aa r X i v : . [ m a t h . AG ] A p r Derived Categories and Birational Geometry
Yujiro KawamataOctober 22, 2018
This paper is concerned with a surprizing parallelism between minimal modelprogram and semi-orthogonal decompositions of derived categories found byBondal and Orlov ([3]).The bounded derived category of coherent sheaves on a smooth projectivevariety has a finiteness property called saturatedness ( § § § §
3. The cases include smoothvarieties due to [13] and [3], and toric varieties [9]. We provide some prop-erties of the desired category, and give a possible definition of the correctcategory for certain three dimensional cases by calculating the behavior ofthe category under the divisorial contractions of smooth threefolds in §
4. Weconclude this paper with a short section of open questions.The author would like to thank the referee for a careful reading and usefulsuggestions. 1
Saturated categories and semi-orthogonaldecompositions
Let X be a projective variety over a field k . The largest derived categoryfor X is the unbounded derived category of quasi-coherent sheaves denotedby D (QCoh( X )) (we refer to [6] for basic definitions). There are functorssuch as f ∗ , f ∗ , f ! for morphisms f : X → Y , and the Grothendieck dualitytheorem holds ([12]). This category is k -linear in the sense that the set ofmorphisms Hom( a, b ) for a, b ∈ D (QCoh( X )) has the structure of a k -vectorspace. It is, however, infinite dimensional and we might hope to work withcategories for which the Hom space has finite dimension.A candidate is the bounded derived category of coherent sheaves denotedby D b (Coh( X )). If X is smooth, then D b (Coh( X )) is of finite type in thesense that P p ∈ Z dim Hom( a, b [ p ]) is finite. But if X is singular, then thehomological dimension is infinite. Indeed, if x ∈ X is a singular point, thenthere are infinitely many p such that Hom( O x , O x [ p ]) = 0, where O x denotesa skyscraper sheaf of length 1 at x .An object c in a triangulated category D having arbitrary coproductsis said to be compact if Hom( c, ` λ a λ ) ∼ = ` λ Hom( c, a λ ). An object c ∈ D (QCoh( X )) is compact if and only if it is perfect in the sense that it islocally isomorphic to a bounded complex of locally free coherent sheaves. LetPerf( X ) = D (QCoh( X )) c be the triangulated category of perfect complexes.It is a full subcategory of D b (Coh( X )), and they coincide if X is smooth.If a ∈ Perf( X ) and b ∈ D b (Coh( X )), then P p ∈ Z dim Hom( a, b [ p ]) < ∞ .Serre duality holds Hom( a, b ) ∼ = Hom( b, a ⊗ ω • X ) ∗ for a ∈ Perf( X ) and b ∈ D b (Coh( X )), where ω • X is a dualizing complex.For example, if X is smooth, then ω • X = ω X [dim X ]. Therefore, Perf( X ) issimilar to homology while D b (Coh( X )) cohomology, because Serre duality issimilar to Poincar´e duality. In particular, if X is smooth, then the functor S X : D b (Coh( X )) → D b (Coh( X )) defined by S X ( a ) = a ⊗ ω X [dim X ] is a Serre functor in the sense thatHom( a, b ) ∼ = Hom( b, S X ( a )) ∗ (2.1)for a, b ∈ D b (Coh( X )). 2t is important to consider singular varieties in minimal model theory.Since the sets Hom( O x , O x [ p ]) do not vanish for p ≥ p < x ∈ X , there cannot be a Serre functor in the category D b (Coh( X )) for a singular variety X . So we would like to have somethinglike intersection homology for a singular variety X that lies between Perf( X )and D b (Coh( X )) and has a Serre functor.A k -linear triangulated category D of finite type is said to be saturated ifany cohomological functor F : D op → ( k -vect) of finite type is representable.That is, if P p ∈ Z dim F ( a [ p ]) < ∞ for any a ∈ D , then there exists b ∈ D such that F ( a ) ∼ = Hom( a, b ). A saturated category has always a Serre functor,because the dual of the left hand side of (2.1) defines a cohomological functor.The category D b (Coh( X )) for smooth X is saturated ([4]). For example, anobject a ∈ D in a k -linear triangulated category is said to be exceptional ifHom( a, a [ p ]) = 0 for p = 0 while Hom( a, a ) ∼ = k . In this case, the subcategory h a i generated by a is saturated because it is equivalent to D b (Coh(Spec k )).A triangulated full subcategory B of a triangulated category A is said tobe right admissible if the embedding j ∗ : B → A has a right adjoint functor j ! : A → B . Let C = B ⊥ be the right orthogonal defined by C = { c ∈ A | Hom( b, c ) = 0 ∀ b ∈ B } . Then an arbitrary a ∈ A has a unique presentation by a distinguished triangle b → a → c → b [1]for b = j ∗ j ! a ∈ B and c ∈ C . We write this as A = h C, B i and call it a semi-orthogonal decomposition of A . We note that the formHom( a, b ) is similar to a bilinear form on A , but it is not symmetric. Thusthe orthogonality is only one-sided. Indeed, D b (Coh( X )) is indecomposablewith respect to Hom as long as X is irreducible ([5]). We denote by h D, C, B i for h D, h C, B ii , etc. For example, if A is of finite type and B is saturated,then B is always right admissible. Conversely, if A is saturated and B isright admissible, then B is saturated.If A has a Serre functor S A , then B has also a Serre functor given by S B = j ! S A j ∗ , becauseHom B ( a, b ) ∼ = Hom A ( j ∗ a, j ∗ b ) ∼ = Hom A ( j ∗ b, S A j ∗ a ) ∗ ∼ = Hom B ( b, j ! S A j ∗ a ) ∗ a, b ∈ B . j ∗ has also a left adjoint j ∗ defined by j ∗ = S − B j ! S A , becauseHom A ( a, j ∗ b ) ∼ = Hom A ( j ∗ b, S A a ) ∗ ∼ = Hom B ( b, j ! S A a ) ∗ ∼ = Hom B ( j ! S A a, S B b )for a ∈ A and b ∈ B . Lemma 2.1.
Assume that X is singular. Then Perf ( X ) is not saturated.Proof. Assume the contrary and let x ∈ X be a singular point. Assume thatthe contravariant functor Hom( • , O x ) on Perf( X ) is represented by an object c ∈ Perf( X ): Hom( • , O x ) ∼ = Hom( • , c ) . Let x ′ = x be another point and b a coherent sheaf supported at x ′ such that b ∈ Perf( X ) as an object in D b (Coh( X )). Then we haveHom( b, c ) ∼ = Hom( b, O x ) = 0 . Hence Supp( c ) = { x } . SinceHom( O X , c ) ∼ = Hom( O X , O x ) ∼ = k we conclude that c is a sheaf of length 1 supported at x , but the latter is notcontained in Perf( X ). The minimal model program consists of three kinds of basic operations,namely, Mori fiber spaces, divisorial contractions and flips (see for exam-ple [10]). The latter two are birational maps which decrease the canonicaldivisor K . The following are the simplest examples of these operations forsmooth projective varieties. The point is that semi-orthogonal decomposi-tions of the derived categories are parallel to the decompositions of canonicaldivisors. Example 3.1.
In this example, we consider only smooth projective varieties X , Y , etc. For simplicity, we write D ( X ) for the bounded derived categoryof coherent sheaves D b (Coh( X )), etc.41) ([13]) Let f : X → Y be a projective space bundle associated to avector bundle of rank r . This is a Mori fiber space. The functor f ∗ : D ( Y ) → D ( X ) is fully faithful, and there is a semi-orthogonal decomposition D ( X ) = h D ( Y ) − r +1 , . . . , D ( Y ) − , D ( Y ) i where the D ( Y ) i denote the subcategories f ∗ D ( Y ) ⊗ O X ( i ) of D ( X ) for thetautological line bundle O X (1).(2) ([13]) Let C be a smooth subvariety of Y of codimension r ≥ f : X → Y the blowing up along C . This is a divisorial contraction. Theexceptional divisor E is a projective space bundle over C as in (1). Let f E : E → C be the induced morphism and j : E → X the embedding. Thenthe functors f ∗ : D ( Y ) → D ( X ) and j ∗ f ∗ E : D ( C ) → D ( X ) are fully faithful,and there is a semi-orthogonal decomposition D ( X ) = h j ∗ f ∗ E D ( C ) − r +1 , . . . , j ∗ f ∗ E D ( C ) − , f ∗ D ( Y ) i where j ∗ f ∗ E D ( C ) i = j ∗ f ∗ E D ( C ) ⊗O X ( − iE ). We have a corresponding equalityof canonical divisors K X = f ∗ K Y + ( r − E. (3) ([3]) Let E be a subvariety of X which is isomorphic to a projec-tive space P r − and such that the normal bundle N E/X is isomorphic to O P r − ( − s . Let f : Y → X be the blowing up along E . Then there isa blowing down f + : Y → X + of the exceptional divisor F = f − ( E ) toanother direction such that E + = f + ( F ) is isomorphic to P s − and suchthat the normal bundle N E + /X + is isomorphic to O P s − ( − r . If r > s , thenthis is a flip, while if r = s , then it is a flop. If r ≥ s , then the functor f ∗ f + ∗ : D ( X + ) → D ( X ) is fully faithful, and there is a semi-orthogonaldecomposition D ( X ) = hO E ( s − r ) , . . . , O E ( − , f ∗ f + ∗ D ( X + ) i where the subcategories hO E ( i ) i generated by the sheaves O E ( i ) are denotedby O E ( i ) for simplicity. In particular, we have an equivalence of triangu-lated categories D ( X ) ∼ = D ( X + ) for the flop case. We have a correspondingequality of canonical divisors f ∗ K X = f + ∗ K X + + ( r − s ) F.
5t is important to deal with singular varieties in the minimal model pro-gram. Therefore, we have to define good derived categories for such varieties.The simplest case is the one with quotient singularities. For a variety X withonly quotient singularities, we can naturally associate a smooth Deligne-Mumford stack X . The set of points of the stack X is the same as that of thevariety X , but the points on X have automorphism groups corresponding tothe stabilizer groups of the points on X . The sheaves on X have actions bythese groups. Let D b (Coh( X )) be the bounded derived category of coherentsheaves on X . For example, if X = M/G is the quotient of a smooth variety M by a finite group G , then D b (Coh( X )) = D b (Coh G ( M )) is the boundedderived category of G -equivariant coherent sheaves on M . The followingexample suggests that the above D b (Coh( X )) is the correct answer to ourproblem for varieties with only quotient singularities (cf. [8] and [9] for morejustifications). Example 3.2. ([7]) Let X be a 4 dimensional smooth projective variety, E a subvariety which is isomorphic to a projective plane P and such thatthe normal bundle N E/X is isomorphic to O P ( − ⊕ O P ( − f : X → X be the blowing up along E . Then the exceptional divisor F = f − ( E ) contains a subvariety E which is isomorphic to P and the normalbundle N E /X is isomorphic to O P ( − . Let f : Y → X be the furtherblowing up along E . Then there is a blowing down f +1 : Y → X +1 of theexceptional divisor F = f − ( E ) to another direction such that E +1 = f +1 ( F )is isomorphic to P . The strict transform F ′ of F on X +1 is isomorphic to P and the normal bundle N F ′ /X +1 is isomorphic to O P ( − f +2 : X +1 → X + be the blowing down of F ′ . The image Q = f +2 ( F ′ ) is an isolated quotientsingularity with stabilizer group Z / f +2 f +1 f − f − is a flop. Namely, we have an equality f ∗ f ∗ K X = f + ∗ f + ∗ K X + . Correspondingly, we have an equivalence of derivedcategories f +2 ∗ f +1 ∗ f ∗ f ∗ : D b (Coh( X )) ∼ = D b (Coh( X + ))where we consider the associated stack X + instead of the underlying vari-ety with a quotient singularity X + . But the functor π ∗ : D b (Coh( X + )) → D b (Coh( X + )) induced by the projection π : X + → X + is not an equivalence.For example, f +2 ∗ f +1 ∗ f ∗ f ∗ (Ω E ( − O Q (1)and π ∗ O Q (1) ∼ = 0, where O Q (1) is a skyscraper sheaf of length 1 supportedat Q on which the stabilizer group acts non-trivially. Thus an ordinary sheaf6 E ( −
1) on X corresponds to an equivariant sheaf O Q (1) on the imaginarystack X + which disappears on the real variety X + .The following example shows that the semi-orthogonal decompositions ofderived categories are governed by the inequalities of canonical divisors andnot by the directions of morphisms. This fact suggests a distinguished statusof the canonical divisors in the theory of derived categories. We note thatthe derived categories contain almost all information on varieties in a similarway like the motives (cf. [14]). Example 3.3. ([9]) Let X be a smooth projective variety of dimension n which contains a divisor E being isomorphic to a projective space P n − andsuch that the normal bundle is isomorphic to O P n − ( − k ) for an integer k > f : X → Y be the blowing down of E . Then Y has an isolated quotientsingularity Q whose stabilizer group is isomorphic to Z /k . Let Y be theassociated smooth Deligne-Mumford stack, and let Z = X × Y Y be the fiberproduct with projections π : Z → X and ˜ f : Z → Y . We have an equality K X = f ∗ K Y + n − kk E. If n > k , then we have K X > f ∗ K Y . Correspondingly, the functor π ∗ ˜ f ∗ : D b (Coh( Y )) → D b (Coh( X )) is fully faithful, and there is a semi-orthogonaldecomposition D b (Coh( X )) = hO E ( − n + k ) , . . . , O E ( − , π ∗ ˜ f ∗ D b (Coh( Y )) i . On the other hand, if n < k , then we have K X < f ∗ K Y . Correspondingly,the functor ˜ f ∗ π ∗ : D b (Coh( X )) → D b (Coh( Y )) is fully faithful, and there isa semi-orthogonal decomposition D b (Coh( Y )) = hO Q ( − n ) , . . . , O Q ( − k + 1) , ˜ f ∗ π ∗ D b (Coh( X )) i where O Q ( i ) denotes a skyscraper sheaf of length 1 at Q with a suitable actionby the stabilizer group. In particular, if n = k , then there is an equivalence π ∗ ˜ f ∗ : D b (Coh( Y )) ∼ = D b (Coh( X )).We note that the functors are given by the pull-backs and the push-downsas in the case of flips. Indeed, divisorial contractions and flips are very similaroperations from the view point of the minimal model program.The above picture extends for Q -factorial toric varieties ([9]):7 heorem 3.4. Let f : X − → Y be a toric divisorial contraction or flipbetween Q -factorial projective toric varieties, and let X and Y be their as-sociated smooth Deligne-Mumford stacks. Then there is a semi-orthogonaldecomposition D b ( Coh ( X )) ∼ = h C, D b ( Coh ( Y )) i which is described in detail in terms of toric fans for X and Y . We note that Q -factorial toric varieties have only quotient singularities.We have similar results for toric Mori fiber spaces. The theorem has a logversion in the case where the coefficients of the boundary are of the form1 − m for some positive integers m . In particular, toric flops induce derivedequivalences. We refer to [9] for details. As a corollary, we obtain the McKaycorrespondence for abelian quotient singularities: Corollary 3.5.
Let X be a projective variety with only quotient singularitieswhose stabilizer groups are abelian groups whose orders are prime to thecharacteristic of the base field, and let f : Y → X be a projective crepantresolution, i.e., Y is smooth, f is projective and birational, and K Y = f ∗ K X .Then there is an equivalence of triangulated categories D b ( Coh ( Y )) ∼ = D b ( Coh ( X )) . Proof.
We may replace X by its local model and assume that X is toric by[5]. Then there exists a toric crepant Q -factorial terminalization f ′ : Y ′ → X .Let H be an f -ample divisor on Y and H ′ its strict transform on Y ′ . Since Y ′ is toric, H ′ is linearly equivalent to a toric divisor which is denoted by H ′ again. We proceed by MMP with respect to ( Y ′ , ǫH ′ ) over X for a smallpositive number ǫ . Since the pair is toric, the process is a toric MMP. Afterfinitely many steps, we reach a log minimal model that is isomorphic to Y .Therefore, f : Y → X is also toric. The MMP over X starting from ( Y, B )for a suitable toric boundary B ends at X . Therefore, Y and the smoothDeligne-Mumford stack over X are derived equivalent by the theorem (seealso [8]).There is a warning. The derived category may have semi-orthogonal de-compositions beyond the minimal model program. For example, the derivedcategory of a projective space has a complete semi-orthogonal decompositionto exceptional objects by [1]. This fact is extended to an arbitrary Q -factorialprojective toric variety ([9]). The derived categories of some Fano manifolds8ave interesting semi-orthogonal decompositions which reflect the geometryof these manifolds ([11]). Even minimal varieties such as Enriques surfaceshave derived semi-orthogonal decompositions. Therefore, MMP is only apreparation as in the case of the classification of surfaces. After that, deeperdecompositions may be possible like in the case of decompositions of motives. -folds We would like to define a correct category for an arbitrary variety whichappears in the minimal model program, or in the Mori category. Thus let X be a projective variety with only terminal singularities, and f : Y → X aresolution of singularities. As a working hypothesis, we look for a minimalsaturated subcategory D = D ( X ) of D b (Coh( Y )) which contains f ∗ Perf( X ): f ∗ Perf( X ) ⊂ D ( X ) ⊂ D b (Coh( Y ))if it exists and unique up to equivalence. If X is smooth, then we have D ( X ) = D b (Coh( X )).As a first step, we consider a divisorial contraction f : Y → X of asmooth 3 dimensional variety to a singular variety. The morphism f is anisomorphism outside a prime divisor E on Y , and P = f ( E ) is the singularpoint of X . There are three cases, where two of them has already answersdescribed in §
3. The above working hypothesis seems to produce the samecategories in these cases. E is isomorphic to a smooth quadric surface P × P with normal bundle O E ( − , − E are numerically equivalent toeach other on X .Let C = hO E ( − , − , O E (0 , − i ⊂ D b (Coh( Y ))be the subcategory generated by a sequence of exceptional objects, and let D = ⊥ C . Since C is equivalent to D b (Coh( P )), it is admissible, and D issaturated. Since f ∗ c = 0 for any c ∈ C , we have f ∗ p ∈ D for p ∈ Perf( X ),thus f ∗ Perf( X ) ⊂ D .If k = C , then there are two small resolutions g i : Y i → X ( i = 1 ,
2) in theanalytic category, with analytic divisorial contractions f i : Y → Y i such that9 i ( E ) is isomorphic to P . We can check that D = f ∗ D b (Coh( Y )), and thelatter should be the correct category since Y is smooth. Since g is crepant,it follows that S D ( d ) ∼ = d [3] if d ∈ D and f ∗ d = 0. We can also prove thisfact for general k , because these objects are concentrated on the divisor E and the global structure of X is irrelevant. Lemma 4.1. D is minimal in the sense that D has no semi-orthogonaldecomposition relative to X , i.e., a semi-orthogonal decomposition such thatone of the factors contains f ∗ Perf ( X ) .Proof. Suppose there is still a semi-orthgonal decomposition D = h C ′ , D ′ i such that f ∗ Perf( X ) ⊂ D ′ . Since C ′ ⊂ ( f ∗ Perf( X )) ⊥ , we have f ∗ c = 0 for c ∈ C ′ . Then we have for c ∈ C ′ and d ∈ D ′ Hom( c, d ) ∼ = Hom( d, c [3]) ∗ ∼ = 0 . Hence D is decomposable; D = C ′ ⊕ D ′ .Assuming that k = C , let a y = f ∗ O y ∈ D for a point y ∈ Y . SinceHom( a y , a y ) ∼ = k , we have either a y ∈ C ′ or a y ∈ D ′ . Since f ∗ a y = 0, weconclude that a y ∈ D ′ . But let c ′ ∈ C ′ be a non-zero object, and write c ′ = f ∗ c . If we take a point y in the support of c , then we have Hom( c, O y [ p ]) = 0for some p , hence Hom( c ′ , a y [ p ]) = 0. But this is a contradiction. For general k , we can still define a y because it is supported on E , and the above argumentworks.If we take C = hO E ( − , − , O E ( − , i ⊂ D b (Coh( Y ))instead of C and define D = ⊥ C , then D and D are equivalent, because Y and Y are related by a standard flop.There is another candidate for an admissible subcategory which is moresymmetric than C or C . Let˜ C = hO E ( − , , O E (0 , − i ⊂ D b (Coh( Y ))and let ˜ D = ⊥ ˜ C be the right orthogonal. We have again f ∗ Perf( X ) ⊂ ˜ D .The generators of ˜ C are exceptional objects,Hom( O E ( − , , O E (0 , − ∼ = Hom( O E (0 , − , O E ( − , ∼ = k and all the other sets of morphisms between their shifts vanish. Thus ˜ C is not equivalent to C or C . However, we can prove that ˜ C is indeed notadmissible: 10 emma 4.2. ˜ C is not saturated.Proof. Since O E ( − ,
0) is exceptional, it generates a saturated subcategoryof ˜ C . Therefore, it is sufficient to prove that its left orthogonal˜ C ′ = ⊥ hO E ( − , i ⊂ ˜ C is not saturated. We decompose the other object O E (0 , −
1) by a distin-guished triangle c → O E (0 , − → O E ( − , → c [1] . where c ∈ ˜ C ′ . Then ˜ C ′ is generated by c , and we haveHom( c , c [ p ]) ∼ = ( k if p = 0 ,
30 otherwise.Thus ˜ C ′ is equivalent to a category generated by an object O M in the derivedcategory D b (Coh( M )) for a Calabi-Yau 3-fold M . Such an object is called a spherical object . Since D b (Coh( M )) is indecomposable and its Serre functoris isomorphic to a shift functor [3], hO M i is not saturated, hence neither is˜ C ′ . We can also check this fact directly. Let F : ( ˜ C ′ ) op → ( k -vect) be acohomological functor such that F ( c ) ∼ = k and F ( c [ p ]) = 0 for p = 0. Then F is not representable by any object c ∈ ˜ C ′ . Indeed, if F ( c ) ∼ = Hom( c, c )for arbitrary c ∈ ˜ C ′ , then P p ( − p dim Hom( c [ p ] , c ) should be even, acontradiction. E is isomorphic to a projective plane P , and the normal bundle of E isisomorphic to O E ( − C = hO E ( − i ⊂ D b (Coh( Y )) . Since O E ( −
1) is an exceptional object, C is an admissible subcategory. Wenote that ( O E ( − , O E ( − O E ( − , O E ( − ∼ = Hom( O E ( − , O E ( − ∗ = 0 . X be the smooth Deligne-Mumford stack associated to the variety X which has only a quotient singularity P , and let π : X → X be the projection.We know that D ∼ = D b (Coh( X )) by [9], and the latter is the correct category.We shall identify these categories in the following.We have f ∗ O P (1) = 0. But O P (1) is not an exceptional object, becauseHom ( O P (1) , O P (1)) = 0. We know also that S D ( d ) ∼ = d [6] if d ∈ D and f ∗ d = 0, but S D ( O P (1)) ∼ = O P [3] = O P (1)[3] . Lemma 4.3. D is minimal.Proof. Suppose that there is a semi-orthgonal decomposition D = h C ′ , D ′ i such that f ∗ Perf( X ) ⊂ D ′ . Let c ∈ C ′ be a non-zero object. Since f ∗ c = 0, c is supported at the point P . Since Hom( O X , c [ n ]) = 0 for any n , H n ( c )should be of the form O P (1) c n . Any object d ∈ D ′ has a finite resolutionwhose terms are of the form O a n X ⊕ O X (1) b n near P . If b n = 0 for some n and for any choice of such a resolution, then we have Hom( d, c [ m ]) = 0 forsome m , a contradiction. Therefore, b n = 0 for all n , and d ∈ f ∗ Perf( X ), acontradiction to Lemma 2.1. E is isomorphic to a singular quadric surface, and the normal bundle of E isisomorphic to O E ( − l be a ruling. Then O E (1) ∼ = O E (2 l ). There is an exact sequence0 → O E ( − l ) → O E ( − l ) → O E ( − l ) → . The Serre functor of D b (Coh( Y )) is given by ⊗O Y ( E )[3], and we haveHom( O E ( − l ) , O E ( − l )[2]) ∼ = Hom( O E ( − l ) , O E ( − l )[1]) ∗ = 0hence O E ( − l ) is not an exceptional object.Let C = hO E ( − i and D = ⊥ C . Since O E ( −
1) is an exceptional object, C is admissible. Wehave O E ( − l ) ∈ D . The above sequence implies that j ! O E ( − l ) ∼ = O E ( − l )[ − j ! is the right adjoint to j ∗ : D → D b (Coh( Y )). Therefore, S D ( O E ( − l )) ∼ = O E ( − l )[2] . We expect that if c ∈ D is right orthogonal to f ∗ Perf( X ), then S D ( c ) ∼ = c [2]. We note that the shift number 2 is different from usual 3 = dim X . Wecan only prove a partial result: Lemma 4.4.
Let c ∈ D such that c ∈ ( f ∗ Perf ( X )) ⊥ . Assume in additionthat there exists d ∈ D b ( Coh ( E )) such that c = i ∗ d for i : E → Y . Then S D ( c ) ∼ = c [2] .Proof. By a generalization of [1], d has a two sided resolution whose termsare direct sums of the sheaves O E , O E ( − l ) and O E ( − l ). SinceHom E ( O E , d [ n ]) ∼ = Hom Y ( O Y , c [ n ]) = 0for any n , the terms do not contain O E . On the other hand, sinceHom Y ( c, O E ( − l )[ n ]) = 0for any n , the terms do not contain O E ( − l ) either. Since S D ( O E ( − l )) ∼ = O E ( − l )[2], we have our assertion.Let B be the full subcategory of D b (Coh( Y )) consisting of objects whosesupports are contained in E . Then B is generated by O E -modules as atriangulated category. But we note that the inclusion functor D b (Coh( E )) → B is not fully faithful because there are more extensions in B .In order to prove the above expectation, one would need more geometricargument. Anyway, if it is true, then the minimality of the category D willfollow from its indecomposability. Lemma 4.5. D is indecomposable.Proof. Assume that D = C ′ ⊕ D ′ . Since Hom( O Y , O Y ) ∼ = k , O Y ∈ D isindecomposable. We may assume that O Y ∈ D ′ .Let x ∈ E be an arbitrary point, and let d = j ! O x for the right adjointfunctor j ! of j : D → A = D b (Coh( Y )). We have a distinguished triangle d → O x → Hom • ( O x , O E ( − ∗ ⊗ O E ( − → d [1] . O E ( − ⊕ O E ( − d, d ) ∼ = Hom( d, O x ) ∼ = Hom( O x , O x ) ∼ = k because Hom( O E ( − k ] , O x ) = 0for k >
0. Therefore, d is indecomposable. SinceHom( O Y , d ) ∼ = Hom( O Y , O x ) = 0we have d ∈ D ′ .Let c ∈ C ′ be an arbitrary object such that c = 0. Then the support of c is contained in E . Thus there exists x and p such thatHom( c, d [ p ]) ∼ = Hom( c, O x [ p ]) = 0a contradiction. Based on the above observation, we would like to ask the following questions:(1) Let X be a projective variety with only terminal singularities, and f : Y → X a resolution of singularities. Does there exist a minimal satu-rated subcategory of D b (Coh( Y )) which contains f ∗ Perf( X )? Are two suchminimal saturated subcategories equivalent?(2) More generally, let X be a smooth projective variety. Does the cat-egory D b (Coh( X )) have finite length with respect to semi-orthogonal de-compositions? Does a Jordan-H¨older type theorem hold for semi-orthogonaldecompositions of saturated categories?(3) It would be nice if we have more method for testing the admissibilityof a subcategory in general situation (cf. Lemma 4.2).(4) Can a geometric argument be given to establish the expectation fromSubsection 4.3? (added by a referee) References [1] A. A. Beilinson.
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