Deforming motivic theories I: Pure weight perfect Modules on divisorial schemes

Deforming motivic theories I: Pure w eigh t p erfect Mo dules on divisorial sc hemes T oshiro Hirano uc hi ∗ and Satoshi Mo c h i zuki † Abstract In this pap er, w e int ro duce a notion of w eigh t r pseudo-coheren t Mo dules asso ciat ed to a regular closed immersion i : Y ֒ → X of co dimension r , and prov e that there is a canonical d eriv ed Morita equiv alence b etw een the DG-categ ory of p erfect complexes on a d i- visorial sc h eme X whose cohomolog ical supp ort are in Y and the DG-cate gory of boun ded complexes o f we igh t r pseu do-co heren t O X - Mo dules supp orted on Y . The theorem implies that there is th e canonical isomorp hism b et w een the B ass-Thomason-T r obaugh non- connected K -theory [TT90], [Sc h 06] (resp. the Keller-W eib el cyclic homology [Kel98], [W ei96 ]) for the immersion and the Sc h lic h ting non- connected K -theo ry [Sch04] asso ciated to (resp. that of ) the exact catego ry of w eigh t r p seudo-coheren t Mod ules. F or the connected K -t heory case, th is result is just Exercise 5.7 in [TT90]. As its appli- cation, we will decide on a generator of the top ological filtration on the n on-connecte d K -theo ry (resp. cyclic h omolo gy theory) for affine Cohen-Macaula y sc hemes. 1 In tro duction Since the w ord “ mo t ive t heory” is an ambiguous w o rd, in this In tro duction, as mot ive theory , we restrictedly mean axiomatic studying ( co)homology the- ories ov er algebraic v arieties b y enriching morphisms b etw een algebraic v a ri- eties with adequate equiv alence relat io ns. T raditionally , to construct mot ivic ∗ Suppo rted by the JSP S F ellowships for Y oung Scientists. † This resear c h is supp orted by JSPS cor e-to-core progra m 1 8005 1 categories, w e used t o c ho ose certain classes of algebraic cycles as morphisms spaces and consider v arious equiv alence r elations on them, for example ratio- nal, nume rical and algebraic relations and so on. In pra ctice, the difficult y of handling a motivic theory is concen tr ating on mo ving alg ebraic cycles suit- ably in an a ppro priate equiv alence relation class (see the proficien t surv ey [Lev06]). A problem of this type is so-called “mo ving lemma” and solving by delib erating on geometry ov er a base (field). In this pap er, we giv e a first step of building up a motivic theory whic h do es not rely up on geometry o ve r a base by replacing (mo duli spaces of ) algebraic cycles with (ro ughly sp eaking, mo duli non-commutativ e spaces of ) pseudo-coheren t complexes and consid- ering an equiv alence relation on them as the deriv ed Morita equiv alences (Compare [Kon07] § 4, [T au07]). The aim of this pap er is to introduce the notion of ( Thomason - T r ob augh ) weight on the class of p erfect Mo dules o n sc hemes inspired by the w ork of Thomason and T robaugh in [TT90]. T o explain this more precisely , let X b e a divisorial sche me (in the sense of [BGI71 ], cf. Def. 3.12) and i : Y ֒ → X a regular closed immersion of co dimension r . A pseudo-coheren t O X -Mo dule is said to b e of ( Tho mason-T r o b augh ) weight r supp orted on Y if it is of T or- dimension ≦ r and supp orted on Y . Here the w ord “we igh t” is coming from the w eigh t of the Adams op erations in [G S87] and a more systematic study will b e done in [Mo c08]. W e den ote by Wt r ( X on Y ) the exact category of pseudo-coheren t O X -Mo dules of w eight r supp orted on the subspace Y and P erf ( X on Y ) the exact catego ry o f p erfect complexes on X whose cohomological supp ort are in Y . W e shall prov e the follow ing theorem: Theorem (Th. 4.3) . Ther e is a c anonic al derive d Morita e quivalenc e b etwe en the e xact c ate gory of b ounde d c om plexes of Wt r ( X on Y ) and Perf ( X on Y ) . As alluded to a bov e, it can b e considered as one of a v ariant of “mo ving lemma”. It might sound a new flav ored theory , but the metho ds o f prov ing Theorem 4.3 ar e classical, standard and almost all o f them were established b y G r othendiec k sc ho ol. F or example, V erdier’s coherator theory ( Pro p. 3.6), Ilusie’s global resolution theorem (Th. 3.1 5 ), Grothendiec k’s lo cal cohomol- ogy theory (Lem. 5.9) a nd so on. Th e theorem implies that there is a canon- ical isomorphism b et w een the Bass-Thomason-T roba ug h no n- connected K - theory K B ( X on Y ) [TT90], [Sc h06 ] (resp. the Keller-W eib el cyclic homol- ogy H C ( X on Y ) [Kel98], [W ei96]) and the Sc hlich ting non-connected K - theory [Sch04] asso ciated to (resp. that of ) t he exact category of w eigh t r 2 pseudo-coheren t O X -Mo dules K S ( Wt ( X on Y )) (resp. H C ( Wt ( X on Y ))). That is, w e hav e isomorphisms K B q ( X on Y ) ≃ K S q ( Wt r ( X on Y )) , H C q ( X on Y ) ≃ H C q ( Wt r ( X on Y )) , for each q ∈ Z . F or t he connected K -theory t his result is nothing ot her than Exercise 5.7 in [TT90]. F or Grot hendiec k groups ( q = 0), there is a detailed pro of if X is the sp ectrum of a Cohen-Macaulay lo cal ring and Y is the closed p oin t of X ([RS03 ], Prop. 2 ). F or K -theory , as mentioned in Exercise 5 .7, this problem is related with the works [Ger74], [G ra76] and [Lev88]. Namely the problem ab out describing the homotopy fib er of K B ( X ) → K B ( X r Y ) (or rather than K Q ( X ) → K Q ( X r Y )) b y using t he K -theory of a certain exact category . As describ ed in [Ger74 ], there is a n example due to Deligne whic h suggests difficulty of the problem f or a general closed immersion. Con vers ely , the example indicate that for an appropriate sche me X , there is a go o d class of pseudo-coheren t O X -Mo dules. That is, Mo dules of pure w eigh t. This concept is in timately related to W eibel’s K -dimensional conjecture [W ei80] (see Conj. 6.4), G ersten ’s conjecture [Ger73] and its consequences. These sub jects will b e treated in [HM08], [Mo c08]. Notice that there are differen t notions of pure weigh t b y Grays o n [G ra95] a nd W alk er [W al00] and these t wo notions are compatible in a particular situation [W al96]. In a future work, the authors hop e to compare the W alk er w eight with the Thomason-T robaugh one by utilizing the ( e quidimension a l ) bivariant alg e b r aic K -the ory [GW00]. No w w e explain the structure of the pap er. In § 2, w e describe to our motiv at io nal picture. After reviewing the fundamen ta l facts in § 3, w e will define the notion of w eight and state the main theorem in § 4. The pro of of the main t heorem will b e giv en in § 5. F inally w e will giv e applications of the main theorem in § 6. Convention. Throughout this pap er, we use the letter X to denote a sc heme. A c o mplex means a c hain complex whose b oundary morphism is increase lev el of term b y one. F or fundamental notations of chain complexes, for example mapping cone and mapping cylinder etc..., w e f ollo w the b o ok [W ei94]. F or an additiv e category A , w e denote b y Ch ( A ) the catego ry of ch ain complexes in A . The w o r d “ O X -Mo dule” means a sheaf on X whic h is a sheaf of mo dules o ver t he sheaf of rings O X . W e denote b y Mod ( X ) t he ab elian cat ego ry of O X -Mo dules and Qcoh ( X ) the category of quasi-coheren t O X -Mo dules. An algebr aic ve ctor bund le o v er the sc heme X is a lo cally free O X -Mo dule of 3 finite rank and we denote b y P ( X ) the category of a lgebraic vec tor bundles. In particular a line bund le is a n algebraic ve ctor bundle of r a nk one (= an in vertible sheaf ). F or the terminolog ies of algebraic K -theory , we fo llo w to the notations in [Sc h07 ]. F or example, for a complicial biW aldhausen category C , w e denote its asso ciated deriv ed category by T ( C ) a nd for an exact category E , w e denote its a ssociated deriv ed cat ego ry T ( Ch ( E )) b y D ( E ). Finally for t he A 1 -motivic theory , w e follo w the notations in [MVW06]. A ckn o w le dgme nts. The second author is tha nkful to Masana Harada, Charles A. W eib el for giving sev eral commen ts to Exercise 5 .7 in [TT90 ], Mar co Sc hlic h ting fo r teac hing ab out elemen tary questions of negativ e K -theory via e-mail, Paul Balmer fo r bringing him to the preprin t [Bal07] and Mark E. W alker for sending the thesis [W al9 6 ] to him. 2 Conjectu ral picture In this section, w e will give a conjectural p ersp ectiv e of de f o rming mo tivi c the ori e s . This section is logically indep enden t of the others. 2.1 Analogies b et w een m ultiplicativ e and additiv e mo- tivic theories As in the In tr oduction, as a motiv e theory , w e prefer to mean axiomatic studying of (co)homology theories o v er alg ebraically geometric ob jects b y enric hing morphisms b et we en algebraically geometric o b jects with adequate equiv alence r elat io ns. So there should b e many motivic theories dep ending on our treating of algebraically geometric ob jects a nd (co)homology theo- ries. F or example, if w e deal with W eil cohomolog y theories, the classical motiv e theory is fitting for our purp ose [K le68]. If w e handle A 1 -homotopy in v ariant (co)homo lo gy t heories, the motivic homotopy theory in the sense of V o ev o dsky is appropriate [V o e00]. If w e consider cohomology theories whic h has the Gersten resolution, the Blo c h-Ogus(-Gabb er) theory [BO74], [CHK97] is suitable. Moreov er there are other mo t ivic theories for example [KS02], [KL07]. It migh t b e b eliev ed that there is “the” motiv e theory whic h is omniscien t and unifying eve ry motivic theories. But as in the follow ing example, there are motivic theories whic h are not seemed to b e compatible with eac h ot her. 4 Example 2.1. If we prefer to giv e a motivic in terpretatio n of the Ho dge decomp osition using t he cyclic homolog y theory lik e as [W ei97], or if we like to understand what is the motiv e asso ciated with the additiv e g roup G a lik e as a generalized 1-motive [La u96], [Ber08], w e shall no t realize them in V o- ev o dsky’s motivic world. F or the cyclic homolo gy theory is not A 1 -homotopy in v ariant and G a is contractible in his motivic category . But we ha ve the analogies table b et w een additive and multiplicativ e w orlds as in [Lo d03]. W e w ould like to extend the table to mo t ivic stag e. F or example, the Blo ch the- orem [Blo86] and the Ho dge decomp osition as in the table b elo w. × (V o ev o dsky’s motivic theory) + (a dditiv e motivic theory) K n ( X ) Q ∼ → L p + q = n H p M ( X , G ⊗ q m ) Q H n ( X , C ) ∼ → L p + q = n H p ( X , G a ⊗ G ⊗ q m ) Of course, the righ t hand side abov e is conjectural des cription. (But see [BE03], [Rul07], [Par07] and [P a r08]). I n these analo g ical line, following [FT85] and [FT87], w e like to call the cyclic ho mo lo gy theory the additive algebr aic K -the ory . W e shall also notice the fact that there are real mathematical prob- lems stretc hing a w ay b oth additive and m ultiplicative worlds . F o r example, V orst’s conjecture [V or79]. Actually the conjec ture is pro v ed in a special case b y frequen tly utilizing both multiplicativ e and additiv e motivic tec h- niques [CHW06]. In the next subsection w e pro p ose another similarly kind problems. 2.2 Motivic mo dules and W eil recipro cit y la w Classically there is the following pro blem. Problem 2.2. Let G 1 , . . . , G r b e comm utativ e group v arieties o v er a base field k . T hen w e ha ve the corresp o ndence Z 0 ( G 1 × k G 2 × k · · · × k G r ) ↔ M L/k : finite extension G 1 ( L ) ⊗ Z G 2 ( L ) ⊗ Z · · · ⊗ Z G r ( L ) where Z 0 (?) means the group of zero cycles. The problem is the following: What ar e the suitable e quivalenc e r elations ma k i n g assignment ab ove iso m or- phism. Z 0 ( G 1 × k G 2 × k · · ·× k G r ) / ∼ ∼ → M L/k : finite extension G 1 ( L ) ⊗ Z G 2 ( L ) ⊗ Z · · · ⊗ Z G r ( L ) / ∼ . 5 Historical Note 2.3. If w e assume all G 1 , . . . , G r are semi-ab elian v arieties, then there are suitable candidates fo r equiv alence relations ab ov e. (i) In the left hand side, the suitable equiv alence relation should come from the tensor pro ducts as 1- motiv es in the sense of [Del74]. That is, the left hand side should b e replaced with Γ(Sp ec k , G 1 ⊗ · · · ⊗ G r ) = Z 0 ( G 1 × k G 2 × k · · · × k G r ) / ∼ where tensor pro duct are tak en as 1-mo t iv es. (ii) In the rig h t hand side, Kazuy a K a to prop osed that the suitable equiv a- lence relation should b e the following tw o relations. • Pro jection form ula f o r norms. • W eil recipro cit y la w for semi-ab elian v arieties. W e will write the left hand side mo dulo equiv alence relations ab ov e as K ( k , G 1 , . . . , G r ) and called it Milnor K -gr oup asso c iate d with G 1 , . . . , G r (see for example [Som90], [Kah92]). The naming coming from the following isomorphism. K ( k , r z }| { G m , . . . , G m ) ∼ → K M r ( k ) . Observ ations 2.4. (i) (At least aft er tensoring with Q ,) the tensor pro duct as 1 -motiv es is equal to the tensor pro duct in the A 1 -motivic category DM( k ) (see fo r example [Org04], [BK0 7]). (ii) The pro jection form ula relation ab o v e is one of the consequence o f presheaf with transfer, that is, there is the followin g statemen t (see for example [Org04]): Every c ommutative gr oup va ri e ty over a field k is c onsider e d as a functor qpsmcor ( k ) → Ab wher e qpsmcor ( k ) is the c ate gory of quasi-pr oje ctive smo oth varieties whose morphisms ar e finite s urje ctive c orr esp ondenc es. (iii) If all G i are 1-dimensional semi-ab elian v arieties, we ha v e the following form ula , Γ(Sp ec k , G 1 ⊗ , · · · ⊗ G r ) ∼ → K ( k , G 1 , . . . , G r ) . 6 This is the affirmativ e answ er for question ab ov e (see for G m case [SV00] and fo r elliptic curve case [Mo c06]). (iv) The r eason why semi-ab elian v arieties are fit in V o ev o dsky’s theory is that semi-ab elian v arieties are A 1 -homotopy inv arian t preshea v es with tra ns- fers. So w e shall sa y A 1 -homotopy in v ariant preshea v es with transfers as a motivic mo dules . Then we can re-write the statemen t in Historical Notes 2.3 (i) a s follows. In the left h and sid e , the suitable e quivalen c e r e l a tion should c ome fr om the tensor pr o ducts as m otivic mo dules. In the observ atio n and § 2.1, w e are in terested in the following question. What is a go o d notion of m otivic m o dules including G a ? Remark 2.5. (i) (cf. [RO06]) The category of motiv es is rein terpreted in the con text of stable motivic ho motop y theory b y R¨ ondings and Østvær as follo ws. Let MZ ∈ SH A 1 ( k ) b e the motivic Eilenberg-Maclane sp ectrum. Then MZ is considered as a r ing ob ject in SH A 1 ( k ) in the nat ural wa y and w e hav e the following identit y: Mo d ( MZ ) ∼ → DM( k ) where w e assume t hat c hara cteristic of k is zero. This means D M ( k ) is actually “the category of motivic mo dules” in some sense. Notice tha t if a presheaf of ab elian groups on the category o f quasi-pro jectiv e smooth sc hemes has a n a ction of MZ , this means that F can extend to a presheaf on qpsmcor ( k ). (ii) Sev eral authors are a ttempting to describ e Γ(Sp ec k , G 1 ⊗ · · · ⊗ G r ) as generators and relations. In this p oin t, relatio ns are relat ed with the func- tional equations of sp ecial functions a sso ciated with G i . F or example, if all G i are equal to G m , the sp ecial function is the p olylogarithms [Gon9 4] and so on. Therefore it is quite surprised that the r elations of K ( k , G 1 , . . . , G r ) do es not dep end on the G i . The W eil recipro cit y la w is implicitly con tro l- ling the functional equations of sp ecial functions asso ciated with G i . So it is imp ortan t that we shall ask what is a meaning of the W eil recipro cit y law in the con text of V o evodsky’s mo t ivic theory . W e can state a generalization of the W eil recipro cit y la w whic h is called Motivic r e cipr o city law . Let k b e a field whic h satisfies the resolution of singularit y assumption. 7 Theorem 2.6 ([Mo c06]) . F or a fiel d extension of tr an s c endental de gr e e one K/ k , the c omp osition of (1) M(Sp ec k )(1)[1] Σ N k ( v ) /k (1)[1] → ˜ Y v : plac e of K /k M(Sp ec k ( v ))(1)[1 ] ˜ Q ∂ v → M(Sp ec K ) is the z er o map in the pr o-c ate gory of D M ( k ) . If w e tak e the Hom DM( k ) (? , Z ( n + 1)[ n + 1]) for t he sequence (1), w e can easily repro v e the W eil recipro city law for Milnor K- groups. Corollary 2.7 ([Sus82]) . The c omp osition of K M n +1 ( K ) ⊕ ∂ v → M v : plac e of K/k K M n ( k ( v )) Σ N k ( v ) /k → K M n ( k ) is the z er o map. The crucial p oin t o f pro ving the mot ivic recipro cit y law is the existence of functorial G ysin triangles whic h is pro v ed by D´ eglise [Deg06] and A 1 - homotop y inv ariance is indisp ensable in his construction of the triangle. On the other hand, R ¨ ulling prov ed the W eil recipro cit y la w for the de Rham- Witt complexes whic h is not an A 1 -homotopy in v ariant theory [Rul07]. W e w ould like to explain this recipro cit y la w also in the con t ext of an alien motivic theory . In this wa y , w e sometimes hav e intereste d in the problems whic h stretc hing aw ay sev eral mot ivic theories and sometimes intend to ana- lyze r elationship of sev eral motivic theories, for example, their analogies and differences. The main theme of deformin g motivic the ories is inv estigating the relationship b et w een v ario us motivic theories. In particular, V o ev o dsky’s motivic theory and an alien (additiv e) motivic theory . 2.3 Ho w to describ e deforming motivic theories I Next we intend to illustrate how to describ e deforming motivic t heories. As in [Ha n95], [R O0 6] and [BV07], the tr ia ngulated category o f motivic shea v es shall b e the connected comp onen ts of the ∞ - cat ego ry of that in some sense. Here the word “ ∞ -categor y” means (quasi-) DG -category o r S - category in the sense of T¨ oen and V ezzosi [TV04]. W e first start to cons ider ho w to men tion a n alien motivic theory a s follo ws. 8 Example 2.8. (i) (T o y mo del) Let V be a finite dimensional vec tor space o ver a field k with an inner pro duct and W its sub ve ctor space. Then w e ha ve an isomorphism (2) V /W ∼ → W ⊥ where W ⊥ is the orthogonal subspace of W in V . (ii) Let k b e a p erfect field, V the deriv ed catego r y of complexes of Nisnevic h shea v es transfer ov er k b ounded from a bov e and W the lo calizing sub category generated b y the complexes of the fo rm Z tr ( X × A 1 ) → Z tr ( X ) for smo oth sc hemes X o v er k . Then w e hav e the equiv alence (2 ) where W ⊥ is the full sub category of those complexes whose cohomology shea v es a r e A 1 - homotop y inv arian t in V (see [V o e00], Prop. 3 .2.3). The sign ⊥ is justified in the conte xt o f (generalized) top oi theory or Bousfiled lo calization t heory as b elo w. (iii) (cf. [BGV72], IV) Let C b e a small category with a Grot hendiec k top ol- ogy τ . W e denote the category of preshea ves on C by V and the category of τ -lo cal contractible preshea v es on C by W . Then we hav e an equiv alence (2) where W ⊥ is the full sub category of τ -shea ves in V . Namely , a n ob ject F in W ⊥ is satisfying the decen t condition (or rather than sa ying the ortho g onal condition) as follows: Hom( U , F ) ∼ → Hom( h X , F ) where h X is the functor represen ted by an ob j ect X in C and U is a crible in τ ( X ). As in [Hir03], [TV05], r eplacing C as ab o ve with a more higher cate- gorical (or ra t her than sa y homotopical) ob ject in some sense, the argumen t ab o ve still works fine by replacing t he decen t condition with the hyper one (F o r precise statemen t, consult with [TV05]). F or DG-categories case, see [Dri04] and [T au07 ] App endix. No w we would b etter consider the reason wh y a h yp er descen t conditio n is not seemed to b e in v olv ed in V o ev osky’s A 1 -homotopy theory . T o do so, let us recall the following Lemma 2.10: Definition 2.9. Let ( I , x, y ) b e a triple consisting of I ∈ qpsmcor ( k ) and differen t k -rational p oin ts x : Sp ec k → I , y : Sp ec k → I . 9 (i) Tw o maps f , g : X → Y in qpsmcor ( k ) are said to b e I -homotopi c if there is a map H : X × I → Y suc h that H ◦ x × id X = f and H ◦ y × id Y = g (or H ◦ x × id X = g and H ◦ y × id Y = f ). (ii) A functor F : qpsmcor ( k ) → C is said t o b e I -hom otopy inva riant if for an y I -ho motopic maps f , g : X → Y , F ( f ) = F ( g ). F o r A 1 -homotopy in v ariant, w e mean that A 1 -homotopy in v ariant for t he triple ( A 1 , 0 , 1). Lemma 2.10. F or any pr eshe af F on qpsmcor ( k ) , the fol lowing c ond i tion s ar e e quivalent. (i) F or any sc heme X , the pr oje ction X × A 1 → X induc es an isomorp h ism F ( X × A 1 ) ∼ → F ( X ) . (ii) F is A 1 -homotopy invariant. Notice that the condition (i) (resp. (ii) ) ab ov e is a descen t condition for ob jects (=0- morphisms ) (resp. morphisms (= 1-morphisms)) in some sense. Therefore the condition (i) is seemed to b e stronger than the condition (ii). W e designate that to prov e (i) from the condition (ii), w e are using the sp ecial feature of A 1 . Namely the existence of the m ultiplicatio n A 1 × A 1 ∋ ( x, y ) 7→ xy ∈ A 1 and this feature is axiomized b y V o ev o dsky as the site with in terv al theory [V o e96], [MV99]. In the authors view p oin t, this is the reason wh y w e are able to shortcut to construct the motivic homotopy category without using a h yp er descen t theory and there is no reason that to establish an alien motivic category , we can av o id using a higher top oi theory . So we prop ose the follow ing. Conjecture 2.11 (V ery obscure ve rsion) . T o build up an alien motivic c at- e g ory, we ne e d to cho ose a mo duli sp ac e V of al g ebr aic al ly ge om e tric obje cts which c ould b e r epr esente d by ∞ -c ate gory or hom otopic al c ate gory as a gen- er a tor class an d a mo duli sp ac e W of r elations sp ac e. Then we c an define an alien motivic c ate gory by the quotient sp ac e V /W and somewhat hyp er de- sc ent the ory imp li es that it is e quiva lent to W ⊥ . Her e W ⊥ is f ul l subsp ac e of V c onsisting of the o b j e cts which satisfy ortho gonal c ondition in some sense. Ob viously the conjecture has t wo faces. One fa ce is the problem of es- tablishing the g eneral f rame w orks of a higher or generalized to po i theory 10 fitting for our purp ose. F or example, presheav es with transfer theory a nd site with in t erv al theory can b e considered a s a shea v e theory ov er general- ized Gro thendiec k top ology and are suitable for describing A 1 -motivic theory . The other face is the problem o f finding the go o d class of V and W ab o ve. T o attac k the first face, w e need drastically axiomat ic consideration. T o study the second one, w e need lo ok squarely at real man y examples. The authors are starting from atta c king to the second one. After getting many imp ortan t examples, they in tend to conte mplate the first o ne [HM08]. 2.4 Thomason categories and b iv arian t algebraic K - theory It is a complicial biW aldhausen category closed under the fo r ma t ion of t he canonical homotop y pus h-outs and pull-bac ks in the sens e of [TT90] that mak es sense of its deriv ed category and algebraic ( resp. additiv e) K -theory and w e like to call it a Tho m ason c ate gory . A morphism b et w een Thomason categories is complicial exact functor in the sense of op . cit. In this pap er, w e examine V in Conjecture 2.11 a s the category of Thomason categories whic h is a homotopical category in the sense of [DHKS04] by declaring the class of w eak equiv alences as de rive d Morita e quivalenc es , that is, morphisms whic h induce equiv alences of derive d categories. The reasons why w e prefer to tak e algebraically geometric o b jects as Thomason categories are the following: • There is a functor from the categor y of sc hemes to that of Thomason cat e- gories: X 7→ Perf ( X ), where Perf ( X ) is the category of p erfect complexes of globally finite T or-amplitude (cf. [TT90], § 2.2). • F or an appropriate sc heme X , from Perf ( X ) (and its tensor structure), w e can recov er the sc heme X completely [Bal02]. Th at is, Perf ( X ) do es not lose the geometric information o f X . • Moreo v er in the category of Thomason categories, we ha v e ob jects lik e P er f ( X on Y ) and P erf r ( X ) (cf. Def. 3.9) whic h are deriv ed from sc hemes and a bsent f r o m the categor y of sc hemes. • Since the alg ebraic K - theory is ∞ -categorical in v arian t (see [Sc h0 2 ], [T o e03], [TV04] and [BM07]), we prefer to the cat ego ry of Thomason categories than that of triangulated categories. Next w e need to consider ho w to en ric h the category V and c ho ose a 11 relation space W . Inspired from the work [W al96 ] and encouraged by the w orks [Kon0 7] § 4 and [T a u07], the authors in tend to enriching V with the biv ariant alg ebraic K -theory . T o men tion the reason wh y we like t o se- lect the biv ariant K -theory as morphisms spaces of V , w e will start from the follo wing Lemma 2.12. Let D b e a tensor triangulated category and M : qpsm cor ( K ) → D a functor preserving copro ducts and tensor pro d- ucts. Here t he tensor pro ducts in qpsmcor ( k ) are the usual pro ducts o v er Sp ec k . F rom no w on, for P 1 -homotopy in v ariant, w e mean that P 1 -homotopy in v ariant for the triple ( P 1 , 0 , 1). Lemma 2.12 (Compare [CHK97]) . The fol lowing c onditions a r e e quivalent. (a) M is P 1 -homotopy invariant. (b) The fol lowing diag r am is c ommutative. M( A 1 r { 0 } ) M( i ) / / M( p ) ' ' P P P P P P P P P P P M( P 1 ) M(Sp ec k ) M( ∞ ) 8 8 r r r r r r r r r r wher e i an d p ar e the natur al inclusion and the structur e map r esp e ctively. (c) (Rigidity) M(1) = M( ∞ ) : M(Sp ec k ) → M ( P 1 ) is c o incide d. Pr o of. (a) ⇒ (b): The map H : ( A 1 r { 0 } ) × P 1 ∋ ( t, [ x 0 : x 1 ]) 7→ [ tx 0 : tx 1 ] ∈ P 1 giv es P 1 -homotopy b et w een i and ∞ ◦ p . Therefore w e get the results. (b) ⇒ (c): Considering the followin g diagram, w e get the r esu lt. M(Sp ec k ) M(0) / / id   M( A 1 r { 0 } ) M( i )   M( p ) w w o o o o o o o o o o o o M(Sp ec k ) M( ∞ ) / / M( P 1 ) . (c) ⇒ (a) : Let f , g : X → Y b e ma ps in qpsmcor ( k ) and H : X × P 1 → Y are their P 1 -homotopy , that is, H ◦ 0 × id X = f and H ◦ 1 × id X = g . Then 12 w e hav e the iden tit y: M( f ) = M( H ◦ 0 × id X ) = M( H ◦ τ ◦ ∞ × id X ) = M ( H ) M( τ ) M( ∞ ) ⊗ M(id X ) = M( H ) M ( τ ) M(1) ⊗ M(id X ) = M ( H ◦ τ ◦ 1 × id X ) = M( H ◦ 1 × id X ) = M( g ) where τ : P 1 ∋ [ x 0 : x 1 ] 7→ [ x 1 : x 0 ] ∈ P 1 . F o r the imp orta nce of the comm uta tiv e diagram in Lemma 2 .12 (b), t he readers shall consult with [CHK97] a nd this topic will b e treated in [HM08]. It is clos ely related to the existenc e of the G ersten resolution for M. W e also notice that the additiv e K -theory , additiv e higher Cho w groups and the additiv e gro up G a are P 1 -homotopy inv arian t as functors on the categor y of algebraic v arieties ( see for example [Qui73], [TT90], [Kel99], [KL07]). But K 0 is not a functor on qpsmcor ( k ). As in § 2 .2, w e sometime hop e to extend the notio n of motivic mo dules to mak e functors ab o ve b elong to the class of generalized motivic mo dules. Imitating W alker’s a rgumen t, w e prefer to replace qpsmcor ( k ) with K naiv e 0 ( qpsm ( k )) whic h is the category of quasi- pro j ective smo oth sc hemes o v er k enric hing with the biv ariant K - theory (F or precise definition, see [W al96], [Sus03 ]). W e lik e to call P 1 -homotopy inv ari- an t preshea v es of ab elian gr oups o n K naiv e 0 ( qpsm ( k )) ge n er alize d motivic mo dules . Now it is imp ortant that we recall the fo llo wing core theorem of the A 1 -motivic theory . Let us assume that k is a p erfect field. Theorem 2.13. F or an A 1 -homotopy in v ariant pr eshe af with tr ansfer F , w e have the fol lowing. (i) F or any p , H p Nis (? , F Nis ) c an b e c onsider e d as an A 1 -homotopy invariant Nisnevich she af with tr ansfer in the natur al way. That is, Nisnevich motivic mo dules ar e c l o se d under taking c ohomolo gy. (ii) H p Nis (? , F Nis ) ∼ → H p Zar (? , F Zar ) for any p . No w Beilinson a nd V ologo dsky p erceiv ed tha t Theorem 2.1 3 is a conse- quence of the existence of the Gersten resolution of F [BV07] and W a lker pro ved tha t fo r A 1 -homotopy in v ariant preshea v es on K 0 ( qpsm ( k )), similar theorem ab ov e are v erified [W a l96]. Therefore the touc hstone of a notion of generalized motivic mo dules are fo llowing. • F or a generalized motivic mo dule, do es it ha ve the Gersten resolution ? 13 • Do es the (equidimensional) biv arian t K -theory ha ve the exp ected prop er- ties lik e as the F riedlander-V o ev o dsky theory [FV00] ? In this pap er, the authors prepare to at t ac k to t he second problem ab o ve . More precisely sa ying, in this pap er and [Mo c08], the authors will observe that the roll of a base of our motivic theory , analyze ho w to av o id to the geometry o ve r the ba se (see § 2.5). Symbolically , let us denote ⋆ the in visible base fo r our motivic theory . If t here exist a bigraded biv ariant K -theory for sc hemes, in particular w e can consider K p,q ( X , ⋆ ) for a sche me X . The second author b eliev e that K p,q ( X , ⋆ ) migh t b e K p ( Wt q ( X )) and f o r a regular no etherian affine sc heme X , the isomorphism K p ( Wt q ( X )) ∼ → K p ( M q ( X )) could b e considered as a v aria n t of F riedlander-V o ev o dsky duality theorem [FV00]. 2.5 Ho w to describ e deforming motivic theories I I T o compare with tw o mo t ivic theories, t he author intend to parametrize the relation space W in Conjecture 2.11. Namely for example we consider mo duli space of motivic theories V /W ( t ). Example 2.14. Let R b e a comm utative discrete v aluation ring and π its uniformizer. W e put K = R [1 /π ] a nd k = R /π R . Then w e can consider the p a r ametrize d Susulin functor b y using the follo wing pa r a metrized cosimplicial sc heme ∆ • . W e define a parametrized cosimplicial sc heme ∆ • b y [ n ] 7→ Sp ec R [ T 0 , . . . , T n ] / (Σ T i − π ) . Ob viously ∆ • | Spec K is usual one and ∆ • | Spec k is app eared in [BE03]. The attempt in Example 2.1 4 is just a naiv e construction of deformatio n space of motivic theories parametrized b y Sp ec R whose fib er ov er Sp ec K is the A 1 -motivic theory and o ver Sp ec k is an alien o ne. But we are confronte d with the follow ing serious problems. What is the motivic the ory of total sp ac e? Why do es the total sp ac e the ory wo rk fine? 14 T o solv e the second problem ab o ve , w e need to assure that w e can build up a motivic theory without relying up on the geometry ov er a ba se. Therefore our deforming motivic theories is starting from ex amining the Thomason- T robaugh w eigh t. 3 Preliminary 3.1 T or-d imension W e br iefly review the definition and fundamen ta l prop erties of T or-dimens i on of Mo dules. Definition 3.1. Let L b e an O X -Mo dule. (i) L is flat if the functor ? ⊗ O X L : Mo d ( X ) → Mo d ( X ) defined by M 7→ M ⊗ O X L is exact. (ii) A T or-dimens i o n of L is the minimal in t eger n such that there is a resolution of L , 0 → F n → F n − 1 → · · · → F 0 → L → 0 , where all F i are flat. W e write as Td( L ) = n . No w we list some well-kno wn f acts on T or-dimension. Lemma 3.2 ([BGI71], Exp. I, 5.8.3, [DG 63], 6 .5 .7.1) . L et L b e an O X - Mo dule. (i) If L is a flat and finitely pr esente d O X -Mo d ule, then L is an alge br aic ve ctor bund le. (ii) The fol lowing c ondition s ar e e quivalent. (a) Td( L ) ≦ d . (b) F or any O X -Mo d ule K and any n > d , we have T or O X n ( L , K ) = 0 . (c) F or any O X -Mo d ule K , we have T or O X d +1 ( L , K ) = 0 . (d) If ther e is an exact se quenc e 0 → K d → F d − 1 → F d − 2 → · · · → F 0 → L → 0 wher e al l F i ar e flat, then K d is also flat. (iii) F or any short exact se quenc e of O X -Mo d ules 0 → L → L ′ → L ′′ → 0 , 15 we h a ve a formula Td( L ′ ) ≦ max { Td( L ) , Td( L ′′ ) } . (iv) F or any x ∈ X and quasi-c oher ent O X -Mo d ules L , K , we have T O R O X n ( L , K ) x ∼ → T or O X,x n ( L x , K x ) . As its c onse quenc e, we hav e the fol lowing f o rmula. Td( L ) ≦ sup x ∈ X Td O X,x ( L x ) . W e define a similar T or-dimension for un b ounded complexes. Definition 3.3 ([TT90], Def. 2.2 .1 1) . Let E • b e a complex of O X -Mo dules. (i) E • has ( glob al ly ) finite T or-a m plitude if t here are integers a ≦ b a nd for all O X -Mo dule F , H k ( E • ⊗ L O X F ) = 0 unless a ≦ k ≦ b . (In t he situation, w e say that E • has T or-ampli tude c o n taine d in [ a, b ]). (ii) E • has lo c al ly finite T or-amplitude if X is cov ered b y op ens U suc h that E • | U has finite T or-amplitude. Remark 3.4. ( i) If the sc heme X is quasi-compact, then ev ery lo cally finite T or-a mplitude complex E • of O X -Mo dules is g lo bally finite T or-amplitude. (ii) F or three v ertexes of a distinguished triangle in the derive d category of Mo d ( X ), if tw o of these three ve rtexes are globally finite T or- a mplitude then the third v ertex is also. 3.2 The coherator W e briefly review the the ory of “coherator” from [BG I7 1 ], I I and [TT 90] App endix B. There are t w o ab elian categories Qcoh ( X ) and Mo d ( X ) and the cano nical inclusion functor φ X : Qcoh ( X ) ֒ → Mo d ( X ) whic h is exact, closed under extensions, reflects exactness, pr eserv es and reflects infinite di- rect sums. The problem is that in general φ X do es not preserv e injectiv e ob jects in Qcoh ( X ). But for coheren t sc hemes, there is a go o d theory for Qcoh ( X ). W e are starting from reviewing the definition of coherence of sc hemes. Definition 3.5 ([D GSV72], VI) . The sc heme X is said to b e quasi-s e p ar ate d if the diagonal map X → X × X is quasi-compact or equiv alently if inter- section of an y pair o f affine op en sets in X is quasi-compact. It is said to b e c o her ent if it is quasi-compact and quasi-separated. 16 Prop osition 3.6 ([BGI71], I I, 3.2; [TT90], App endix B) . L et X b e a c oher- ent scheme. Then we h ave the fol lowing: (i) φ X has the right adjoint functor Q X : Mo d ( X ) → Qcoh ( X ) wh i c h is said to b e coherator and the c anon i c al adjunction map id → Q X φ X is an isomorphism. In p articular Qcoh ( X ) has enough inje ctive a n d clo s e d under limit. (ii) Q X pr e serves limit. (iii) F or an y E • ∈ D ( Qcoh ( X )) and F • ∈ D ( Mo d ( X )) , the c anonic al ad- junction maps E • → R Q X φ X E • and φ X RQ X F • → F • ar e quasi - i s omorphisms. 3.3 P erfect and pseudo-coheren t complexes W e review the notion of pseudo-coheren t and p erfect complexes. F or a com- plex of O X -Mo dules E • on X , p erfection and pseudo-coherence dep end only on the quasi-isomorphism class of E • and are lo cal prop erties on X . So fir st w e define the strict v ersion of them and next w e define them as b eing lo cal prop erties. Definition 3.7 ([BGI71], Exp. I; [TT90], § 2.2) . Let E • b e a complex of O X -Mo dules. (i) E • is strictly p erfe ct (resp. strictly pseudo - c oh e r ent ) if it is a b ounded complex (resp. bounded ab o ve complex) of algebraic vector bundles. (ii) E • is p erfe ct (resp. n -pseudo-c oher ent ) if it is lo cally quasi-isomorphic (resp. n - quasi-isomorphic) to strictly p erfect complexes. More precisely , for an y p oint x ∈ X , there is a neighborho o d U in X , a strictly p erfect complex F • , and a quasi-isomorphism (resp. an n -quasi-isomorphism) F • ∼ → E • | U . E • is said to b e pseudo-c oher ent if it is n - pse udo-coheren t fo r all in teger n . Lemma 3.8 ([TT90], § 2.2) . L et E • b e a c omplex of O X -Mo d ules on X . (i) If E • is strictly pseudo-c oher ent, then it is pseudo-c oher ent. (ii) In ge n er al, a pse udo - c oh e r ent c omplex may not b e lo c a l ly quasi-isomo rp h ic to a strictly pseudo-c oher ent c omplex. But if E • is pseudo-c oher ent c om- plex of quasi-c ohe r ent O X -Mo d ules , then E • is lo c al ly q uasi - isomorphic to a strictly p s eudo-c oher ent c omplex. (iii) If E • is a pseudo-c oher ent, then al l c ohomolo gy she af H i ( E • ) is quasi- c o her ent. I n p articular, a pseudo-c oher ent O X -Mo d ule is a quasi-c oher ent 17 O X -Mo d ule. Mor e over if we assume X is quasi-c omp act and E • is pseudo- c o her ent, then E • is c ohomolo gic al ly b ounde d ab ove. (iv) Mor e over if we assume X is no etherian, we have the fol lowing e quivalent c o nditions. (a) E • is ps e udo-c oher ent. (b) E • is c oho m olo gic al ly b ounde d ab ove and al l the c ohomolo gy she af H k ( E • ) ar e c oher ent O X -Mo d ules . In p articular, a ps eudo-c oher ent O X -Mo d ule is c oher ent. (v) T he c omplex E • is p erfe ct i f and only if E • is pseudo-c oher ent and has lo c al ly finite T or-amp l i tude. (vi) Pseudo-c ohe r enc e and p erfe ction h ave 2 out of 3 pr op erties. Namely, let E • , F • and G • b e the thr e e vertexes of a distinguishe d triangle in the derive d c a te gory of Mo d ( X ) an d if two of these thr e e vertexes ar e pseudo- c oh e r ent (r e s p. p erfe ct) then the thir d vertex is also. (vii) F or any c om plexes of O X -Mo d ules F • and G • , F • ⊕ G • is pseudo- c o her ent (r esp. p erfe ct) if and only if F • and G • ar e . (viii) A strictly b ounde d c o m plex of p erfe ct O X -Mo d ules E • is p erfe ct. Definition 3.9. (i) F or an y O X -Mo dule F , w e denote its supp ort b y Supp F := { x ∈ X ; F x 6 = 0 } . (ii) ([Tho97], 3 .2 ) F or a complex of O X -Mo dules E • , the c oho m olo gic al s up- p o rt o f E • is the subspace Supph E • ⊂ X those p oints x ∈ X at whic h the stalk complex of O X,x -mo dule E • x is not acyclic. (iii) F or an y closed subset Y of X , w e denote b y P erf ( X on Y ) (resp. P er f qc ( X o n Y ), sP erf ( X on Y )) the complicial biW aldhausen category of globally finite T o r - amplitude p erfect complexes (r esp. globally finite T o r- amplitude p erfect complexes o f quasi-coherent O X -Mo dules, strictly p erfect complexes) whose cohomolog ical supp ort on Y . Here, the cofibrations ar e the degree-wise split mono morphisms , and the w eak equiv alences are the quasi-isomorphisms. Put P er f r ( X ) := [ Codim Y ≧ r P er f ( X on Y ) . Lemma 3.10. (i) F or any short exact se quenc e of O X -Mo d ules 0 → F → G → H → 0 , 18 we h a ve Supp G = Supp F ∪ Supp H . (ii) F or a c omplex of O X -Mo d ules E • , we have Supph E • = ∪ n ∈ Z Supp H n ( E • ) . Lemma 3.11. F or a c oher ent sche m e X and its clo se d se t Y , the c anonic al inclusion functor Perf qc ( X on Y ) ֒ → Perf ( X on Y ) in duc es an e quivalen c e of c ate gories b etwe en thir derive d c ate gories. Pr o of. The inv erse functor of T ( Pe rf qc ( X o n Y )) → T ( Perf ( X on Y )) is giv en by the coherat o r (Prop. 3.6, (iii)). 3.4 Divisorial sc hemes Since p erfect and pseudo-coheren t complexes are w ell- b ehav ior o n divi s orial sc hemes, we briefly review the definition a nd fundamen ta l pr o perties of divi- sorial sche mes. Definition 3.12 ( [BG I71], I I, 2.2.5; [TT90], Def. 2.1.1) . A coheren t sc heme X is said t o b e diviso ria l if it has an ampl e fam ily of line bund les . That is it has a f a mily of line bundles {L α } whic h satisfies the follo wing condition (see op. cit. for another equiv alen t conditions): F o r any f ∈ Γ( X , L ⊗ n α ), we put the op en set X f := { x ∈ X | f ( x ) 6 = 0 } . Then { X f } is a basis for the Zariski top olog y of X where n runs ov er all p ositiv e in t eger, L α runs ov er the fa mily of line bundles a nd f runs o v er all global sections of all of L ⊗ n α . Example 3.13. (i) A quasi-pro jectiv e sc heme o ve r affine sche me is divisorial. So classical algebraic v arieties ar e divisorial. Since ev ery sc heme is lo cally affine, ev ery sche me is lo cally divisorial. (ii) A separated regular no etherian sc heme is divisorial. (iii) ([TT90], Exerc. 8.6) Let k b e an field and X an A n k with double origin. Then X is regular no etherian but is not divisorial. Lemma 3.14 ([D G61], I I, 5.5 .8 a nd [BGI71], I I, 2.2.3.1) . F or a line bund le L on X and a se ction f ∈ Γ( X , L ) , the c anoni c al op en immersio n X f → X is affi n e. 19 Theorem 3.15 (Global resolution theorem, [BGI71], I I; [TT90 ], Prop. 2.3 .1 ) . L et X b e a diviso rial scheme. Then we have the fol lowing. (i) A n y pseudo-c oher ent c omplex of quasi-c oher ent O X -Mo d ules is glob al ly quasi-isomorphic to a strictly pseudo-c oher ent c omplex. (ii) A ny p erfe ct c om plex is isomorphic to a strictly p erfe ct c om plex in D ( Mo d ( X )) . 3.5 Regular closed immersion There are sev eral definitions of regular immersion (see [DG67] and [BG I7 1], VI I). Both definitions are equiv alen t if a total sche me is no etherian. W e adopt the definition in [BG I71] a nd for readers conv enience, w e briefly review t he notation and fundamen tal prop erties of regular closed immersion. Definition 3.16. Let u : L → O X b e a morphism of O X -Mo dules from an algebraic ve ctor bundle L to O X . A Koszul c omple x asso ciated to u is the strictly p erfect complex Kos • ( u ) defined as follows: F or n > 0, we put Kos − n ( u )(= Ko s n ( u )) := n ^ L , and d n ( x 1 ∧ · · · ∧ x n ) := n X r =1 ( − 1) r − 1 u ( x r ) x 1 ∧ · · · ∧ b x r ∧ · · · ∧ x n . Definition 3.17 ([BG I71], VI I, 1.4) . (i) An O X -Mo dule ho momo r phism u : L → O X from an algebraic vector bundle L to O X is said to b e r e gular if Kos • ( u ) is a resolution of O X / Im u . (ii) An ideal sheaf I on X is r e gular if lo cally on X , there is a regular map u : L → O X suc h that Im u = I . More precisely , this means that if there is an op en co v ering { U i } i ∈ I of X and for each i ∈ I , there is a regular ma p u i : L | U i → O U i suc h that Im u i = I | U i . (iii) A closed immersion Y ֒ → X is said to b e r e gular if the defining ideal of Y is regular . W e put N X/ Y := I / I 2 and call it the c onormal she af o f the regular closed immersion. Lemma 3.18 ([DG 67 ]) . L et Y ֒ → X b e a r e gular close d immersi o n whose defining id e al is I . (i) The ide al she af I satisfies the fol lowin g c ondition s: 20 (a) I is of fi nite typ e. (b) F or e ach n , I n / I n +1 is a lo c al ly fr e e O X / I -Mo dule of finite typ e. (c) A c anonic al map Sym O X / I ( N X/ Y ) → Gr I ( O X ) is an isomorphism of O X / I -A lgebr a. Her e Sym O X / I ( N X/ Y ) is the sym- metric algebr a ass o cia te d to N X/ Y , Gr I ( O X ) = L n ≧ 0 I n / I n +1 is the gr ad e d algebr a asso ciate d to an I -adic filtr a tion in O X and the c anon ic al map is define d by the universa l pr op erty of symmetric algebr a. (ii) I f the scheme X is no etherian, then I is r e gular in the sense of [DG67]. That is, for any p oint x ∈ X ther e is an op en neigh b orho o d U of x , and a r e gular se quenc e f 1 , . . . , f r ∈ Γ( U, I ) which gener ates I | U . 4 W eigh t on pseu do-coherent Mo dules Definition 4.1. A pseudo-coheren t O X -Mo dule F is of weight r if it is of T or-dimension ≦ r and there is a r egula r close d immersion Y ֒ → X of co dimension r in X such that the supp ort of F is in Y . W e denote by Wt r ( X ) the category o f pseudo-coheren t O X -Mo dules of w eight r . F or a regular closed immersion Y ֒ → X of co dimension r , w e denote b y Wt r ( X on Y ) the category of pseudo coheren t O X -Mo dules of w eight r supp orted on the subspace Y . Immediately , a pseudo-coheren t O X -Mo dule of w eight 0 is just an algebraic v ector bundle. Lemma 4.2. The c ate gory Wt r ( X on Y ) is close d under extens i o ns and di- r e ct summand in the ab elian c ate gory Mo d ( X ) . In p articular, Wt r ( X on Y ) is an idemp otent c o mplete exact c ate gory. Pr o of. The assertion follows from Lemma 3.2, (iii), Lemma 3.10, (i), and Lemma 3.8, (vii). A pseudo-coheren t O X -Mo dule F o f we igh t r has globally finite T or - amplitude. Th us it is p erfect b y Lemma 3.8, (v) and w e hav e an inclusion functor Wt r ( X on Y ) ֒ → Perf ( X on Y ). Moreo v er w e hav e the natural in- clusion functor Ch b ( Wt r ( X on Y )) ֒ → Perf ( X on Y ) by Lemma 3.8, (viii). No w, we state our main theorem. 21 Theorem 4.3. L e t X b e a divisorial scheme and Y ֒ → X a r e gular clos e d immersion of c o dimens ion r . Th en the inclusion Ch b ( Wt r ( X on Y )) ֒ → P er f ( X on Y ) induc es a derive d Morita e quivalenc e. No w consider the inclusion functor Wt r ( X on Y ) → Ch b ( Wt r ( X on Y )) whic h sends F in Wt r ( X on Y ) to t he complex whic h is F in degree 0 and 0 in other degrees. W e denote by K S ( Ch b ( Wt r ( X on Y )); qis) the K -theory sp ectrum of the W aldhausen catego r y asso ciated to Ch b ( Wt r ( X on Y )) whose w eak equiv alences are the quasi-isomorphisms. The inclusion a b ov e induces a ho motop y equiv alence K S ( Wt r ( X o n Y )) ∼ → K S ( Ch b ( Wt r ( X on Y )); qis) b y non-connected v ersion of the Gillet- W aldhausen theorem in [Sc h04 ]. There- fore we g et the following corollary . Corollary 4.4. In the notation ab ove, we have the identities K S ( Wt r ( X on Y )) ∼ → K S ( X on Y ) ∼ → K B ( X on Y ) , H C ( Wt r ( X on Y )) ∼ → H C ( X on Y ) . Pr o of. F or the K - theory case, it is follow ed from t he observ at io n ab o ve and the Sc hlich ting appro ximatio n theorem a nd the comparison theorem in [Sc h06]. F o r the cyclic homology case, it is fo llow ed fro m the deriv ed inv ari- ance b y [Kel99 ]. 5 Pro of of the main theorem First w e consider the follow ing t w o categor ies. Let B b e the category o f p erfect complexes in Ch − ( Wt r ( X on Y )) a nd C the catego ry of p erfect complexes of quasi-coheren t O X -Mo dules supp orted on Y . By Lemma 3.8, the categories B a nd C are closed under extensions and direct summand in Ch ( Mo d ( X )). Therefore, they are idemp oten t complete exact cate- gories. Note that any p erfect complex ha s globally finite T o r-amplitude on X (Rem. 3.4 and Lem. 3.8, (v)). F rom Lemma 3.8 , (iii), w e hav e the f o llo wing natural exact inclusion f unctors Ch b ( Wt r ( X o n Y )) α → B β → C γ → Perf ( X on Y ) . W e shall prov e α , β and γ induce category equiv alences b et wee n their asso- ciated deriv ed categories by using the fo llo wing criterion. 22 Lemma 5.1 ([TT90], 1.9 .7 and [Tho93]) . L et i : X → Y b e a ful ly faithful c o mplicial exact functor b etwe en c om plicial biWaldhausen c ate gories which close d under the formation of c an o nic al h omotopy pul lb acks and pushouts and assume their we ak e quivalenc e classes ar e just quasi-isomorphism classes. If i satisfies the c ond i tion ( DE ) or ( DE ) op b e low, then i ind uc es c ate gory e quiv- alenc es b e twe en their deriv e d c ate go ries. ( DE ) F or an y obje ct Y in Y , ther e is an obje ct X in X an d a we ak e quiva- lenc e i ( X ) → Y . ( DE ) op F or any obje ct Y in Y , ther e is an obje ct X in X a n d a we ak e quiv- alenc e Y → i ( X ) . W e shall prov e that α induces categor y equiv alence b et w een their deriv ed categories. T o do so first we review the follo wing lemma. Lemma 5.2 ([BS01], 2.6) . L et E b e an id e mp otent c ompl e te exa ct c ate gories and f : X • → Y • a quasi-is o morphism b etwe en b ounde d ab ove c omplexe s in Ch ( E ) . Assume X • or Y • is strictly b o und e d. Say the other one as Z • . Then ther e is a sufficiently smal l N such that Z • → τ ≧ N Z • is a quasi-isomorphis m . Lemma 5.3. The inclusion α : Ch b ( Wt r ( X on Y )) ֒ → B satisfies the c on- dition ( DE ) op in L emma 5.1. In p articular, we have an e quivalen c e of c ate- gories T ( C h b ( Wt r ( X on Y ))) ∼ → T ( B ) . Pr o of. Let E b e t he category of pseudo-coherent O X -Mo dules of T or- dimens ion ≦ r . It is closed under extensions (Lem. 3.2, (iii)) and direct summand (Lem. 3.8, (vii)). In particular, it is an idemp oten t complete exact cate- gory . W e denote b y D the category of p erfect complexes in Ch − ( E ) whose cohomological supp ort is in Y . Fix a complex P • in B . By the glo bal resolu- tion theorem (Th. 3.15), P • is quasi-isomorphic to a strict p erfect complex. Since we ha v e an inclusion sP erf ( X on Y ) ⊂ D , P • is quasi-isomorphic to a b ounded complex in D . No w applying Lemma 5.2 to E , there exists a n in teger N suc h that the canonical map P • → τ ≧ N P • is a quasi-isomorphism. Since Supp(Im d N − 1 ) is in Y , τ ≧ N P • is actually in Ch b ( Wt r ( X on Y )). The assertion follo ws from it. Prop osition 5.4. The inclusion functor β : B ֒ → C satisfies the c on d ition ( DE ) in L emma 5.1. T o prov e Prop osition 5.4, we need the f o llo wing lemmas. 23 Lemma 5.5. (i) L et I b e the definition ide al of Y . Then O X / I p is of weight r for any non-ne ga tive inte ger p . (ii) L et F b e a pseudo-c oher ent O X -Mo d ule of w e ight r and L an algebr aic ve ctor bund le. Then, L ⊗ O X F is of weight r . Pr o of. (i) First w e notice that O X / I is in Wt r ( X on Y ) by K o szul reso- lution. Next since I n / I n +1 is lo cally isomorphic to direct sum of O X / I , w e learn that I n / I n +1 is a lso in Wt r ( X on Y ) b y Lemma 3.2 (iv). Using Lemma 4.2 for 0 → I n +1 / I n + p → I n / I n + p → I n / I n +1 → 0 , the d ´ evissage argumen t show s that I n / I n + p is also in Wt r ( X on Y ) for a ny non-negativ e in teger n and p ositiv e in teger p . (ii) Since L is flat, we ha v e an inequality Td ( L ⊗ O X F ) ≦ r . W e also ha v e a form ula Supp L ⊗ O X F = Supp L ∩ Supp F ⊂ Y . Therefore L ⊗ O X F is of w eight r . Lemma 5.6 ([TT90], Lem. 1.9.5) . L et A b e an ab elian c ate go ry and D a ful l sub ad d itive c ate gory of A . L et C b e a ful l sub c ate gory of Ch ( A ) s atisfies the fol lo wing c onditions: (a) C is close d under quasi-isom orphisms. That is, any c omplex quasi-isomorphic to an o bje ct in C is also in C . (b) Every c omplex i n C is c ohomolo gic al ly b ounde d ab ove. (c) Ch b ( D ) is c o n taine d in C . (d) C c ontains the mapping c one of any map fr om an obje ct in C h b ( D ) to an obje ct in C . Final ly, Supp ose the fol lowing c ond ition, so “ D has enough obje cts to r e- solve”: (e) F or any inte ger n , any C • in C such that H i ( C • ) = 0 for an y i ≧ n and any ep imorphism in A , A ։ H n − 1 ( C • ) , then ther e exists a D in D and a morphism D → A such that the c omp osite D ։ H n − 1 ( C • ) is an epimorphism in A . Then, for an y D • in Ch − ( D ) ∩ C , any C • in C , a n d any morphism x : D • → C • , ther e e x ist a E • in Ch − ( D ) ∩ C , a d e gr e e-wise split mon omorphism a : D • → E • and a quasi-isomorphism y : E • ∼ → C • such that x = y ◦ a . Mor e over if x : D • → C • is an n -quasi-is o morphism for some inte ge r n , then one may c h o ose E • ab ove so that a k : D k → E k is an is omorphism for k ≧ n . 24 Lemma 5.7. L et X b e a divisorial s cheme w h ose ampl e family o f lin e bund les is {L α } an d E • a p erfe ct c omplex on X . Then ther e ar e line bund le s L α k in the am ple family, inte g ers m k and se ctions f k ∈ Γ( X , L ⊗ m k α k ) ( 1 ≦ k ≦ m ) such that (a) F or e ach k , X f k is affine. (b) { X f k } 1 ≦ k ≦ m is an op en c over of X . (c) F or e a ch k , E • | X f k is quasi-isomorph ic to a strictly p e rf e ct c omp lex. Pr o of. Since E • is p erfect, w e can take a n affine op en cov ering { U i } i ∈ I of X such that E • | U i is quasi-isomorphic t o a strictly p erfect complex for each i ∈ I . Since {L α } is a n ample family , for each x ∈ X, there are an i x ∈ I , a line bundle L α x in the a mple f amily , an in teger m x and a section f x ∈ Γ( X , L ⊗ m x α x ) suc h that x ∈ X f x ⊂ U i x . Since U i x is affine, X f x is affine b y Lemma 3.14 . No w { X f x } x ∈ X is an affine op en co v ering of X and has a finite sub cov ering b y quasi-compactness of X . Lemma 5.8 ([TT90], Lem. 1.9.4 , (b)) . L et E • b e a s trictly pseudo - c oh e r ent c o mplex on X such that H i ( E • ) = 0 for i ≧ m . Then Ker d m − 1 is an algebr aic ve ctor bund le. In p articular H m − 1 ( E • ) is of finite typ e. Pr o of of Pr op. 5. 4. Let {L α } b e an ample family of line bundles on X and I the defining ideal of Y . W e denote b y D the additiv e category generated by all the L ⊗ m α ⊗ O X O X / I p with in teger m and p ositiv e in teger p . By Lemma 5.5, D ⊂ Wt r ( X on Y ). W e in tend to apply Lemma 5 .6 to A = Qcoh ( X on Y ) the category of quasi-coheren t O X -Mo dules whose supp ort on Y . T o do so, w e hav e to c hec k the a ssu mptions in Lemma 5.6. Only non-trivial assumption is “ having enough ob jects to resolv e” condition. Let C • b e a complex in C suc h that H i ( C • ) = 0 for i ≧ n , and F ։ H n − 1 ( C • ) an epimorphism in A . By Lemma 5.7, there ar e line bundles L α k in tegers m k and their sections f k ∈ Γ( X , L ⊗ m k α k ) (1 ≦ k ≦ m ) suc h tha t they satisfy the follow ing conditions. (a) F o r eac h k , X f k is affine. (b) { X f k } 1 ≦ k ≦ m is an o pen cov er of X . (c) F or eac h k , C • | X f k is quasi-isomorphic t o a strictly p erfect complex. Fix an in teger k . Since H n − 1 ( C • ) | X f k is o f finite t yp e by Lemma 5.8, there is sub O X f k -Mo dule of finite ty p e G ⊂ F | X f k suc h t ha t the comp o sition G ֒ → F | X f k ։ H n − 1 ( C • ) | X f k is an epimorphism. No w since G and I | X f k are O X f k -Mo dules of finite type (Lemma 3 .18, (i)), we ha v e ( I | X f k ) p k G = 0 for some p k . Therefore G is considered as O X / I p k | X f k -Mo dule of finite type. 25 Hence w e ha v e an epimorphism ( O X / I p k | X f k ) ⊕ t k ։ G . W e hav e an O X - Mo dules homomorphism ( O X / I p k ) ⊕ t k → F ⊗ O X L ⊗ m k s k α k for some integer s k ([DG60], 9.3.1 and [DG 64], 1.7.5). Therefore considering t he same argumen t for ev ery k , w e get a morphism m M k =1 ( O X / I p k ⊗ O X L ⊗− m k s k α k ) ⊕ t k → F whose comp osition with F ։ H n − 1 ( C • ) is an epimorphism in Qcoh ( X on Y ). Finally , w e shall prov e that γ induces category equiv a lence b etw een their deriv ed catego ries. Now w e consider the following exact inclusion functor s: C γ 1 → Perf qc ( X on Y ) γ 2 → Pe rf ( X on Y ) . Lemma 3.1 1 assert that γ 2 induces a homoto p y equiv alence o n sp ectra. Thu s, it is enough to sho w that the inclusion functor γ 1 induces an equiv alence of categories b etw een their derive d categories. More stro ng ly we show the follo wing: Lemma 5.9. The lo c al c o h omolo gic al functor R Γ Y = lim − → E X T ( O X / I p , ?) : T ( P erf qc ( X on Y )) → T ( C ) gives inverse functor of the inclusion functor γ 1 . Pr o of. Let us consider the functor Γ Y := lim − → HO M ( O X / I p , ?) : Qcoh ( X ) → Qcoh ( X on Y ) . Since I is of finite type, for any O X -Mo dule M in Qcoh ( X on Y ) , we ha v e the iden tity (3) Γ Y M = M . This iden tity and the existenc e of the canonical nat ural transformation Γ Y → id imply that Γ Y is a righ t adj o in t functor of the inclusion Qcoh ( X on Y ) ֒ → Qcoh ( X ). Therefore w e learn that Qcoh ( X on Y ) has enough injectiv e ob- jects and for an y complex E • in C suc h that each comp onen ts are injectiv e quasi-coheren t O X -Mo dules, we hav e the iden tity R Γ Y E • = E • b y (3 ) . Com- bining the ob vious fact that γ 1 is fully faithful, w e conclude that R Γ Y giv es an inv erse functor of γ 1 . 26 6 Applicatio n s In this section, w e assume that A is the Cohen-Macaulay ring of Krull di- mension d and X = Sp ec A . By the very definition, the ring A satisfies the follo wing condition (cf. [Bou98 ], § 2.5, Prop. 7): F or any ideal J in A suc h that its high t h t J = r , there is an A - regular sequence x 1 , . . . , x r con tained in J . In this case, a coheren t A -mo dule of we igh t d is just a mo dule o f finite length and finite pro jectiv e dimension. Prop osition 6.1. F or any inte ger 0 ≦ r ≦ d , Wt r ( X ) is cl o s e d under extensions in Mo d ( X ) . In p articular Wt r ( X ) is an idemp otent c o m plete exact c ate gory in the natur al way. Pr o of. Let us consider the short exact sequence F ֌ G ։ H in Mo d ( X ) suc h that F and H ar e in Wt r ( X ). Th en w e learn that G is of T or-dimension ≦ r and Co dim Supp G ≧ r . Therefore there is an A - regular sequence x 1 , . . . , x r suc h that Supp G ⊂ V ( x 1 , . . . , x r ). Hence w e conclude that G is in Wt r ( X ). Theorem 6.2. F or any inte ger 0 ≦ r ≦ d , the c anon i c al inclusion f unc tor Ch b ( Wt r ( X )) ֒ → P erf r ( X ) is a derive d Morita e q uivalenc e. Pr o of. W e can write the categor ies Ch b ( Wt r ( X )) a nd Perf r ( X ) as follows. Ch b ( Wt r ( X )) = lim Y ⊂ X Ch b ( Wt r ( X on Y )) , P er f r ( X ) = lim Y ⊂ X P er f ( X on Y ) , where the limits taking ov er the regular closed immersion of co dimension ≧ r . Hence w e get the result b y Theorem 4.3 and contin uit y of functor T . Corollary 6.3. F or any inte ge r 0 ≦ r ≦ d , we h a v e the c a nonic al homo topy e q uivalenc e of sp e ctr a and mixe d c om p lexes K S ( Wt r ( X )) ∼ → K S ( P erf r ( X )) , H C ( Wt r ( X )) ∼ → H C ( Perf r ( X )) . 27 Pr o of. Since b oth theories are deriv ed inv ariant, the statemen t is just a corol- lary of Theorem 6.2. No w moreov er w e a ssume that A is lo cal and let m b e its maximal ideal. Then since A is Cohen-Macaula y , Y := V ( m ) ֒ → X is a regular closed im- mersion. Therefore by Theorem 4.3, w e learn that K S ( X o n Y ) is homotop y equiv alen t to K S ( Wt ( X o n Y )). Now recall that W eib el’s K -dimensional conjecture. Conjecture 6.4 ( K -dimensional conjecture) . F or any no etherian sch eme Z of fin ite Krul l-dimensio n n , and inte ger q > n , we have K B − q ( Z ) = 0 . This conjecture is recen tly pro ved for sc hemes whic h is essen tially of fi- nite type ov er a field of characteristic 0 [CHSW08]. According to the pap er [Bal07], if for an y lo cal ring O Z,z of Z , w e hav e K B − q (Sp ec O Z,z on { z } ) = 0 for q > dim O Z,z , then the conjecture ab o v e is true for Z . Therefore for an y Cohen-Macaula y sc heme, the conjecture is r educ ed to v anishing of K S − q ( Wt ( X on Y )) for q > d . References [BS01] P . Ba lmer, M. 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