Derived categories of coherent sheaves and triangulated categories of singularities

Dmitri Orlov

Algebra Section, Steklov Mathematical Institute RAN, Gubkin str. 8, Moscow

119991, RUSSIA. orlov@mi.ras.ru

Dedicated to Yuri Ivanovich Manin on the occasion of his 70th birthday

Summary. In this paper we establish an equivalence between the category of graded D-branes of type B in Landau-Ginzburg models with homogeneous superpotential W and the triangulated category of singularities of the fiber of W over zero. The main result is the theorem which shows that the graded triangulated category of singularities of the cone over a projective variety is connected via a fully faithful functor to the bounded derived category of coherent sheaves on the base of the cone. This implies that the category of graded D-branes of type B in Landau-Ginzburg models with homogeneous superpotential W is connected via a fully faithful functor to the derived category of coherent sheaves on the projective variety defined by the equation W = 0.

Introduction^{1}

With any algebraic variety X one can naturally associate two triangulated categories: the bounded derived category D^{b}(coh(X)) of coherent sheaves and the triangulated subcategory 𝔓𝔢𝔯𝔣(X) ⊂ D^{b}(coh(X)) of perfect complexes on X. If the variety X is smooth, then these two categories coincide. For singular varieties this is no longer true. In [22] we introduced a new invariant of a variety X - the triangulated category D_{Sg}(X) of the singularities of X - as the quotient of D^{b}(coh(X)) by the full subcategory of perfect complexes 𝔓𝔢𝔯𝔣(X). The category D_{Sg}(X) captures many properties of the singularities of X.

Similarly we can define a triangulated category of singularities D_{Sg}(A) for any noetherian algebra A. We set D_{Sg}(A) = D^{b}(mod−A)/𝔓𝔢𝔯𝔣(A), where

1This work was done with a partial financial support from the Weyl Fund, from grant RFFI 05-01-01034, from grant CRDF Award No RUM1-2661-MO-05.

D^{b}(mod−A) is the bounded derived category of finitely generated right Amodules and 𝔓𝔢𝔯𝔣(A) is its triangulated subcategory consisting of objects which are quasi-isomorphic to bounded complexes of projectives. We will again call 𝔓𝔢𝔯𝔣(A) the subcategory of perfect complexes, but usually we will write D^{b}(proj−A) instead of 𝔓𝔢𝔯𝔣(A) since this category can also be identified with the derived category of the exact category proj−A of finitely generated right projective A-modules (see, e.g. [19]).

The investigation of triangulated categories of singularities is not only connected with a study of singularities but is mainly inspired by the Homological Mirror Symmetry Conjecture [20]. More precisely, the objects of these categories are directly related to D-branes of type B (B-branes) in Landau-Ginzburg models. Such models arise as a mirrors to Fano varieties [15]. For Fano varieties one has the derived categories of coherent sheaves (B-branes) and given a symplectic form one can propose a suitable Fukaya category (Abranes). Mirror symmetry should interchange these two classes of D-branes. Thus, to extend the Homological Mirror Symmetry Conjecture to the Fano case, one should describe D-branes in Landau-Ginzburg models.

To specify a Landau-Ginzburg model in general one needs to choose a target space X, and a holomorphic function W on X called a superpotential. The B-branes in the Landau-Ginzburg model are defined as W-twisted ℤ_{2}- periodic complexes of coherent sheaves on X. These are chains {⋯ →^{d} P_{0} →^{d} P_{1} →^{d} P_{0} →^{d} P_{1} →^{d} P_{0} ⋯ } of coherent sheaves in which the composition of differentials is no longer zero, but is equal to multiplication by W (see, e.g. [17, 22, 23]). In the paper [22] we analysed the relationship between the categories of B-branes in Landau-Ginzburg models and triangulated categories of singularities. Specifically, we showed that for an affine X the product of the triangulated categories of singularities of the singular fibres of W is equivalent to the category of B-branes of (X, W).

In this paper we consider the graded case. Let A = ⨁_{i} A_{i} be a graded noetherian algebra over a field k. We can define the triangulated category of singularities D_{Sg}^{gr}(A) of A as the quotient D^{b}(gr−A)/D^{b}(grproj−A), where D^{b}(gr−A) is the bounded derived category of finitely generated graded right A-modules and D^{b}(grproj−A) is its triangulated subcategory consisting of objects which are isomorphic to bounded complexes of projectives.

The graded version of the triangulated category of singularities is closely related to the category of B-branes in Landau-Ginzburg models (X, W) equipped with an action of the multiplicative group k^{∗} for which W is semiinvariant. The notion of grading on D-branes of type B was defined in the papers [16, 28]. In the presence of a k^{∗}-action one can construct a category of graded B-branes in the Landau-Ginzburg model (X, W) (Definition 30 and Subsection 3.3). Now our Theorem 39 gives an equivalence between the category of graded B-branes and the triangulated category of singularities D_{Sg}^{gr}(A), where A is such that Spec(A) is the fiber of W over 0.

This equivalence allows us to compare the category of graded B-branes and the derived category of coherent sheaves on the projective variety which is defined by the superpotential W. For example, suppose X is the affine space 𝔸^{N} and W is a homogeneous polynomial of degree d. Denote by Y ⊂ ℙ^{N−1} the projective hypersurface of degree d which is given by the equation W = 0. If d = N, then the triangulated category of graded B-branes DGrB(W) is equivalent to the derived category of coherent sheaves on the Calabi-Yau variety Y. Furthermore, if d < N (i.e. Y is a Fano variety), we construct a fully faithful functor from DGrB(W) to D^{b}(coh(Y)), and, if d > N (i.e. Y is a variety of general type), we construct a fully faithful functor from D^{b}(coh(Y)) to DGrB(W) (see Theorem 40).

This result follows from a more general statement for graded Gorenstein algebras (Theorem 16). It gives a relation between the triangulated category of singularities D_{Sg}^{gr}(A) and the bounded derived category D^{b}(qgr A), where qgr A is the quotient of the abelian category of graded finitely generated Amodules by the subcategory of torsion modules. More precisely, for Gorenstein algebras we obtain a fully faithful functor between D_{Sg}^{gr}(A) and D^{b}(qgr A), and the direction of this functor depends on the Gorenstein parameter a of A. In particular, when the Gorenstein parameter a is equal to zero, we obtain an equivalence between these categories. Finally, the famous theorem of Serre, which identifies D^{b}(qgr A) with D^{b}(coh(Proj (A))) when A is generated by its first component, allows to apply this result to geometry.

I am grateful to Alexei Bondal, Anton Kapustin, Ludmil Katzarkov, Alexander Kuznetsov, Tony Pantev and Johannes Walcher for very useful discussions.

1 Triangulated categories of singularities for graded algebras.

1.1 Localization in triangulated categories and semiorthogonal decomposition.

Recall that a triangulated category D is an additive category equipped with the additional data:

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

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

X ⟶^{u} Y ⟶^{v} Z ⟶^{w} X[1],

which must satisfy a certain set of axioms (see [27], also [12, 19, 21]).

A functor F : D ⟶ D′ between two triangulated categories is called exact if it commutes with the translation functors, i.e. F ∘ [1] ≅ [1] ∘ F, and transforms exact triangles into exact triangles.

With any pair N ⊂ D, where N is a full triangulated subcategory, in a triangulated category D, we can associate the quotient category D/N. To construct it let us denote by Σ(N) a class of morphisms s in D fitting into an exact triangle

X ⟶^{s} Y ⟶ N ⟶ X[1]

with N ∈ N. It can be checked that Σ(N) is a multiplicative system. Define the quotient D/N as the localization D[Σ(N)^{−1}] (see [10, 12, 27]). It is a triangulated category. The translation functor on D/N is induced from the translation functor in the category D, and the exact triangles in D/N are the triangles isomorphic to the images of exact triangles in D. The quotient functor Q : D ⟶ D/N annihilates N. Moreover, any exact functor F : D ⟶ D′ between triangulated categories, for which F (X) ≃ 0 when X ∈ N , factors uniquely through Q. The following lemma is obvious.

Lemma 1. Let N and N′ be full triangulated subcategories of triangulated categories D and D′ respectively. Let F : D → D′ and G : D′ → D be an adjoint pair of exact functors such that F (N) ⊂ N′ and G(N′) ⊂ N. Then they induce functors

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

which are adjoint as well. Moreover, if the functor F : D → D′ is fully faithful, then the functor \overline{F} : D/N ⟶ D′/N′ is also fully faithful.

Now recall some definitions and facts concerning admissible subcategories and semiorthogonal decompositions (see [7, 8]). Let N ⊂ D be a full triangulated subcategory. The right orthogonal to N is the full subcategory N^{⊥} ⊂ D consisting of all objects M such that Hom(N, M) = 0 for any N ∈ N. The left orthogonal ^{⊥}N is defined analogously. The orthogonals are also triangulated subcategories.

Definition 2. Let I : N ↪ D be an embedding of a full triangulated subcategory N in a triangulated category D. We say that N is right admissible (respectively left admissible) if there is a right (respectively left) adjoint functor Q : D → N . The subcategory N will be called admissible if it is right and left admissible.

Remark 3. For the subcategory N the property of being right admissible is equivalent to requiring that for each X ∈ D there is an exact triangle N → X → M, with N ∈ N , M ∈ N^{⊥}.

Lemma 4. Let N be a full triangulated subcategory in a triangulated category D. If N is right (respectively left) admissible, then the quotient category D/N is equivalent to N^{⊥} (respectively ^{⊥}N). Conversely, if the quotient functor Q : D ⟶ D/N has a left (respectively right) adjoint then D/N is equivalent to N^{⊥} (respectively ^{⊥}N).

If N ⊂ D is a right admissible subcategory, then we say that the category D has a weak semiorthogonal decomposition ⟨N^{⊥}, N ⟩. Similarly if N ⊂ D is a left admissible subcategory, we say that D has a weak semiorthogonal decomposition ⟨N ,^{⊥}N ⟩.

Definition 5. A sequence of full triangulated subcategories (N_{1}, …, N_{n}) in a triangulated category D will be called a weak semiorthogonal decomposition of D if there is a sequence of left admissible subcategories D_{1} = N_{1} ⊂ D_{2} ⊂ ⋯ ⊂ D_{n} = D such that N_{p} is left orthogonal to D_{p−1} in D_{p}. We will write D = ⟨N_{1}, …, N_{n}⟩. If all N_{p} are admissible in D then the decomposition D = ⟨N_{1}, …, N_{n}⟩ is called semiorthogonal.

The existence of a semiorthogonal decomposition on a triangulated category D clarifies the structure of D. In the best scenario, one can hope that D has a semiorthogonal decomposition D = ⟨N_{1}, …, N_{n}⟩ in which each elementary constituent N_{p} is as simple as possible, i.e. is equivalent to the bounded derived category of finite dimensional vector spaces.

Definition 6. An object E of a k-linear triangulated category T is called exceptional if Hom(E, E[p]) = 0 when p ≠ 0, and Hom(E, E) = k. An exceptional collection in T is a sequence of exceptional objects (E_{0}, …, E_{n}) satisfying the semiorthogonality condition Hom(E_{i}, E_{j}[p]) = 0 for all p when i > j.

If a triangulated category D has an exceptional collection (E_{0}, …, E_{n}) which generates the whole D then we say that the collection is full. In this case D has a semiorhtogonal decomposition with N_{p} = ⟨E_{p}⟩. Since E_{p} is exceptional each of these categories is equivalent to the bounded derived category of finite dimensional vector spaces. In this case we write D = ⟨E_{0}, …, E_{n}⟩.

Definition 7. An exceptional collection (E_{0}, …, E_{n}) is called strong if, in addition, Hom(E_{i}, E_{j}[p]) = 0 for all i and j when p ≠ 0.

1.2 Triangulated categories of singularities for algebras.

Let A = ⨁_{i≥0} A_{i} be a noetherian graded algebra over a field k. Denote by

mod−A and gr−A the category of finitely generated right modules and the category of finitely generated graded right modules respectively. Note that morphisms in gr−A are homomorphisms of degree zero. These categories are abelian. We will also use the notation Mod−A and Gr−A for the abelian categories of all right modules and all graded right modules and we will often omit the prefix ”right”. Left A-modules are will be viewed as right A^{∘}-modules and A−B bimodules as right A^{∘}−B-modules, where A^{∘} is the opposite algebra.

The twist functor (p) on the category gr−A is defined as follows: it takes a graded module M = ⊕_{i} M_{i} to the module M(p) for which M(p)_{i} = M_{p+i} and takes a morphism f : M ⟶ N to the same morphism viewed as a morphism between the twisted modules f(p) : M(p) ⟶ N(p).

Consider the bounded derived categories D^{b}(gr−A) and D^{b}(mod−A). They can be endowed with natural structures of triangulated categories. The categories D^{b}(gr−A) and D^{b}(mod−A) have full triangulated subcategories consisting of objects which are isomorphic to bounded complexes of projectives. These subcategories can also be considered as the derived categories of the exact categories of projective modules D^{b}(grproj−A) and D^{b}(proj−A) respectively (see, e.g. [19]). They will be called the subcategories of perfect complexes. Observe also that the category D^{b}(gr−A) (respectively D^{b}(mod−A)) is equivalent to the category D_{gr−A}^{b}(Gr−A) (respectively D_{mod−A}^{b}(Mod−A)) of complexes of arbitrary modules with finitely generated cohomologies (see [5]). We will tacitly use this equivalence throughout our considerations.

Definition 8. We define triangulated categories of singularities D_{Sg}^{gr}(A) and D_{Sg}(A) as the quotient D^{b}(gr−A)/D^{b}(grproj−A) and D^{b}(mod−A)/D^{b}(proj−A) respectively.

Remark 9. As in the commutative case [22, 23], the triangulated categories of singularities D_{Sg}^{gr}(A) and D_{Sg}(A) will be trivial if A has finite homological dimension. Indeed, in this case any A-module has a finite projective resolution, i.e. the subcategories of perfect complexes coincide with the full bounded derived categories of finitely generated modules.

Homomorphisms of (graded) algebras f : A → B induce functors between the associated derived categories of singularities. Furthermore, if B has a fi-

nite Tor-dimension as an A-module then we get the functor ⊗^{L}_{A} B between the bounded derived categories of finitely generated modules which maps perfect complexes to perfect complexes. Therefore, we get functors between triangulated categories of singularities

⊗^{L}_{A} B : D_{Sg}^{gr}(A) ⟶ D_{Sg}^{gr}(B) and ⊗^{L}_{A} B : D_{Sg}(A) ⟶ D_{Sg}(B).

If, in addition, B is finitely generated as an A-module, then these functors have right adjoints induced from the functor which sends a complex of B-modules to itself considered as a complex of A-modules.

More generally, suppose _{A}\underline{M}_{B} is a complex of graded A − B bimodules which as a complex of graded B-modules is quasi-isomorphic to a perfect complex. Suppose that _{A}\underline{M} has a finite Tor-amplitude as a left A-module. Then

we can define the derived tensor product functor ⊗^{L}_{A} \underline{M}_{B} : D^{b}(gr−A) ⟶ D^{b}(gr−B). Moreover, since \underline{M}_{B} is perfect over B this functor sends perfect complexes to perfect complexes. Therefore, we get an exact functor

⊗^{L}_{A} \underline{M}_{B} : D_{Sg}^{gr}(A) ⟶ D_{Sg}^{gr}(B).

In the ungraded case we also get the functor ⊗^{L}_{A} \underline{M}_{B} : D_{Sg}(A) ⟶ D_{Sg}(B).

1.3 Morphisms in categories of singularities.

In general, it is not easy to calculate spaces of morphisms between objects in a quotient category. The following lemma and proposition provide some information about the morphism spaces in triangulated categories of singularities.

Lemma 10. For any object T ∈ D_{Sg}^{gr}(A) (respectively T ∈ D_{Sg}(A)) and for any sufficiently large k, there is a module M ∈ gr−A (respectively M ∈ mod−A), depending on T and k, and such that T is isomorphic to the image of M[k] in the triangulated category of singularities. If, in addition, the algebra A has finite injective dimension, then for any sufficiently large k the corresponding module M satisfies Ext_{A}^{i}(M, A) = 0 for all i > 0. Proof. The object T is represented by a bounded complex of modules \underline{T} . Choose a bounded above projective resolution \underline{P} →^{∼} \underline{T} and a sufficiently large k ≫ 0. Consider the stupid truncation σ^{≥−k+1}\underline{P} of \underline{P} . Denote by M the cohomology module H^{−k+1}(σ^{≥−k+1}\underline{P} ). Clearly T ≅ M[k] in D_{Sg}^{gr}(A) (respectively D_{Sg}(A)).

If now A has finite injective dimension, then morphism spaces Hom(\underline{T}^{·}, A[i]) in D^{b}(gr−A) (respectively D^{b}(mod−A)) are trivial for all but finitely many i ∈ ℤ. So if M corresponds to T and a sufficiently large k, then we will have Ext_{A}^{i}(M, A) = 0 for all i > 0. □

Proposition 11. Let M be an A-module such that Ext_{A}^{i}(M, A) = 0 for all i > 0. Then for any A-module N we have

Hom_{D_{Sg}(A)}(M, N) ≅ Hom_{A}(M, N)/R

where R is the subspace of elements factoring through a projective, i.e. e ∈ R iff e = βα with α : M → P and β : P → N, where P is projective. If M is a graded module, then for any graded A-module N

Hom_{D_{Sg}^{gr}(A)}(M, N) ≅ Hom_{gr−A}(M, N)/R.

Proof. We will only discuss the graded case. By the definition of localization any morphism from M to N in D_{Sg}^{gr}(A) can be represented by a pair

M ⟶^{a} \underline{T} ⟵^{s} N,  (1)

of morphisms in D^{b}(gr−A), such that the cone \underline{C} (s) is a perfect complex. Consider a bounded above projective resolution \underline{Q} → N and its stupid truncation σ^{≥−k}\underline{Q} for sufficiently large k. There is an exact triangle

E[k] ⟶ σ^{≥−k}\underline{Q} ⟶ N ⟶^{s′} E[k + 1],

where E denotes the module H^{−k}(σ^{≥−k}\underline{Q} ). Choosing k to be sufficiently large we can guarantee that Hom(\underline{C} (s), E[i]) = 0 for all i > k. Using the triangle

\underline{C} (s)[−1] ⟶ N ⟶^{s} \underline{T} ⟶ \underline{C} (s),

we find that the map s′ : N → E[k + 1] can be lifted to a map \underline{T} → E[k + 1]. The map \underline{T} → E[k + 1] induces a pair of the form

M ⟶^{a′} E[k + 1] ⟵^{s′} N,  (2)

and this pair gives the same morphism in D_{Sg}(A) as the pair (1). Since Ext^{i}(M, P) = 0 for all i > 0 and any projective module P, we obtain

Hom(M, (σ^{≥−k}\underline{Q} )[1]) = 0.

Hence, the map a′ : M → E[k + 1] can be lifted to a map f which completes the diagram

\xymatrix{M \ar[rr]^{f} \ar[dr]_{a′} &  & N \ar[dl]^{s′} \\  & E[k + 1]}

Thus, the map f is equivalent to the map (2) and, as consequence, to the map (1). Hence, any morphism from M to N in D_{Sg}^{gr}(A) is represented by a morphism from M to N in the category D^{b}(gr−A).

Now if f is the 0-morphism in D_{Sg}^{gr}(A), then without a loss of generality we can assume that the map a is the zero map. In this case we will have a′ = 0 as well. This implies that f factors through a morphism M → σ^{≥−k}\underline{Q} . By the assumption on M any such morphism can be lifted to a morphism M → Q^{0}. Hence, if f is the 0-morphism in D_{Sg}^{gr}(A) then it factors through Q^{0}. The same proof works in ungraded case (see [22]). □

Next we describe a useful construction utilizing the previous statements. Let \underline{M} and \underline{N} be two bounded complexes of (graded) A-modules. Assume that Hom(\underline{M} , A[i]) in the bounded derived categories of A-modules are trivial except for a finite number of i ∈ ℤ. By Lemma 10 for sufficiently large k there are modules M, N ∈ gr−A (resp. M, N ∈ mod−A) such that \underline{M} and \underline{N} are isomorphic to the images of M[k] and N[k] in the triangulated category of singularities. Moreover it follows immediately from the assumption on \underline{M} and the construction of M, that for any sufficiently large k we have Ext_{A}^{i}(M, A) = 0 whenever i > 0. Hence, by Proposition 11, we get

Hom_{D_{Sg}^{gr}(A)}(\underline{M} , \underline{N} ) ≅ Hom_{D_{Sg}^{gr}(A)}(M, N) ≅ Hom_{A}(M, N)/R

where R is the subspace of elements factoring through a projective module. This procedure works in the ungraded situation as well.

2 Categories of coherent sheaves and categories of singularities.

2.1 Quotient categories of graded modules.

Let A = ⨁_{i≥0} A_{i} be a noetherian graded algebra. We suppose that A is con-

nected, i.e. A_{0} = k. Denote by tors−A the full subcategory of gr−A which consists of all graded A–modules which are finite dimensional over k.

An important role will be played by the quotient abelian category qgr A = gr−A/ tors−A. It has the following explicit description. The objects of qgr A are the objects of gr−A (we denote by πM the object in qgr A which corresponds to a module M). The morphisms in qgr A are given by

Hom_{qgr}(πM, πN) := lim_{M′^{⟶}} Hom_{gr} (M′, N)  (3)

where M′ runs over submodules of M such that M/M′ is finite dimensional.

Given a graded A-module M and an integer p, the graded A-submodule ⨁_{i≥p} M_{i} of M is denoted by M_{≥p} and is called the p-th tail of M. In the same way we can define the p-th tail \underline{M}_{≥p} of any complex of modules \underline{M} . Since A is noetherian, we have

Hom_{qgr}(πM, πN) = lim_{p→∞} Hom_{gr} (M_{≥p}, N).

We will also identify M_{p} with the quotient M_{≥p}/M_{≥p+1}.

Similarly, we can consider the subcategory Tors−A ⊂ Gr−A of torsion modules. Recall that a module M is called torsion if for any element x ∈ M one has xA_{≥p} = 0 for some p. Denote by QGr A the quotient category Gr−A/ Tors−A. The category QGr A contains qgr A as a full subcategory. Sometimes it is convenient to work in QGr A instead of qgr A.

Denote by Π and π the canonical projections of Gr−A to QGr A and of gr−A to qgr A respectively. The functor Π has a right adjoint Ω and, moreover, for any N ∈ Gr−A

ΩΠN ≅ ⨁_{n=−∞}^{∞} Hom_{QGr}(ΠA, ΠN(n)).  (4)

For any i ∈ ℤ we can consider the full abelian subcategories Gr−A_{≥i} ⊂ GrA and gr−A_{≥i} ⊂ grA which consist of all modules M such that M_{p} = 0 when p < i. The natural projection functor Π_{i} : Gr−A_{≥i} ⟶ QGr −A has a right adjoint Ω_{i} satisfying

Ω_{i}Π_{i}N ≅ ⨁_{n=i}^{∞} Hom_{QGr}(ΠA, Π_{i}N(n)).

Since the category QGr A is an abelian category with enough injectives there is a right derived functor

RΩ_{i} : D^{+}(QGr A) ⟶ D^{+}(Gr−A_{≥i})

defined as

RΩ_{i}M ≅ ⨁_{k=i}^{∞} R Hom_{QGr}(ΠA, M(k)).  (5)

Assume now that the algebra A satisfies condition ”χ” from [1, Sec. 3]. We recall that by definition a connected noetherian graded algebra A satisfies condition ”χ” if for every M ∈ gr−A the grading on the space Ext_{A}^{i}(k, M) is right bounded for all i. In this case it was proved in [1, Prop. 3.14] that the restrictions of the functors Ω_{i} to the subcategory qgr A give functors ω_{i} : qgr A ⟶ gr−A_{≥i} which are right adjoint to π_{i}. Moreover, it follows from [1, Th. 7.4] that the functor ω_{i} has a right derived

Rω_{i} : D^{+}(qgr A) ⟶ D^{+}(gr−A_{≥i})

and all R^{j}ω_{i} ∈ tors−A for j > 0.

If, in addition, the algebra A is Gorenstein (i.e. if it has a finite injective dimension n and D(k) = R Hom_{A}(k, A) is isomorphic to k(a)[−n]) we obtain the right derived functor

Rω_{i} : D^{b}(qgr A) ⟶ D^{b}(gr−A_{≥i})

between bounded derived categories (see [30, Cor. 4.3]). It is important to note that the functor Rω_{i} is fully faithful because π_{i}Rω_{i} is isomorphic to the identity functor ([1, Prop. 7.2]).

2.2 Triangulated categories of singularities for Gorenstein algebras

The main goal of this section is to establish a connection between the triangulated category of singularities D_{Sg}^{gr}(A) and the derived category D^{b}(qgr A), in the case of a Gorenstein algebra A.

When the algebra A has finite injective dimension as right and as left module over itself ( i.e. A is a dualizing complex for itself) we get two functors

D := R Hom_{A} (−, A) : D^{b}(gr−A )^{∘} ⟶ D^{b}(gr−A^{∘}),  (6)
D^{∘} := R Hom_{A^{∘}}(−, A) : D^{b}(gr−A^{∘})^{∘} ⟶ D^{b}(gr−A ),  (7)

which are quasi-inverse triangulated equivalences (see [29, Prop. 3.5]).

Definition 12. We say that a connected graded noetherian algebra A is Gorenstein if it has a finite injective dimension n and D(k) = R Hom_{A}(k, A) is isomorphic to k(a)[−n] for some integer a, which is called the Gorenstein parameter of A. (Such algebra is also called AS-Gorenstein, where ”AS” stands for ”Artin-Schelter”.)

Remark 13. It is known (see [30, Cor. 4.3]) that any Gorenstein algebra satisfies condition ”χ” and for any Gorenstein algebra A and for any i ∈ ℤ we have derived functors

Rω_{i} : D^{b}(qgr A) ⟶ D^{b}(gr−A_{≥i})

which are fully faithful.

Now we describe the images of the functors Rω_{i}. Denote by D_{i} the subcategories of D^{b}(gr−A) which are the images of the composition of Rω_{i} and the natural inclusion of D^{b}(gr−A_{≥i}) to D^{b}(gr−A). All D_{i} are equivalent to D^{b}(qgr −A). Further, for any integer i denote by S_{<i}(A) (or simple S_{<i}) the full triangulated subcategory of D^{b}(gr−A) generated by the modules k(e) with e > −i. In other words, the objects of S_{<i} are complexes \underline{M} for which the tail \underline{M}_{≥i} is isomorphic to zero. Analogously, we define S_{≥i} as the triangulated subcategory which is generated by the objects k(e) with e ≤ −i. In other words, the objects of S_{≥i} are complexes of torsion modules from gr−A_{≥i}. It is clear that S_{<i} ≅ S_{<0}(−i) and S_{≥i} ≅ S_{≥0}(−i).

Furthermore, denote by P_{<i} the full triangulated subcategory of D^{b}(gr−A) generated by the free modules A(e) with e > −i and denote by P_{≥i} the triangulated subcategory which is generated by the free modules A(e) with e ≤ −i. As above we have P_{<i} ≅ P_{<0}(−i) and P_{≥i} ≅ P_{≥0}(−i).

Lemma 14. Let A = ⨁_{i≥0} A_{i} be a connected graded noetherian algebra. Then

the subcategories S_{<i} and P_{<i} are left and respectively right admissible for any i ∈ ℤ. Moreover, there are weak semiorthogonal decompositions

D^{b}(gr−A) = ⟨S_{<i}, D^{b}(gr−A_{≥i})⟩,  D^{b}(tors−A) = ⟨S_{<i}, S_{≥i}⟩,  (8)
D^{b}(gr−A) = ⟨D^{b}(gr−A_{≥i}), P_{<i}⟩,  D^{b}(grproj−A) = ⟨P_{≥i}, P_{<i}⟩. (9)

Proof. For any complex \underline{M} ∈ D^{b}(mod−A) there is an exact triangle of the form

\underline{M}_{≥i} ⟶ \underline{M} ⟶ \underline{M} /\underline{M}_{≥i}.

By definition the object \underline{M} /\underline{M}_{≥i} belongs to S_{<i} and the object \underline{M}_{≥i} is in the left orthogonal ^{⊥}S_{<i}. Hence, by Remark 3, S_{<i} is left admissible. Moreover, \underline{M}_{≥i} also belongs to D^{b}(gr−A_{≥i}), i.e. D^{b}(gr−A_{≥i}) ≅ ^{⊥}S_{<i} in the category D^{b}(gr−A). If \underline{M} is a complex of torsion modules then \underline{M}_{≥i} belongs to S_{≥i}. Thus, we obtain both decompositions of (8).

To prove the existence of the decompositions (9) we first note that, due to the connectedness of A, any finitely generated graded projective A-module is free. Second, any finitely generated free module P has a canonical split decomposition of the form

0 ⟶ P_{<i} ⟶ P ⟶ P_{≥i} ⟶ 0,

where P_{<i} ∈ P_{<i} and P_{≥i} ∈ P_{≥i}. Third, any bounded complex of finitely generated A-modules \underline{M} has a bounded above free resolution \underline{P} → \underline{M} such that P^{−k} ∈ P_{≥i} for all k ≫ 0. This implies that the object \underline{P} _{<i} ∈ P_{<i} from the exact sequence of complexes

0 ⟶ \underline{P} _{<i} ⟶ \underline{P} ⟶ \underline{P} _{≥i} ⟶ 0,

is a bounded complex. Since \underline{P} is quasi-isomorphic to a bounded complex, the complex \underline{P} _{≥i} is also quasi-isomorphic to some bounded complex \underline{K} from D^{b}(gr−A_{≥i}). Thus, any object \underline{M} has a decomposition

\underline{P} _{<i} ⟶ \underline{M} ⟶ \underline{K} ,

where \underline{P} _{<i} ∈ P_{<i} and \underline{K} ∈ D^{b}(gr−A_{≥i}). This proves the decompositions (9). □

Lemma 15. Let A = ⨁_{i≥0} A_{i} be a connected graded noetherian algebra which

is Gorenstein. Then the subcategories S_{≥i} and P_{≥i} are right and respectively left admissible. Moreover, for any i ∈ ℤ there are weak semiorthogonal decompositions

D^{b}(gr−A_{≥i}) = ⟨D_{i}, S_{≥i}⟩, D^{b}(gr−A_{≥i}) = ⟨P_{≥i}, T_{i}⟩,  (10)

where the subcategory D_{i} is equivalent to D^{b}(qgr A) under the functor Rω_{i}, and T_{i} is equivalent to D_{Sg}^{gr}(A).

Proof. The functor Rω_{i} is fully faithful and has the left adjoint π_{i}. Thus, we obtain a semiorthogonal decomposition

D^{b}(gr−A_{≥i}) = ⟨D_{i},^{⊥}D_{i}⟩,

where D_{i} ≅ D^{b}(qgr A). Furthermore, the orthogonal^{⊥}D_{i} consists of all objects \underline{M} satisfying π_{i}(\underline{M} ) = 0. Thus, ^{⊥}D_{i} coincides with S_{≥i}. Hence, S_{≥i} is right admissible in D^{b}(gr−A_{≥i}) which is right admissible in whole D^{b}(gr−A). This implies that S_{≥i} is right admissible in D^{b}(gr−A) as well.

The functor D from (6) establishes an equivalence of the subcategory P_{≥i}(A)^{∘} with the subcategory P_{<−i+1}(A^{∘}) which is right admissible by Lemma 14. Hence, P_{≥i}(A) is left admissible and there is a decomposition of the form

D^{b}(gr−A_{≥i}) = ⟨P_{≥i}, T_{i}⟩

with some T_{i}.

Now applying Lemma 1 to the full embedding of D^{b}(gr−A_{≥i}) to D^{b}(gr−A) and using Lemma 4 we get a fully faithful functor from T_{i} ≅ D^{b}(gr−A_{≥i})/P_{≥i} to D_{Sg}^{gr}(A) = D^{b}(gr−A)/D^{b}(grproj−A). Finally, since this functor is essentially surjective on objects it is actually an equivalence. □

Theorem 16. Let A = ⨁_{i≥0} A_{i} be a connected graded noetherian algebra which

is Gorenstein with Gorenstein parameter a. Then the triangulated categories D_{Sg}^{gr}(A) and D^{b}(qgr A) are related as follows:

(i) if a > 0, there are fully faithful functors Φ_{i} : D_{Sg}^{gr}(A) ⟶ D^{b}(qgr A) and semiorthogonal decompositions

D^{b}(qgr A) = ⟨πA(−i − a + 1), …, πA(−i), Φ_{i}D_{Sg}^{gr}(A)⟩,

where π : D^{b}(gr−A) ⟶ D^{b}(qgr A) is the natural projection; (ii) if a < 0, there are fully faithful functors Ψ_{i} : D^{b}(qgr A) ⟶ D_{Sg}^{gr}(A) and

semiorthogonal decompositions

D_{Sg}^{gr}(A) = ⟨qk(−i), …, qk(−i + a + 1), Ψ_{i}D^{b}(qgr A)⟩,

where q : D^{b}(gr−A) ⟶ D_{Sg}^{gr}(A) is the natural projection; (iii)if a = 0, there is an equivalence D_{Sg}^{gr}(A) ⟶^{∼} D^{b}(qgr A).

Proof. Lemmas 14 and 15 gives us that the subcategory T_{i} is admissible in D^{b}(gr−A) and the right orthogonal T_{i}^{⊥} has a weak semiorthogonal decomposition of the form

T_{i}^{⊥} = ⟨S_{<i}, P_{≥i}⟩.  (11)

Now let us describe the right orthogonal to the subcategory D_{i}. First, since A is Gorenstein the functor D takes the subcategory S_{≥i}(A) to the subcategory S_{<−i−a+1}(A^{∘}). Hence, D sends the right orthogonal S_{≥i}^{⊥}(A) to the left orthogonal^{⊥}S_{<−i−a+1}(A^{∘}) which coincides with the right orthogonal P_{<−i−a+1}^{⊥}(A^{∘}) by Lemma 14. Therefore, the subcategory S_{≥i}^{⊥} coincides with ^{⊥}P_{≥i+a}. On the other hand, by Lemmas 14 and 15 we have that

⊥P_{≥i+a} = S_{≥i}^{⊥} ≅ ⟨S_{<i}, D_{i}⟩.

This implies that the right orthogonal D_{i}^{⊥} has the following decomposition

D_{i}^{⊥} = ⟨P_{≥i+a}, S_{<i}⟩.  (12)

Assume that a ≥ 0. In this case, the decomposition (12) is not only semiorthogonal but is in fact mutually orthogonal, because P_{≥i+a} ⊂ D^{b}(gr−A_{≥i}). Hence, we can interchange P_{≥i+a} and S_{<i}, i.e.

D_{i}^{⊥} = ⟨S_{<i}, P_{≥i+a}⟩.

Thus, we obtain that D_{i}^{⊥} ⊂ T_{i}^{⊥} and, consequently, T_{i} is a full subcategory of D_{i}. Moreover, we can describe the right orthogonal to T_{i} in D_{i}. Actually, there is a decomposition

P_{≥i} = ⟨P_{≥i+a}, P_{i}^{a}⟩,

where P_{i}^{a} is the subcategory generated by the modules A(−i−a+1), …, A(−i). Moreover, these modules form an exceptional collection. Thus, the category D_{i} has the semiorthogonal decomposition

D_{i} = ⟨A(−i − a + 1), …, A(−i), T_{i}⟩,

Since D_{i} ≅ D^{b}(qgr A) and T_{i} ≅ D_{Sg}^{gr}(A) we obtain the decomposition

D^{b}(qgr A) ≅ ⟨πA(−i − a + 1), …, πA(−i), Φ_{i}D_{Sg}^{gr}(A)⟩,

where the fully faithful functor Φ_{i} is the composition D_{Sg}^{gr}(A) →^{∼} T_{i} ↪

D^{b}(gr−A) →^{π} D^{b}(qgr A).

Assume now that a ≤ 0. In this case, the decomposition (11) is not only semiorthogonal but is in fact mutually orthogonal, because the algebra A is Gorenstein and R Hom_{A}(k, A) = k(a)[−n] with a ≤ 0. Hence, we can interchange P_{≥i} and S_{<i}, i.e.

T_{i}^{⊥} = ⟨P_{≥i}, S_{<i}⟩.

Now we see that T_{i}^{⊥} ⊂ D_{i−a}^{⊥} and, consequently, D_{i−a} is the full subcategory of T_{i}. Moreover, we can describe the right orthogonal to D_{i−a} in T_{i}. Actually, there is a decomposition of the form

S_{<i−a} = ⟨S_{<i}, k(−i), …, k(−i + a + 1)⟩.

Therefore, the category T_{i} ≅ D_{Sg}^{gr}(A) has a semiorthogonal decomposition of the form

T_{i} = ⟨k(−i), …, k(−i + a + 1), D_{i−a}⟩,  (13)

Since D_{i−a} ≅ D^{b}(qgr A) and T_{i} ≅ D_{Sg}^{gr}(A) we obtain the decomposition

D_{Sg}^{gr}(A) ≅ ⟨qk(−i), …, qk(−i + a + 1), Ψ_{i}D^{b}(qgr A)⟩,

where the fully faithful functor Ψ_{i} can be defined as the composition D^{b}(qgr A) →^{∼} D_{i−a} ↪ D^{b}(gr−A) →^{q} D_{Sg}^{gr}(A). If a = 0, then we get equivalence. □

Remark 17. It follows from the construction that the functor Ψ_{i+a} from the bounded derived category D^{b}(qgr A) to D_{Sg}^{gr}(A) is the composition of the functor Rω_{i} : D^{b}(qgr A) ⟶ D^{b}(gr−A_{≥i}), which is given by formula (5), natural embedding D^{b}(gr−A_{≥i}) ↪ D^{b}(gr−A), and the projection D^{b}(gr−A) →^{q} D_{Sg}^{gr}(A).

Let us consider two limiting cases. The first case is when the algebra A has finite homological dimension. In this case the triangulated category of singularities D_{Sg}^{gr}(A) is trivial and, hence, the Gorenstein parameter a is nonnegative and the derived category D^{b}(qgr A) has a full exceptional collection σ = (πA(0), …, πA(a − 1)). More precisely we have the following:

Corollary 18. Let A = ⨁_{i≥0} A_{i} be a connected graded noetherian algebra which

is Gorenstein with Gorenstein parameter a. Suppose that A has finite homological dimension. Then, a ≥ 0 and the derived category D^{b}(qgr A) has a full strong exceptional collection σ = (πA(0), …, πA(a − 1)) . Moreover, the category D^{b}(qgr A) is equivalent to the derived category D^{b}(mod− Q(A)) of finite

(right) modules over the algebra Q(A) := End_{gr−A} ( ⨁_{i=0}^{a−1} A(i)) of homomor-

phisms of σ. Proof. Since A has finite homological dimension the category D_{Sg}^{gr}(A) is trivial. By Theorem 16 we get that a ≥ 0 and that D^{b}(qgr A) has a full exceptional

collection σ = (πA(0), …, πA(a − 1)) . Consider the object P_{σ} = ⨁_{i=0}^{a−1} πA(i)

and the functor

Hom(P_{σ}, −) : qgr A ⟶ mod− Q(A),

where Q(A) = End_{qgr A} ( ⨁_{i=0}^{a−1} πA(i)) = End_{gr−A} ( ⨁_{i=0}^{a−1} A(i)) is the algebra

of homomorphisms of the exceptional collection σ. It is easy to see that this functor has a right derived functor

R Hom(P_{σ}, −) : D^{b}(qgr A) ⟶ D^{b}(mod− Q(A))

(e.g. as a composition Rω_{0} and Hom ( ⨁_{i=0}^{a−1} A(i), −)). The standard reasoning

(see e.g. [6] or [7]) now shows that the functor R Hom(P_{σ}, −) is an equivalence. □

Example 19. As an application we obtain a well-known result (see [4]) asserting the existence of a full exceptional collection in the bounded derived category of coherent sheaves on the projective space ℙ^{n}. This result follows immediately if we take A = k[x_{0}, …, x_{n}] with its standard grading. More generally, if we take A to be the polynomial algebra k[x_{0}, …, x_{n}] graded by deg x_{i} = a_{i}, then we get a full exceptional collection (O, …, O(∑_{i=0}^{n} a_{i} − 1)) in the bounded derived category of coherent sheaves on the weighted projective space ℙ(a_{0}, …, a_{n}) considered as a smooth orbifold (see [3, 2]). It is also true for noncommutative (weighted) projective spaces [2].

Another limiting case is when the algebra A is finite dimensional over the base field (i.e. A is a Frobenius algebra). In this case the category qgr A is trivial and, hence, the triangulated category of singularities D_{Sg}^{gr}(A) has a full exceptional collection (compare with [13, 10.10]). More precisely we get the following:

Corollary 20. Let A = ⨁_{i≥0} A_{i} be a connected graded noetherian algebra which

is Gorenstein with Gorenstein parameter a. Suppose that A is finite dimensional over the field k. Then, a ≤ 0 and the triangulated category of singularities D_{Sg}^{gr}(A) has a full exceptional collection (qk(0), …, qk(a + 1)) , where q : D^{b}(gr−A) ⟶ D_{Sg}^{gr}(A) is the natural projection. Moreover, the triangulated category D_{Sg}^{gr}(A) is equivalent to the derived category D^{b}(mod− Q(A))

of finite (right) modules over the algebra Q(A) = End_{gr−A} ( ⨁_{i=a+1}^{0} A(i)) .

Proof. Since A is finite dimensional the derived category D^{b}(qgr A) is trivial. By Theorem 16 we get that a ≤ 0 and D_{Sg}^{gr}(A) has a full exceptional collection (qk(0), …, qk(a + 1)) . Unfortunately, this collection is not strong. However, we can replace it by the dual exceptional collection which is already strong (see Definition 7). By Lemma 15 there is a weak semiorthogonal decomposition D^{b}(gr−A_{≥0}) = ⟨P_{≥0}, T_{0}⟩, where T_{0} is equivalent to D_{Sg}^{gr}(A). Moreover, by formula (13) we have the following semiorthogonal decomposition for T_{0} :

T_{0} = ⟨k(0), …, k(a + 1)⟩.

Denote by E_{i} where i = 0, …, −a−1 the modules A(i+a+1)/A(i+a+1)_{≥a}. These modules belong to T_{0} and form a full exceptional collection

T_{0} = ⟨E_{0}, …, E_{−(a+1)}⟩.

Furthermore, this collection is strong and the algebra of homomorphisms of

this collection coincides with the algebra Q(A) = End_{gr−A} ( ⨁_{i=a+1}^{0} A(i)) . As

in the previous proposition consider the object E = ⨁_{i=0}^{−(a+1)} E_{i} and the functor

R Hom(E, −) : T_{0} = D_{Sg}^{gr}(A) ⟶ D^{b}(mod− Q(A)).

Again the standard reasoning from ([6, 7]) shows that the functor R Hom(E, −) is an equivalence of triangulated categories. □

Example 21. The simplest example here is A = k[x]/x^{n+1}. In this case the triangulated category of singularities D_{Sg}^{gr}(A) has a full exceptional collection and is equivalent to the bounded derived category of finite dimensional representations of the Dynkin quiver of type A_{n} : \underbrace{• − • − ⋯ − •}_{n}, because in this

case the algebra Q(A) is isomorphic to the path algebra of this Dynkin quiver. This example is considered in detail in the paper [26].

Remark 22. There are other cases when the triangulated category of singularities D_{Sg}^{gr}(A) has a full exceptional collection. It follows from Theorem 16 that if a ≤ 0 and the derived category D^{b}(qgr A) has a full exceptional collection, then D_{Sg}^{gr}(A) has a full exceptional collection as well. It happens, for example, in the case when the algebra A is related to a weighted projective line – an orbifold over ℙ^{1} (see e.g. [11]).

2.3 Categories of coherent sheaves for Gorenstein varieties.

Let X be an irreducible projective Gorenstein variety of dimension n and let L be a very ample line bundle. Denote by A the graded coordinate algebra ⨁_{i≥0} H^{0}(X, L^{i}). The famous Serre theorem [25] asserts that the abelian category of coherent sheaves coh(X) is equivalent to the quotient category qgr A.

Assume that the dualizing sheaf ω_{X} is isomorphic to L^{−r} for some r ∈ ℤ and assume also that H^{j}(X, L^{k}) = 0 for all k ∈ ℤ when j ≠ 0, n. (For example, if X is a complete intersection in ℙ^{N} then it satisfies these conditions.) In this case, Theorem 16 allows us to compare the triangulated category of singularities D_{Sg}^{gr}(A) with the bounded derived category of coherent sheaves D^{b}(coh(X)). To apply that theorem we need the following lemma.

Lemma 23. Let X be a projective Gorenstein (irreducible) variety of dimension n. Let L be a very ample line bundle such that ω_{X} ≅ L^{−r} for some r ∈ ℤ and H^{j}(X, L^{k}) = 0 for all k ∈ ℤ when j ≠ 0, n. Then the algebra A = ⨁_{i≥0} H^{0}(X, L^{i}) is Gorenstein with Gorenstein parameter a = r.

Proof. Consider the projection functor Π : Gr−A → QGr A and its right adjoint Ω : QGr A → Gr−A which is given by the formula (4)

ΩΠN ≅ ⨁_{n=−∞}^{∞} Hom_{QGr}(ΠA, ΠN(n)).

The functor Ω has a right derived RΩ that is given by the formula

R^{j}Ω(ΠN) ≅ ⨁_{n=−∞}^{∞} Ext_{QGr}^{j}(ΠA, ΠN(n))

(see, e.g. [1, Prop. 7.2]). The assumptions on X and L imply that R^{j}Ω(ΠA) ≅ 0 for all j ≠ 0, n. Moreover, since X is Gorenstein and ω_{X} ≅ L^{−r}, Serre duality for X yields that

R^{0}Ω(ΠA) ≅ ⨁_{i=−∞}^{∞} H^{0}(X, L^{i}) ≅ A  and  R^{n}Ω(ΠA) ≅ ⨁_{i=−∞}^{∞} H^{n}(X, L^{i}) ≅ A^{∗}(r),

where A^{∗} = Hom_{k}(A, k). As X is irreducible, the algebra A is connected. Since Π and RΩ are adjoint functors we have

R Hom_{Gr}(k(s), RΩ(ΠA)) ≅ R Hom_{QGr}(Πk(s), ΠA) = 0

for all s. Furthermore, we know that R Hom_{A}(k, A^{∗}) ≅ R Hom_{A}(A, k) ≅ k. This implies that R Hom_{A}(k, A) ≅ k(r)[−n − 1]. This isomorphism implies that the affine cone SpecA is Gorenstein at the vertex and the assumption on X now implies that SpecA is Gorenstein scheme ([14, V, §9,10]). Since SpecA has a finite Krull dimension, the algebra A is a dualizing complex for itself, i.e. it has a finite injective dimension. Thus, the algebra A is Gorenstein with parameter r. □

Theorem 24. Let X be an irreducible projective Gorenstein variety of dimension n. Let L be a very ample line bundle such that ω_{X} ≅ L^{−r} for some r ∈ ℤ. Suppose H^{j}(X, L^{k}) = 0 for all k ∈ ℤ when j ≠ 0, n. Set A := ⨁_{i≥0} H^{0}(X, L^{i}).

Then, the derived category of coherent sheaves D^{b}(coh(X)) and the triangulated category of singularities D_{Sg}^{gr}(A) are related as follows:

(i) if r > 0, i.e. if X is a Fano variety, then there is a semiorthogonal decom-

position

D^{b}(coh(X)) = ⟨L^{−r+1}, …, O_{X}, D_{Sg}^{gr}(A)⟩,

(ii) if r < 0, i.e. if X is a variety of general type, then there is a semiorthogonal decomposition

D_{Sg}^{gr}(A) = ⟨qk(r + 1), …, qk, D^{b}(coh(X))⟩,

where q : D^{b}(gr−A) ⟶ D_{Sg}^{gr}(A) is the natural projection, (iii)if r = 0, i.e. if X is a Calabi-Yau variety, then there is an equivalence

D_{Sg}^{gr}(A) ⟶^{∼} D^{b}(coh(X)).

Proof. Since L is very ample Serre’s theorem implies that the bounded derived category D^{b}(coh(X)) is equivalent to the category D^{b}(qgr A), where A = ⨁_{i≥0} H^{0}(X, L^{i}). Since H^{j}(X, L^{k}) = 0 for j ≠ 0, n and all k ∈ ℤ, Lemma 23 implies that A is Gorenstein. Now, the theorem immediately follows from Theorem 16. □

Corollary 25. Let X be an irreducible projective Gorenstein Fano variety of dimension n with at most rational singularities. Let L be a very ample line bundle such that ω_{X}^{−1} ≅ L^{r} for some r ∈ ℕ. Set A = ⨁_{i≥0} H^{0}(X, L^{i}). Then the category D^{b}(coh(X)) admits a semiorthogonal decomposition of the form

D^{b}(coh(X)) = ⟨L^{−r+1}, …, O_{X}, D_{Sg}^{gr}(A)⟩.

Proof. The Kawamata-Viehweg vanishing theorem (see, e.g. [18, Th. 1.2.5]) yields H^{j}(X, L^{k}) = 0 for j ≠ 0, n and all k. Hence, we can apply Theorem 24(i). □

Corollary 26. Let X be a Calabi-Yau variety. That is, X is an irreducible projective variety with at most rational singularities, with trivial canonical sheaf ω_{X} ≅ O_{X}, and such that H^{j}(X, O_{X}) = 0 for j ≠ 0, n. Let L be a some very ample line bundle on X. Set A = ⨁_{i≥0} H^{0}(X, L^{i}). Then there is an equivalence

D^{b}(coh(X)) ≅ D_{Sg}^{gr}(A).

Proof. The variety X has rational singularities hence it is Cohen-Macaulay. Moreover, X is Gorenstein, because ω_{X} ≅ O_{X}. The Kawamata-Viehweg vanishing theorem ([18, Th. 1.2.5]) yields H^{j}(X, L^{k}) = 0 for j ≠ 0, n and all k ≠ 0. Since by assumption H^{j}(X, O_{X}) = 0 for j ≠ 0, n, we can apply Theorem 24 (iii). □

Proposition 27. Let X ⊂ ℙ^{N} be a complete intersection of m hypersurfaces D_{1}, …, D_{m} of degrees d_{1}, …, d_{m} respectively. Then X and L = O_{X}(1) satisfy the conditions of Theorem 24 with Gorenstein parameter r = N +1−∑_{i=1}^{m} d_{i}.

Proof. Since the variety X is a complete intersection it is Gorenstein. The canonical class ω_{X} is isomorphic to O(∑ d_{i} − N − 1). It can be easily proved by induction on m that H^{j}(X, O_{X}(k)) = 0 for all k and j ≠ 0, n, where n = N − m is the dimension of X. Indeed, The base of the induction is clear. For the induction step, assume that for Y = D_{1} ∩ ⋯ ∩ D_{m−1} these conditions hold. Then, consider the short exact sequence

0 ⟶ O_{Y} (k − d_{m}) ⟶ O_{Y} (k) ⟶ O_{X}(k) ⟶ 0.

Since the cohomologies H^{j}(Y, O_{Y} (k)) = 0 for all k and j ≠ 0, n + 1 we obtain that H^{j}(X, O_{X}(k)) = 0 for all k and j ≠ 0, n. □

Theorem 24 can be extended to the case of quotient stacks. To do this we will need an appropriate generalization of Serre’s theorem [25]. The usual Serre theorem says that if a commutative connected graded algebra A = ⨁_{i≥0} A_{i} is generated by its first component, then the category qgr A is equivalent to the category of coherent sheaves coh(X) on the projective variety X = Proj A. (Such equivalence holds for the categories of quasicoherent sheaves Qcoh(X) and QGr A too.)

Consider now a commutative connected graded k-algebra A = ⨁_{i≥0} A_{i} which is not necessary generated by its first component. The grading on A induces an action of the group k^{∗} on the affine scheme SpecA. Let 0 be the closed point of SpecA that corresponds to the ideal A_{+} = A_{≥1} ⊂ A. This point is invariant under the action.

Denote by ℙroj A the quotient stack [(SpecA\0)/k^{∗}] . (Note that there is a natural map ℙroj A → Proj A, which is an isomorphism if the algebra A is generated by A_{1}.)

Proposition 28. (see also [2]) Let A = ⊕_{i≥0} A_{i} be a connected graded finitely

generated algebra. Then the category of (quasi)coherent sheaves on the quotient stack ℙroj (A) is equivalent to the category qgr A (respectively QGr A). Proof. Let 0 be the closed point on the affine scheme SpecA which corresponds to the maximal ideal A_{+} ⊂ A. Denote by U the complement SpecA\0. We know that the category of (quasi)coherent sheaves on the stack ℙroj A is equivalent to the category of k^{∗}-equivariant (quasi)coherent sheaves on U. The category of (quasi)coherent sheaves on U is equivalent to the quotient of the category of (quasi)coherent sheaves on SpecA by the subcategory of (quasi)coherent sheaves with support on 0 (see [9]). This is also true for the categories of k^{∗}-equivariant sheaves. But the category of (quasi)coherent k^{∗}- equivariant sheaves on SpecA is just the category gr−A (resp. Gr−A) of graded modules over A, and the subcategory of (quasi)coherent sheaves with support on 0 coincides with the subcategory tors−A (resp. Tors−A). Thus, we obtain that coh(ℙroj A) is equivalent to the quotient category qgr A = gr−A/ tors−A (and Qcoh(ℙroj A) is equivalent to QGr A = Gr−A/ Tors−A). □

Corollary 29. Assume that the noetherian Gorenstein connected graded algebra A from Theorem 16 is finitely generated and commutative. Then instead the bounded derived category D^{b}(qgr A) in Theorem 16 we can substitute the category D^{b}(coh(ℙroj A)), where ℙroj A the quotient stack [(SpecA\0)/k^{∗}] .

3 Categories of graded D-branes of type B in Landau-Ginzburg models.

3.1 Categories of graded pairs.

Let B = ⨁_{i≥0} B_{i} be a finitely generated connected graded algebra over a field k. Let W ∈ B_{n} be a central element of degree n which is not a zero-divisor, i.e. W b = bW for any b ∈ B and bW = 0 only for b = 0. Denote by J the two-sided ideal J := W B = BW and denote by A the quotient graded algebra B/J.

With any such element W ∈ B_{n} we can associate two categories: an exact category GrPair(W) and a triangulated category DGrB(W).^{2} Objects of 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} ∈ gr−B are finitely generated free graded right B-modules, p_{1} is a map of degree 0 and p_{0} is a map of degree n (i.e a map from P_{0} to P_{1}(n)) such that the compositions p_{0}p_{1} and p_{1}(n)p_{0} are the left multiplications by the element W. A morphism f : \overline{P} → \overline{Q} in the category GrPair(W) is a pair of morphisms f_{1} : P_{1} → Q_{1} and f_{0} : P_{0} → Q_{0} of degree 0 such that f_{1}(n)p_{0} = q_{0}f_{0} and q_{1}f_{1} = f_{0}p_{1}. The morphism f = (f_{1}, f_{0}) is nullhomotopic if there are two morphisms s : P_{0} → Q_{1} and t : P_{1} → Q_{0}(−n) such that f_{1} = q_{0}(n)t + sp_{1} and f_{0} = t(n)p_{0} + q_{1}s. Morphisms in the category DGrB(W) are the classes of morphisms in GrPair(W) modulo null-homotopic morphisms.

In other words, objects of both categories are quasi-periodic infinite sequences

\underline{K} := \{⋯ ⟶ K^{i} ⟶^{k^{i}} K^{i+1} ⟶^{k^{i+1}} K^{i+2} ⟶ ⋯ \},

of morphisms in gr−B of free graded right B-modules so that the composition k^{i+1}k^{i} of any two consecutive morphisms is equal to multiplication by W. The quasi-periodicity property here means that \underline{K} [2] = \underline{K} (n). In particular

2One can also construct a differential graded category the homotopy category of which is equivalent to DGrB .

K^{2i−1} ≅ P_{1}(i · n), K^{2i} ≅ P_{0}(i · n), k^{2i−1} = p_{1}(i · n), k^{2i} = p_{0}(i · n).

A morphism f : \underline{K} ⟶ \underline{L} in the category GrPair(W) is a family of morphisms f^{i} : K^{i} ⟶ L^{i} in gr−B which is quasi-periodic, i.e f^{i+2} = f^{i}(n), and which commutes with k^{i} and l^{i}, i.e. f^{i+1}k^{i} = l^{i}f^{i}.

Morphisms in the category DGrB(W) are morphisms in GrPair(W) modulo null-homotopic morphisms, and we consider only quasi-periodic homotopies, i.e. such families s^{i} : K^{i} ⟶ L^{i−1} that s^{i+2} = s^{i}(n).

Definition 30. The category DGrB(W) constructed above will be called the category of graded D-branes of type B for the pair (B = ⨁_{i≥0} B_{i}, W).

Remark 31. If B is commutative, then we can consider the affine scheme SpecB. The grading on B corresponds to an action of the algebraic group k^{∗} on SpecB. The element W can be viewed as a regular function on SpecB which is semi-invariant with respect to this action. This way, we get a singular Landau-Ginzburg model (SpecB, W) with an action of torus k^{∗}. Thus, Definition 30 is a definition of the category of graded D-branes of type B for this model (see also [16, 28]).

It is clear that the category GrPair(W) is an exact category (see [24] for the definition) with monomorphisms and epimorphisms being the componentwise monomorphisms and epimorphisms. The category DGrB(W) can be endowed with a natural structure of a triangulated category. To exhibit this structure we have to define a translation functor [1] and a class of exact triangles.

The translation functor as usually is defined as a functor that takes an object \underline{K} to the object \underline{K} [1], where K[1]^{i} = K^{i+1} and d[1]^{i} = −d^{i+1}, and takes a morphism f to the morphism f[1] which coincides with f componentwise.

For any morphism f : \underline{K} → \underline{L} from the category GrPair(W) we define a mapping cone \underline{C} (f) as an object

\underline{C} (f) = \{⋯ ⟶ L^{i} ⊕ K^{i+1} ⟶^{c^{i}} L^{i+1} ⊕ K^{i+2} ⟶^{c^{i+1}} L^{i+2} ⊕ K^{i+3} ⟶ ⋯ \}

such that

c^{i} = (\begin{matrix}l^{i} & f^{i+1} \\ 0 & −k^{i+1} \end{matrix}) .

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

Now we define a standard triangle in the category DGrB(W) as a triangle of the form

\underline{K} ⟶^{f} \underline{L} ⟶^{g} \underline{C} (f) ⟶^{h} \underline{K} [1].

for some f ∈ GrPair(W).

Definition 32. A triangle \underline{K} →\underline{L} →\underline{M} →\underline{K} [1] in DGrB(W) will be called an exact (distinguished) triangle if it is isomorphic to a standard triangle.

Proposition 33. The category DGrB(W) endowed with the translation functor [1] and the above class of exact triangles becomes a triangulated category.

We omit the proof of this proposition which is more or less the same as the proof of the analogous result for a usual homotopic category (see, e.g. [12]).

3.2 Categories of graded pairs and categories of singularities

With any object \underline{K} as above, one associates a short exact sequence

0 ⟶ K^{−1} ⟶^{k^{−1}} K^{0} ⟶ Coker k^{−1} ⟶ 0.  (14)

We can attach to an object \underline{K} the right B-module Coker k^{−1}. It can be easily checked that the multiplication by W annihilates it. Hence, the module Coker k^{−1} is naturally a right A-module, where A = B/J with J = W B = BW. Any morphism f : \underline{K} → \underline{L} in GrPair(W) induces a morphism between cokernels. This construction defines a functor Cok : GrPair(W) ⟶ gr−A. Using the functor Cok we can construct an exact functor between triangulated categories DGrB(W) and D_{Sg}^{gr}(A).

Proposition 34. There is a functor F which completes the following commutative diagram

GrPair(W) ⟶^{Cok} gr−A
↓  ↓
DGrB(W) ⟶^{F} D_{Sg}^{gr}(A).

Moreover, the functor F is an exact functor between triangulated categories. Proof. We have the functor GrPair(W) ⟶ D_{Sg}^{gr}(A) which is the composition of Cok and the natural functor from gr−A to D_{Sg}^{gr}(A). To prove the existence of a functor F we need to show that any morphism f : \underline{K} → \underline{L} which is null-homotopic goes to the 0-morphism in D_{Sg}^{gr}(A). Fix a homotopy s = (s^{i}) with s^{i} : K^{i} → L^{i−1}. Consider the following decomposition of f:

\xymatrix{K^{−1} \ar[d]_{(s^{−1},f^{−1})} \ar[r]^{k^{−1}} & K^{0} \ar[d]^{(s^{0},f^{0})} \ar[r] & Coker k^{−1} \ar[d] \\ L^{−2} ⊕ L^{−1} \ar[d]_{pr} \ar[r]^{u^{−1}} & L^{−1} ⊕ L^{0} \ar[d]^{pr} \ar[r] & L^{0} ⊗_{B} A \ar[d] & where & u^{−1} = (\begin{matrix}−l^{−2} & id \\ 0 & l^{−1} \end{matrix}) , \\ L^{−1} \ar[r]^{l^{−1}} & L^{0} \ar[r] & Coker l^{−1}}

This yields a decomposition of F (f) through a locally free object L^{0} ⊗_{B} A. Hence, F (f) = 0 in the category D_{Sg}^{gr}(A). By Lemma 36, which is proved below, the tensor product \underline{K} ⊗_{B} A is an acyclic complex. Hence, there is an exact sequence 0 → Coker k^{−1} → K^{1}⊗_{B} A → Coker k^{0} → 0. Since K^{1}⊗_{B} A is free, we have Coker k^{0} ≅ Coker k^{−1}[1] in D_{Sg}^{gr}(A). But, Coker k^{0} = F (\underline{K} [1]). Hence, the functor F commutes with translation functors. It is easy to see that F takes a standard triangle in DGrB(W) to an exact triangle in D_{Sg}^{gr}(A). Thus, F is exact. □

Lemma 35. The functor Cok is full. Proof. Any map g : Coker k^{−1} → Coker l^{−1} between A-modules can be considered as the map of B-modules and can be extended to a map of short exact sequences

0 ⟶ K^{−1} ⟶^{k^{−1}} K^{0} ⟶ Coker k^{−1} ⟶ 0
f^{−1}↓  ↓^{f^{0}}  ↓^{g}
0 ⟶ L^{−1} ⟶^{l^{−1}} L^{0} ⟶ Coker l^{−1} ⟶ 0,

because K^{0} is free. This gives us a sequence of morphisms f = (f^{i}), i ∈ ℤ, where f^{2i} = f^{0}(in) and f^{2i−1} = f^{−1}(in). To prove the lemma it is sufficient to show that the family f is a map from \underline{K} to \underline{L} , i.e f^{1}k^{0} = l^{0}f^{0}. Consider the sequence of equalities

l^{1}(f^{1}k^{0} − l^{0}f^{0}) = f^{2}k^{1}k^{0} − W f^{0} = f^{2}W − W f^{0} = f^{0}(2)W − W f^{0} = 0.

Since l^{1} is an embedding, we obtain that f^{1}k^{0} = l^{0}f^{0}. □

Lemma 36. For any sequence \underline{K} ∈ GrPair(W) the tensor product \underline{K} ⊗_{B} A is an acyclic complex of A-modules and the A-module Coker k^{−1} satisfies the condition

Ext_{A}^{i}(Coker k^{−1}, A) = 0 for all i > 0.

Proof. It is clear that \underline{K} ⊗_{B} A is a complex. Applying the Snake Lemma to the commutative diagram

0 ⟶ K^{i−2} ⟶^{k^{i−2}} K^{i−1} ⟶ Coker k^{i−2} ⟶ 0
W ↓  ↓^{W}  ↓^{0}
0 ⟶ K^{i} ⟶^{k^{i}} K^{i+1} ⟶ Coker k^{i} ⟶ 0,

we obtain an exact sequence

0 → Coker k^{i−2} ⟶ K^{i} ⊗_{B} A ⟶^{k_{i}|_{W}} K^{i+1} ⊗_{B} A ⟶ Coker k^{i} → 0.

This implies that \underline{K} ⊗_{B} A is an acyclic complex.

Further, consider the dual sequence of left B-modules \underline{K} ^{∨}, where \underline{K} ^{∨} ≅ Hom_{B}(\underline{K} , B). By the same reasons as above A⊗_{B} \underline{K} ^{∨} is an acyclic complex. On the other hand, the cohomologies of the complex {(K^{0})^{∨} ⟶ (K^{−1})^{∨} ⟶ (K^{−2})^{∨} ⟶ ⋯ } are isomorphic to Ext_{A}^{i}(Coker k^{−1}, A). And so, by the acyclicity of A ⊗_{B} \underline{K} ^{∨}, they are equal to 0 for all i > 0. □

Lemma 37. If F \underline{K} ≅ 0, then \underline{K} ≅ 0 in DGrB(W).

Proof. If F \underline{K} ≅ 0, then the A-module Coker k^{−1} is perfect as a complex of A-modules. Let us show that Coker k^{−1} is projective in this case. Indeed, there is a natural number m such that Ext_{A}^{i}(Coker k^{−1}, N) = 0 for any A-module N and any i ≥ m. Considering the exact sequence

0 ⟶ Coker k^{−2m−1} ⟶ K^{−2m}⊗_{B}A ⟶ ⋯ ⟶ K^{−1}⊗_{B}A ⟶ K^{0}⊗_{B}A ⟶ Coker k^{−1} ⟶ 0

and taking into account that all A-modules K^{i}⊗_{B}A are free, we find that for all modules N Ext_{A}^{i}(Coker k^{−2m−1}, N) = 0 when i > 0. Hence, Coker k^{−2m−1} is a projective A-module. This implies that Coker k^{−1} is also projective, because it is isomorphic to Coker k^{−2m−1}(−mn).

Since Coker k^{−1} is projective there is a map f : Coker k^{−1} → K^{0} ⊗_{B} A which splits the epimorphism pr : K^{0} ⊗_{B} A → Coker k^{−1}. It can be lifted

to a map from the complex {K^{−1} ⟶^{k^{−1}} K^{0}} to the complex {K^{−2} ⟶^{W} K^{0}}. Denote the lift by (s^{−1}, u). Consider the following diagram

K^{−1} ⟶^{k^{−1}} K^{0} ⟶ Coker k^{−1}
s^{−1}↓  ↓^{u}  ↓^{f}
K^{−2} ⟶^{W} K^{0} ⟶ K^{0} ⊗_{B} A
k^{−2}↓  ↓^{id}  ↓^{pr}
K^{−1} ⟶^{k^{−1}} K^{0} ⟶ Coker k^{−1}.

Since the composition pr f is identical, the map (k^{−2}s^{−1}, u) from the pair

{K^{−1} ⟶^{k^{−1}} K^{0}} to itself is homotopic to the identity map. Hence, there is a map s^{0} : K^{0} → K^{−1} such that

id_{K^{−1}} − k^{−2}s^{−1} = s^{0}k^{−1}  and  k^{−1}s^{0} = id_{K^{0}} − u.

Moreover, we have the following equalities

0 = (uk^{−1} − W s^{−1}) = (uk^{−1} − s^{−1}(n)W) = (u − s^{−1}(n)k^{0})k^{−1}.

This gives us that u = s^{−1}(n)k^{0}, because there are no maps from Coker k^{−1} to K^{0}. Finally, we get the sequence of morphisms s^{i} : K^{i} ⟶ K^{i−1}, where s^{2i−1} = s^{−1}(in), s^{2i} = s^{0}(in), such that k^{i−1}s^{i} + k^{i}s^{i+1} = id. Thus the identity morphism of the object \underline{K} is null-homotopic. Hence, the object \underline{K} is isomorphic to the zero object in the category DGrB_{0}(W). □

Theorem 38. The exact functor F : DGrB(W) ⟶ D_{Sg}^{gr}(A) is fully faithful.

Proof. By Lemma 36 we have Ext_{A}^{i}(Coker k^{−1}, A) = 0 for i > 0. Now, Proposition 11 gives an isomorphism

Hom_{D_{Sg}^{gr}(A)}(Coker k^{−1}, Coker l^{−1}) ≅ Hom_{gr−A}(Coker k^{−1}, Coker l^{−1})/R,

where R is the subspace of morphisms factoring through projective modules. Since the functor Cok is full we get that the functor F is also full.

Next we show that F is faithful. The reasoning is standard. Let f : \underline{K} → \underline{L} be a morphism for which F (f) = 0. Include f in an exact triangle \underline{K} ⟶^{f} \underline{L} ⟶^{g} \underline{M} . Then the identity map of F \underline{L} factors through the map F \underline{L} ⟶^{F g} F \underline{M} . Since F is full, there is a map h : \underline{L} → \underline{L} factoring through g : \underline{L} → \underline{M} such that F h = id. Hence, the cone \underline{C} (h) of map h goes to zero under the functor F. By Lemma 37 the object \underline{C} (h) is the zero object as well, so h is an isomorphism. Thus g : \underline{L} → \underline{M} is a split monomorphism and f = 0. □

Theorem 39. Suppose that the algebra B has a finite homological dimension. Then the functor F : DGrB(W) ⟶ D_{Sg}^{gr}(A) is an equivalence. Proof. We know that F is fully faithful. To prove the theorem we need to show that each object T ∈ D_{Sg}^{gr}(A) is isomorphic to F \underline{K} for some \underline{K} ∈ DGrB(W).

The algebra B has a finite homological dimension and, as consequence, it has a finite injective dimension. This implies that A = B/J has a finite injective dimension too. By Lemma 10 any object T ∈ D_{Sg}^{gr}(A) is isomorphic to the image of an A-module M such that Ext_{A}^{i}(M, A) = 0 for all i > 0. This means that the object D(M) = R Hom_{A}(M, A) is a left A-module. We can consider a projective resolution \underline{Q} → D(M). The dual of \underline{Q} is a right projective A-resolution

0 ⟶ M ⟶ {P^{0} ⟶ P^{1} ⟶ ⋯ }.

Consider M as B-module and chose an epimorphism K^{0} ↠ M from free Bmodule K^{0}. Denote by k^{−1} : K^{−1} → K^{0} the kernel of this map.

The short exact sequence 0 → B →^{W} B → A → 0 implies that for a projective A-module P and any B-module N we have equalities Ext_{B}^{i}(P, N) = 0 when i > 1. This also yields that Ext_{B}^{i}(M, N) = 0 for i > 1 and any Bmodule N, because M has a right projective A-resolution and the algebra B has finite homological dimension. Therefore, Ext_{B}^{i}(K^{−1}, N) = 0 for i > 0 and any B-module N, i.e. B-module K^{−1} is projective. Since A is connected and finitely generated, any graded projective module is free. Hence, K^{−1} is free.

Since the multiplication on W gives the zero map on M, there is a map k^{0} : K^{0} → K^{−1}(n) such that k^{0}k^{−1} = W and k^{−1}(n)k^{0} = W. This way, we get a sequence \underline{K} with

K^{2i} ≅ K^{0}(i · n), K^{2i−1} = K^{−1}(i · n), k^{2i} = k^{0}(i · n), k^{2i−1} = k^{−1}(i · n).

and this sequence is an object of DGrB(W) for which F \underline{K} ≅ T. □

3.3 Graded D-branes type B and coherent sheaves.

By a Landau-Ginzburg model we mean the following data: a smooth variety X equipped with a symplectic Kähler form ω, a closed real 2-form B, which is called B-field, and a regular nonconstant function W on X. The function W is called the superpotential of the Landau-Ginzburg model. Since for the definition of D-branes of type B a symplectic form and B-field are not needed we do not fix them.

With any point λ ∈ 𝔸^{1} we can associate a triangulated category DB_{λ}(W). We give a construction of this categories under the condition that X = Spec(B) is affine (see [17, 22]). The category of coherent sheaves on an affine scheme X = Spec(B) is the same as the category of finitely generated B-modules. The objects of the category DB_{λ}(W) 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 B-modules

and the compositions p_{0}p_{1} and p_{1}p_{0} are the multiplications by the element (W − λ) ∈ B. The morphisms in the category DB(W) are the classes of morphisms between pairs modulo null-homotopic morphisms, where a morphism f : \overline{P} → \overline{Q} between pairs 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 null-homotopic 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.

We define a category of D-branes of type B (B-branes) on X = Spec(B) with the superpotential W as the product DB(W) = ∏_{λ∈𝔸^{1}} DB_{λ}(W).

It was proved in the paper [22, Cor. 3.10] that the category DB_{λ}(W) for smooth affine X is equivalent to the triangulated category of singularities D_{Sg}(X_{λ}), where X_{λ} is the fiber over λ ∈ 𝔸^{1}. Therefore, the category of Bbranes DB(W) is equivalent to the product ∏_{λ∈𝔸^{1}} D_{Sg}(X_{λ}). For non-affine X the category ∏_{λ∈𝔸^{1}} D_{Sg}(X_{λ}) can be considered as a definition of the category of D-branes of type B. Note that, in the affine case, X_{λ} is Spec(A_{λ}), where A_{λ} = B/(W − λ)B and, hence, the triangulated categories of singularities D_{Sg}(X_{λ}) is the same that the category D_{Sg}(A_{λ}).

Assume now that there is an action of the group k^{∗} on the Landau-Ginzburg model (X, W) such that the superpotential W is semi-invariant of the weight d. Thus, X = Spec(B) and B = ⨁_{i} B_{i} is a graded algebra. The superpotential W is an element of B_{d}. Let us assume that B is positively graded and connected. In this case, we can consider the triangulated category of graded B-branes DGrB(W), which was constructed in subsection 3.1 (see Definition 30).

Denote by A the quotient graded algebra B/W B. We see that the affine variety Spec(A) is the fiber X_{0} of W over the point 0. Denote by Y the quotient stack [(Spec(A) \ 0)/k^{∗}] , where 0 is the point on Spec(A) corresponding to the ideal A_{+}. Theorems 16, 39 and Proposition 28 allow us to establish a relation between triangulated category of graded B-branes DGrB(W) and the bounded derived category of coherent sheaves on the stack Y.

First, Theorem 39 gives us the equivalence F between the triangulated category of graded B-branes DGrB(W) and the triangulated category of singularities D_{Sg}^{gr}(A). Second, Theorem 16 describes the relationship between the category D_{Sg}^{gr}(A) and the bounded derived category D^{b}(qgr A). Third, the category D^{b}(qgr A) is equivalent to the derived category D^{b}(coh(Y)) by Proposition 28. In the particular case, when X is the affine space 𝔸^{N} with the standard action of the group k^{∗}, we get the following result.

Theorem 40. Let X be the affine space 𝔸^{N} and let W be a homogeneous polynomial of degree d. Let Y ⊂ ℙ^{N−1} be the hypersurface of degree d which is given by the equation W = 0. Then, there is the following relation between the triangulated category of graded B-branes DGrB(W) and the derived category of coherent sheaves D^{b}(coh(Y)) :

(i) if d < N, i.e. if Y is a Fano variety, there is a semiorthogonal decompo-

sition

D^{b}(coh(Y)) = ⟨O_{Y} (d − N + 1), …, O_{Y} , DGrB(W)⟩,

(ii) if d > N, i.e. if X is a variety of general type, there is a semiorthogonal decomposition

DGrB(W) = ⟨F^{−1}q(k(r + 1)), …, F^{−1}q(k), D^{b}(coh(Y))⟩,

where q : D^{b}(gr−A) ⟶ D_{Sg}^{gr}(A) is the natural projection, and F : DGrB ⟶^{∼} D_{Sg}^{gr}(A) is the equivalence constructed in Proposition 34.

(iii)if d = N, i.e. if Y is a Calabi-Yau variety, there is an equivalence

DGrB(W) ⟶^{∼} D^{b}(coh(Y)).

Remark 41. We can also consider a weighted action of the torus k^{∗} on the affine space 𝔸^{N} with positive weights (a_{1}, …, a_{N}), a_{i} > 0 for all i. If the superpotential W is quasi-homogeneous then we have the category of graded B-branes DGrB(W). The polynomial W defines an orbifold (quotient stack) Y ⊂ ℙ^{N−1}(a_{1}, …, a_{N}). The orbifold Y is the quotient of Spec(A)\0 by the action of k^{∗}, where A = k[x_{1}, …, x_{N}]/W. Proposition 28 gives the equivalence between D^{b}(coh(Y)) and D^{b}(qgr A). And Theorem 16 shows that we get an analogue of Theorem 40 for the weighted case as well.

References

1. M. Artin, . J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), no. 2, p. 248–287.

2. D. Auroux, L. Katzarkov, D. Orlov, Mirror symmetry for weighted projective planes and their noncommutative deformations, accepted to Annals of Math.

3. D. Baer, Tilting sheaves in representation theory of algebras, Manuscripta Math. 60 (1988), no. 3, p. 323–247.

4. A. Beilinson, Coherent sheaves on ℙ^{n} and problems in linear algebra, Funct. Anal. Appl. 12 (1978), no. 3, p. 68–69.

5. P. Berthelot, A. Grothendieck, L. Illusie , Théorie des intersections et théoreme de Riemann-Roch, Lectere Notes in Mathematics, vol. 225, Springer, 1971.

6. A. Bondal, Representation of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR 53 (1989), no. 1, p. 25–44.

7. A. Bondal, M. Kapranov, Enhanced triangulated categories, Matem. Sb. 181 (1990), no. 5, p. 669–683.

8. A. Bondal, D. Orlov, Semiorthogonal decomposition for algebraic varieties, preprint MPIM 95/15, 1995, arXiv:math.AG/9506012.

9. P. Gabriel, Des Catégories Abéliennes, Bull. Soc. Math. Fr. 90 (1962), p. 323– 448.

10. P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag New York, New York, 1967.

11. W. Geigle, H. Lenzing, A class of weighted projective curves arising in representation theorey of finite dimensional algebras, Singularities, representation of algebras, and vector bundles (Proc. Symp., Lambrecht/Pfalz/FRG 1985), Lect. Notes Math., vol. 1273, 1987, p. 265–297.

12. S. Gelfand, Y. Manin, Homological algebra, algebra v, Encyclopaedia Math. Sci., vol. 38, Springer-Verlag, 1994.

13. D. Happel, On the derived categories of a finite-dimensional alghebra, Comment. Math. Helv 62 (1987), p. 339–389.

14. R. Hartshorne, Residues and Duality, Lecture Notes in Mathematics, vol. 20, Springer, 1966.

15. K. Hori, C. Vafa, Mirror Symmetry, arXiv:hep-th/0002222.

16. K. Hori, J. Walcher, F-term equation near Gepner points,

arXiv:hep-th/0404196.

17. A. Kapustin, Y. Li, D-branes in Landau-Ginzburg models and algebraic geometry, J. High Energy Physics, JHEP 12 (2003), no. 005, arXiv:hep-th/0210296.

18. Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model program, Adv. Stud. in Pure Math. 10 (1987), p. 283–360.

19. B. Keller, Derived categories and their uses, vol. 1, ch. Chapter of the Handbook of algebra, Elsevier, 1996.

20. M. Kontsevich, Homological algebra of mirror symmetry, Proceedings of ICM, Zurich 1994 (Basel), Birkhauser, 1995, p. 120–139.

21. A. Neeman, Triangulated categories, Ann. of Math. Studies, vol. 148, Princeton University Press, 2001.

22. D. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Trudy Steklov Math. Institute 246 (2004), p. 240–262.

23. D. Orlov, Triangulated categories of singularities and equivalences between Landau-Ginzburg models, to appear in Matem. Sbornik, 2006,

arXiv:math.AG/0503630.

24. D. Quillen, Higher Algebraic K-theory I, Springer Lecture Notes in Math., vol. 341, Springer-Verlag, 1973.

25. J. P. Serre, Modules projectifs et espaces fibrés à fibre vectorielle, Séminaire Dubreil–Pisot, vol. 23, Paris, 1958.

26. A. Takahashi, Matrix factorizations and representations of quivers 1, arXiv:math.AG/0506347.

27. J. L. Verdier, Categories derivées, SGA 4 1/2, Lecture Notes in Math., vol. 569, Springer-Verlag, 1977.

28. J. Walcher, Stability of landau-ginzburg branes, arXiv:hep-th/0412274.

29. A. Yekutieli, Dualizing complexes over noncommutative graded algebras, J. Algebra 153 (1992), p. 41–84.

30. A. Yekutieli, J. J. Zhang, Serre duality for noncommutative projective schemes, Proc. Amer. Math. Soc. 125 (1997), p. 697–707.