheaf P we know that

\underline{Ext}^{i}(F, P) = \underline{Ext}^{i}(F, O_{X}) ⊗ P = 0

for any i > n_{0} . Using the spectral sequence from local to global Ext’s we obtain that

Ext^{i}(F, P) = 0

for i > n_{0} + n where n is dimension of X . Since P^{·} is a bounded complex there exists m depending only on P^{·} such that

Hom^{i}(F, P^{·}) = 0

for i > m . □ Lemma 1.19. Let X as above and Gorenstein. The the following conditions on a coherent sheaf F are equivalent.

1) The sheaves \underline{Ext}^{i}(F, O_{X}) are trivial for all i > 0 .

2) There is a right locally free resolution

0 ⟶ F ⟶ {Q^{0} ⟶ Q^{1} ⟶ Q^{3} ⋯ }

Proof. 1) ⇒ 2). Denote by F^{∨} the sheaf \underline{Hom}(F, O_{X}) . Consider a left locally free resolution of F^{∨} . Applying to it the functor \underline{Hom}(·, O_{X}) we obtain a right locally free resolution of F , because O_{X} is dualizing complex.

2) ⇒ 1). Consider a brutal truncation σ^{≤k}Q^{·} for sufficiently large k . Denote by G the nontrivial cohomology H^{k}(σ^{≤k}Q^{·}) . For any i > 0 we have isomorphisms

\underline{Ext}^{i}(F, O_{X}) ≅ \underline{Ext}^{i+k+1}(G, O_{X}) = 0

The last equality follows from Remark 1.17. □ Lemma 1.20. Let X be as above and Gorenstein. Let F be a coherent sheaf which is perfect as a complex. Suppose that \underline{Ext}^{i}(F, O_{X}) = 0 for all i > 0 . Then F is locally free. Proof. If F is perfect then F^{∨} = \underline{Hom}(F, O_{X}) is also perfect. Hence, it has bounded locally free resolution P^{·} ⟶ F . Since O_{X} is a dualizing complex we obtain a bounded right resolution of F ⟶ P^{·} . Thus F is locally free. □ Proposition 1.21. Let X be as above and Gorenstein. Let F and G be coherent sheaves such that \underline{Ext}^{i}(F, O_{X}) = 0 for all i > 0 . Fix N such that Ext^{i}(P, G) = 0 for i > N and for any locally free sheaf P . Then

Hom_{D_{Sg}(X)}(F, G[N]) ≅ Ext^{N}(F, G)/R

where R is the subspace of elements factoring through locally free, i.e. e ∈ R iff e = βα with α : F → P and β ∈ Ext^{N}(P, G) where P is locally free. Proof. By the definition of a localization any morphism from F to G[N] in D_{Sg}(X) can be represented by a pair of morphisms in D^{b}(coh(X)) of the form

(2)  F ⟵^{s} A ⟶^{a} G[N]

such that the cone C(s) is a perfect complex. By Lemma 1.19 there is a right locally free resolution F ⟶ Q^{·} . We consider a brutal truncation σ^{≤k}Q^{·} for sufficiently large k such that Hom(E[−k − 1], C(s)) = 0, where E = H^{k}(σ^{≤k}Q^{·}) . Such k exists by Lemma 1.18. Using the triangle

E[−k − 1] ⟶ F ⟶ σ^{≤k}Q^{·} ⟶ E[−k]

we find that the map F → C(s) can be lifted to a map σ^{≤k}Q^{·} → C(s) . Therefore, there is a map E[−k − 1] → A and we get a morphism in D_{Sg}(X) of the form

(3)  F ⟵^{s′} E[−k − 1] ⟶^{e} G[N]

which is equivalent to the morphism (2). Since Ext^{i}(P, G) = 0 for i > N and for any locally free sheaf P we get

Hom(σ^{≤k}Q^{·}[−1], G[N]) = 0.

Hence, there is a map f which completes the following diagram

\xymatrix{ & E[−k − 1] \ar@{-->}[dr]^{e} \ar[dl]_{s′} \\ F \ar[rr]^{f} & & G[N]} .

This gives us that the map f is equivalent to the map (3) and as consequence to the map (2). Thus, any morphism from F to G[N] in D_{Sg}(X) is equivalent to a morphism from F to G[N] in D^{b}(coh(X)) . Now if f is equivalent to 0-morphism then repeating the procedure above we find that f factors through a morphism σ^{≤k}Q^{·} → G[N] . By the condition on G any such morphism can be lifted to a morphism Q^{0} → G[N] . Hence, if f is equivalent to the 0-morphism in D_{Sg}(X) then it factors through Q^{0} . □ Remark 1.22. If X is affine then N can be taken equal to 0 . Proposition 1.23. Let X be as above and Gorenstein. Then any object A ∈ D_{Sg}(X) is isomorphic to the image of a coherent sheaf F such that \underline{Ext}^{i}(F, O_{X}) = 0 for all i > 0 . Proof. An object A is a bounded complex of coherent sheaves. Let us take locally free bounded above resolution P^{·} →^{∼} A which exists by Lemma 1.5. Consider a brutal truncation σ^{≥−k}P^{·} for sufficiently large k ≫ 0 . Denote by G the cohomology H^{−k}(σ^{≥−k}P^{·}) . Since A is bounded and X is Gorenstein we know that R\underline{Hom}^{i}(A, O_{X}) = 0 for sufficiently large i . This implies that iff k ≫ 0 then \underline{Ext}^{i}(G, O_{X}) = 0 for all i > 0 . Moreover, we get that A ≅ G[k + 1] in D_{Sg}(X) .

By Lemma 1.19 there is a right locally free resolution G → Q^{0} → Q^{1} → · · · . Consider Im d_{k−1} = Ker d_{k} ⊂ Q^{k} and denote it by F . Applying again Lemma 1.19 we obtain that \underline{Ext}^{i}(F, O_{X}) = 0 for all i ≥ 0 . And we have an isomorphism A ≅ G[k + 1] ≅ F in D_{Sg}(X) . □ Corollary 1.24. Let X be as above, proper and Gorenstein and let U ⊂ X be open subscheme such that Sing(X) ⊂ U . Then dim_{k} Hom(A, B) < ∞ for any two objects A, B ∈ D_{Sg}(U) . Proof. We know that for any two objects in D^{b}(coh(X)) the space of morphisms is finite dimensional. Now the statement of the corollary immediately follows from Propositions 1.21 and 1.23. □

2. Reduction of dimension

Let X be a separated regular noetherian scheme of finite Krull dimension. Any such scheme has enough locally free sheaves (see [1], II). Let f : X → 𝔸^{1} be a flat morphism. Consider the scheme Y = X × 𝔸^{2} and a morphism g = f + xy to 𝔸^{1} , where x, y are coordinates on 𝔸^{2} . Denote by X_{0} = X/f and Y_{0} = Y/g the fibers of f and g respectively over the point 0 .

Consider the scheme Z = Y_{0}/x . There are natural maps i : Z → Y_{0} and q : Z → X_{0} where the former is a closed embedding and the latter is an 𝔸^{1} -bundle. All the schemes, introduced above, are separated noetherian schemes of finite Krull dimension that have enough locally free sheaves.

Consider the composition functor Ri_{∗}q^{∗} : D^{b}(coh(X_{0})) → D^{b}(coh(Y_{0})) and denote it by Φ_{Z} . The aim of this paragraph is to prove the following theorem.

Theorem 2.1. The functor Φ_{Z} : D^{b}(coh(X_{0})) → D^{b}(coh(Y_{0})) defined by formula

Φ_{Z}(·) = Ri_{∗}q^{∗}(·)

induces a functor \overline{Φ}_{Z} : D_{Sg}(X_{0}) ⟶ D_{Sg}(Y_{0}) which is an equivalence.

At first, we consider partial compactifications of Y and Y_{0} . Denote by \overline{Y} the scheme X × ℙ^{2} and let \overline{Y}_{0} ⊂ \overline{Y} be a closed subscheme which is given by equation fz^{2} + xy = 0 , where now x, y, z are projective coordinates on ℙ^{2} . There is a flat map π : \overline{Y}_{0} → X which is a conic bundle over X . The scheme \overline{Y}_{0} is a partial compactification of Y_{0} such that Sing(\overline{Y}_{0}) = Sing(Y_{0}) . Hence, by Proposition 1.14 we have an equivalence D_{Sg}(\overline{Y}_{0}) ≅ D_{Sg}(Y_{0}) of triangulated categories of singularities.

Consider the cartesian square

Z̃ ⟶^{\overline{i}} \overline{Y}_{0}
p↓ ↓^{π}
X_{0} ⟶^{j} X

Here Z̃ = X_{0} ×_{X} \overline{Y}_{0} is a fiber product. The scheme Z̃ is a union of two components \overline{Z}_{1} ∪ \overline{Z}_{2} . Each \overline{Z}_{i} is isomorphic to ℙ^{1} × X_{0} and their intersection is isomorphic to X_{0} . The component \overline{Z}_{1} is a partial compactification of Z = 𝔸^{1} × X_{0} . We denote by i_{1}, i_{2} the closed embeddings of \overline{Z}_{1}, \overline{Z}_{2} in \overline{Y}_{0} and we denote by W ⊂ \overline{Y}_{0} the intersection \overline{Z}_{1} ∩ \overline{Z}_{2} . We know that W is isomorphic to X_{0} . Moreover, Sing(\overline{Y}_{0}) is contained in W and coincides with Sing(X_{0}) under the isomorphism W ≅ X_{0} . Lemma 2.2. The closed embedding i_{1} : \overline{Z}_{1} ↪ \overline{Y}_{0} is regular and \overline{Z}_{1} is a Cartier divisor in \overline{Y}_{0} . The restriction of the line bundle O_{\overline{Y}_{0}}(\overline{Z}_{1}) on \overline{Z}_{1} is isomorphic to O_{\overline{Z}_{1}}(−1) = O_{ℙ^{1}}(−1)⊠O_{X_{0}} . Proof. Consider a Cartier divisor in \overline{Y}_{0} given by the equation x = 0 . It is the union of \overline{Z}_{1} and D , where D is a component of \overline{Y}_{0} \ Y_{0} . Hence, D does not meet the singularities of \overline{Y}_{0} . This implies that D is a Cartier divisor. Therefore, \overline{Z}_{1} is a Cartier divisor too and i_{1} is a regular embedding.

Consider the divisor Z̃ = \overline{Z}_{1} ∪\overline{Z}_{2} . It is equal to π^{−1}(X_{0}) and its restriction on \overline{Z}_{1} is trivial. The restriction of O(\overline{Z}_{2}) on \overline{Z}_{1} is O_{ℙ^{1}}(1) ⊠ O_{X_{0}} because the intersection \overline{Z}_{2} ∩ \overline{Z}_{1} = W is isomorphic to X_{0} . Hence, the restriction of O(\overline{Z}_{1}) on \overline{Z}_{1} is O_{ℙ^{1}}(−1) ⊠ O_{X_{0}} □ Denote by p_{1} the projection of \overline{Z}_{1} on X_{0} and consider the commutative diagram

\overline{Z}_{1} ⟶^{i_{1}} \overline{Y}_{0}
p_{1}↓ ↓^{π}
X_{0} ⟶^{j} X

Proposition 2.3. The functor Ri_{1∗}p_{1}^{∗} : D^{b}(coh(X_{0})) → D^{b}(coh(\overline{Y}_{0})) is fully faithful. Proof. The functor Φ_{\overline{Z}_{1}} = Ri_{1∗}p_{1}^{∗} has a right adjoint functor Φ_{\overline{Z}_{1}∗} = Rp_{1∗}i_{1}^{♭} where i_{1}^{♭}(·) ≅ Li_{1}^{∗}(· ⊗ O(Z_{1}))[−1] (see, for example, [8] Cor.7.3).

At first, note that the functor p_{1}^{∗} : D^{b}(coh(X_{0})) ⟶ D^{b}(coh(\overline{Z_{1}})) is fully faithfull, because by the projection formula we have an isomorphism

Rp_{1∗}p_{1}^{∗}(A) ≅ A ⊗ Rp_{1∗}O_{\overline{Z}_{1}} ≅ A ⊗ O_{X_{0}} ≅ A

for any A ∈ D^{b}(coh(X_{0})) .

Consider the canonical transformation of functors id → Φ_{\overline{Z}_{1}∗}Φ_{\overline{Z}_{1}} . We have to show that this transformation is an isomorphism of functors. Take an object A ∈ D^{b}(coh(X_{0})) and consider an exact triangle

(4)  A ⟶ Φ_{\overline{Z}_{1}∗}Φ_{\overline{Z}_{1}}(A) ⟶ C

To show that C = 0 it is sufficient to check that Rj_{∗}C = 0 , because j is a closed embedding. Since the functor p_{1}^{∗} is fully faithful, the triangle (4) is the image of the triangle

(5)  p_{1}^{∗}A ⟶ i_{1}^{♭}Ri_{1∗}p_{1}^{∗}A ⟶ B

under the functor Rp_{1}^{∗} , where the former morphism is a canonical map induced by the natural transformation id → i_{1}^{♭}Ri_{1∗} . Applying the functor Ri_{1∗} to the exact triangle (5) we obtain the following triangle

Ri_{1∗}p_{1}^{∗}A ⟶ Ri_{1∗}i_{1}^{♭}Ri_{1∗}p_{1}^{∗}A ⟶ Ri_{1∗}B

This triangle is splited by the canonical morphism Ri_{1∗}i_{1}^{♭}Ri_{1∗}p_{1}^{∗}A → Ri_{1∗}p_{1}^{∗}A which is obtained by the tensoring product of the object Ri_{1∗}p_{1}^{∗}A with a map α from the following exat triangle

i_{1∗}O_{\overline{Z}_{1}}(\overline{Z}_{1})[−1] ⟶^{α} O_{\overline{Y}_{0}} ⟶ O_{\overline{Y}_{0}}(Z_{1}) ⟶ i_{1∗}O_{\overline{Z}_{1}}(\overline{Z}_{1})

Therefore, the object Ri_{∗}B is isomorphic to Ri_{∗}p_{1}^{∗}A(\overline{Z_{1}}) = Ri_{∗}(p_{1}^{∗}A ⊗ O_{\overline{Z}_{1}}(−1)) . Now we have the following sequence of isomorphisms

Rj_{∗}C ≅ Rj_{∗}Rp_{1∗}B ≅ Rπ_{∗}Ri_{1∗}B ≅ Rπ_{∗}Ri_{1∗}(p_{1}^{∗}A ⊗ O_{\overline{Z}_{1}}(−1)) ≅ Rj_{∗}Rp_{1∗}(p_{1}^{∗}A ⊗ O_{\overline{Z}_{1}}(−1)) ≅ Rj_{∗}(A ⊗ Rp_{1∗}O_{\overline{Z}_{1}}(−1)) ≅ 0

Thus the functor Φ_{\overline{Z}_{1}} = Ri_{1∗}p_{1}^{∗} is fully faithful. □

Corollary 2.4. The functor Φ_{\overline{Z}_{1}} : D^{b}(coh(X_{0})) → D^{b}(coh(\overline{Y}_{0})) induces a functor \overline{Φ}_{Z} : D_{Sg}(X_{0}) ⟶ D_{Sg}(\overline{Y}_{0}) which is fully faithful. Proof. The functors p_{1}^{∗} and i_{1}^{♭} = Li_{1}^{∗}(· ⊗ O(\overline{Z}_{1}))[−1] take perfect complexes to perfect complexes as functors of inverse images. The functors Ri_{1∗} and Rp_{1∗} also preserve perfect complexes, because both morphisms i_{1} and p_{1} are finite Tor-dimension, proper, and of finite type.

Thus, we get a functor \overline{Φ}_{\overline{Z}_{1}} : D_{Sg}(X_{0}) → D_{Sg}(Y_{0}) and this functor has the right adjoint \overline{Φ}_{\overline{Z}_{1}∗} . As the composition Φ_{\overline{Z}_{1}∗}Φ_{\overline{Z}_{1}} is isomorphic to the identity functor, the composition \overline{Φ}_{\overline{Z}_{1}∗}\overline{Φ}_{\overline{Z}_{1}} is also isomorphic to the identity functor. □

Thus, we obtained the functor \overline{Φ}_{\overline{Z}_{1}} : D_{Sg}(X_{0}) ⟶ D_{Sg}(\overline{Y}_{0}) and showed that this functor is fully faithful. To complete the proof of the theorem we have to check that this functor is an equivalence. We show that any object E ∈ D_{Sg}(\overline{Y}_{0}) satisfying condition \overline{Φ}_{\overline{Z}_{1}∗}E = 0 is zero object. The property to be an equivalence will easy follow from this fact. Lemma 2.5. Any object A ∈ D^{b}(coh(\overline{Z}_{1})) such that Rp_{1∗}A = 0 is isomorphic to an object of the form p_{1}^{∗}B ⊗ O_{\overline{Z}_{1}}(−1) for some B ∈ D^{b}(coh(X_{0})) . Proof. Denote by B the object Rp_{1∗}(A(1)) . We have a natural map p_{1}^{∗}B ⊗ O_{\overline{Z}_{1}}(−1) → A . Denote by C the cone of this map. The object C satisfies the following conditions

Rp_{1∗}C = 0  and  Rp_{1∗}(C(1)) = 0,

the latter condition is following from the fact that p_{1}^{∗} is fully faithful. Now any sheaf O_{\overline{Z}_{1}}(n) has a resolution of the form

O_{\overline{Z}_{1}}(−1)^{⊕n} ⟶ O_{\overline{Z}_{1}}^{⊕(n+1)} ⟶ O_{\overline{Z}_{1}}(n).

Tensoring this sequence with the object C and applying the functor Rp_{1∗} we get that Rp_{1∗}(C(n)) = 0 for all n . Hence, the object C is zero, because the sheaf O\overline{Z}(1) is relatively ample. □

Actually, this Lemma shows us that the category D^{b}(coh(\overline{Z}_{1})) has a semiorthogonal decomposition of the form

⟨p_{1}^{∗}D^{b}(coh(X_{0})) ⊗ O_{\overline{Z}_{1}}(−1), p_{1}^{∗}D^{b}(coh(X_{0}))⟩

(for definition see [2]). It can be proved for any ℙ^{1} -bundle and moreover for the projectivization of any bundle. The proof for smooth base can be found in [16] and it works for any base.

The second Lemma is also almost evident. Lemma 2.6. Let i : Z ↪ Y be a closed embedding of a Cartier divisor. Let E be a sheaf on Y such that its restriction to the complement U = X \ Z is locally free and Li^{∗}E is isomorphic to a local free sheaf on Z . Then E is local free on Y .

Proof. Let us consider a morphism E(−Z) → E . By the assumption it is a monomorphism and the cokernel of this map is a locally free sheaf on Z . To prove that E is locally free it is sufficient to show that for any closed point y ∈ Y we have the equalities

Ext^{i}(E, O_{y}) = 0

for all i > 0 . The sheaf E is locally free on U . Hence, we only need to consider the case y ∈ Z . We have the following long exact sequence of Ext’s

⋯ ⟶ Ext^{i}(E|_{Z}, O_{y}) ⟶ Ext^{i}(E, O_{y}) ⟶ Ext^{i}(E(−Z), O_{y}) ⟶ Ext^{i+1}(E|_{Z}, O_{y}) ⟶ ⋯

where E|_{Z} is a short notation for i_{∗}i^{∗}E = Ri_{∗}Li^{∗}E .

Moreover, all maps Ext^{i}(E, O_{y}) → Ext^{i}(E(−Z), O_{y}) are zero, because y ∈ Z . Now, since O_{y} = i_{∗}O_{y} , we have isomorphisms

Ext^{i}(E|_{Z}, O_{y}) = Hom_{Z}^{i}(Li^{∗}E|_{Z}, O_{y})

The complex Li^{∗}E|_{Z} has only 0 and −1 cohomology sheaves which are locally free on Z . Hence,

Hom_{Z}^{i}(Li^{∗}E|_{Z}, O_{y}) = 0

for all i ≠ 0, 1 .

This gives us that Ext^{i}(E(−Z), O_{y}) = 0 for all i ≠ 0 . Therefore, E(−Z) is locally free, and E is locally free as well. □

Using these two lemmas we can prove the following proposition. Proposition 2.7. Assume that an object E ∈ D_{Sg}(\overline{Y}_{0}) satisfies the condition \overline{Φ}_{\overline{Z}_{1}∗}E = 0 . Then E = 0 in D_{Sg}(\overline{Y}_{0}) . Proof. At first, note that all schemes X_{0}, \overline{Z}_{1}, \overline{Y}_{0} are Gorenstein. By Proposition 1.23 we can assume that E is a sheaf and \underline{Ext}^{i}(E, O_{X}) = 0 for all i ≠ 0 . Note that any such E is locally free on the complement X \ Sing(X) . In addition, for such E we have Li_{1}^{∗}E ≅ i_{1}^{∗}E is a sheaf.

Denote by L the relatively ample line bundle on \overline{Y}_{0} obtained by the restriction of the line bundle O\overline{Y} (1) on \overline{Y} = ℙ^{2} × X . We can see that the object E ⊗ L^{⊗n} is isomorphic to E in the category D_{Sg}(\overline{Y}_{0}) , because there is an inclusion E → E ⊗ L^{⊗n} such that the support of the cokernel does not intersect Sing(\overline{Y}_{0}) and, hence, it is perfect by Lemma 1.10.

Take the sheaf i_{1}^{∗}E and denote by F the sheaf p_{1∗}i_{1}^{∗}E on X_{0} . Tensoring E with L^{n} if it is necessary we can reduce the situation to the case

R^{i}p_{1∗}i_{1}^{∗}E = 0 for all i > 0 and p_{1}^{∗}F ⟶^{α} i_{1}^{∗}E is surjective.

By Lemma 2.5 the kernel of α is a sheaf of the form p_{1}^{∗}G ⊗ O_{\overline{Z}_{1}}(−1), where G is a sheaf on X_{0} . By the assumption, the sheaf F is perfect as a complex on X_{0} . Moreover, the sheaf G ≅ p_{1∗}i_{1}^{∗}(E ⊗L^{−1}) is also perfect as a complex on X_{0} . Hence, the sheaf i_{1}^{∗}E on \overline{Z}_{1} is a perfect complex on \overline{Z}_{1} . On the other hand, we know that the sheaf E has a right local free resolution. Hence the sheaf i_{1}^{∗}E as the restriction with respect to a regular embedding of a divisor also has a right local free resolution. By Lemma 1.19 this implies that

\underline{Ext}^{i}(i_{1}^{∗}E, O_{\overline{Z}_{1}}) = 0 for all i > 0.

By Lemma 1.20 this means that i_{1}^{∗}E is locally free on \overline{Z}_{1} . Now apply Lemma 2.6 we get that E is locally free on whole \overline{Y}_{0} . Hence, it is isomorphic to the zero object in D_{Sg}(\overline{Y}_{0}) . □ Proof of Theorem 2.1 We already know that \overline{Φ}_{\overline{Z}_{1}} is fully faithful. Take an object A ∈ D_{Sg}(\overline{Y}_{0}) and consider the natural map \overline{Φ}_{\overline{Z}_{1}} \overline{Φ}_{\overline{Z}_{1}∗}A → A . Denote by C its cone. Apply the functor \overline{Φ}_{\overline{Z}_{1}∗} to the obtained exact triangle. Since \overline{Φ}_{\overline{Z}_{1}} is fully faithful, we get that \overline{Φ}_{\overline{Z}_{1}∗}C = 0 . By Proposition 2.7 the object C is the zero object. Hence, the functor \overline{Φ}_{\overline{Z}_{1}∗} is also fully faithful and, consequently, it is an equivalence. It remains only to note that the functor \overline{Φ}_{Z} : D_{Sg}(X_{0}) → D_{Sg}(Y_{0}) is the composition of the functor \overline{Φ}_{\overline{Z}_{1}} : D_{Sg}(X_{0}) → D_{Sg}(\overline{Y}_{0}) and the functor \overline{J}^{∗} : D_{Sg}(\overline{Y}_{0}) → D_{Sg}(Y_{0}) , where J : Y_{0} ↪ \overline{Y}_{0} is the open embedding. Both of these functors are equivalence. Hence, \overline{Φ}_{Z} is an equivalence too. □

3. Triangulated category of B-branes in Landau-Ginsburg model

3.1. Kontsevich’s proposal and categories of pairs. A mathematical definition of the categories of B-branes in Landau-Ginsburg models is proposed by M.Kontsevich (see also [11]).

By a Landau-Ginsburg model we mean the following data: a smooth variety (or regular scheme) X and a regular function W on X such that the morphism W : X ⟶ 𝔸^{1} is flat (for the definition of B-branes we don’t need a symplectic form on X which have to be in a LG model too).

With any point w_{0} ∈ 𝔸^{1} we can associate a differential ℤ/2ℤ-graded category DG_{w_{0}}(W) , an exact category P air_{w_{0}}(W) , and a triangulated category DB_{w_{0}}(W) . We give constructions of these categories under the condition that X = Spec(A) is affine (see [11]). The general definition is more sophisticated.

Since the category of coherent sheaves on an affine scheme X = Spec(A) is the same as the category of finitely generated A-modules we will frequently go from sheaves to modules and back. Note that under this equivalence locally free sheaves are the same as projectives modules.

Objects of all these categories are ordered pairs

\overline{P} := ( \xymatrix{P_{1} \ar[r]^{p_{1}} & P_{0} \ar[l]^{p_{0}}} )

where P_{0}, P_{1} are finitely generated projective A-modules and the compositions p_{0}p_{1} and p_{1}p_{0} are the multiplications by the element (W − w_{0}) ∈ A .

Morphisms from \overline{P} to \overline{Q} in the category DG_{w_{0}}(W) form ℤ/2ℤ-graded complex

ℍom(\overline{P} , \overline{Q}) = ⨁_{i,j} Hom(P_{i}, Q_{j})

with a natural grading (i−j) mod 2, and with a differential D acting on homogeneous elements of degree k as

Df = q ∘ f − (−1)^{k}f ∘ p.

The space of morphisms Hom(\overline{P} , \overline{Q}) in the category P air_{w_{0}}(W) is the space of homogeneous of degree 0 morphisms in DG_{w_{0}}(W) which commutes with the differential. The space of morphisms in the category DB_{w_{0}}(W) is the space of morphisms in P air_{w_{0}}(W) modulo null-homotopic morphisms, i.e.

Hom_{P air_{w_{0}}(W)}(\overline{P} , \overline{Q}) = Z^{0}(ℍom(\overline{P} , \overline{Q})), Hom_{DB_{w_{0}}(W)}(\overline{P} , \overline{Q}) = H^{0}(ℍom(\overline{P} , \overline{Q})).

Thus a morphism f : \overline{P} → \overline{Q} in the category P air_{w_{0}}(W) is a pair of morphisms f_{1} : P_{1} → Q_{1} and f_{0} : P_{0} → Q_{0} such that f_{1}p_{0} = q_{0}f_{0} and q_{1}f_{1} = f_{0}p_{1} . The morphism f is nullhomotopic if there are two morphisms s : P_{0} → Q_{1} and t : P_{1} → Q_{0} such that f_{1} = q_{0}t + sp_{1} and f_{0} = tp_{0} + q_{1}s .

It is clear that the category P air_{w_{0}}(W) is an exact category with respect to componentwise monomorphisms and epimorphisms (see definition in [17]).

The category DB_{w_{0}}(W) can be endowed with a natural structure of a triangulated category. To determine it we have to define a translation functor [1] and a class of exact triangles.

The translation functor can be defined as a functor that takes an object \overline{P} to the object

\overline{P} [1] = [figure: (P_{0}; −p_{0}; P_{1}; −p_{1}] ),

i.e. it changes the order of the modules and signs of the morphisms, and takes a morphism f = (f_{0}, f_{1}) to the morphism f[1] = (f_{1}, f_{0}) . We see that the functor [2] is the identity functor.

For any morphism f : \overline{P} → \overline{Q} from the category P air_{w_{0}}(W) we define a mapping cone C(f) as an object

C(f) = [figure: (Q_{1} ⊕ P_{0}; c_{1}; Q_{0} ⊕ P_{1}; c_{0}] )

such that

c_{0} = (\begin{matrix}q_{0} & f_{1} \\ 0 & −p_{1} \end{matrix}) , c_{1} = (\begin{matrix}q_{1} & f_{0} \\ 0 & −p_{0} \end{matrix}) .

There are maps g : \overline{Q} → C(f), g = (id, 0) and h : C(f) → \overline{P} [1], h = (0, −id) .

Now we define a standard triangle in the category DB_{w_{0}}(W) as a triangle of the form

\overline{P} ⟶^{f} \overline{Q} ⟶^{g} C(f) ⟶^{h} \overline{P} [1].

for some f ∈ P air_{w_{0}}(W) . Definition 3.1. A triangle \overline{P} →\overline{Q}→\overline{R}→\overline{P} [1] in DB_{w_{0}}(W) will be called an exact triangle if it is isomorphic to a standard triangle. Proposition 3.2. The category DB_{w_{0}}(W) endowed with the translation functor [1] and the above class of exact triangles becomes a triangulated category. Proof.The proof is the same as the proof of the analogous result for a usual homotopic category (see, for example, [5] ot [12]). □ Definition 3.3. We define a category of B-branes on X with the superpotential W as a disjoint

union DB(W) = ∐_{w∈𝔸^{1}} DB_{w}(W) .

Note that since X is regular, the set of points on 𝔸^{1} with singular fibers is finite (see [7], III,Cor.10.7). It will be shown that the category DB_{w}(W) is trivial if the fiber over point w is smooth. Hence, the category DB(W) is a disjoint union of finitely many numbers of categories.

Actually, the construction of the category DB_{w_{0}}(W) fits into a general construction of the stable category associated with an exact category. We briefly recall the definition of an exact category, which was introduced by Quillen in [17].

An exact category E is an additive category together with a choice of a class of sequences {F ↣ E ↠ G} said to be exact. This determines two classes of morphisms: the admissible epimorphisms E ↠ G and the admissible monomorphisms F ↣ E . The exact category is to satisfy the following axioms: Any sequence isomorphic to an exact sequence is exact. In any exact sequence F ↣^{i} E ↠^{p} G , the map i is a kernel of p and p is a cokernel of i . The class of admisible monomorphisms is closed under composition and is closed under cobase change by pushout along an arbitrary map F → F′ . Dually, the class of admissible epimorphisms is closed under composition and under base change by pullback along an arbitrary map G′ → G .

This definition is equivalent to the original definition of Quillen (see [14]).

An object I ∈ E is injective (resp. P is projective) if the sequence

Hom(E, I) → Hom(F, I) → 0 (resp. Hom(P, E) → Hom(P, G) → 0)

is exact for each admissible monomorphism (resp. epimorphism). We say that E has enough injectives, if for each F ∈ E there is an admissible monomorphism in an injective.

If E has enough injectives then we can define the stable category \underline{E} as a category which has the same objects as E and a morphism in E is the equivalence class \overline{f} of a morphism f of E modulo the subgroup of morphisms factoring through an injective in E (see, for example, [6, 13, 14]).

If E also has enough projectives (i.e. for each G ∈ E there is an admissible epimorphism P ↠ G with projective P ), and the classes of projectives and injectives coincide, then E is called a Frobenius category. The stable category \underline{E} associated with a Frobenius category has a natural structure of a triangulated category ([6]).

In our case, the category P air_{w_{0}}(W) is an exact category with respect to componentwise monomorphisms and epimorphisms. Moreover, it can be shown that P air_{w_{0}}(W) is a Frobenius category and the class of injectives consists exactly of homotopic to zero pairs. Hence, the category DB_{w_{0}}(W) is nothing more than the stable category associated to the exact category P air_{w_{0}}(W) and has the natural structure of a triangulated category. It gives another proof of Proposition 3.2.

3.2. Categories of pairs and categories of singularities. Further we suppose w_{0} = 0 . Denote by X_{0} the fiber of f : X → 𝔸^{1} over the point 0 . With any pair \overline{P} we can associate a short exact sequence

(6)  0 ⟶ P_{1} ⟶^{p_{1}} P_{0} ⟶ Coker p_{1} ⟶ 0

We can attach to an object \overline{P} the sheaf Coker p_{1} . This is a sheaf on X . But the multiplication by W annihilates it. Hence, we can consider Coker p_{1} as a sheaf on X_{0} . Any morphism f : \overline{P} → \overline{Q} in P air_{0}(W) gives a morphism between cokernels. This way we get a functor Cok : P air_{0}(W) ⟶ coh(X_{0}) . Lemma 3.4. The functor Cok is full. Proof. Any map g : Coker p_{1} → Coker q_{1} can be extended to a map of exact sequences

0 ⟶ P_{1} ⟶^{p_{1}} P_{0} ⟶ Coker p_{1} ⟶ 0
f_{1}↓ ↓^{f_{0}} ↓^{g}
0 ⟶ Q_{1} ⟶^{q_{1}} Q_{0} ⟶ Coker q_{1} ⟶ 0,

because P_{1} and P_{0} are projective. To prove the lemma we need to show that f = (f_{1}, f_{0}) is a map of pairs, i.e f_{1}p_{0} = q_{0}f_{0} . We have the sequence of equalities

q_{1}(f_{1}p_{0} − q_{0}f_{0}) = f_{0}q_{0}p_{0} − q_{1}q_{0}f_{0} = f_{0}W − W f_{0} = 0

Since q_{1} is an embedding, we obtain that f_{1}p_{0} = q_{0}f_{0} . □ Lemma 3.5. For any pair \overline{P} the coherent sheaf Coker p_{1} on X_{0} satisfies the condition

Ext^{i}(Coker p_{1}, O_{X_{0}}) = 0

for all i > 0 .

Proof. First note that X_{0} is Gorenstein as a full intersection. Consider the restriction of the sequence (6) on X_{0} . We obtain an exact sequence

0 → Coker p_{1} ⟶ P_{1}/W ⟶^{p_{1}|_{W}} P_{0}/W ⟶ Coker p_{1} → 0.

This gives us a periodic resolution of the sheaf Coker p_{1}

0 → Coker p_{1} ⟶ P_{1}/W ⟶^{p_{1}|_{W}} P_{0}/W ⟶ P_{1}/W ⟶ ⋯

Since X_{0} is affine and Gorenstein the existance of such resolution implies that

Ext^{i}(Coker p_{1}, O_{X_{0}}) = 0

for all i > 0 ( by Lemma 1.19). □

Now we show that the functor Cok induces an exact functor between triangulated categories DB_{0}(W) and D_{Sg}(X_{0}) . Proposition 3.6. There is a functor F which completes the following commutative diagram

P air_{0}(W) ⟶^{Cok} coh(X_{0})
↓ ↓
DB_{0}(W) ⟶^{F} D_{Sg}(X_{0}).

Moreover, the functor F is an exact functor between triangulated categories. Proof. We have a functor P air_{0}(W) → D_{Sg}(X_{0}) which is the composition of Cok and the natural functor from coh(X) to D_{Sg}(X_{0}) . To prove the existance of a functor F we need to show that any morphism \overline{f} = (f_{1}, f_{0}) : \overline{P} → \overline{Q} which is homotopic to 0 goes to 0-morphism in D_{Sg}(X_{0}) . Fix a homotopy (t, s) where t : P_{1} → Q_{0} and s : P_{0} → Q_{1} . Consider the following decomposition of \overline{f} :

[figure: P_{1}; (t,f_{1}); p_{1}; P_{0}; p_{0}; (s,f_{0}); Coker p_{1}; Q_{0} ⊕ Q_{1}; pr; c_{1}; Q_{1} ⊕ Q_{0}; pr; c_{0}; Q_{0}/W; where; c_{0} = (\begin{matrix}−q_{0} & id \\ 0 & q_{1} \end{matrix}) , c_{1} = (\begin{matrix}−q_{1} & id \\ 0 & q_{0} \end{matrix}) .; Q_{1}; q_{1}; Q_{0}; q_{0}; Coker q_{1}]

This gives the decomposition of F (\overline{f}) through a locally free object Q_{0}/W . Hence, F (\overline{f}) = 0 in the category D_{Sg}(X_{0}) . We leave to the reader the proof that the functor F is exact. □ Lemma 3.7. If F \overline{P} = 0 then \overline{P} = 0 in DB_{0}(W) .

Proof. If F \overline{P} = 0 then Coker p_{1} is a perfect complex. This implies that it is locally free by Lemmas 1.20 and 3.5. Hence, there is a map f : Coker p_{1} → P_{0}/W which splits the epimorphism pr : P_{0}/W → Coker p_{1} . It can be lifted to a map from {P_{1} ⟶^{p_{1}} P_{0}} to {P_{0} ⟶^{W} P_{0}} . Consider a diagram

P_{1} ⟶^{p_{1}} P_{0} ⟶ Coker p_{1}
t↓ ↓^{u} ↓^{f}
P_{0} ⟶^{W} P_{0} ⟶ P_{0}/W
p_{0}↓ ↓_{id} ↓^{pr}
P_{1} ⟶^{p_{1}} P_{0} ⟶ Coker p_{1}

Since the composition prf is identical, the map (p_{0}t, u) from the pair {P_{1} ⟶^{p_{1}} P_{0}} to itself is homotopic to the identical map of this pair. Hence, there is a map s : P_{0} → P_{1} such that

id_{P_{1}} − p_{0}t = sp_{1}  and  p_{1}s = id_{P_{0}} − u.

Moreover, we have the following equalities

0 = (up_{1} − W t) = (up_{1} − tW) = (u − tp_{0})p_{1}.

This gives us that (u − tp_{0}) = 0 , because no maps from Coker p_{1} to P_{0} . Finally, we obtain two morphisms t and s such that

id_{P_{1}} = p_{0}t + sp_{1}  and  id_{P_{0}} = p_{1}s + tp_{0}.

Hence, the pair \overline{P} is isomorphic to the zero object in the category DB_{0}(W) . □ Theorem 3.8. The functor F : DB_{0}(W) ⟶ D_{Sg}(X_{0}) is an exact equivalence. Proof. By Lemma 3.5 the coherent sheaves Coker p_{1} and Coker q_{1} satisfy the condition of Proposition 1.21 with N = 0 . This gives an isomorphism

Hom_{D_{Sg}(X_{0})}(Coker p_{1}, Coker q_{1}) ≅ Hom_{coh(X_{0})}(Coker p_{1}, Coker q_{1})/R

where R is the subspace of morphisms factoring through a locally free sheaf. Since Cok is full we get that the functor F is also full.

Now we prove that F is faithful. It is a standard consideration. Let f : \overline{P} → \overline{Q} be a morphism for which F (f) = 0 . Suppose that f sits in an exact triangle

\overline{P} ⟶^{f} \overline{Q} ⟶^{g} \overline{R}

Then the identity map of F \overline{Q} factors through the map F \overline{Q} ⟶^{F g} F \overline{R} . Since F is full, there is a map h : \overline{Q} → \overline{Q} factoring through g : \overline{Q} → \overline{R} such that F h = id . Hence, the cone C(h) of map h goes to zero under the functor F . By Lemma 3.7 the object C(h) is the zero object as well, so h is an isomorphism. Thus g : \overline{Q} → \overline{R} is a split monomorphism and f = 0 .

To complete the proof that F is an equivalence we need to show that every object A ∈ D_{Sg}(X_{0}) is isomorphic to F \overline{P} for some \overline{P} . By Proposition 1.23 any object A ∈ D_{Sg}(X_{0}) is isomorphic to the image of a coherent sheaf F such that \underline{Ext}^{i}(F, O_{X}) = 0 for all i > 0 . Consider an epimorphism P_{0} → F of sheaves on X with locally free P_{0} . Denote by p_{1} : P_{1} → P_{0} the kernel of this map. Since the multiplication on W gives the zero map on F , there is a map p_{0} : P_{0} → P_{1} such that p_{0}p_{1} = W and p_{1}p_{0} = W . We get a pair

\overline{P} := ( \xymatrix{P_{1} \ar[r]^{p_{1}} & P_{0} \ar[l]^{p_{0}}} )

And we need only to show that P_{1} is locally free. But it is locally free on the complement X \ X_{0} and its restriction to X_{0} (in the derived sense) is locally free sheaf because Coker p_{1} and P_{0}/W are locally free on X_{0} and there is an exact sequence

0 → Coker p_{1} → P_{1}/W → P_{0}/W → Coker p_{1} → 0.

Hence, by Lemma 2.6 P_{1} is locally free as well. □ Corollary 3.9. The category of B-branes on smooth X with a superpotensial W is equivalent to the disjoint union ∐_{w∈𝔸^{1}} D_{Sg}(X_{w}) , and this union is finite.

3.3. Some simple calculations. In this paragraph we give a description of the category of B-branes in the Landau-Ginsburg model with a superpotential W′ = z_{0}^{n} + z_{1}^{2} + ⋯ + z_{2k}^{2} which is given on ℂ^{2k+1} . This superpotential has only one singular point over 0 . By Theorem 3.8 the category of B-branes DB_{0}(W′) is equivalent to the triangulated category of singularities D_{Sg}(Y_{0}) , where Y_{0} is the fiber over 0 , i.e Y_{0} is given by the equation W′ = 0 . By Theorem 2.1 this category is equivalent to the category D_{Sg}(X_{0}) , where X_{0} = Spec(ℂ[z]/z^{n}) is the fiber over 0 of the superpotential W = z^{n} on ℂ . Thus, it is enough to describe the category D_{Sg}(X_{0}) .

Objects. Denote by A the algebra ℂ[z]/z^{n} . By Proposition 1.23 any object is coming from a finite-dimensional module M over the algebra A . Any A-module is nothing more as a vector space with an operator L such that L^{n} = 0 . Hence any module is a direct sum of the modules V_{i} for i = 1, ..., n , where V_{i} = A/z^{i} . This is the Jordan decomposition in Jordan blocks. Moreover, the module V_{n} = A is free, hence it is equal to the zero object in D_{Sg}(X_{0}) . So indecomposible objects in the category D_{Sg}(X_{0}) are

V_{1}, V_{2}, ...., V_{n−1}.

All other objects are finite direct sums of V_{μ}, μ = 1, ...n − 1 .

Morphisms. For each pair V_{μ}, V_{ν} we fix a morphism

να_{μ} : V_{μ} → V_{ν}

which is coming from the natural projection if μ ≥ ν and from the injection that sends 1 ∈ V_{μ} = A/z^{μ} to z^{ν−μ} if ν ≥ μ . All other morphisms are linear combination of compositions of να_{μ} . There are the following relations:

(7)

1) _{μ}α_{μ} = id_{μ},
2) _{ν}α_{λ}_{λ}α_{μ} = _{ν}α_{μ}  if ν ≥ λ ≥ μ or ν ≤ λ ≤ μ,
3) _{ν}α_{λ}_{λ}α_{μ} = 0  if λ ≥ μ + ν or λ + n ≤ μ + ν,
4) _{ν}α_{λ}_{λ}α_{μ} = _{ν}α_{κ}_{κ}α_{μ} if λ + κ = μ + ν.

Using this relation we can see that the space Hom(V_{μ}, V_{ν}) has a basis formed by the morphisms of the form _{ν}α_{λ}_{λ}α_{μ} , where max(μ, ν) ≤ λ < μ + ν . Denote by depth V_{μ} the integer number equals to min(μ, n − μ) . We obtain that

dim Hom(V_{μ}, V_{ν}) = min(depth V_{μ}, depth V_{ν}).

Moreover, the ring End(V_{μ}) is isomorphic to ℂ[x]/x^{d} , where d = depth V_{μ} .

Translation functor. To see the translation functor it is convinient to draw the following pictures

[figure: ∘^{V_{1}}; ∘^{V_{2}}; ⋯; ∘^{V_{k}}; ∘^{V_{1}}; ⋯; ∘^{V_{k−1}}; −; −; ∘^{V_{k}}; ∘^{V_{n−1}}; ∘^{V_{n−2}}; ⋯; ∘^{V_{k+1}}; ∘^{V_{n−1}}; ⋯; ∘^{V_{k+1}}; n=2k+1; n=2k]

The translation functor [1] is the reflection with respect to the horisonal line, i.e. it sends V_{μ} to V_{n−μ} and takes _{ν}α_{μ} to _{(n−ν)}α_{(n−μ)} . It is easy to see that all relations (7) are preserved.

Exact triangles. For the convenience, we will write V_{−μ} instead V_{n−μ} considering all integers modulo n . We also put V_{0} = 0 . At first, any triangle of the form

(8)  V_{μ} ⟶^{_{ν}α_{μ}} V_{ν} ⟶^{_{(ν−μ)}α_{ν}} V_{(ν−μ)} ⟶^{h} V_{(−μ)},

where h = _{(−μ)}α_{(ν−μ)} if ν − μ ≥ 0 and h = −_{(−μ)}α_{(ν−μ)} if ν − μ < 0 , is exact. If now f : V_{μ} → V_{ν} is another morphism which is a composition of some α’s then using the relations (7) we can represent it as _{ν}α_{λ}_{λ}α_{μ} where λ > max(μ, ν) and λ < μ + ν . In this case the triangle of the form

(9)  V_{μ} ⟶^{f} V_{ν} ⟶^{g} V_{(λ−μ)} ⊕ V_{(ν−λ)} ⟶^{h} V_{(−μ)}

is exact, where g = (_{(λ−μ)}α_{ν} ,_{(ν−λ)}α_{ν})^{t} and h = (_{(−μ)}α_{(λ−μ)}, −_{(−μ)}α_{(ν−λ)}) . Note that the triangle (8) is a particular case of the triangle (9) for λ = max(μ, ν).

All other exact triangles are isomorphic to the triangles defined above.

Acknoledgments

I am grateful to Alexei Bondal, Anton Kapustin, Ludmil Katzarkov, and Maxim Kontsevich for very useful discussions. I also thank Nikolai Tyurin for reading a preliminary draft of the paper and making a number of valuable comments. I would like to thank Max-Planck-Institut für Mathematik for the hospitality during the writing of this paper. This work is done under partial financial support of the Russian Foundation for Basic Research (grant no. 99-01-01144), grant in support of leading scientific schools (no. 00-15-96085) and grant INTAS-OPEN-2000-269. The research describing in this work was made possible in part by CRDF Award No RM1-2405-MO-02. It is also a pleasure for me to express my gratitude to the Foundation for Promotion of National Science.

Finally, I want to thank my nonformal adviser Anrei Nikolaevish Tyurin without whom this work could not be written at all.

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