Base change for semiorthogonal decompositions

BASE CHANGE F OR SEMIOR THOGONAL DECO MPOSI TIONS ALEXANDER KUZNETSOV Abstra ct. Let X b e an algebraic v ariet y o ve r a base scheme S and φ : T → S a base c hange. Give n an admissible subcategory A in D b ( X ), the b ounded derived ca tegory of coherent s heav es on X , w e construct under some technical conditions an admissible sub category A T in D b ( X × S T ), called the base change of A , in such a w a y that th e follow ing base change theorem holds: if a semiorthogonal decomp osition of D b ( X ) is giv en th en the b ase c hanges of its components form a semiorthogonal decomp osition of D b ( X × S T ). As an intermedia te step we construct a compatible system of semiorthogonal decomp ositions of the unb ounded derived category of q uasicoheren t shea ve s on X and of the category of p erfe ct complexes on X . As an application w e prov e that the pro jection functors of a semiorthogonal decomp osition are ke rnel functors. 1. Introduction An imp ortant approac h to the noncomm utat iv e algebraic geometry is t o consider triangulated cate - gories with go o d prop erties as sub stitutes for noncommutativ e v arieties. Giv en su ch category w e consider it as the b oun ded deriv ed category of coheren t shea ves on a w ould-b e v ariety and tr y to do some ge- ometry . Note ho w ev er that eve n the simplest geometric fu nctors b etw een deriv ed categories often d o not preserv e b ounded n ess or coherence — the pullbac k functor preserv es b oundedness only if the corr e- sp ond in g morphism has finite T o r -d imension and the p ushforward functor preserv es coherence only if the corresp ondin g map is prop er. So, to d o noncommutativ e geometry we need some unb ounded and quasi- coheren t v ersions of triangulated categories under consideration. On e goal of this pap er is the follo wing: giv en a go o d triangulated ca tegory A (co nsidered as a b ounded derived category of coheren t shea ve s) to define a ca tegory A q c whic h is a su bstitute for the unboun ded derive d ca tegory of quasicoheren t sheav es and a category A − , a substitute for the b ounded ab ov e deriv ed catego ry of co herent shea v es. A straigh tforward approac h to constru ct A q c w ould b e just to consider the clo sure of A und er col imits. Ho wev er it is n ot clear ho w to define a tr iangulated structure there. So, instead, w e assume that the catego ry A is giv en as an admissible sub category in D b ( X ), the b ounded derived category of coheren t shea ves on some algebraic v ariet y X , and consider the minimal triangulated sub cat egory ˆ A ⊂ D q c ( X ) con taining A an d closed under arbitrary direct sums. Defined this wa y the catego ry ˆ A inherits a triangu- lated s tr ucture automatically , but there arises a question of dep endence of ˆ A on the c h oice of the v ariet y X and of the emb edding A → D b ( X ). W e pro ve that it is actually indep endent of these c hoices under some te c h nical condition. Another, and in f act the most imp ortant goal of the p ap er, is to define a base change for triangulated catego ries. Assume that S is an algebraic v ariet y and A is a go o d triangulate d categ ory o v er S (whic h can b e under s to o d, for example, as that A is a mo dule category ov er the tensor triangulate d categ ory D perf ( S ) of p erfect complexes on S ). Giv en a base c h ange φ : T → S we would like to defi n e a triangulated category A T o ver T to b e considered as the base c h ange of A . Again, an a bstract approac h is to o complica ted, so we assume that A is giv en as an S -linear adm issible sub category in D b ( X ) ( S -linear means closed under tensoring with p ullbac ks of p erfect complexes on S ), where X is an alg ebraic v ariet y o v er S , and construct A T as a certain triangulated sub category in D b ( X × S T ). On ce aga in there arises an issue of I was partially sup p orted by RFFI gra nts 07-01-00051, 07-01-9221 1, and 08-01-00297, NS h 1987.20 08.1, Russian Presi- dential grant for young scien tists N o. MD-2712.2009.1 . 1 dep end ance on the chosen em b edd ing A → D b ( X ), and aga in w e sho w that the result is indep end ent of the choic e. The most im p ortan t tec hnical notion used in the pap er is that of a semiorthogo nal decomp osi- tion. Actually , we start not with an admissible s ub category A ⊂ D b ( X ) but with a s emiorthogo- nal decomp osition D b ( X ) =  A 1 , A 2 , . . . , A m  . Then we consider a c hain of triangulated categories D perf ( X ) ⊂ D b ( X ) ⊂ D − ( X ) ⊂ D q c ( X ) (here D − ( X ) is the deriv ed categ ory of b ounded ab o ve co mplexes with coheren t cohomology) and ask whether there exist semiorthogonal d ecomp ositions of th ese catego ries compatible with the initial decomp ositio n. It tu r ns out that the categories A perf i = A i ∩ D perf ( X ) alwa ys giv e a semiorthogonal decomp osition of D perf ( X ), while the catego ries ˆ A i (the minimal triangulated sub - catego ries of D q c ( X ) con taining A perf i and closed under a rbitrary direct sums) and A − i = ˆ A i ∩ D − ( X ) alw ays f orm semiorthogonal deco mp ositions of D q c ( X ) and D − ( X ) resp ectiv ely . Ho wev er, for compatibil- it y of the last t wo decomp ositions with the initial decomp ositio n of D b ( X ), we need a tec hnical condition to b e satisfied, namely the righ t cohomologica l amplitude of the pro jection functors of the initial d ecom- p osition s h ould be fi nite (this condition holds automatically if X is smo oth). Similarly , in a situatio n of a base change w e s tart with a semiorthogonal decomp osit ion of D b ( X ). Ho wev er, here w e need some a dditional assumptions from the ve ry b eginning. First of a ll the deco mp o- sition of D b ( X ) should b e S -linear, and second, th e base c hange φ : T → S should b e faithful for the pro jection f : X → S . The latter condition m ore or less by definition (see [K1]) is equiv alent to the base c han ge isomorphism f ∗ φ ∗ ∼ = φ ∗ f ∗ , where the pro jections of X T = X × S T to X and T b y an abu se of notation are d enoted b y φ and f resp ectiv ely . The semiorthogonal decomp osition of D b ( X T ) is constructed in sev eral steps. First, w e consider t he semiorthogonal decomp osition of D perf ( X ) constructed ab o v e. Then we define the s ub category A p iT of D perf ( X T ) to b e the closed u nder direct summands triangulated sub categ ory generate d by ob jects of the form φ ∗ F ⊗ f ∗ G w ith F ∈ A perf i and G ∈ D perf ( T ). It turn s out that acting this w a y we alw ays obtain a semiorthogo nal decomp osition of D perf ( X T ). F urther, we define the cate gory ˆ A iT to b e the minimal triangulated sub categ ory of D q c ( X T ) cont aining A p iT and closed und er arbitrary d irect su ms, and A − iT = ˆ A iT ∩ D − ( X T ). T h us we obtain semiorthogonal decomp ositions of D q c ( X T ) and D − ( X T ). Finally w e consider sub cat egories A iT = A − iT ∩ D b ( X T ) ⊂ D b ( X T ). But to pro ve that they form a semiorthogonal decomp osition w e again n eed the assu m ption of fi n iteness of cohomological amplitude of the pro jectio n functors of the initial semiorthogo nal d ecomp osition of D b ( X ). W e pro v e that the pro jection fu nctors of the obtained decomp osition of D b ( X T ) also ha ve fin ite cohomological amplitude. W e show that the constru cted semiorthogonal decomp osit ions of D q c ( X ) and D q c ( X T ) are compatible with resp ect to the p ushforward and the pullbac k f unctors via th e pr o jection φ : X T → X . I t follo w s, that the semiorthogonal decomp ositions of D b ( X ) and D b ( X T ) are compatible w ith r esp ect to φ ∗ whenev er φ is pr op er, and with resp ect to φ ∗ whenev er φ has finite T o r -dimen s ion. It sh ou ld b e men tioned, that seemingly to o complicated p ro cedure of constru cting A iT is pr obably inevitable. The straigh tforw ard approac h of taking f or A iT the sub categ ory of D b ( X T ) generated by ob jects of the form φ ∗ F ⊗ f ∗ G with F ∈ A i and G ∈ D b ( T ) doesn’t giv e the desired result ev en when b oth φ an d f h a ve finite T or -dimension. Ind eed, assume that A i = D b ( X ) and X is smo oth. Th en D b ( X ) = D perf ( X ) and it is clear that defined this w ay sub category of D b ( X T ) is just the cate gory of p erfect complexes D perf ( X T ), n ot the whole D b ( X ) as one w ould w ish. So, one defin itely needs to add something to this catego ry to obtain the righ t answe r. It seems th at to add all colimits and th en to in tersect with D b ( X T ) is the simplest p ossible solution. And consid ering p erfect complexes as an in termediate step b oth remo v es man y tec hnical problems and giv es an additional information. As an application of the obtained results w e pr o ve the follo wing. Assume that D b ( X ) =  A 1 , . . . , A m  is a semiorthogonal decomp osition the p ro jectio n functors of whic h h av e fi nite cohomological amplitude. 2 W e pro v e then that these functors are isomorphic to k er n el functors Φ K i giv en b y some explicit k er n els K i ∈ D b ( X × X ). In particular, if A ⊂ D b ( X ) is an admissible sub ca tegory and the pro jection fun ctor to A has finite cohomologica l amplitude then it is isomorphic to a ke rnel functor. In a sp ecial case, wh en A ∼ = D b ( Y ) for a smo oth pro jectiv e v ariet y Y this follo ws from the Orlov’s Theorem on represen tabilit y of fully faithful functors [O1]. Ind eed, in this case the emb edding functor D b ( Y ) → D b ( X ) as w ell a s its adjoin t are g iv en by appropr iate kernels on X × Y , so the pro jecti on f unctor is gi v en by the co nv olution of these k ern els. Thus, our result can b e considered as a generaliza tion of Orlo v’s Theorem. The pap er is organize d as follo ws. In Sectio n 2 w e remind the main tec hnical notions used in the pap er — semiorthogonal decomp ositions, cohomological amplitude, homotop y colimits e.t.c. W e also discu ss sev eral notions and facts related to app ro xim ation of un b ound ed quasicoheren t complexes b y p erfect ones. In Section 3 we in vesti gate when a semiorthogonal decomp osition of a triangulated category T ′ induces a semiorthogonal decomp osition of its full tr iangulated sub cat egory T ⊂ T ′ . In Section 4 w e construct extensions of a semiorthogonal decomposition of D b ( X ) to D perf ( X ) ⊂ D − ( X ) ⊂ D q c ( X ). In Section 5 we define the base c hange for an admissible su b category and pro v e the faithfu l base c hange Theorem. In Section 6 w e sho w that extensions ˆ A , A − and the base c hange A T of A do not dep end on the c hoice of X a nd of th e embedd ing A → D b ( X ) inv olv ed in the definitions. In S ection 7 we pro v e that the pr o jectio n functors of a semiorthogonal decomp osition can b e represen ted as kernel fun ctors. Ac knowledgemen ts: I wo uld like to thank A. Bondal, D. Kaledin, D. Orlo v and L. P ositselski for v ery helpful d iscussions. My sp ecial thanks to A. Elagin who suggested a sig nificant impr o vemen t of the previous v ersion of this paper. 2. Preliminar ies 2.1. Notation. All alge braic v arieties are assu med to b e quasipro jective . F or an algebraic v ariet y X , we denote b y D b ( X ) the b ounded deriv ed category of coherent shea ves on X , b y D − ( X ) the b ounded ab ov e derived category of coherent sh ea ve s on X , and b y D q c ( X ) the unboun ded deriv ed category of quasicoherent shea v es on X . Reca ll that an ob j ect F ∈ D q c ( X ) is a p erfect complex if it is lo cally q u asiisomorphic to a b ound ed complex of lo cally fr ee shea ves of finite r ank. Recall th at p erfect complexes a re precisely compact objects in D q c ( X ), i.e. if P is p erfect then Hom ( P , ⊕ α F α ) ∼ = ⊕ α Hom ( P , F α ) for an y system F α ∈ D q c ( X ). W e denote b y D perf ( X ) th e full sub cate gory of D q c ( X ) consisting of p erf ect complexes. Note that D perf ( X ) is a triangulated sub categ ory in D b ( X ). Giv en an ob ject F ∈ D q c ( X ) we denote b y H i ( F ) the i -th co homology sheaf of F . F or F , G ∈ D q c ( X ), w e denote b y R H om ( F , G ) the local R H om -complex and b y F ⊗ G the deriv ed tensor pr o duct. Similarly , for a map f : X → Y , we denote b y f ∗ : D q c ( X ) → D q c ( Y ) the deriv ed pushf orward functor and by f ∗ : D q c ( Y ) → D q c ( X ) the deriv ed pullbac k functor. W e refer to [KSch] for the definition of t hese functors. W e also denote b y f ! : D q c ( Y ) → D q c ( X ) the righ t adjoin t fun ctor of f ∗ (usually it is referred to as the t wisted pullbac k functor). It exists by [N2] (see also [KSc h ]). If the morphism f is smo oth th en f ! ( F ) ∼ = f ∗ ( F ) ⊗ ω X/ Y [dim X − dim Y ] again b y [N2]. Giv en a class E of ob jects in a triangulated category T we den ote b y  E  the minimal strictly full triangulated sub categ ory in T conta ining all ob jects in E and clo sed un der taking direct su m mands. W e sa y that E generates T if T =  E  . 2.2. Semiorthogonal decomp ositions. Giv en a class E of ob jects in a triangulated category T we denote the ri ght a nd t he left o rthogonal to E by E ⊥ = { T ∈ T | Hom ( E [ k ] , T ) = 0 for all E ∈ E and all k ∈ Z } , ⊥ E = { T ∈ T | Hom ( T , E [ k ]) = 0 f or all E ∈ E and all k ∈ Z } . 3 It is clear that b ot h E ⊥ and ⊥ E are triangulated sub categ ories in T closed under taking direct summands. The classes E 1 , E 2 ⊂ T are ca lled semio rthogonal if E 1 ⊂ E ⊥ 2 , or equiv alen tly E 2 ⊂ ⊥ E 1 . Lemma 2.1. If classes E 1 and E 2 ar e semiorth o gonal then the sub c ate gories  E 1  and  E 2  ar e semiorth o g- onal as wel l. Pro of: W e ha ve E 1 ⊂ E ⊥ 2 , hence  E 1  ⊂ E ⊥ 2 , hence E 2 ⊂ ⊥  E 1  , hence  E 2  ⊂ ⊥  E 1  .  Definition 2.2 ([BK, BO1, BO2]) . A s emiorthog onal decomp osition of a triangulated catego ry T is a sequence of full triangulated s u b categories A 1 , . . . , A m in T suc h that A i ⊂ A ⊥ j for i < j and for every ob ject T ∈ T there exists a c hain of morphisms 0 = T m → T m − 1 → · · · → T 1 → T 0 = T such that the cone of the morphism T k → T k − 1 is con tained in A k for eac h k = 1 , 2 , . . . , m . In other words, there exists a d iagram 0 T m / / T m − 1 ~ ~ | | | | | | | | / / . . . / / T 2 / / T 1          / / T 0          T A m _ _ . . . A 2 ] ] A 1 ] ] (1) where a ll triangle s are distinguish ed (d ashed arr o ws ha ve d egree 1) and A k ∈ A k . Th us, every ob ject T ∈ T a dmits a decreasing “filtration” w ith factors in A 1 , . . . , A m resp ectiv ely . Lemma 2.3. If T =  A 1 , . . . , A m  is a semiort ho gonal de c omp osition and T ∈ T then the diagr am (1) for T i s unique and functorial (for any morphism T → T ′ ther e exists a unique c ol le ction of morphisms T i → T ′ i , A i → A ′ i c ombining into a morphism of diagr am (1) for T i nto diagr am (1) for T ′ ). Pro of: Note that T 1 ∈  A 2 , . . . , A m  b y (1). It follo w s from the semiorthogonalit y that Hom ( T 1 , A ′ 1 [ k ]) = 0 for all k ∈ Z . T herefore an y map T 0 = T → T ′ = T ′ 0 extends in a unique w ay to a map of the triangle T 1 → T 0 → A 1 in to th e triangle T ′ 1 → T ′ 0 → A ′ 1 . In particular, we obtain a map T 1 → T ′ 1 as w ell a s a map A 1 → A ′ 1 and proceed b y induction.  W e d enote b y α k : T → T the fu nctor T 7→ A k . W e call α k the k -th projec tion functor of the semiorthogonal deco mp osition. Definition 2.4 ([B K, B]) . A full triangulated su b category A of a triangulated category T is called right admissible if for the inclusion f unctor i : A → T th er e is a r igh t adjoint i ! : T → A , a nd left admissible if there is a left adjoin t i ∗ : T → A . Su b category A is called admissible if it is b oth right and left admissible. Lemma 2.5 ([B]) . If T =  A , B  is a semiortho gonal de c omp osition then A i s left admissible and B is right admissible. Conversely, if A ⊂ T i s left admissible then T =  A , ⊥ A  is a se miortho gonal de c omp osition, and if B ⊂ T i s right admissible th en T =  B ⊥ , B  is a semiortho gonal de c omp osition. Definition 2.6. W e will say that a semiorthogo nal decomp osition T =  A 1 , . . . , A m  is a strong semio rthogonal decomp osition if for eac h k the c ategory A k is admissible in  A k , . . . , A m  . Note that A k is left adm issible in  A k , . . . , A m  b y Lemma 2.5. So the additional condition in the def- inition is the r ight admissibilit y . Not e also that if A k is right admissible in T then it is also admissible in  A k , . . . , A m  (th u s a semiorthogonal d ecomp osition with admissib le comp onents is a strong semiorthog- onal decomp osition), and that in the ca se when T = D b ( X ) with X b eing smo oth and pro jectiv e any semiorthogonal deco mp osition is strong. 4 2.3. S -linearity . Let f : X → S b e a morph ism of alge braic v arieties. A triangulated sub ca tegory A ⊂ D q c ( X ) is called S -li nea r (see [K1]) if it is sta ble with resp ect to tensoring b y p ullbac ks of p erfect complexes on S . In other w ords, if A ⊗ f ∗ F ∈ A for an y A ∈ A , F ∈ D perf ( S ). Lemma 2.7. A p air of S -line ar sub c ate gories A , B ⊂ D q c ( X ) is semiortho gonal if and only if the e quality f ∗ R H om ( B , A ) = 0 holds for any A ∈ A , B ∈ B . Pro of: First we note that for any ob ject 0 6 = G ∈ D q c ( S ) there exists a n onzero map P → G fr om a p erfect c omplex P ∈ D perf ( S ). Indeed, r ep resen t G b y a complex of quasicoheren t shea v es and assume that H i ( G ) 6 = 0. Let Z i = Ker ( G i → G i +1 ) so that we ha v e an epimorphism Z i → H i ( G ). It is clear that there exists a lo cally free sheaf P of fin ite rank and a map P → Z i suc h that the comp osition P → Z i → H i ( G ) is n onzero. Then the comp osition P → Z i ⊂ G i induces the required morph ism P [ − i ] → G (it is nonzero since the induced morph ism of the cohomolog y H i ( P [ − i ]) = P → H i ( G ) is nonzero). F urther note th at RHom ( P, f ∗ R H om ( B , A )) ∼ = RHom ( f ∗ P , R H om ( B , A )) ∼ = RHom ( B ⊗ f ∗ P , A ) f or an y P ∈ D perf ( S ). So, if A and B are semiorthogo nal then RHom ( B ⊗ f ∗ P , A ) = 0 since B is S -linear and the ab o v e observ ation sh o ws that f ∗ R H om ( B , A ) = 0. Th e inv erse is evident.  Let f : X → S and g : Y → S b e algebraic morphism s, and assu me that A ⊂ D q c ( X ), B ⊂ D q c ( Y ) are S -linear triangulated sub categories. A functor Φ : A → B is called S -linea r if there is given a functorial isomorphism Φ( F ⊗ f ∗ G ) ∼ = Φ( F ) ⊗ g ∗ G f or all F ∈ A , G ∈ D perf ( S ). Lemma 2.8. If T ⊂ D q c ( X ) is an S -line ar triangulate d sub c ate gory and T =  A 1 , . . . , A m  is an S -line ar semiortho gonal de c omp osition then its pr oje ction func tors α i : T → T ar e S -line ar. Pro of: T ak e an y G ∈ D perf ( S ) and consider th e endofunctor of T giv en b y tensoring with f ∗ G . It preserve s all A i hence by Lemma 3.1 b elo w it comm utes with the pr o jection fu nctors. This giv es the required fu nctorial isomorp h ism.  2.4. F aithful base changes. L et f : X → S and φ : T → S b e algebraic morphisms . Let X T = X × T S b e the fib er pro duct. By an abus e of notat ion d enote the pr o jecti ons X T → T and X T → X also b y f and φ resp ectiv ely . It is easy to see that there is a canonica l morphism of functors φ ∗ f ∗ → f ∗ φ ∗ . Recall that th e cartesian square X T φ / / f   X f   T φ / / S is called exact (see [K1]) if this morphism of functors is an isomorphism. By [K1] the square is exact if either f or φ is flat, and the squ are is exact if and only if the transp osed squ are is exact . A map φ : T → S considered as a change of base is called fa ithful fo r f : X → S (see [K1]) if the corresp ondin g cartesian square is e xact. Th u s an y change of base is faithful for a flat f and s imilarly a flat change of base is faithful for an y f . 2.5. T runcations. Giv en a complex C • its stupid truncations are defined as ( σ ≤ m C ) n = ( C n , if n ≤ m 0 , if n > m and ( σ ≥ m C ) n = ( C n , if n ≥ m 0 , if n < m It is clear that σ ≥ m C → C → σ ≤ m − 1 C is a distinguished triangle in the derived ca tegory . The adv an tage of the stu p id truncations which we will use sub sequen tly in the p ap er is that when applied to a complex of lo cally free shea v es (a p erfect complex) they prod uce a p erfect complex as well . 5 Similarly , the canonical truncations (also kno wn as smart truncations ) are defined as ( τ ≤ m C ) n =        C n , if n < m Ker ( d : C m → C m +1 ) , if n = m 0 , if n > m ( τ ≥ m C ) n =        C n , if n > m Cok er ( d : C m − 1 → C m ) , if n = m 0 , if n < m Again, in the deriv ed catego ry we ha v e a distinguish ed triangle τ ≤ m C → C → τ ≥ m +1 C . The adv an tage of the canonical tr u ncations is that th ey descend to fun ctors on the der ived category . Note also that H n ( τ ≤ m C ) ∼ = ( H n ( C ) , if n ≤ m 0 , if n > m and H n ( τ ≥ m C ) ∼ = ( H n ( C ) , if n ≥ m 0 , if n < m 2.6. Cohomological amplitude. Let D [ p,q ] q c ( X ) denote the full su b category of D q c ( X ) consisting of all complexes F ∈ D q c ( X ) with H i ( F ) = 0 for i 6∈ [ p, q ]. Let T ⊂ D q c ( X ) b e a triangulate d su b category . W e say that ( a, b ) is th e c ohomological amplitude of a triangulated functor Φ : T → D q c ( Y ) if Φ( T ∩ D [ p,q ] q c ( X )) ⊂ D [ p + a,q + b ] q c ( Y ) for all p, q ∈ Z . In particular, w e say that Φ has finite left (resp. righ t) cohomological amplitud e if a > −∞ (resp. b < ∞ ). If b oth a and b are finite w e sa y that Φ has finite cohomologic al amplitude. Lemma 2.9. Every exact functor Φ : D perf ( X ) → D q c ( Y ) has finite c ohomolo gic al amplitude. Pro of: The same as in [K3], Prop osition 2.5. A smo othness of X is not requ ired since w e consid er only p erfect c omplexes on X .  Let X and Y b e algebraic v arieties. Consider the pr o duct X × Y and let p : X × Y → X and q : X × Y → Y b e the pro jections. Recall (see [K1], 10.39) that an ob j ect K ∈ D b ( X × Y ) has finite T o r -ampli t ude over X if the fun ctor F 7→ K ⊗ p ∗ F has finite cohomological amplitude. Similarly , an ob ject K ∈ D b ( X × Y ) ha s finit e Ext-amplitude over Y if th e functor F 7→ R H om ( K, q ! F ) has finite cohomologi cal amplitude. Lemma 2.10. If K ∈ D b ( X × Y ) has finite T o r -amplitude over X then the fu nctor Φ K ( F ) = q ∗ ( K ⊗ p ∗ F ) has finite c ohomol o gic al amplitude. Similarly, if K ∈ D b ( X × Y ) has finite Ext -amplitude over Y then the functor Φ ! K ( G ) = q ∗ R H om ( K, q ! G ) has finite c ohomol o gic al amplitude. Pro of: I t suffices to note that the p u shforward fun ctor has fi nite c ohomologica l amplitude (it is equal to (0 , d ) where d is the maxim um of the dimensions of fib ers).  2.7. Homotop y colimits. Recall (see [BN]) th e definition of homotop y colimits in triangulated cat- egories. Let F 1 → F 2 → F 3 → . . . b e a sequence of ob jects of a triangulated cate gory ha ving coun table direct sums. Its homotop y colimit , hocolim F i , is defined as a cone of the canonical m or- phism ⊕ F i id − shift / / ⊕ F i , where shift denotes the map ⊕ F i → ⊕ F i defined on F i as the comp osition F i → F i +1 ⊂ ⊕ F j . Thus we h a ve a distinguished triangle L F i id − shift / / L F i / / ho colim F i . In what follo ws w e only consid er h omotopy colimits o v er the set of p ositiv e integ ers. Colimits o ver other partially ord ered sets are not consid ered at all. Lemma 2.11. If a functor Φ c ommutes with c ountable dir e ct sums, that is the c anonic al morphism ⊕ i Φ( F i ) → Φ ( ⊕ i F i ) is an isomorph ism then Φ c ommutes with ho motopy c olimits in th e sense that ther e is a nonc anonic al isomorphism ho colim Φ( F i ) ∼ = Φ( ho colim F i ) . In p articular, homotop y c olimits c ommute with tensor pr o ducts, pul lb acks and pushforwar ds. 6 Pro of: By the assu m ptions we hav e a diagram L i Φ( F i ) id − shift / / ∼ =   L i Φ( F i ) / / ∼ =   ho colim Φ( F i ) Φ( L i F i ) id − shift / / Φ( L i F i ) / / Φ( ho colim F i ) whic h is eviden tly comm utativ e. It follo ws that there is an isomorphism hocolim Φ( F i ) ∼ = Φ( ho colim F i ). F or the second claim w e use th e f act that countable d ir ect su ms commute w ith tensor pr o ducts, pullbac ks (eviden t) and p ushforwards ([BV], 3.3.4).  R emark 2.12 . Note that by [BV] 3 .3.4 tensor pro d ucts, pullbac ks and p ushforward comm ute with arbi- trary d irect su ms (not only w ith countable) . W e will use s ubsequently th is fact . No w assu m e that the triangulated category und er consideration is the unboun ded deriv ed category D ( A ), w here A is an ab elian category with exa ct coun table colimits. Lemma 2.13. If F 1 → F 2 → F 3 → . . . is a dir e ct system of c omplexes in A and F is the c omplex obtaine d by taking termwise c olimits o f the ab ove dir e ct system th en ho colim F i ∼ = F . Pro of: C onsider the sequ ence of complexes ⊕ F i id − shift / / ⊕ F i / / F . Since A has exact co limits, it is termwise exact. Therefore F is isomorphic to the cone of th e map ⊕ F i id − shift / / ⊕ F i .  Lemma 2.14 ([BN]) . If { F i } is a dir e ct system i n D ( A ) then we have H n ( ho colim F i ) ∼ = lim − → H n ( F i ) . Pro of: T he long exact sequ ence of cohomolo gy shea ves of the triangle defining hocolim F i giv es · · · → ⊕ i H n ( F i ) → ⊕ i H n ( F i ) → H n ( ho colim F i ) → ⊕ i H n +1 ( F i ) → ⊕ i H n +1 ( F i ) → . . . Since th e ca tegory A has exact colimits the last map ab ov e is injectiv e. It follo ws that H n ( ho colim F i ) ∼ = Cok er ( ⊕ i H n ( F i ) → ⊕ i H n ( F i )) ∼ = lim − → H n ( F i ), the last isomorphism b eing the defin ition of the colimit.  Lemma 2.15. If { F i } is a dir e ct system and ther e is given a morphism of this dir e ct system to F then ther e e xi sts a ma p hocolim F i → F c omp atible with the maps F i → F . Mor e over, if lim − → H t ( F i ) = H t ( F ) for e ach t ∈ Z then hocolim F i ∼ = F . Pro of: W e hav e a canonical map ⊕ F i → F . Its comp osition with ⊕ F i id − shift / / ⊕ F i v anishes since the map is induced by a map of the direct system { F i } to F . Hence it can b e f actored through a map ho colim F i → F . On the t -th cohomology it giv es the map H t ( ho colim F i ) = lim − → H t ( F i ) → H t ( F ) in duced b y the map of the direct system {H t ( F i ) } to H t ( F ). If it is an isomorphism for all t then the map ho colim F i → F is a quasiisomorphism.  2.8. Appro ximation. W e sa y that a d irect system { F i } in D ( A ) app ro xi mates F ∈ D ( A ) if there is gi v en a morphism from the direct system to F suc h that for an y n ≥ 0 the map τ ≤ n τ ≥− n F k → τ ≤ n τ ≥− n F is an isomorph ism f or k ≫ 0. The follo wing is a n immediate corollary of Lemma 2.1 5. Lemma 2.16. If a dir e ct system { F i } appr oximates F in D ( A ) then ho colim F i ∼ = F . Recall (see [K3]) that a direct system { F i } in D ( A ) is said to b e stabilizing in finite deg rees if for any n ∈ Z the map τ ≥ n F i → τ ≥ n F i +1 is an isomorphism f or i ≫ 0. Let B ⊂ A b e an ab elian sub categ ory and let D − B ( A ) denote the full sub cate gory in D − ( A ), the b ound ed ab o v e deriv ed category of A , consisting of all ob jects with cohomology in B . 7 Lemma 2.17. If a dir e ct system { F i } in D − B ( A ) stabilizes in finite de gr e es th en ho colim F i ∈ D − B ( A ) . Pro of: F ollo ws immediately from Lemma 2. 14.  The follo wing easy Lemma s ho w s th at every ob ject of D − ( X ) can b e appro ximated by a stabilizing in finite d egrees d ir ect system of p erfect complexes. This fact will b e u s ed su bsequentl y in the pap er. Lemma 2.18. F or every F ∈ D − ( X ) ther e is a stabilizing in finite de gr e es dir e ct system of p erfe ct c omplexes F k ∈ D perf ( X ) which ap pr oximates F . In p articular, hocolim F k ∼ = F . Pro of: Cho ose a lo cally free resolution for F and denote by F k its stup id tr uncation at degree − k . Th en F k is a p erfect complex and F k form a stabiliz ing in finite d egrees direct system. Moreov er, for an y n ∈ Z w e ha ve τ ≥− n F k ∼ = τ ≥− n F for k ≫ 0, h ence F k appro ximates F . By Lemma 2.16 w e ha ve F ∼ = ho colim F k .  W e are a lso in terested in appro ximation of arbitrary unbou n ded quasicoheren t complexes. C ertainly arbitrary ob jects of D q c ( X ) can’t b e represen ted as homotop y colimits of p erfect complexes. There is ho wev er the f ollo wing implicit appro ximation result. Lemma 2.19. The minimal ful l triangula te d sub c ate gory of D q c ( X ) close d under arbitr ary dir e ct sums and c ontaining D perf ( X ) i s D q c ( X ) . Pro of: Let R ⊂ D q c ( X ) b e th e minimal full triangulated sub categ ory closed un der arbitrary direct sums and con taining D perf ( X ). By the Bousfield lo calizat ion Theorem (see [N1], Lemma 1.7) there is a semiorthogonal decomp osition D q c ( X ) = hR ⊥ , Ri (the category R ⊥ is the catego ry of R -local ob jects). But R ⊥ ⊂ ( D perf ( X )) ⊥ and the latter category is zero (e.g. by the argumen t in the pro of of Lemma 2.7), hence R = D q c ( X ).  W e conclud e this s ection with the follo wing simple r esult whic h will b e us ed later. Lemma 2.20. L et φ : Y → X b e a quasipr oje ctive morphism and assume th at L is a line bund le on Y ample over X . If F ∈ D [ p,q ] ( Y ) then fo r any k ≫ 0 ther e is a dir e ct system G m in D [ p,q ] ( X ) such that φ ∗ ( F ⊗ L k ) ∼ = ho colim G m . Pro of: T aking the smart truncations of F at p and q we ca n assume that F is a complex suc h that F t = 0 unless t ∈ [ p, q ]. Sin ce L is ample ov er X for k ≫ 0 th e higher dir ect images of F t ⊗ L k v anish hence φ ∗ ( F ⊗ L k ) is isomorphic to the complex · · · → 0 → R 0 φ ∗ ( F p ⊗ L k ) → · · · → R 0 φ ∗ ( F q ⊗ L k ) → 0 → . . . Since φ is quasipro ject iv e, eac h sheaf R 0 φ ∗ ( F t ⊗ L k ) is a qu asicoheren t sheaf w h ic h can represent ed as a coun table union of co herent subsh ea ves. Cho ose suc h represen tation R 0 φ ∗ ( F t ⊗ L k ) = ∪ C t i and ta k e G t m = ∪ i ≤ m C t i + d ( ∪ i ≤ m C t − 1 i ) . Then it is clear th at G m form a direct system o f c omplexes the term wise colimit of whic h is the ab ov e complex. Hence φ ∗ ( F ⊗ L k ) ∼ = ho colim G m b y Lemm a 2.13 .  3. Inducing a semior thogonal d ecomposition Let T and T ′ b e triangulated categories an d assume that we are giv en semiorthogonal decomp ositi ons T =  A 1 , . . . , A m  and T ′ =  A ′ 1 , . . . , A ′ m  . A triangulated functor Φ : T → T ′ is compatible with the semiorthogonal deco mp ositions if Φ( A i ) ⊂ A ′ i for all 1 ≤ i ≤ m . Let α i : T → T and α ′ i : T ′ → T ′ b e the pro jection fu n ctors of the semiorthogonal decomp ositions. 8 Lemma 3.1. If the functor Φ is c omp atible with the semiortho gonal de c omp ositions then it c ommutes with the pr oje c tion f unctors, that is we have an isomorphism of f u nctors Φ ◦ α i ∼ = α ′ i ◦ Φ . Pro of: T ak e an y T ∈ T a nd let 0 T m / / T m − 1 / /          T m − 2 / / } } | | | | | | | | . . . / / T 2 / / T 1 / /          T 0          T α m ( T ) ] ] α m − 1 ( T ) a a . . . α 2 ( T ) [ [ α 1 ( T ) [ [ b e the filtration o f T with factors in A i . Applying the fu nctor Φ w e obtain a diagram 0 = Φ( T m ) / / Φ( T m − 1 ) / / ~ ~ ~ ~ ~ ~ ~ ~ ~ Φ( T m − 2 ) } } z z z z z z z z → · · · → Φ( T 2 ) / / Φ( T 1 ) / /          Φ( T 0 )= Φ( T )          Φ( α m ( T )) ^ ^ Φ( α m − 1 ( T )) a a . . . Φ( α 2 ( T )) \ \ Φ( α 1 ( T )) \ \ Since Φ( α i ( T )) ∈ Φ ( A i ) ⊂ A ′ i w e see that this diag ram giv es the filtration o f Φ( T ) with factors in A ′ i , hence we get isomorphisms Φ( α i ( T )) ∼ = α ′ i (Φ( T )). S ince suc h fi ltration is fu n ctorial b y Lemma 2.3 , the obtained isomo rphism s are functorial as w ell.  Lemma 3.2. A ssume that T =  A 1 , . . . , A m  and T =  A ′ 1 , . . . , A ′ m  ar e se miortho gonal de c omp ositions such that A ′ i ⊂ A i for al l 1 ≤ i ≤ m . Then A ′ i = A i for al l i . Pro of: The iden tit y functor T → T is compatible w ith th ese semiorthogonal decomp osit ions, hence their pro jections functors are isomorph ic by Lemma 3.1. In particular, f or an y i and an y A ∈ A i w e hav e A ∼ = α i ( A ) ∼ = α ′ i ( A ) ∈ A ′ i , wh ere α i and α ′ i are the pro jection f unctors, h ence A i ⊂ A ′ i .  Lemma 3.3. If Φ : T → T ′ is a ful ly faithful functor and T ′ =  A ′ 1 , . . . , A ′ m  is a semiortho gonal de c omp osition then th er e exists at most one semiortho gonal de c omp osition of T c omp atible with Φ , which is given by A i = Φ − 1 ( A ′ i ) . Pro of: Let T =  A 1 , . . . , A m  b e a s emiorthogonal d ecomp osition compatible with Φ . Then w e ha ve A i ⊂ Φ − 1 ( A ′ i ). On the other hand, let A ∈ Φ − 1 ( A ′ i ). T hen α ′ j (Φ( A )) = 0 for all j 6 = i . Hence b y Lemma 3.1 we ha ve Φ( α j ( A )) = 0 for all j 6 = i . Bu t since Φ is fully faithful, it follo ws th at α j ( A ) = 0 f or all j 6 = i , so A ∈ A i . T h us we are forced to ha ve A i = Φ − 1 ( A ′ i ).  In general the coll ection of su b categories A i = Φ − 1 ( A ′ i ) do es not giv e a semiorthogonal decomp osition. Actually , it is easy to see th at this collection is semiorthog onal (b y faithfulness of Φ), ho we v er it can b e not fu ll. The simplest example is the functor Φ : D b ( k ) → D b ( P 1 ) wh ich tak es k to O P 1 . If one considers the semiorthogonal d ecomp ositio n D b ( P 1 ) =  A ′ 1 , A ′ 2  with A ′ i =  O P 1 ( i )  then Φ − 1 ( A ′ i ) = 0 for i = 1 , 2. Nev ertheless, if the sub categories A i = Φ − 1 ( A ′ i ) form a semiorthogonal d ecomp osition of T we will sa y that this decomp osition is induced on T b y the semiorthogonal decomp osition of T ′ via Φ . Lemma 3.4. L et Φ : T → T ′ b e a ful ly faithful functor and T ′ =  A ′ 1 , . . . , A ′ m  a semiortho gonal de c omp osition. It induc es a semiortho gonal de c omp osition on T if and only if the image of Φ is stable under the pr oje ction functors of the sem iortho g onal de c omp osition of T ′ . Pro of: Th e “only if ” p art follo w s immediately fr om Lemma 3.1 . F or the if part we only ha v e to pro v e th at ev ery ob ject T of T can b e d ecomp osed with resp ect to the collection of sub categories A i = Φ − 1 ( A ′ i ). So, le t T ′ = Φ( T ) and let 0 = T ′ m → T ′ m − 1 → · · · → T ′ 1 → T ′ 0 = T ′ b e its filtration w ith f actors in A ′ i . Note that the f actors are give n b y α ′ i ( T ′ ) ∼ = α ′ i (Φ( T )). Since the image of Φ is s table u nder α ′ i , it foll o w s that α ′ i ( T ′ ) ∼ = Φ( A i ) for some ob jects A i ∈ A i . Let us c hec k that these are the comp onents of T . T o do 9 this we hav e to constru ct a filtration 0 = T m → T m − 1 → · · · → T 1 → T 0 = T su c h that its factors a re isomorphic to A i . W e do it inductiv ely . First of a ll, w e put T 0 = T . No w assume that T i is co nstructed in su c h a wa y that Φ( T i ) ∼ = T ′ i . T hen w e co mp ose this iso morphism with the map T ′ i → α ′ i ( T ′ ) ∼ = Φ( A i ). Since Φ is f ully faithful, the resu lted map comes fr om a map T i → A i in T . W e tak e T i +1 to b e the cone of this morphism sh if ted by − 1. App lying to the triangle T i +1 → T i → A i the functor Φ we conclude that Φ( T i +1 ) ∼ = T ′ i +1 . Applying this pro cedur e m t imes w e construct T m . Note that Φ( T m ) ∼ = T ′ m = 0. Since Φ is fully faithful, it follo ws that T m = 0, so the d esired filtration of T is constructed.  Lemma 3.5. L et Φ : T → T ′ b e a ful ly f aithful emb e dding, and assume that T ′ =  A ′ 1 , . . . , A ′ m  and T =  A 1 , . . . , A m  ar e semiortho gonal de c omp ositions c omp atible with Φ . L et Ψ ′ : T ′ → T ′ b e an endofunctor, such that T and al l A ′ i ar e stable under Ψ ′ . Then every A i is also stable under Ψ ′ . Pro of: Since T is stable u nder Ψ ′ and Φ is fully faithful, the r estriction of Ψ ′ to T defines an endofun ctor Ψ : T → T , suc h that Φ ◦ Ψ = Ψ ′ ◦ Φ. Since A i = Φ − 1 ( A ′ i ) w e h a ve to c hec k that Φ(Ψ( A i )) ⊂ A ′ i . But Φ(Ψ( A i )) = Ψ ′ (Φ( A i )) ⊂ Ψ ′ ( A ′ i ) ⊂ A ′ i since A ′ i is Ψ ′ -stable.  4. Extensions of a semior t hogonal deco mposition Let X be an algebraic v ariet y and assume that we are giv en a semiorthogonal decomp osit ion of D b ( X ). In this section w e construct a compati ble system of semiorthogonal decomp ositions of the cate gories D perf ( X ) ⊂ D − ( X ) ⊂ D q c ( X ). 4.1. P erfect complexes. First of all w e note that an y strong semiorthogonal decomp osit ion (see Defe - nition 2.6 ) of D b ( X ) induces a semiorthogonal d ecomp osition of the category of p erfect complexes. Prop osition 4.1. L et D b ( X ) =  A 1 , . . . , A m  b e a str ong semiortho gonal de c omp osition. Then th er e is a unique semiortho gonal de c omp osition of the c ate gory D perf ( X ) c omp atible with the natur al emb e dding D perf ( X ) → D b ( X ) . Pro of: The existence of a semiorthogonal decomp osit ion of D perf ( X ) compatible with that of D b ( X ) follo ws from [O2], 1.10 and 1.1 1. Moreo v er, it follo ws from Lemma 3.3 that the comp onents of this decomp osition are giv en b y A perf i = A i ∩ D perf ( X ) (2) and that the d ecomp osition is uniqu e.  4.2. Un b ounded quasicoheren t complexes. No w we are going to sh o w that an y (not necessarily strong) semiorthogonal decomp osition of D perf ( X ) induces a semiorthogonal decomp osition of the u n- b ound ed d eriv ed cat egory of quasicoheren t shea ves D q c ( X ). Prop osition 4.2. L et D perf ( X ) =  A perf 1 , . . . , A perf m  b e a semiortho gonal de c omp osition. Then ther e is a unique semiortho gonal de c omp osition D q c ( X ) =  ˆ A 1 , . . . , ˆ A m  c omp atible with the natur al emb e dding D perf ( X ) → D q c ( X ) and with clo se d under arbitr ary dir e ct sums c omp onents. The pr oje ction functors ˆ α i of this de c omp osition c ommute with dir e ct sums and homo topy c olimits. Mor e over, if the initial de c omp osition of the c ate gory D perf ( X ) is induc e d by a semior tho gonal de c om- p osition D b ( X ) =  A 1 , . . . , A m  of D b ( X ) the pr oje ction functors of which have finite right c ohom o- lo gic al amp litude then the obtaine d de c omp osition of D q c ( X ) is c omp atible with the natur al emb e dding D b ( X ) → D q c ( X ) as wel l. 10 Pro of: Define the sub cate gory ˆ A i ⊂ D q c ( X ) to b e the sub category of D q c ( X ) obtained by iterated addition of cones to the closure of A perf i in D q c ( X ) u nder all direct sums. Let u s c hec k that the categories ˆ A i form a s emiorthogonal d ecomp osition of D q c ( X ). First of all, if j > i , A l j ∈ A perf j , A k i ∈ A perf i then Hom ( ⊕ l A l j , ⊕ k A k i ) ∼ = Y l Hom ( A l j , ⊕ k A k i ) ∼ = Y l M k Hom ( A l j , A k i ) = 0 (in the second isomorphism w e used the fact that A l j are p erfect complexes, hence compact ob jects o f D q c ( X )). Addition of cones do es not sp oil semiorthogonalit y (see Lemma 2.1 ), hence the collect ion of sub categories ˆ A 1 , . . . , ˆ A m is semiorthogonal. Note also that a d irect sum of cones is a cone of direct sums b y [KS c h ], 1 0.1.19, so ˆ A i is a closed un der all direct su ms triangulated sub category of D q c ( X ). No w consider the tr iangulated sub cat egory  ˆ A 1 , . . . , ˆ A m  generated in D q c ( X ) by the sub cat egories ˆ A 1 , . . . , ˆ A m . It is clear that it is a triangulated s u b category of D q c ( X ) close d under all d irect sums. Moreo v er, it con tains  A perf 1 , . . . , A perf m  = D perf ( X ). Hence it coincides w ith D q c ( X ) by Lemma 2.19. This means that  ˆ A 1 , . . . , ˆ A m  = D q c ( X ). The uniqu en ess of suc h semiorthogonal decomp osition is eviden t b y Lemma 3.2. The compatibilit y w ith the em b eddin g D perf ( X ) → D q c ( X ) and closedness under arbitrary direct s ums are eviden t. Commutativit y of ˆ α i with arbitrary d irect sums follo ws immediately and for homotop y colimits w e apply Lemma 2. 11. F urther, to c h ec k that the constru cted semiorthogonal decomp osition of D q c ( X ) is compatible with the semiorthogonal d ecomp osition of D b ( X ) we ha v e to chec k that for an y A ∈ A i ⊂ D b ( X ) w e ha v e ˆ α i ( A ) ∼ = A . Indeed, choose a lo cally free resolution P • → A , and take A n = σ ≥− n ( P • ), the stupid truncation of the co mplex P • at d egree − n , so that w e ha v e a distinguished triangle σ ≥− n P • → A → σ ≤− n − 1 P • . Note that the direct system σ ≥− n P • appro ximates A in the sense of paragraph 2.8, hence by L emma 2.16 w e ha v e an isomorphism hocolim ( σ ≥− n P • ) ∼ = A . Th erefore ˆ α i ( A ) ∼ = ˆ α i ( ho colim ( σ ≥− n P • )) ∼ = ho colim ˆ α i ( σ ≥− n P • ) ∼ = ho colim α i ( σ ≥− n P • ) , the last isomorphism is due to the fact that σ ≥− n P • is a p erfect complex. S o, it suffices to c hec k that ho colim α i ( σ ≥− n P • ) ∼ = A . In deed, app lying α i to the ab o v e triangle we obtain α i ( σ ≥− n P • ) → A → α i ( σ ≤− n − 1 P • ) . Let ( a i , b i ) be the cohomological amplitude of t he functor α i . Since σ ≤− n − 1 P • ∈ D ≤− n − 1 ( X ) w e ha v e α i ( σ ≤− n − 1 P • ) ∈ D ≤− n − 1+ b i ( X ), hence α i ( σ ≥− n P • ) appr o ximates A , so ho colim α i ( σ ≥− n P • ) ∼ = A .  4.3. Bounded abov e coheren t complexes. Th e next step is the follo wing. Prop osition 4.3. L et D perf ( X ) =  A perf 1 , . . . , A perf m  b e a semiortho gonal de c omp osition. Then th er e is a uniq u e semiortho gonal de c omp osition of D − ( X ) c omp atible with this de c omp osition of D perf ( X ) and with the de c omp osition of D q c ( X ) c onstructe d in P r op osition 4.2 with r esp e ct to the na tur al emb e ddings D perf ( X ) → D − ( X ) → D q c ( X ) . Its c omp onents a r e close d under homotop y c olimits of st abilizing in finite de gr e es dir e ct systems. Pro of: W e ha v e to c hec k that D − ( X ) is stable under the pr o jecti on functors ˆ α i . Th en by Lemma 3.4 it w ould f ollo w that the sub categories A − i = ˆ A i ∩ D − ( X ) (3) 11 giv e a semiorthogonal decomp osition, whic h is evid ently compatible w ith those of D perf ( X ) and D q c ( X ). So, w e tak e an y F ∈ D − ( X ). By Lemma 2.18 there exists a stabilizing in fi nite degree s direct system of p erfect c omplexes F k suc h th at F ∼ = ho colim F k . It follo ws that ˆ α i ( F ) ∼ = ˆ α i ( ho colim F k ) ∼ = ho colim α i ( F k ) (the second isomorphism follo ws from Prop osition 4.2). But b y Lemm a 2.9 the direct system α i ( F k ) also stabilizes in finite degree s, so it follo ws from Lemma 2.17 that ho colim α i ( F k ) ∈ D − ( X ). The last claim is clear since b oth ˆ A i and D − ( X ) are closed un d er homotop y colimits of stabilizing in finite d egrees d ir ect systems.  4.4. S -linearity . Assume that X is a sc h eme ov er S , that is w e are giv en a map f : X → S . Re- call that any strong semiorthogonal decomp osition of D b ( X ) by Prop osition 4.1 indu ces a compatible semiorthogonal deco mp osition of D perf ( X ), which in i ts turn by P rop ositions 4 .2 and 4.3 induces c om- patible semio rthogonal decomp ositions of D q c ( X ) and D − ( X ). Lemma 4.4. If the initial semiortho gonal de c omp osition of the c ate gory D b ( X ) is S -line ar then the induc e d semiortho gonal de c omp osition of D perf ( X ) is S -line ar. Similarly, if the semiortho gonal de c om- p osition o f the c ate gory D perf ( X ) is S -line ar then the induc e d semiortho gonal de c omp ositions of D q c ( X ) and D − ( X ) ar e S -line ar as wel l. Pro of: T ake an y G ∈ D perf ( S ). Th en Ψ G ( H ) := H ⊗ f ∗ G is an en dofunctor of D q c ( X ) which p re- serv es D − ( X ), D b ( X ) and D perf ( X ) as w ell as the initial semiorthogo nal decomp osition. It follo ws from Lemma 3.5 th at th e semiorthogonal decomp osition (2) of D perf ( X ) is stable un der Ψ G . No w let us c hec k that eac h comp onent ˆ A i of the semiorthogonal decomp osition of D q c ( X ) is s table un der Ψ G . Indeed, b y definition ˆ A i is the smallest triangulated sub cate gory of D q c ( X ) con taining A perf i and closed under arbitrary direct sums . Bu t the functor Ψ G comm u tes with direct sums (see [BV], 3.3.4) and is exact whic h implies the claim. Again applying Lemma 3.5 w e conclude that the semiorthogonal decomp osi- tion (3) of D − ( X ) is also stable under Ψ G . Since this is true for all G ∈ D perf ( S ), w e see that all these decomp ositions are S -linear.  Actually , f or the comp onen ts of semiorthogonal d ecomp ositions of D q c ( X ) an d D − ( X ) we h a ve a stronger resu lt. Lemma 4.5. If D − ( X ) =  A − i , . . . , A − m  is an S -line ar semiortho gonal de c omp osition with c omp onents close d under homotopy c olimits of stabilizing in finite de gr e es dir e ct systems th en A − i ⊗ f ∗ D − ( S ) ⊂ A − i . Similarly, if D q c ( X ) =  ˆ A i , . . . , ˆ A m  is an S -line ar semiortho g onal de c omp osition with c omp onents close d under arbitr ary dir e ct su ms then ˆ A i ⊗ f ∗ D q c ( S ) ⊂ ˆ A i . Pro of: T ake an y G in D − ( S ). Applying Lemma 2.18 c ho ose a s tabilizing in fin ite degrees direct system of p erfect complexes G k appro ximating G so that G ∼ = ho colim G k . Then for an y F ∈ A − i w e h a ve F ⊗ f ∗ G ∼ = F ⊗ f ∗ ( ho colim G k ) ∼ = ho colim ( F ⊗ f ∗ G k ). Since the fun ctors ⊗ a nd f ∗ are right exact, it follo ws that th e direct system F ⊗ f ∗ G k stabilizes in finite degree s. Hence its homotop y colimit b elongs to A − i since A − i is S -linear an d closed und er h omotop y colimits of stabilizing in finite d egrees direct systems. F or the second claim recall th at b y Lemma 2.1 9 the catego ry D q c ( S ) can b e obtained by iterated addition of cones to the closure of D perf ( S ) under arbitrary direct sums. F urther , w e kno w by Lemma 4.4 that ˆ A i ⊗ f ∗ G ⊂ ˆ A i for any p erf ect G . Since f ∗ and ⊗ comm u te w ith d irect su ms, i t follo ws that the same is true for G b eing arb itrary direct sum of p erfect complexes. Fin ally , since f ∗ and ⊗ are exact and ˆ A i is triangulated, the same em b edding h olds for arbitrary G .  12 5. Change of a base Let f : X → S b e an algebraic map. C on s ider a base c hange φ : T → S and d enote by X T = X × S T the fib er p ro duct. Denote the pro jectio ns X T → T and X T → X b y f and φ resp ectiv ely , so that w e ha ve a cartesia n diagram X T φ / / f   X f   T φ / / S (4) Throughout this secti on we assum e that the b ase c hange φ is faithful for f : X → S (see paragraph 2.4 for the definition). 5.1. Base c hange f or p erfect complexes. Let D perf ( X ) =  A perf 1 , . . . , A perf m  b e an S -linear semiortho- gonal deco mp osition. Let A p iT denote t he minimal tr iangulated sub cate gory of D perf ( X T ) c losed under taking d irect sum mands and con taining all ob jects of the form φ ∗ F ⊗ f ∗ G with F ∈ A perf i , G ∈ D perf ( T ): A p iT =  φ ∗ A perf i ⊗ f ∗ D perf ( T )  . (5) Note that the su b category A p iT ⊂ D perf ( X T ) is T -linear, since the generat ing class φ ∗ A perf i ⊗ f ∗ D perf ( T ) is T -linear, and the p ro cess o f a dding cones and direct su mmands preserves T -linearit y . Prop osition 5.1. We have D perf ( X T ) =  A p 1 T , . . . , A p mT  , a T -line ar semiortho gonal de c omp osition c omp atible with the functor φ ∗ : D perf ( X ) → D perf ( X T ) . Pro of: Because of Lemma 2.7 and Lemma 2.1 to v erify semiorthogonalit y it suffices to chec k that f ∗ R H om ( φ ∗ F i ⊗ f ∗ G, φ ∗ F j ⊗ f ∗ G ′ ) = 0 for any F i ∈ A perf i , F j ∈ A perf j and an y G, G ′ ∈ D perf ( T ) if i > j . But f ∗ R H om ( φ ∗ F i ⊗ f ∗ G, φ ∗ F j ⊗ f ∗ G ′ ) ∼ = f ∗ φ ∗ R H om ( F i , F j ) ⊗ G ∗ ⊗ G ′ ∼ = φ ∗ f ∗ R H om ( F i , F j ) ⊗ G ∗ ⊗ G ′ = 0 (for the first isomorphism w e use p erfectness of F i , F j , G and G ′ , for the second w e us e f aithfulness of the base c h ange φ , and for the third — S -linearit y of the initial semiorthogonal decomp osit ion of D perf ( X ) and Lemma 2.7 for it ). It remains to c h ec k th at the sub cat egories A p iT generate D perf ( X T ). T ake an y ob ject H ∈ D perf ( X T ). Then by Lemma 5.2 b elo w it can b e obtained by consecutiv e taking cones and direct summand s starting from the collectio n of ob jects φ ∗ F t ⊗ f ∗ G t , where F t ∈ D perf ( X ), G t ∈ D perf ( T ), and t = 1 , . . . , N . On the other hand, ev ery ob ject F t can be d ecomp osed with resp ect to the semiorthog onal decomp osition D perf ( X ) =  A perf 1 , . . . , A perf m  , in other w ords, it can b e obtained by consecutiv e taking cones from a collect ion of ob jects A t i ∈ A perf i , i = 1 , . . . , m . It follo ws that H can b e obtained by consecutiv e taking cones and direct summands starting from the collection of ob jects φ ∗ A t i ⊗ f ∗ G t , and it remains to note that φ ∗ A t i ⊗ f ∗ G t ∈ A p iT b y d efinition. The s econd cl aim follo ws immediately fr om (5).  Lemma 5.2. The c ate gory D perf ( X T ) c oincides with the minimal triangulate d sub c ate gory of D q c ( X ) close d under taking dir e ct summands and c ontaining the class of obje cts φ ∗ D perf ( X ) ⊗ f ∗ D perf ( T ) := { φ ∗ F ⊗ f ∗ G | F ∈ D perf ( X ) , G ∈ D perf ( T ) } . Pro of: T ake an y ob ject H ∈ D perf ( X ) and construct a lo cally free resolution P • → H in whic h all shea v es P k ha ve form P k ∼ = φ ∗ F ⊗ f ∗ G , where F and G are lo cally free shea ves on X and T resp ectiv ely (this can b e done since φ is qu asipro jectiv e). Then its stupid truncation σ ≥ n ( P • ) ∈  φ ∗ D perf ( X ) ⊗ f ∗ D perf ( T )  for all n , and for n ≪ 0 the ob ject H is a direct summand of σ ≥ n ( P • ). Ind eed, since H is a p erfect 13 complex it is qu asiisomorphic to a b oun ded complex of lo cally fr ee shea ve s of finite rank. Assume that this complex is bou n ded from the left by degree l ∈ Z . T ake n ≤ l − d im X and consider the triangle σ ≥ n P • → P • → σ ≤ n − 1 P • . Note that since P • is quasiisomorphic to H and H is quasiisomorphic to a complex of lo cally fr ee shea ves sup p orted in d egrees ≥ l it follo ws that the complex computing E xt i ( P • , σ ≤ n − 1 P • ) is supp orted in degrees ≤ n − 1 − l . The hypercohomology sequence then sho w s that Ext i ( P • , σ ≤ n − 1 P • ) = 0 f or i > n − 1 − l + dim X . But n − 1 − l + dim X ≤ − 1 for n ≤ l − dim X , hence Hom ( P • , σ ≤ n − 1 P • ) = 0. In particular, the ab o v e t riangle sp lits, hence P • is a direct summand of σ ≥ n P • and w e are done since P • is qu asiisomorph ic to H .  5.2. Base c hange for unbounded quasicoherent complexes. W e start with an S -linear semiorthog- onal decomp ositio n D perf ( X ) =  A perf 1 , . . . , A perf m  . Let D perf ( X T ) =  A p 1 T , . . . , A p mT  b e th e T -linear semiorthogonal decomp osition constructed in Prop osition 5.1. Then using Prop osition 4.2 w e construct semiorthogonal decomp osit ions D q c ( X ) =  ˆ A 1 , . . . , ˆ A m  and D q c ( X T ) =  ˆ A 1 T , . . . , ˆ A mT  . By Lemma 4.4 these deco mp ositions are S and T -linear. Prop osition 5.3. The functors φ ∗ : D q c ( X T ) → D q c ( X ) and φ ∗ : D q c ( X ) → D q c ( X T ) ar e c omp atible with the ab ove semiortho gonal de c omp ositions. M or e over, ˆ A iT = { H ∈ D q c ( X T ) | φ ∗ ( H ⊗ f ∗ G ) ∈ ˆ A i for al l G ∈ D perf ( T ) } . (6) Pro of: Recall that b oth ˆ A i and ˆ A iT are obtained from ˆ A perf i and ˆ A p iT b y add ition of arbitrary direct sums and iterated add ition of cones and b ot h are closed u nder arbitrary direct sums triangulated c ate- gories. Since b oth φ ∗ and φ ∗ comm u te with arbitrary direct sums and are exac t, it suffices to c hec k that φ ∗ ( A perf i ) ⊂ ˆ A iT and that φ ∗ ( A p iT ) ⊂ ˆ A i . Th e first is eviden t by definition of ˆ A iT . F or the second tak e an y F ∈ A perf i , G ∈ D perf ( T ). Th en φ ∗ ( φ ∗ F ⊗ f ∗ G ) ∼ = F ⊗ φ ∗ f ∗ G ∼ = F ⊗ f ∗ φ ∗ G . But F ⊗ f ∗ φ ∗ G ∈ ˆ A i b y Lemm a 4.5. T o pro ve (6 ) w e note that the LHS is con tained in the RHS b y the T -linearit y of ˆ A iT and co mpatibilit y with φ ∗ . Con v ersely , assum e that H is in the RHS but not in ˆ A iT so that ˆ α j T ( H ) 6 = 0 for some j . Since the semiorthogonal d ecomp osition  ˆ A 1 T , . . . , ˆ A mT  is T -linear, the functors ˆ α j T are T -linear by Lemma 2.8, hence ˆ α j T ( H ⊗ f ∗ L k ) ∼ = ˆ α j T ( H ) ⊗ f ∗ L k for an y line bu ndle L on T and any k ∈ Z . By Lemma 5.4 b elo w we ha v e ho colim φ ∗ ( ˆ α j T ( H ) ⊗ f ∗ L k i ) 6 = 0 for s ome s equence L k 1 → L k 2 → L k 3 → . . . if L is amp le o v er S . It remains to n ote that ˆ α j ( ho colim φ ∗ ( H ⊗ f ∗ L k i )) ∼ = ho colim ˆ α j ( φ ∗ ( H ⊗ f ∗ L k i )) ∼ = ho colim φ ∗ ( ˆ α j T ( H ⊗ f ∗ L k i )) 6 = 0 (the first isomorphism is by Prop osition 4.2, the second is by L emm a 3.1) so ho colim φ ∗ ( H ⊗ f ∗ L k i ) 6∈ ˆ A i . But this means that φ ∗ ( H ⊗ f ∗ L k ) 6∈ ˆ A i for some k ∈ Z since ˆ A i is closed under homotop y colimits. So, H is not in the RHS of (6), a con tradiction.  Lemma 5.4. L et φ : Y → X b e a quasipr oje ctive morphism and let L b e a line bu nd le o n Y ample over X . L et F ∈ D q c ( Y ) . Then F ∈ D [ p,q ] q c ( Y ) if a nd only if for any se qu enc e of m aps L k 1 → L k 2 → L k 3 → . . . with k i → ∞ we have ho colim φ ∗ ( F ⊗ L k i ) ∈ D [ p,q ] q c ( X ) . In p articular F = 0 if and only if for any se quenc e L k 1 → L k 2 → L k 3 → . . . with k i → ∞ we have ho colim φ ∗ ( F ⊗ L k i ) = 0 . Pro of: As φ is quasipro jectiv e we can r epresen t φ as π 1 ◦ j 1 , wh ere j 1 : Y → Y is an op en em b edding and π 1 : Y → X is a pr o jecti v e morphism. F u r thermore, any op en embedd ing j 1 : Y → Y can b e represente d as a comp osition of an affin e op en emb ed ding j : Y → e Y and of a pro jectiv e morp hism π 2 : e Y → Y (w e tak e for e Y the blo wup of the ideal of the closed subset Y \ Y in Y ). Put π = π 1 ◦ π 2 . Thus φ = π ◦ j , 14 where j is an affin e op en em b ed ding and π is pro jectiv e. Since j is an affine op en em b edd ing the f unctors j ∗ and j ∗ are exact and j ∗ j ∗ ∼ = id , hence w e ha ve F ∈ D [ p,q ] q c ( Y ) if and on ly if j ∗ F ∈ D [ p,q ] q c ( e Y ). Th us the claim of the Lemma reduces to the case when φ is pr o jectiv e. So, assum e that φ is pro jectiv e. F or any nonzero coherent sheaf H on X w e kn o w that H t ( φ ∗ ( H ⊗ L k )) is zero for t 6 = 0 and k ≫ 0. Therefore for any quasicoherent sh eaf H on X w e ha v e lim − → H t ( φ ∗ ( H ⊗ L k i )) = 0 for t 6 = 0 if k i → ∞ . So, the hypercohomology sp ectral sequence and Lemma 2.14 imply that H t ( ho colim φ ∗ ( F ⊗ L k i )) ∼ = lim − → H 0 ( φ ∗ ( H t ( F ) ⊗ L k i )) . It follo w s immediatel y that F ∈ D [ p,q ] q c ( Y ) imp lies ho colim φ ∗ ( F ⊗ L k i ) ∈ D [ p,q ] q c ( X ). As for the other implication it suffices to c hec k that for an y quasicoheren t sheaf H 6 = 0 on Y ther e exists a sequence of maps L k 1 → L k 2 → L k 3 → . . . with k i → ∞ suc h that lim − → H 0 ( φ ∗ ( H ⊗ L k i )) 6 = 0. Sin ce tensoring with a line bun d le and the colimit are exact fun ctors on the ab elian cate gory Qcoh ( X ), wh ile H 0 φ ∗ is left exact, it f ollo ws th at it suffices to pro v e the ab ov e for an y nonzero s ubsheaf of H . Thus we can assume that H is coheren t. Th en using amp leness of L w e can find m and a s ection s of L m suc h that th e map H → H ⊗ L m giv en by s is an emb edding. No w consider the sequence L m → L 2 m → L 3 m → . . . with all maps giv en by s . Then all the maps in the sequen ce H 0 ( φ ∗ ( H ⊗ L m )) → H 0 ( φ ∗ ( H ⊗ L 2 m )) → H 0 ( φ ∗ ( H ⊗ L 3 m )) → . . . are em b eddings. Moreo ve r, H 0 ( φ ∗ ( H ⊗ L im )) 6 = 0 for i ≫ 0. Hence the limit is nonzero and we a re done.  5.3. Base c hange for b ounded ab o v e coheren t complexes. As ab o v e we start with an S -linear semiorthogonal decomp osition D perf ( X ) =  A perf 1 , . . . , A perf m  . Let D perf ( X T ) =  A p 1 T , . . . , A p mT  b e the T -linear semiorthogonal decomp osition constru cted in Prop osition 5.1. Let D q c ( X ) =  ˆ A 1 , . . . , ˆ A m  and D q c ( X T ) =  ˆ A 1 T , . . . , ˆ A mT  b e the S and T -linear semiorthogonal d ecomp ositions constructed in Prop osition 4.2 from the ab o v e decomp ositions of D perf ( X ) and D perf ( X T ) resp ectiv ely . Finally , let D − ( X ) =  A − 1 , . . . , A − m  and D − ( X T ) =  A − 1 T , . . . , A − mT  b e the S and T -linear semiorthogonal decom- p ositions c onstructed in Pr op osition 4.3. Lemma 5.5. The functors φ ∗ : D − ( X T ) → D q c ( X ) and φ ∗ : D − ( X ) → D − ( X T ) ar e c omp atible with the ab ove semiortho gonal de c omp ositions. Pro of: F ollo ws immediate ly from P rop osition 5.3 since A − i = ˆ A i ∩ D − ( X ) an d A − iT = ˆ A iT ∩ D − ( X T ).  5.4. Base c hange for b ounded coheren t complexes. This time we start with an S -linear strong semiorthogonal decomp osition D b ( X ) =  A 1 , . . . , A m  . Let D perf ( X ) =  A perf 1 , . . . , A perf m  b e the in- duced S -linear semiorthogonal decomp osition of D perf ( X ). F urther, consider the T -linear semiorthog- onal d ecomp osition D perf ( X T ) =  A p 1 T , . . . , A p mT  of Prop osition 5.1, and let D q c ( X ) =  ˆ A 1 , . . . , ˆ A m  and D q c ( X T ) =  ˆ A 1 T , . . . , ˆ A mT  b e the S and T -linear semiorthogonal d ecomp ositions constructed in Prop osition 4.2 from the abov e d ecomp ositions of D perf ( X ) and D perf ( X T ) resp ectiv ely . F urther, let D − ( X ) =  A − 1 , . . . , A − m  and D − ( X T ) =  A − 1 T , . . . , A − mT  b e the S and T -linear semiorthogonal decom- p ositions c onstructed in Pr op osition 4.3. Finally , we define A iT = A − iT ∩ D b ( X T ) . (7) Theorem 5.6. L et D b ( X ) =  A 1 , . . . , A m  b e an S - line ar str ong semiortho gonal d e c omp osition the pr o- je ction functors of which have finite c ohomol o gic al amplitude and assume that the b ase change φ is faithful for f . Then the sub c ate gories A iT ⊂ D b ( X T ) define d in (7) form a T -line ar semiortho gonal d e c omp osition D b ( X T ) =  A 1 T , . . . , A mT  . The pr oje ction functors of this semiort ho gonal de c omp osition have the same c ohomol o gic al amplitude as the pr oje ction functors of the initial semiortho gonal de c omp osition. Mor e over, the functors φ ∗ : D b ( X T ) → D q c ( X ) and φ ∗ : D b ( X ) → D − ( X T ) ar e c omp atible with th e semiortho gonal de c omp ositions of D q c ( X ) and D − ( X T ) r esp e ctively. 15 Pro of: T ak e any H ∈ D [ p,q ] ( X T ). W e hav e to c hec k that α − iT ( H ) is b ounded. Let ( a i , b i ) b e the co- homologica l amplitud e of α i . Let us sh o w that α − iT ( H ) ∈ D [ p + a i ,q + b i ] ( X T ). T his will p r o ve b oth that the categ ories A iT form a semiorthogonal decomp osit ion of D b ( X T ) and that the cohomologi cal am- plitude of th e pr o jection functors is the same as that of α i . Usin g Lemma 5.4 we see that it suffices to c hec k that f or k ≫ 0 we ha ve φ ∗ ( α − iT ( H ) ⊗ L k ) ∈ D [ p + a i ,q + b i ] q c ( X ), wh er e L is a line bundle on X T ample o ver X . W e can take L = f ∗ M where M is a line bun dle on T amp le o v er S . Note that φ ∗ ( α − iT ( H ) ⊗ f ∗ M k ) ∼ = φ ∗ ( α − iT ( H ⊗ f ∗ M k )) ∼ = ˆ α i ( φ ∗ ( H ⊗ f ∗ M k )) by Lemma 5.5 a nd L emma 2.9. F ur- ther, note that by Lemma 2.20 for k ≫ 0 we hav e φ ∗ ( H ⊗ f ∗ M k ) ∼ = ho colim G m for a certain d irect system G m with G m ∈ D [ p,q ] ( X ). Therefore ˆ α i ( φ ∗ ( H ⊗ f ∗ M k )) = ˆ α i ( ho colim G m ) ∼ = ho colim α i ( G m ) since ˆ α i comm u tes with homotop y co limits. Finally , α i ( G m ) ∈ D [ p + a i ,q + b i ] ( X ), h ence ho colim α i ( G m ) ∈ D [ p + a i ,q + b i ] ( X ) b y Lemma 2.14 , hence ˆ α i ( φ ∗ ( H ⊗ f ∗ M k )) ∈ D [ p + a i ,q + b i ] ( X ) as it w as required. Finally , it remains to c hec k that the sub ca tegories (7) are T -linear, and also that φ ∗ ( A iT ) ⊂ ˆ A i and φ ∗ ( A i ) ∈ A − iT . Th e first is clear since A − iT is T -linear and the other t wo claims follo w fr om Lemma 5.5.  The se miorthogonal decomp osition of D b ( X T ) c onstructed in Theorem 5.6 will b e r eferred t o as th e induced decomp ositi on of D b ( X T ) with resp ect to th e base change φ . Note that the d efinition of its comp onent A iT dep end s only on A i (i.e. do esn’t dep end on the c hoice of a semiorthogonal decomp osition con taining A i as a comp onen t). Indeed, sp elling out (6), (3), and (7) we obtain the follo win g Corollary 5.7. If A ⊂ D b ( X ) is an S -line ar ad missible sub c ate gory such that the c orr esp onding pr oje ction functor has finite c ohomolo gic al amplitude and φ : T → S is a b ase change faith ful for f : X → S then the c ate gory A T = { F ∈ D b ( X T ) | φ ∗ ( F ⊗ f ∗ G ) ∈ ˆ A for al l G ∈ D perf ( T ) } , (8) ( wher e ˆ A is the minimal close d under arbitr ary dir e ct sums triangulate d sub c ate gory of D q c ( X ) c onta in- ing A ) is a T -line ar admissible sub c ate gory in D b ( X T ) such that the c orr esp onding pr oje ction functor has finite c ohomolo gic al amplitude. Mor e over, we have φ ∗ ( A ) ⊂ A T if φ has finite T or -dimension and φ ∗ ( A T ) ⊂ A if φ is pr oje ctiv e. 5.5. Exterior pro duct of semiorthogonal decomp ositions. No w assume that w e ha ve t wo algebraic v arieties o ver the same base, say f : X → S and g : Y → S and S -linear str on g semiorthogonal decomp ositions of their deriv ed cat egories D b ( X ) =  A 1 , . . . , A m  and D b ( Y ) =  B 1 , . . . , B n  . Assume that th eir pro j ection functors h a ve fi nite co homological amplitude. Assume al so that th e cartesian square X × S Y p / / q   X f   Y g / / S (9) is exact, so that g is a faithful base change for f a nd f is a faithful base c h ange for g . Applying Theorem 5.6 w e obtain a pair o f semiorthogonal decomp ositio ns of D b ( X × S Y ): D b ( X × S Y ) =  A 1 Y , . . . , A mY  and D b ( X × S Y ) =  B 1 X , . . . , B nX  . Let A i ⊠ S B j := A iY ∩ B j X ⊂ D b ( X × S Y ) (10) W e call the cat egory A i ⊠ S B j the exterio r p ro duct (o v er S ) of A i and B j . Consider any complete order on the set { ( i, j ) } 1 ≤ i ≤ m, 1 ≤ j ≤ n extending the natural partial order. 16 Theorem 5.8. The exterior pr o ducts sub c ate gories A i ⊠ S B j ⊂ D b ( X × S Y ) form a semiortho gonal de c omp osition of the c ate gory D b ( X × S Y ) : D b ( X × S Y ) =  A i ⊠ S B j  1 ≤ i ≤ m, 1 ≤ j ≤ n . Mor e over, we have the fol lowing semiortho gonal de c omp ositions A iY =  A i ⊠ S B 1 , . . . , A i ⊠ S B n  and B j X =  A 1 ⊠ S B j , . . . , A m ⊠ S B j  . Pro of: Let C p ij =  p ∗ A perf i ⊗ q ∗ B perf j  ⊂ D perf ( X × S Y ) b e the minimal triangulated sub cate gory of D perf ( X × S Y ) closed un der taking direct su mmands and con taining ob jects of the form p ∗ A ⊗ q ∗ B with A ∈ A perf i , B ∈ A p j . The arguments of Prop ositio n 5.1 show that D perf ( X × S Y ) =  C p ij  1 ≤ i ≤ m, 1 ≤ j ≤ n is a semiorthogonal deco mp osition. Moreo v er, it is cle ar from th e constr u ction that we h a ve semiorthogonal decomp ositions A p iY =  C p i 1 , . . . , C p in  and B p j X =  C p 1 j , . . . , C p mj  . Extending these decomp ositi ons to D q c ( X × S Y ) as in Prop osition 4.2 we obtain s emiorthogonal decom- p ositions D q c ( X × S Y ) =  ˆ C ij  1 ≤ i ≤ m, 1 ≤ j ≤ n as w ell as ˆ A iY =  ˆ C i 1 , . . . , ˆ C in  and ˆ B j X =  ˆ C 1 j , . . . , ˆ C mj  , where ˆ C ij is obtained from C ij b y add ition of arbitrary d ir ect su ms and iterated addition of cones. Finally , in tersecting with D b ( X × S Y ) w e obtain semiorthogonal decomp ositions D b ( X × S Y ) =  C ij  1 ≤ i ≤ m, 1 ≤ j ≤ n as well as A iY =  C i 1 , . . . , C in  and B j X =  C 1 j , . . . , C mj  , where C ij = ˆ C ij ∩ D b ( X × Y ). S o, it r emains to c hec k that C ij = A iY ∩ B j X . S ince C ij ⊂ A iY ∩ B j X b y construction it suffices to c h ec k only the other inclusion. Indeed, w e ha ve B j X = ⊥  B 1 X , . . . , B j − 1 ,X  ∩  B j +1 ,X , . . . , B nX  ⊥ = ⊥  C it  1 ≤ i ≤ m, 1 ≤ t ≤ j − 1 ∩  C it  ⊥ 1 ≤ i ≤ m, j +1 ≤ t ≤ n , hence A iY ∩ B j X ⊂ A iY ∩ ⊥  C it  1 ≤ t ≤ j − 1 ∩  C it  ⊥ j +1 ≤ t ≤ n = C ij whic h is precisely w hat w e n eed.  5.6. Pro ducts. If S is a p oin t then a ny semiorthogonal decomp osition of D b ( X ) is S -linear. Moreo v er, an y base change T → S is fl at, h ence faithful for f : X → S , and X × S T = X × T is th e pro d uct. Th us giv en a semiorthogonal decomp osition of D b ( X ) w e can construct a compatible semiorthogonal decomp osition of the b oun ded deriv ed categ ory of the p ro duct of X with an y qu asipro jectiv e v ariet y . Explicitly , applying Theorem 5.6 we obtain the follo win g Corollary 5.9. L et D b ( X ) =  A 1 , . . . , A m  b e a str ong semiortho g onal de c omp osition the pr oje ction functors of which have finite c ohomolo gic al amplitude. L et Y b e a quasipr oje ctive variety. Then the sub c ate gories A iY = { F ∈ D b ( X × Y ) | p ∗ ( F ⊗ q ∗ G ) ∈ ˆ A i for any G ∈ D perf ( Y ) } , wher e p : X × Y → X and q : X × Y → Y ar e the pr oje ctions, and ˆ A i is obta ine d fr om A i by ad dition of arbitr ary dir e ct sums and iter ate d addition of c ones, form a Y - line ar semiortho gonal de c omp osition D b ( X × Y ) =  A 1 Y , . . . , A mY  . The pr oje ction functors of this semiorth o gonal de c omp osition also h ave finite c ohomolo gic al amplitude. The functors p ∗ : D b ( X × Y ) → D q c ( X ) and p ∗ : D b ( X ) → D b ( X × Y ) ar e c omp atible with the semiortho gonal de c omp ositions of D q c ( X ) and D b ( X ) r esp e ctively. Similarly , Theorem 5.8 giv es 17 Corollary 5.10. L et D b ( X ) =  A 1 , . . . , A m  and D b ( Y ) =  B 1 , . . . , B n  b e str ong semiortho gonal de- c omp ositions with pr oje ction functors of finite c ohomolo gic al amplitude. Then ther e is a semiortho gonal de c omp osition D b ( X × Y ) =  A i ⊠ B j  1 ≤ i ≤ m, 1 ≤ j ≤ n , wher e A i ⊠ B j = A iY ∩ B j X . Mor e over, we have semiortho gonal de c omp ositions A iY =  A i ⊠ B 1 , . . . , A i ⊠ B n  and B j X =  A 1 ⊠ B j , . . . , A m ⊠ B j  . 6. Corre ctness The goal of this section is to show that the extensions A perf , ˆ A , A − of a triangulat ed cate gory A and its base c hange A T under a base c hange T → S (if A is S -linear) do not dep end on a c hoice of an em b edding A → D b ( X ). The m ost imp ortan t tec hnical notion for this section is that of a splitting f u nctor. 6.1. Splitting functors. An exact fun ctor Φ : T → T ′ is called ri ght splitt i ng if Ker Φ is a r igh t admissib le sub category in T , the restriction of Φ to ( Ker Φ) ⊥ is fu lly faithful, and Im Φ is righ t admiss ible in T ′ (note that Im Φ = Im (Φ | ( Ker Φ) ⊥ ) is a triangulated sub catego ry of T ′ ). Lemma 6.1 ([K2]) . L et Φ : T → T ′ b e an exact functor. Then the fol lowing c onditions ar e e quivalent (1) Φ is right splitting; (2) Φ ha s a right adjoint functor Φ ! and the c omp osition of the c anonic al morph ism of functors id T → Φ ! Φ with Φ gives an isomorphism Φ ∼ = ΦΦ ! Φ ; (3) Φ has a right adjoint func tor Φ ! , ther e ar e semiortho gonal d e c omp ositions T = h Im Φ ! , Ker Φ i , T ′ = h Ker Φ ! , Im Φ i , and the functors Φ and Φ ! give quasiinverse e quiv alenc es Im Φ ! ∼ = Im Φ ; (4) ther e exists a ful l triangulate d left admissible sub c ate gory α : A ⊂ T , a f u l l triangulate d right admissible sub c ate gory B ⊂ T ′ and an e quivalenc e ξ : A → B suc h that Φ = β ◦ ξ ◦ α ∗ , Φ ! = α ◦ ξ − 1 ◦ β ! . There is an analo gous notion of left splitting functors, w hic h enjoy a s imilar set of prop erties. Ho wev er w e will n ot n eed this notion in this paper. 6.2. Extensions. Let X b e a quasipro jectiv e v ariet y . Let α : A → D b ( X ) and β : B → D b ( Y ) b e admissible sub categories, and ξ : A → B an equiv alence. Consider the corresp ondin g righ t splitting functor Φ : D b ( X ) → D b ( Y ), Φ = β ◦ ξ ◦ α ∗ . W e assu me al so that Φ is geometric , meaning that it is isomorphic to a ke rnel functor Φ E : D q c ( X ) → D q c ( Y ) , Φ E ( F ) = q ∗ ( p ∗ F ⊗ E ) with a ke rnel E ∈ D − ( X × Y ). Here p : X × Y → X and q : X × Y → Y are the pro jectio ns. Note that the right adjoint functor Φ ! E of Φ E is giv en b y the formula Φ ! E : D q c ( Y ) → D q c ( X ) , Φ ! E ( G ) = p ∗ R H om ( E , q ! F ) . It follo ws in p articular that Φ E comm u tes with direct sums. In deed, Hom (Φ E ( ⊕ F i ) , G ) ∼ = Hom ( ⊕ F i , Φ ! E ( G )) ∼ = Y Hom ( F i , Φ ! E ( G )) ∼ = Y Hom (Φ E ( F i ) , G ) ∼ = Hom ( ⊕ Φ E ( F i ) , G ) implies Φ E ( ⊕ F i ) ∼ = ⊕ Φ E ( F i ). Recall that if E ∈ D b ( X × Y ) has finite T o r -amp litud e ov er X , fi nite Ext -amplitud e o v er Y , and supp E is pr o jectiv e o v er b oth X and Y then Φ E tak es D b ( X ) to D b ( Y ) a nd Φ ! E tak es D b ( Y ) to D b ( X ) b y [K1]. 18 Theorem 6.2. Assume that an obje ct E ∈ D b ( X × Y ) has finite T o r -amplitude over X , finite Ext - amplitude over Y , and supp E is pr oje ctive over b oth X and Y . Assume also th at the r estriction of the functor Φ E : D q c ( X ) → D q c ( Y ) to D b ( X ) is a right splitting functor giving an e quivalenc e of sub c ate gories A ⊂ D b ( X ) and B ⊂ D b ( Y ) . Then t he fu nctor Φ E : D q c ( X ) → D q c ( Y ) and its r estriction to D − ( X ) ar e right splitting functors giving e qui valenc es ˆ A ∼ = ˆ B and A − ∼ = B − . Pro of: As w e already m en tioned ab o v e the fun ctor Φ E comm u tes with direct sums. Let us c hec k that Φ ! E also commutes w ith direct sum s. T o do this w e c ho ose a closed em b edd ing i : X → X ′ with X ′ b eing smo oth an d consider the fu nctor i ∗ Φ ! E instead. Since i ∗ is a conserv ativ e fun ctor comm uting with d irect sums, it suffices to c hec k that i ∗ Φ ! E comm u tes with direct sums. But it is clear th at i ∗ Φ ! E ∼ = Φ ! ( i × id Y ) ∗ E , so from the wh ole b eginning we can assum e that X is smo oth. Th en the pro jectio n X × Y → Y is smo oth, hence q ! ( F ) ∼ = q ∗ ( F ) ⊗ ω X [dim X ] eviden tly co mm utes w ith direct su ms. F ur ther, E is a p erfect complex by [K1] , 10.46 , hence the f u nctor R Hom ( E , − ) commutes with direct s u ms. Finally , the fu nctor p ∗ comm u tes with direct sums b y [BV], 3.3.4. Th us Φ ! E comm u tes with direct sums. F urther, since the fu nctors Φ E and Φ ! E comm u te w ith direct su ms they comm u te with homotop y colimits b y Lemma 2.11. No w if F ∈ D − ( X ) th en b y Lemma 2.18 there exists a stabilizing in fi n ite degrees direct system of p erfect complexes F k ∈ D b ( X ) su c h that F ∼ = ho colim F k . Th erefore Φ E ( F ) ∼ = Φ E ( ho colim F k ) ∼ = ho colim Φ E ( F k ). But th e functor Φ E has finite co homological amplitud e by Lemma 2.10. Th erefore the direct system Φ E ( F k ) ∈ D b ( Y ) stabilizes in fi nite degrees, hence ho colim Φ E ( F k ) ∈ D − ( Y ) b y Lemma 2. 17. Th us Φ E tak es D − ( X ) to D − ( Y ). Th e same argumen t sho ws that Φ ! E tak es D − ( Y ) to D − ( X ). T o c hec k that Φ E is righ t splitting on D q c ( X ) we hav e to c heck that applying Φ E to the canonical morphism of fu nctors id → Φ ! E Φ E giv es an isomorphism Φ E ∼ = Φ E Φ ! E Φ E . Consider the full sub categ ory T ⊂ D q c ( X ) consisting of all ob jects F ∈ D q c ( X ) f or whic h Φ E ( F ) ∼ = Φ E Φ ! E Φ E ( F ) in D q c ( Y ). W e w an t to show that T = D q c ( X ). Note that D b ( X ) ⊂ T b y the conditions, and hence D perf ( X ) ⊂ T . Moreo ver, since Φ and Φ ! comm u te with direct sums, T is closed und er arbitrary direct s u ms. Finally , sin ce Φ E and Φ ! E are e xact, T is triangulat ed. So, by Lemma 2.19 we ha v e T = D q c ( X ). No w let us c hec k that ˆ B = Φ E ( D q c ( X )). Indeed, the RHS is con tained in the LHS b y Lemma 2.19 since ˆ B is closed u nder arbitrary d irect sums triangulated sub category cont aining Φ E ( D perf ( X )) ⊂ Φ E ( D b ( X )) = B . F or the other emb edding it suffices to c hec k that ˆ B is con tained in the full sub cat- egory T ⊂ D q c ( Y ) consisting of all ob jects G such that the canonical morph ism Φ E Φ ! E ( G ) → G is an isomorphism. Indeed, T con tains B b y conditions of the Prop osition. Moreo ve r, it is closed und er arbi- trary d irect su ms since b oth Φ E and Φ ! E comm u te with direct su ms, and is triangulated sin ce b oth Φ E and Φ ! E are exact. The same argumen t sho ws that ˆ A = Im Φ ! E , so it follo ws that Φ E induces an e quiv alence ˆ A ∼ = ˆ B . Finally , sin ce Φ E and Φ ! E preserve D − and A − = ˆ A ∩ D − ( X ), B − = ˆ B ∩ D − ( Y ), it follo w s that Φ E induces an equiv alence A − ∼ = B − .  R emark 6.3 . One can also c heck that Φ E tak es D perf ( X ) to D perf ( Y ) (this follo ws easily from the fact that Φ ! E comm u tes with direct su ms). If it were also known that Φ ! E tak es D perf ( Y ) to D perf ( X ) then it w ould f ollo w that Φ E induces an equiv alence A perf ∼ = B perf . 6.3. Base c hange. No w assume that f : X → S and g : Y → S are quasipro jectiv e morphisms, α : A → D b ( X ) and β : B → D b ( Y ) are admissible S -linear su b categories, and ξ : A → B is a n S -linear equiv alence. Assume also that φ : T → S is a base change faithful for b oth f and g . Again, consider the co rresp onding right sp litting functor Φ : D b ( X ) → D b ( Y ), Φ = β ◦ ξ ◦ α ∗ . W e assume al so that Φ is geometrically S -linear , meaning that it is isomorph ic to a ke rnel functor Φ E : D q c ( X ) → D q c ( Y ) , Φ E ( F ) = q ∗ ( p ∗ F ⊗ E ) 19 with a k ernel E ∈ D − ( X × S Y ) supp orted on th e fib er prod uct of X and Y o v er S . Here p : X × S Y → X and q : X × S Y → Y are the pro jections. Note that th e right adjoin t functor Φ ! E of Φ E is giv en b y the form ula Φ ! E : D q c ( Y ) → D q c ( X ) , Φ ! E ( G ) = p ∗ R H om ( E , q ! F ) . Consider the follo wing comm utativ e diagram X T φ   X T × T Y T φ   q T / / p T o o Y T φ   X X × S Y q / / p o o Y Define the k ern el E T := φ ∗ E ∈ D − ( X T × T Y T ). Theorem 6.4. Assume that E ∈ D b ( X × S Y ) has finite T o r -amplitude over X , finite Ext - amplitude over Y , and supp E is pr oje ctive over b oth X and Y . Assume also that Φ E : D b ( X ) → D b ( Y ) is a right splitting f u nctor giving a n e quivalenc e of S -line ar sub c ate gories A ⊂ D b ( X ) and B ⊂ D b ( Y ) . Then Φ E T : D b ( X T ) → D b ( Y T ) is a right splitting functor inducing a n e quivalenc e A T ∼ = B T . Pro of: First of all note that E T has finite T o r -amplitude ov er X T , finite Ext -amplitud e o ver Y , and pro jectiv e supp ort ov er b oth X T and Y T b y [K1], 10.47 . Hence as it was mentio ned in the pro of of Theorem 6.2 the f u nctors Φ E , Φ ! E , Φ E T , Φ ! E T comm u te with d irect s ums and homotop y colimits. Moreo v er, b y [K 1] 2.4 the fun ctors Φ ! E and Φ ! E T are righ t adjoin t to Φ E and Φ E T resp ectiv ely , and all these functors preserv e b ounded n ess and coherence. Finally , by [K1] 2 .42, there are canonica l isomor- phisms Φ E T φ ∗ = φ ∗ Φ E , Φ E φ ∗ = φ ∗ Φ E T , Φ ! E T φ ∗ = φ ∗ Φ ! E , Φ ! E φ ∗ = φ ∗ Φ ! E T . (11) Since Φ E is r igh t sp litting on D q c ( X ) b y Theorem 6.2, app lying Φ E to the canonical morphism of functors id → Φ ! E Φ E giv es an isomorphism Φ E ∼ = Φ E Φ ! E Φ E . No w tak e any H ∈ D q c ( X T ). W e wa nt to sho w that Φ E T ( H ) ∼ = Φ E T Φ ! E T Φ E T ( H ) in D q c ( Y T ). By Lemma 5.4 to do this it suffi ces to c heck that φ ∗ (Φ E T ( H ) ⊗ g ∗ L k ) ∼ = φ ∗ (Φ E T Φ ! E T Φ E T ( H ) ⊗ g ∗ L k ) in D q c ( Y ) for an amp le o v er S li ne bund le L on T and an y k ≫ 0. But φ ∗ (Φ E T ( H ) ⊗ g ∗ L k ) ∼ = φ ∗ (Φ E T ( H ⊗ f ∗ L k )) ∼ = Φ E ( φ ∗ ( H ⊗ f ∗ L k )) ∼ = ∼ = Φ E Φ ! E Φ E ( φ ∗ ( H ⊗ f ∗ L k )) ∼ = φ ∗ (Φ E T Φ ! E T Φ E T ( H ⊗ f ∗ L k )) ∼ = φ ∗ (Φ E T Φ ! E T Φ E T ( H ) ⊗ g ∗ L k ) . The first and the fifth isomorphisms are given b y T -linearit y of the functors Φ E T and Φ ! E T , the second and the fourth are given by (11) and the third is b ecause Φ E is righ t splitting. So, w e conclude that Φ E T ∼ = Φ E T Φ ! E T Φ E T , hence Φ E T is a righ t splitting functor. No w let us sho w that Φ E T ( D q c ( X T )) = ˆ B T . Ind eed, let F ∈ D q c ( X T ). Let G b e a perfect complex on T . Th en we ha v e φ ∗ (Φ E T ( F ) ⊗ g ∗ G ) ∼ = φ ∗ (Φ E T ( F ⊗ f ∗ G )) ∼ = Φ E ( φ ∗ ( F ⊗ f ∗ G )) ∈ ˆ B , hence Φ E T ( F ) ∈ ˆ B T b y (6). F urther, s in ce Φ E T is a righ t splitting T -linear fu nctor comm uting with arbitrary dir ect su ms, the categ ory Φ E T ( D q c ( X T )) is a T -linear triangulated sub ca tegory in D q c ( Y T ) closed u nder arb itrary d irect su ms. On the other hand, φ ∗ ( B perf ) ⊂ φ ∗ ( B ) = φ ∗ (Φ E ( D b ( X ))) = Φ E T ( φ ∗ ( D b ( X ))) ⊂ Φ E T ( D q c ( X T )) so it follo ws from the definition of ˆ B T that ˆ B T ⊂ Φ E T ( D q c ( X T )). The same argument s h o w s that Φ ! E T ( D q c ( Y T )) = ˆ A T . 20 Finally , as we alrea dy ment ioned the functors Φ E T and Φ ! E T preserve D b and since A T = ˆ A T ∩ D b ( X T ), B T = ˆ B T ∩ D b ( Y T ), it follo ws that Φ E T induces an equiv alence A T ∼ = B T .  7. Applica tions As an applicatio n we d educe that the pro jection fun ctors of a strong semiorthogonal decomp osition are k ernel fu nctors. Theorem 7.1. L et X b e a quasipr oje ctive variety and D b ( X ) =  A 1 , . . . , A m  a st r ong semiorth o gonal de c omp osition. L et α i : D b ( X ) → D b ( X ) b e the pr oje ction functor to the i -th c omp onent. Assume that e ach α i has finite c ohomo lo gic al ampl itude. Then for every i ther e is an obje ct K i ∈ D b ( X × X ) such that α i ∼ = Φ K i . R emark 7.2 . Note that the condition that the semiorthog onal decomp osition is strong is necessary fo r the pro jection fun ctors to b e represen table by k ernels. Indeed, every functor isomorph ic to Φ K has a righ t adjoint f unctor, h ence if α 1 ∼ = Φ K then α 1 has a r igh t adjoin t fun ctor hence A 1 is right admissible. Pro of: W e consider the semiorthogo nal decomposition D b ( X × X ) =  A 1 X , . . . , A mX  constructed in Corollary 5.9 and let K i b e the comp onen t of ∆ ∗ O X ∈ D b ( X × X ) in A iX . Consider the co rresp onding filtration o f ∆ ∗ O X : 0 T m / / T m − 1 / /          . . . / / T 1 / / T 0          ∆ ∗ O X K m \ \ . . . K 1 [ [ T ak e a n y F ∈ D q c ( X ), pull it bac k to X × X via the p ro jectio n p 1 : X × X → X , then tensor it b y the ab o v e diagram and p u sh forwa rd to X via the pro jection p 2 : X × X → X . W e w ill obtain the f ollo wing diagram in D q c ( X ) 0 p 2 ∗ ( T m ⊗ p ∗ 1 F ) / / _ _ p 2 ∗ ( T m − 1 ⊗ p ∗ 1 F ) / / } } z z z z z z z z z . . . / / p 2 ∗ ( T 1 ⊗ p ∗ 1 F ) / / _ _ p 2 ∗ ( T 0 ⊗ p ∗ 1 F )           p 2 ∗ (∆ ∗ O X ⊗ p ∗ 1 F ) p 2 ∗ ( K m ⊗ p ∗ 1 F ) . . . p 2 ∗ ( K 1 ⊗ p ∗ 1 F ) Note that by Lemma 4.5 we ha ve K i ⊗ p ∗ 1 F ∈ ˆ A iX , hence p 2 ∗ ( K i ⊗ p ∗ 1 F ) ∈ ˆ A i b y Prop ositio n 5.3. On the other han d p 2 ∗ (∆ ∗ O X ⊗ p ∗ 1 F ) ∼ = F , so w e co nclude that p 2 ∗ ( K i ⊗ p ∗ 1 F ) ∼ = ˆ α i ( F ). Restricting t o D b ( X ) and using Lemma 3.1 we obtain an isomorphism Φ K i ∼ = α i on D b ( X ).  This Theorem has a relativ e v arian t. Theorem 7.3. L et f : X → S b e a morphism of quasipr oje ctive varieties and D b ( X ) =  A 1 , . . . , A m  an S -line ar str ong semiortho gonal de c omp osition. L et α i : D b ( X ) → D b ( X ) b e the pr oje ction functor to the i -th c omp onent. Assume that the map f is faithful b ase change for i tself and e ach α i has finite c ohomol o gic al ampl itude. Then for every i ther e is an obje ct K i ∈ D b ( X × S X ) such that α i ∼ = Φ K i . The pr o of is analo gous. 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