Comparison of cobordism theories

COMP ARISON OF COB O RDISM THEORIES MARC LEVINE Abstract. Relying on results of Hopkins-Morel, we sho w that , for X a quasi- pro j ectiv e v ariety o ver a field of c haracteristic zero, the canonical map Ω n ( X ) → MGL ′ 2 n,n ( X ) i s an isomorphism. Contents Int ro duction 1 1. Motivic homotopy theo ry 3 2. MGL cohomolo gy and Bor el-Mo ore homology 6 3. Ω ∗ and MGL ′ 2 ∗ , ∗ 9 4. MGL ∗ , ∗ for fields 11 5. Chern classes, divisors and the b oundary map 13 6. Pro of of the main theorem 20 References 22 Introduction In what fo llows, k will b e a fixed base-field of characteristic zero. W e let Sm /k denote the categor y of smooth, quasi-pro jective v arieties o ver k , and S H ( k ) the Morel-V o evo dsky motivic stable homo to p y ca tegory (see [4, 10]). W e denote the classical stable homotopy c ategory b y S H . V o evodsky [10] has defined the big r aded cohomology theory MGL ∗ , ∗ on Sm /k as the theor y repres e nted b y the algebr aic Thom complex MGL ∈ S H ( k ). This in turn is the T -sp ectrum constructed as the algebraic a nalog of the classical Thom complex MU ∈ S H , replacing B U n with BGL n and the Thom space o f the universal C n -bundle E n → B U n with the Thom space of the univ ersal rank n vector bundle ov er BGL n . Besides the definition of complex cob ordism via stable homotopy theory , one can also descr ib e MU ∗ ( X ) as cob or dism classes of C - oriented pro p e r maps. As an algebraic version of this co nstruction, w e ha ve defined w ith F. Morel the theory X 7→ Ω ∗ ( X ), also called algebraic cob or dism, and sho w th at Ω ∗ is the univ ersal 1991 Mathematics Subje ct Classific ation. Pr imary 14C25, 19E15; Secondary 19E08 1 4F42, 55P42. Key wor ds and phr ases. Al gebraic cobordism, Morel-V oevodsky motivic stable homotop y cat- egory , oriented cohomology . The aut hor gratefully ack nowledge s the support of the Hum boldt F oundation, and the support of the NSF via the grant DMS-0457195. 1 2 MARC LEVINE oriented cohomology theory on Sm /k , in the sense o f [3, definition 5.1.3 ]. This universal pro p er t y gives us a natural trans fo rmation of functors on Sm /k : ϑ MGL ( X ) : Ω ∗ ( X ) → MGL 2 ∗ , ∗ ( X ) . As remarked in the intro duction to [3 ], Hopkins- Morel ha ve constructed a spectra l sequence E p,q 2 ( n ) = H p − q,n − q ( F ) ⊗ L q = ⇒ MGL p + q,n ( F ) ; using this, it is easy to show that ϑ MGL ( k ) is surjective. Cho osing an em b edding σ : k → C , gives the natur al transformation ϑ MU ,σ ( X ) : Ω ∗ ( X ) → MU 2 ∗ ( X ( C )) and the commutativ e diagr am Ω ∗ ( X ) ϑ MGL ( X ) / / ϑ MU ,σ ( ( R R R R R R R R R R R R R R MGL 2 ∗ , ∗ ( X ) Re σ ( X )   MU 2 ∗ ( X ( C )) . As ϑ MU ,σ ( k ) is an isomor phism ([3, theorem 1.2.7]), this sho ws that ϑ MGL ( k ) is an is o morphism as well. The purp ose of this pap er is to extend this to show that ϑ MGL ( X ) is an isomorphism for all X ∈ Sm /k . In fact, we pr ove more. In [2], we hav e shown how one can extend MGL ∗ , ∗ to a bi-gr ade d oriente d duality the ory (MGL ′ ∗ , ∗ , MGL ∗ , ∗ ) on quas i-pro jective k -schemes, Sc h k , and how ϑ MGL extends to a natural transfo r mation ϑ MGL ′ : Ω ∗ → MGL ′ 2 ∗ , ∗ . Here Ω ∗ is the extension of X 7→ Ω dim X −∗ ( X ) to a n oriented Borel-Mo o re ho mology theory on Sch k , as describ ed in [3]. Our main r esult in this pap er is that ϑ MGL ′ ( X ) is an isomorphism for all X ∈ Sc h k . W e review the underlying founda tions o f motivic homotopy theory in § 1, reca ll the co ns truction of MGL-theory a nd its extension to a Bor el-Mo ore homolog y the- ory in § 2 and dis c us s the g eometric theory Ω ∗ and its relation to MGL -theory in § 3, stating our main result in theorem 3.1. The next three sections deal with the pro of of theorem 3.1. In § 4 , we use the Hopkins-More l sp ectra l sequence to get g e n- erators for MGL 2 n,n ( F ) and MGL 2 n − 1 ,n ( F ), for F a finitely gener a ted field ov er k . As outlined a bove, this prov es the r esult for X = Spec F . In § 5, w e lo ok at the bo undary map ∂ : MGL ′ 2 n +1 ,n ( k ( X )) → lim − → W ⊂ X MGL ′ 2 n,n ( W ) and give a formula for ∂ in terms o f the diviso r cla sses defined in [3]. Combining this information with the right-exact lo caliza tion seq uence for Ω ∗ gives a pro of of the main theorem by us ing induction on the dimension of X . Although the existence of the Hopkins-Mo rel spectra l sequence has b een an- nounced some time ago, and its constr uction has been describ ed in lectures (e.g., lectures of M. Hopkins at Ha rv ard in the Spring of 20 06), the details o f the con- struction hav e not yet b een published. The cautious re a der may ther e fore wan t to consider the results of this pap er as conditional, relying on the existence of the Hopkins-Morel sp ectral sequence. COMP ARISON OF COBORDISM THEORIES 3 1. Motivic homotopy theor y W e b egin by r ecalling some ba sic notions in the Morel- V o evodsky mo tiv ic stable homotopy categ o ry S H ( k ); we r efer the reader to [4, 1 0] for details . One starts with the category Sp c ( k ) of sp ac es over k , this b eing the ca tegory of pr esheav es of simplicial sets on Sm /k . The p ointed version, Sp c • ( k ) is the categ o ry of presheav es of pointed simplicial sets on Sm /k . The standa rd o per ations on simplicial s ets, e.g., pro ducts A × C B , co-pro ducts A ∪ C B , quotients A/B := A ∪ B pt , etc., are a ll inherited b y Sp c ( k ), by op era ting on the v alues of the given presheaves. In pa rticular, o ne ha s wedge pr o duct a nd po int ed union in Sp c • ( k ). In a ddition, the internal Hom in s implicial sets H om ( A, B )( n ) := Hom Sp c ( A × ∆ n , B ) gives rise to an internal Hom in Sp c ( k ); we ha ve p o int ed versions as w ell. Letting Sp c denote the category of simplicial sets (and Sp c • po int ed simplicial sets), taking the constant pr esheaf giv es functors Sp c → Sp c ( k ), Sp c • → Sp c • ( k ). W e a ls o hav e the fully faithful functor Sm /k → Sp c ( k ), taking X ∈ Sm / k to the representable presheaf (of sets) Ho m Sm /k ( − , X ), wher e we iden tify a set S with the constant simplicial se t n 7→ S . Similarly , given X ∈ Sm /k a nd a k -p oint x ∈ X , we may consider the pointed sc heme ( X, x ) as an ob ject in Sp c • ( k ). Example 1.1 . The simplicia l susp ension op er ator Σ s : Sp c • ( k ) → Sp c • ( k ) is Σ s W := S 1 ∧ W . Similarly , define Σ gm W := G m ∧ W , wher e G m := A 1 \ { 0 } , po int ed by 1 . F or a ≥ b , define the weighte d spher e S a,b ∈ Sp c • ( k ) by S a,b := Σ a − b s Σ b gm S 0 . W e hav e as well the T -susp ension ope rator Σ T W := ( A 1 / ( A 1 \ { 0 } )) ∧ W The unstable motivic ho motopy c ate gory H ( k ) is formed from Sp c ( k ) by a t wo- step lo ca lization pro cess. First, one intro duces the Nisnev ich top olo gy . Giv en A ∈ Sp c ( k ), and x ∈ X ∈ Sm /k , one ha s the stalk of A at x , A x := lim − → x ∈ U → X A ( U ) as x ∈ U → X runs o ver all Nisnevic h neighbor ho o ds of x in X , i.e., ov e r all ´ etale maps f : U → X together with a lifting of the inclusion x → X to x → U . Decla re a map f : A → B in Sp c ( k ) to a Nisnevich lo c al we ak e quivalenc e if for each x ∈ X ∈ Sm /k , the ma p of simplicial sets f x : A x → B x induces an is omorphism on π 0 and on all homotopy groups π n ( A x , q ) → π n ( B x , f x ( q )), n ≥ 1 . Inv erting all Nisnevich lo cal weak equiv alences forms the categor y H Nis ( k ). Next, o ne defines an ob ject Z of H Nis ( k ) to b e A 1 -lo c al if, for all X ∈ Sm /k , and all n ≥ 0, the map p ∗ : Hom H Nis ( k ) (Σ n s X + , Z ) → H om (Σ n s X × A 1 + , Z ) is an isomo r phism in H Nis ( k ). A f : A → B in H Nis ( k ) is a n A 1 -we ak e qu ivalenc e if p ∗ : Hom H Nis ( k ) (Σ n s B + , Z ) → : Hom H Nis ( k ) (Σ n s A + , Z ) is an isomorphism for a ll A 1 -lo cal Z and a ll n ≥ 0 . Inv erting all the A 1 -weak equiv alences in H Nis ( k ) giv es us the catego ry H ( k ). The same cons truction in the po int ed setting gives us the pointed v ersio n H • ( k ). 4 MARC LEVINE R emark 1.2 . In H • ( k ), we ha ve isomorphis ms ( P 1 , ∞ ) ∼ = A 1 / ( A 1 \ { 0 } ) ∼ = S 1 ∧ G m = S 2 , 1 . Thu s, the T -susp ension oper ator Σ T is isomorphic (in H ( k )) to P 1 ∧ − . F or ea ch ob ject A in H • ( k ) we hav e the bi-gr ade d A 1 homotopy she aves π A 1 a,b ( A ), defined (for a ≥ b ≥ 0) as the Nisnevich sheafification of the presheaf U 7→ Hom H • ( k ) ( U + ∧ S a,b , A ) W e now pass to the stable theory . The motivic stable homotopy c ate gory over k , S H ( k ), is defined as a localiza tion of the catego ry Spt ( k ) of T - sp e ctra : ob jects in Spt ( k ) are sequences E := ( E 0 , E 1 , . . . ) E n ∈ Sp c • ( k ), together with b onding maps ǫ n : E n ∧ T → E n +1 . A morphism f : E → E ′ is a s equence of ma ps f n : E n → E ′ n in Sp c ( k ) that commute with the res pective bonding maps. The construction of Spt ( k ) and S H ( k ) models that of the category of sp ectra, Spt and the classica l stable homotopy category S H : 1. The category of spec tr a, Spt , is the category with ob jects sequences of p ointed simplicial s ets, ( E 0 , E 1 , . . . ), together with b onding maps E n ∧ S 1 → E n +1 , where a morphis m E → E ′ is given by a sequence of maps E n → E ′ n commuting with the bo nding maps. Sending a p ointed s implicial set S ∈ Sp c • to Σ ∞ S := ( S, S ∧ S 1 , S ∧ S 1 ∧ S 1 , . . . ) with identit y bo nding maps defines a functor Σ ∞ : Sp c • → Spt . Defining the susp ension op erator Σ on Spt by Σ( E 0 , E 1 , . . . ) := (Σ E 0 , Σ E 1 , . . . ) , the functor Σ ∞ commutes with the re spe c tiv e susp ension op erator s. 2. Given a morphism f : E → E ′ in Spt , one has the c ofib er se quenc e E f − → E ′ i − → C ( f ) p − → Σ E where C ( f ) is the sequence of mapping cones C ( f n ) := ([0 , 1] × E n / 0 × E n ∪ [0 , 1] × ∗ ) ∪ 1 × E n E ′ n . The map i is the sequence of inclusions E ′ n → C ( f n ) and p is the sequence of quo- tien t maps C ( f n ) → S 1 ∧ E n (collapsing i ( E ′ n )). 3. F or a sp ectrum E = ( E 0 , E 1 , . . . ), the stable homotopy gr oup π s a ( E ) is defined by π s a ( E ) := lim − → n π n + a E n where the colimit is taken us ing the maps sending f : S n + a → E n to S n + a +1 = S n + a ∧ S 1 f ∧ id − − − → E n ∧ S 1 → E n +1 . COMP ARISON OF COBORDISM THEORIES 5 Note that π s a ( E ) is defined for all a ∈ Z . A mor phis m E → E ′ is a stable we ak e quivalenc e if the induced ma p π s a ( E ) → π s a ( E ′ ) is an isomor phism for all a . 4. The stable homo to p y catego r y S H is formed from Spt b y inv erting the sta- ble weak equiv alences; t he suspensio n op erato r on Spt desc e nds to one on S H . The functor Σ ∞ : Sp c • → Spt descends to Σ ∞ : H → S H , commuting with the r esp ective susp ension op er a tors, and the susp ension op era- tor Σ on S H is an equiv alence. S H is a triangulated category with tr a nslation Σ and distinguished triangles given by the images o f cofib er sequences (up to isomor - phism). The construction of the stable homo topy c a tegory S H , suitably modified, giv es us the motivic s ta ble homotopy catego ry S H ( k ). W e g ive here a quick sketc h o f the construction. Given E = ( E 0 , E 1 , . . . ) in Spt ( k ), the b o nding maps give ris e to the inductive system . . . → π A 1 2 n + a,n + b ( E n ) → π A 1 2 n +2+ a,n +1+ b ( E n +1 ) → . . . defined by sending f : U + ∧ S 2 n + a,n + b → E n to the comp osition U + ∧ S 2 n +2+ a,n +1+ b = U + ∧ S 2 n + a,n + b ∧ S 2 , 1 f ∧ id − − − → E n ∧ S 2 , 1 ∼ = E n ∧ T ǫ n − → E n +1 . Define the motivic stable homotopy she af π A 1 a,b ( E ) by π A 1 a,b ( E ) := lim − → n π A 1 2 n + a,n + b ( E n ) . Note that π A 1 a,b ( E ) is defined for all a, b ∈ Z . A morphism f : E → E ′ in Spt ( k ) is a stable A 1 we ak e qu ivalenc e if f induces an isomorphism f ∗ : π A 1 a,b ( E ) → π A 1 a,b ( E ′ ) for all a, b . S H ( k ) is formed from Spt ( k ) by in verting all A 1 stable weak eq uiv a- lences. R emarks 1 .3 . 1. W e hav e the infinite T -susp ens ion functor Σ ∞ T : Sp c • ( k ) → Spt ( k ) sending A to the sequence ( A, A ∧ T , A ∧ T ∧ T , . . . ). W e hav e a s well the o per ations Σ T , Σ s , Σ gm on Spt ( k ), defined by Σ ? ( E 0 , E 1 , . . . ) := (Σ ? E 0 , Σ ? E 1 , . . . ) , and commut ing with Σ ∞ T . 2. Σ ∞ T descends to a functor Σ ∞ T : H • ( k ) → S H ( k ) 6 MARC LEVINE and Σ T , Σ s and Σ gm descend to op erator s on S H ( k ) a nd H • ( k ), with Σ T ∼ = Σ s ◦ Σ gm ∼ = Σ gm ◦ Σ s . 3. Σ T is in vertible o n S H ( k ), the inv erse given by ( E 0 , E 1 , . . . ) 7→ ( pt, E 0 , E 1 , . . . ) Thu s Σ s and Σ gm are also in vertible on S H ( k ). F o r a, b ∈ Z , define the opera tor Σ a,b on S H ( k ) by Σ a,b := Σ a − b s ◦ Σ b gm . 4. Given a morphism f : E → F in Sp c • ( k ), we have the cone Cone( f ) := ([0 , 1] × E / 0 × E ∪ [0 , 1] × ∗ ) ∪ 1 × E F, and the cofib er sequence E f − → F → Cone( f ) → Σ s E defined just as fo r spaces. Given a mor phism f : E → F in Spt ( k ), we hav e the cone Cone( f ) n = Co ne( f n : E n → F n ) and the cofib er sequence E f − → F → Cone( f ) → Σ s E which is just the se q uence of cofiber seq uences E n f n − → F → Co ne( f n ) → Σ s E n . S H ( k ) is a triangulated categor y , with trans lation functor Σ s . The distinguis hed triangles are those isomor phic to the image of a co fibe r sequence in Spt ( k ). Definition 1.4. L et E and F be in S H ( k ). The E -c ohomolo gy gr oups of F , E a,b ( F ), are E a,b ( F ) := Hom S H ( k ) ( F, Σ a,b E ) . F or X ∈ Sm /k , define E a,b ( X ) : E a,b (Σ ∞ T X + ) = Hom S H ( k ) (Σ ∞ T X + , Σ a,b E ) . 2. MGL coho mology and Borel-Moore homol ogy Let p : U → B b e a vector bundle over a scheme B with 0 section 0 B . The Thom sp ac e of U is the spa ce ov er k Th( U ) := U / U \ 0 B ∈ Sp c • ( k ) . There is a canonical isomor phism Th( U ⊕ O B ) ∼ = Th( U ) ∧ T . W e recall the Morel- V o evodsky theor y o f alg ebraic cob ordis m X 7→ MGL ∗∗ ( X ), represented in the Mo rel-V o evods ky motivic stable homoto p y catego ry S H ( k ) by the a lgebraic version, MGL, of the class ical Thom sp ectrum MU. As in classical top ology , we hav e MGL = (MGL 0 , MGL 1 , . . . , MGL n , . . . ) COMP ARISON OF COBORDISM THEORIES 7 where MGL n is the Thom space o f the universal rank n quotient bundle U n ov er BGL n : MGL n := Th( U n → BGL n ) BGL n in turn is just the limit o f Grassmann v arieties BGL n := Gr a ss( ∞ , n ) := lim − → N →∞ Grass( N , n ) where Gra ss( N , n ) is the Grassmannia n of rank n quotien ts of O N , and U n → BGL n is the r epresenting r ank n bundle with universal quotient π N ,n : O N → U n . The inductive system . . . → Grass( N , n ) → Gr ass( N + 1 , n ) → . . . is defined via the pro jections O N +1 → O N on the first N fac tors. W e hav e the closed immersio n i n : BGL n → BGL n +1 representing the sur jection O N ⊕ O π n ⊕ id − − − − → U n ⊕ O on BGL n ; the gluing maps ǫ n : MGL n ∧ T → MGL n +1 are just the maps MGL n ∧ T = Th( U n ) ∧ T = Th( U n ⊕ O ) Th( i n ) − − − − → Th( U n +1 ) = MGL n +1 The resulting bi-g raded co ho mology theory X 7→ MGL is an oriente d theory . There ar e many equiv alent definitions o f this notion; for details we refer the rea der to [8, 9]. F or our purp ose s , we can tak e an oriented theory to be one with a g o o d theory of fir st Chern cla sses for line bundles. In the case of MGL, c 1 ( L ) is defined as follows. T ake a line bundle L → X , with X smoo th a nd quasi-pro jective ov er k . By Jouanoulo u’s trick, and the homotopy in v ariance of MGL ∗ , ∗ , we can assume that X is affine, and hence L is gener ated by g lo bal se ctions. Thus, L is the pull-back of O P N (1) for so me morphism f : X → P N . Mo dulo chec king indep endence of v a rious choices, this reduces us to defining c 1 ( O (1)) ∈ MGL 2 , 1 ( P ∞ ), wher e MGL 2 , 1 ( P ∞ ) is short-ha nd for lim ← − N MGL 2 , 1 ( P N ), the limit begin defined b y a fix e d sequence of linear em b eddings P 1 → P 2 → . . . → P N → . . . . F or this, no te that BGL 1 = P ∞ , U 1 → BGL 1 is O (1), and hence MGL 1 = T h ( O P ∞ (1)). The b o nding maps in MGL give us the sequence of maps MGL 1 ∧ T ∧ n → MGL n +1 . This defines the map in Spt ( k ) ι : Σ ∞ T T h ( O P ∞ (1)) → S 2 , 1 ∧ MGL , giving us the class [ ι ] ∈ MGL 2 , 1 ( T h ( O P ∞ (1))). Comp osing the 0-section s : P ∞ → O P ∞ (1) with the canonical quotient map O P ∞ (1) → T h ( O P ∞ (1)) defines π : P ∞ → T h ( O P ∞ (1)); we set c 1 ( O (1)) := π ∗ ([ ι ]) ∈ MGL 2 , 1 ( P ∞ ) . An orientation on a bi-graded cohomolog y giv es rise to a g o o d theory of push- forward maps for pro jective morphisms (see [8, theorem 4.1.4] for a de ta iled state- men t). F or MGL, this says w e have functorial push-for ward maps f ∗ : MGL a,b ( Y ) → MGL a +2 d,b + d ( X ) 8 MARC LEVINE for ea ch pro jective morphism f : Y → X in Sm /k , where d = co dim f := dim k X − dim k Y . The connectio n with the first Chern map is tha t, for L → X a line bundle on X ∈ Sm /k with zero-sectio n s : X → L , one has c 1 ( L ) = s ∗ ( s ∗ (1 X )) where 1 X ∈ MGL 0 , 0 ( X ) is the unit. In fact, our extension [2, theorem 1.10 ] o f Panin’s theor em gives go o d pro jective push-forward maps for MGL-co homology with suppo r ts. W e describ e the gener al situation. Let SP denote the category of smo oth p airs , this b eing the categ o ry with ob jects ( M , X ), M ∈ Sm /k , X ⊂ M a clos ed subset. A morphism f : ( M , X ) → ( N , Y ) is a morphism f : M → N in Sm /k s uch that f − 1 ( Y ) ⊂ X . W e hav e as well the category SP ′ , with the same ob jects as SP , but where a mo rphism f : ( N , Y ) → ( M , X ) is a pro jective morphism f : N → M in Sm /k with f ( Y ) ⊂ X . F or any T -sp ectrum E , sending ( M , X ) ∈ SP to the E -c ohomolo gy with supp orts E a,b X ( M ) := Hom S H ( k ) (Σ ∞ T M / ( M \ X ) , Σ a,b E ) defines a functor E ∗ , ∗ from SP op to bi- g raded a belia n g r oups. In case E is an oriented r ing T -sp ectrum, ( M , X ) 7→ E ∗ , ∗ X ( M ) defines a bi- graded oriente d ring c ohomolo gy the ory o n SP , in the sense of [2, definition 1.3]. In particular , for E = MGL, we hav e the bi-gra ded oriented r ing cohomolog y theo r y ( M , X ) 7→ MGL ∗ , ∗ X ( M ) on SP . Let E be a a bi-g raded or iented ring cohomology theor y on SP . By [2, theorem 1.10], there are push-forward maps f ∗ : E a,b Y ( N ) → E a +2 d,b + d X ( M ) for each map f : ( N , Y ) → ( M , X ) in S P ′ , where d = co dim f , such that the maps f ∗ define an int e gr ation with supp orts o n E ∗ , ∗ , in the se ns e of [2, definition 1.6 ]. Without lis ting all of this definition here, this means that f 7→ f ∗ is functorial on SP ′ , satisfies a pro jection formula with resp ect to cup pro ducts, and is compa t- ible with pull-back in tr a nsverse cartesian squares. In addition, the ma ps f ∗ are compatible with the b oundary maps in the long ex a ct sequence of triples. Next, we extend the in teg ration on the or ient ed theory E ∗ , ∗ to a bi-gr aded oriente d duality the ory ( H , E ). Let Sc h ′ k be the catego ry of reduced quasi-pro jective schemes over k , with morphisms the pro jective mo rphisms. F or X ∈ Sc h ′ k , choo se a closed immersion X → M with M ∈ Sm /k , and define H a,b ( X ) := E 2 d M − a,d M − b X ( M ) where d M := dim k M (we assume that M is eq ui- dimensional ov er k ). In [2, theorem 3.4], we s how that X 7→ H ∗ , ∗ ( X ) extends to a functor H a,b : Sc h ′ k → Ab , and that the pair ( H ∗ , ∗ , E ∗ , ∗ ) defines a bi-graded or iented duality theor y on Sch k . W e refer the rea der to [2, defin tio n 3.1] for the precise definitio n of this notion, noting that this includes compariso n iso morphisms α M ,X : H ∗ , ∗ ( X ) → E 2 d M −∗ ,d M −∗ X ( M ) for each closed immers ion X → M , M ∈ Sm /k , suc h that, if we a re given a pro jective mor phism f : Y → X , closed immersions Y → N , X → M , N , M ∈ COMP ARISON OF COBORDISM THEORIES 9 Sm /k , and a n extensio n of f to a pro jective morphism F : N → M , then the diagram H ( Y ) α N ,Y / / f ∗   A Y ( N ) F ∗   H ( X ) α M,X / / A X ( M ) commutes. In additio n, H ∗ , ∗ has pull-back maps for op en immersio ns, a b oundar y map δ X,Y : H a,b ( X \ Y ) → H a − 1 ,b ( Y ) for each closed subset Y ⊂ X , externa l pro ducts H ∗ , ∗ ( X ) ⊗ H ∗ , ∗ ( Y ) → H ∗ , ∗ ( X × Y ) and cap pro ducts f ∗ ( − ) ∩ : A a,b X ( M ) ⊗ H p,q ( Y ) → H p − a,q − b ( f − 1 ( X )) for each map f : Y → M and smo oth pair ( M , X ) ∈ SP , such that these o pe rations are compatible with the corresp onding ones on E - cohomolog y with suppo rts via the compariso n isomorphis ms α . Let L → Y be a line bundle on some Y ∈ Sc h k . Using the fact there is a line bundle L → M for s ome M ∈ Sm /k , and a morphism f : Y → M with L ∼ = f ∗ L , the cap pro duct with c 1 ( L ) gives a well-defined first Chern class op erator ˜ c 1 ( L ) : H p,q ( Y ) → H p +2 ,q +1 ( Y ) , independent of the choice of s mo oth en velope Y → M and extensio n of L to L . R emark 2.1 . One can think of a bi-g raded oriented duality theo ry ( H, E ) as a generaliza tion of a Blo ch-Ogus twisted duality theory , the difference b eing that one do es not requir e that c 1 ( L ⊗ M ) = c 1 ( L ) + c 1 ( M ). Replacing this is the formal gr oup law F E ( u, v ) ∈ E 2 ∗ , ∗ ( k )[[ u, v ]], E 2 ∗ , ∗ ( k ) := ⊕ n E 2 n,n ( k ). This is the p ow er series c hara cterized b y the iden tity F E ( c 1 ( L ) , c 1 ( M )) = c 1 ( L ⊗ M ) for each pair of line bundles L , M on a fixed X ∈ Sm / k . See [8, § 3.9] for further details. W e denote the ex tens io n o f MGL ∗ , ∗ to a bi-graded oriented duality theor y by (MGL ′ ∗ , ∗ , MGL ∗ , ∗ ). 3. Ω ∗ and MGL ′ 2 ∗ , ∗ T ogether with F. Mo rel [3], we hav e defined the “g e o metric” theory of algebraic cob ordism Ω ∗ , on S ch k . By [3, theorem 7.1.3], the theory Ω ∗ is the universal oriented Borel-Mo ore homolo gy theory on Sch k , in the sense of [3 , definition 5 .1.3]. In particular, for each n , Ω n is a functor Ω n : Sc h ′ k → Ab , for each l.c.i. morphism g : X ′ → X of r e la tive dimension d , there is a pull-back map g ∗ : Ω n ( X ) → Ω n + d ( X ′ ) , 10 MARC LEVINE functorial in g , there is a n asso ciative and commutativ e external pro duct × : Ω n ( X ) ⊗ Ω n ′ ( X ) → Ω n + n ′ ( X × X ′ ) , and a unit elemen t 1 ∈ Ω 0 ( k ). The ex ternal pro duct makes Ω ∗ ( k ) a commutativ e, graded ring , and Ω ∗ ( X ) a graded Ω ∗ ( k )-mo dule for each X . In fact, there is a cano nical iso morphism L ∗ ∼ = Ω ∗ ( k ), wher e L ∗ is the Laza rd ring, giving us the for mal group law F Ω ( u, v ) ∈ Ω ∗ ( k )[[ u, v ]] (se e [3, theorem 4.3.7]). Define the first Chern cla ss oper ator of a line bundle L → X w ith zer o-section s : X → L as ˜ c 1 ( L )( α ) := s ∗ ( s ∗ ( α )) . Then the lo cally nilp o tent op erator s ˜ c 1 ( L ) : Ω ∗ ( X ) → Ω ∗− 1 ( X ) comm ute with one another (for fixed X ) and satisfy the formal gro up law F Ω (˜ c 1 ( L ) , ˜ c 1 ( M )) = ˜ c 1 ( L ⊗ M ) . F or X ∈ Sc h k and n ≥ 0 an integer, let M n ( X ) b e the free ab elian group on the set of isomorphism classes of pro jective maps f : Y → X , with Y ∈ Sm / k , and Y irre ducible of dimension n over k . Sending X to M n ( X ) bec o mes a functor M n : Sc h ′ k → Ab where, for g : X → X ′ a pro jectiv e morphism and f : Y → X in M n ( X ), we define g ∗ ( f : Y → X ) := g ◦ f : Y → X ′ . A dditionally , for g : X ′ → X a smo oth, quasi-pro jective mor phism of relative dimension d , we define g ∗ : M n ( X ) → M n + d ( X ′ ) by g ∗ ( f : Y → X ) := p 2 : Y × X X ′ → X ′ . Finally , w e hav e an exter na l pro duct × : M n ( X ) ⊗ M n ′ ( X ′ ) → M n + n ′ ( X × X ′ ) by sending ( f : Y → X ) ⊗ ( f ′ : Y ′ → X ′ ) to f × f ′ : Y × Y ′ → X × X ′ (strictly sp eaking, we tak e the sum of the restr ictions of f × f ′ to the irreducible comp onents of Y × Y ′ ). The construction of Ω ∗ gives a na tur al s urjection ρ X : M n ( X ) → Ω n ( X ) compatible with push-forward g ∗ for pro jective g , pull-back g ∗ for smo oth, quasi- pro jective g and the exter nal pr o duct × . Now let ( H, E ) b e an oriented duality theory . F or each Y ∈ Sm /k , we hav e the unit 1 Y ∈ E 0 , 0 ( Y ). If Y has dimension n ov er k , the comparison isomorphis m α Y ,Y : H 2 n,n ( Y ) → E 0 , 0 ( Y ) gives us the fun damental class [ Y ] H ∈ H 2 n,n ( Y ) [ Y ] H := α − 1 Y ,Y (1 Y ) . F or X ∈ Sc h k , we may map M n ( X ) to H 2 n,n ( X ) by se nding f : Y → X to f ∗ ([ Y ] H ); by [2, pr op osition 4.2], this descends to a homomor phism ϑ H ( X ) : Ω ∗ ( X ) → H 2 ∗ , ∗ ( X ) , natural with re s pec t to pro jective push-forward, pull-bac k by op en immers ions and compatible with externa l pro ducts and fir st Chern class op era tors. By [2, COMP ARISON OF COBORDISM THEORIES 11 lemma 4.3], the natural tra ns formation ϑ H , restricted to Sm /k , defines a natur al transformatio n of oriented cohomolo gy theories (in the sense of [3, definition 1.1.2]) ϑ E : Ω ∗ → E 2 ∗ , ∗ ; ϑ E ( X ) := α X,X ◦ ϑ H ( X ) . Here Ω ∗ ( Y ) := Ω n −∗ ( Y ) where n = dim k Y . W e can now sta te our main result. Theorem 3.1. L et k b e a field of char acteristic zer o. Then ϑ MGL ′ ( X ) : Ω ∗ ( X ) → MGL ′ 2 ∗ , ∗ ( X ) is an isomorphism for al l X ∈ Sc h k . The pro of with o ccupy the next three sections. 4. MGL ∗ , ∗ f or fields Let X b e a smo o th irr educible s cheme ov er k , F = k ( X ). By [3, theorem 4.3.7], Ω ∗ ( F ) = L ∗ := L −∗ . Th us, the natural transfor mation ϑ MGL : Ω ∗ → MGL 2 ∗ , ∗ gives a ring homomo r phism ϕ : L ∗ → MGL 2 ∗ , ∗ ( F ) . Next, we wan t to define a gro up homo mo rphism ψ X : Γ( X , O ∗ X ) → MGL 1 , 1 ( X ) . F or this, we note that, in the ho motopy category H ( k ), B G m represents the Pica rd functor on Sm /k , and similar ly , in the p ointed homoto p y catego ry H • ( k ), B G m represents the relative Picard functor on pairs of smo oth v arieties (this follows directly from [5, prop os ition 4 .3.8]). Since Γ( X , O ∗ X ) ∼ = Pic( X × A 1 , X × { 0 , 1 } ) , it suffices to construct a map ψ : B G m → MGL 1 . But B G m ∼ = P ∞ in H • ( k ), and MGL 1 = T h ( O P ∞ (1)), so the comp osition P ∞ zero-section − − − − − − − → O P ∞ (1) π − → T h ( O P ∞ (1)) do es the trick. R emark 4.1 . The c areful reader with note that the map ψ do es not send the base- po int of B G m = P ∞ to the base -po int o f MGL 1 . W e co rrect this b y extending ψ to ˜ ψ : P ∞ ∪ (1:0: ... ) A 1 → MGL 1 , by identifying A 1 with the fib er o f O (1) over (1 : 0 : . . . ). Using the base-p oint 1 ∈ A 1 for P ∞ ∪ (1:0: ... ) A 1 makes ˜ ψ a p ointed map, and the collapse map ( P ∞ ∪ (1:0: ... ) A 1 , 1) → ( P ∞ , (1 : 0 : . . . )) is an isomorphism in H • ( k ). Let H Z ∈ S H ( k ) b e the T -sp ectrum classifying motivic cohomology (see e.g. [6, 9, 10]). W e have the cano nical map ρ H Z : MGL → H Z classifying H Z as an orie nted cohomology theory ([9, theorem 1.0.1]). 12 MARC LEVINE Lemma 4.2. F or X ∈ Sm /k , t he c omp osition ρ H Z ( X ) ◦ ψ X : Γ( X , O ∗ X ) → H 1 , 1 ( X ) is an isomorphism of ab elian gr oups. Pr o of. By [8, theo rem 3.2 .4(1)], the map ψ : B G m → MGL 1 comp osed with the canonical map ι 1 : Σ ∞ T MGL 1 → S 2 , 1 ∧ MGL induces the first Chern class ma p for the oriented theor y MGL via the natural transformatio n Pic( X, A ) ∼ = Hom H • ( k ) ( X/ A, B G m ) ψ ∗ − − → Hom H ( k ) ( X/ A, MGL 1 ) ι ∗ − → Hom S H ( k ) (Σ ∞ T ( X/ A ) , S 2 , 1 ∧ MGL) . As ρ H Z induces a ma p of oriented theories, ρ H Z is compatible with the resp ective Chern classes. Thus, w e see that the comp osition Pic( X, A ) ∼ = Hom H • ( k ) ( X/ A, B G m ) ψ ∗ − − → Hom H • ( k ) ( X/ A, MGL 1 ) ι ∗ − → Hom S H ( k ) (Σ ∞ T ( X/ A ) , S 2 , 1 ∧ MGL) ρ H Z − − → Hom S H ( k ) (Σ ∞ T ( X/ A ) , S 2 , 1 ∧ H Z ) induces the first Chern class map for motivic c ohomology . Since c H Z 1 : Γ( X , O ∗ X ) = Pic( X × A 1 , X × { 0 , 1 } ) → H 2 , 1 ( X × A 1 /X × { 0 , 1 } ) = H 1 , 1 ( X ) is an isomorphism, the result follows.  Using ϕ and the MGL ∗ , ∗ ( k )-mo dule structure on MGL ∗ , ∗ ( X ), ψ X extends to ψ X : Γ( X , O ∗ X ) ⊗ L ∗ → MGL 2 ∗ +1 , ∗ +1 ( X ) . Lemma 4.3. L et X ∈ Sm / k b e irr e ducible. F or F = k ( X ) , the map ϕ F : L ∗ → MGL 2 ∗ , ∗ ( F ) is an isomorphism, and the map ψ F : F × ⊗ L ∗ → MGL 2 ∗ +1 , ∗ +1 ( F ) is surje ct ive. Pr o of. W e use the Hopkins-Morel spectra l sequence for Spec F (4.1) E p,q 2 ( n ) = ⊕ n H p − q,n − q ( F ) ⊗ L q = ⇒ MGL p + q,n ( F ) . Since H a,b ( F ) = 0 if a > b or b < 0, H 0 , 0 ( F ) = Z , H 1 , 1 ( F ) = F × , and L q = 0 if q > 0, this gives us E p,q 2 ( n ) =      0 if p > n L n if p = q = n F × ⊗ L n − 1 if p = n, q = n − 1 . Thu s, elements of E n,n 2 ( n ) and E n,n − 1 ( n ) are p ermanent cycles , giving us surjec- tions E n,n 2 ( n ) → E n,n ∞ ( n ); E n,n − 1 2 ( n ) → E n,n − 1 ∞ ( n ) . In a dditio n, if p + q = 2 n or p + q = 2 n − 1 , the only non-zero E p,q 2 terms are E n,n 2 or E n,n − 1 2 , so we have MGL 2 n,n ( F ) = E n,n ∞ ( n ); MGL 2 n − 1 ,n ( F ) = E n,n − 1 ∞ . COMP ARISON OF COBORDISM THEORIES 13 It follo ws dir ectly fro m the constr uction of the spectr a l seq uence that the comp o- sition L n = E n,n 2 ( n ) → E n,n ∞ ( n ) → MGL 2 n,n ( F ) is ϕ F . Similar ly , the comp osition F × ⊗ L n − 1 = E n,n − 1 2 ( n ) → E n,n − 1 ∞ ( n ) → MGL 2 n − 1 ,n ( F ) is ψ F . This pro ves that ϕ F and ψ F are surjective. T o see that ϕ F is injectiv e, we may assume that k admits a n embedding σ : k → C . Via this embedding, w e have the oriented co homology theory MU ∗ , ∗ σ on Sm /k MU a,b σ ( Y ) := MU a ( Y ( C )) . By [9], this induces a natural trans formation MGL a,b ( Y ) → MU a ( Y ( C )) of oriented coho mology theo ries. Note that MU a,b σ ( F ) = lim − → U MU a ( U ( C )) as U runs over non-empty Za riski open subsets o f X . The comp osition L ∗ → MGL 2 ∗ , ∗ ( k ) → MU 2 ∗ , ∗ σ ( k ) = MU 2 ∗ ( pt ) is the ring homomo rphism classifying the formal group law of MU ∗ . As this is the universal fo r mal group law, this comp osition is a n isomorphism. Now let U ⊂ X be a no n-empty op en subset. Since U ha s a C -p oint, the pull-back map MU ∗ ( pt ) → MU ∗ ( U ( C )) is injectiv e, and thus MU ∗ , ∗ σ ( pt ) → MU ∗ , ∗ σ ( F ) is injectiv e as well. This shows that ϕ F is injectiv e, completing the pro o f.  5. Chern classes, divisors and the boundar y map Let U be an o pe n subscheme of s ome X ∈ Sm /k with closed complement i : D → X , and inclus ion j : U → X . Giv en u ∈ Γ( U, O ∗ ), we have the element ψ ( u ) ∈ MGL 1 , 1 ( U ) . T aking the b o undary in the long exa ct loca lization sequence . . . → MGL 1 , 1 ( X ) j ∗ − → MGL 1 , 1 ( U ) ∂ X,D − − − → MGL 2 , 1 D ( X ) i ∗ − → MGL 2 , 1 ( X ) → . . . gives us the element ∂ X,D ( ψ ( u )) ∈ MGL 2 , 1 D ( X ) . Our goal in this section is to give a form ula for ∂ ( ψ ( u )). T o help with o ur computation, w e int ro duce the effe ctive Pic ar d monoid with supp orts , Pic ef f D ( X ), and the relative fir st Chern class c D 1 ( L ) ∈ MGL 2 , 1 D ( X ) . The s et P ic ef f D ( X ) is the s et of isomor phis m classes of pairs ( L , s ), with L → X a line bundle, and s : X → L a sectio n that is nowhere v a nis hing on X \ D ; an 14 MARC LEVINE isomorphism of such pairs ϕ : ( L, s ) → ( L ′ , s ′ ) is a n isomor phism of line bundles ϕ : L → L ′ with s ′ = ϕ ◦ s . W e make P ic ef f D ( X ) a monoid using tensor pro duct: ( L, s ) · ( L ′ , s ′ ) := ( L ⊗ L ′ , s ⊗ s ′ ); the unit is ( O X , 1). In fact, the same form ula defines a commutativ e pro duct · : Pic ef f D 1 ( X, D 1 ) × Pic ef f D 2 ( X ) → P ic ef f D 1 ∪ D 2 ( X ) . R emark 5.1 . This notion is taken fro m F ulton [1], who calls a pair ( L, s ) a pseudo- divisor . Let Pic ef f ( X ) ⊂ Pic( X ) be the monoid of isomorphis m c lasses of line bundles L on X that admit a non-z ero section. W e have the evident for getful maps Pic ef f D ( X ) → P ic ef f ( X ) → P ic( X ) . W e now des c rib e ho w to lift the Chern class map c 1 : Pic( X ) → MGL 2 , 1 ( X ) to a Chern class map with supp orts c D 1 : Pic ef f D ( X ) → MGL 2 , 1 D ( X ) . Given ( L, s ) ∈ Pic ef f D ( X ), we first supp ose that L is gener a ted by global s ections on X . Extend s to a finite set of ge ne r ating sections s = s 0 , s 1 , . . . , s N , giving us the map f : X → P N . Define h : U × A 1 → P N by h ( u, t ) := ( s 0 ( u ) : ts 1 ( u ) : . . . : ts N ( u )); since s 0 is nowhere zer o on U , h is well-defined. Also h ( u, 1) = f ( u ); h ( u, 0) = (1 : 0 : . . . : 0) , and th us f ∪ h gives a w e ll- defined map of pointed spaces ov er k : f ∪ h : X ∪ U × 1 U × A 1 /U × 0 → ( P N , (1 : 0 : . . . : 0 )) . Comp osing f ∪ h with P N → P ∞ = BGL 1 ψ − → T h ( O (1)) = MGL 1 and noting that the collapse map X ∪ U × 1 U × A 1 /U × 0 → X/U is an isomorphism in H • ( k ) gives us c D 1 ( L ) ∈ MGL 2 , 1 ( X/U ) = MGL 2 , 1 D ( X ) . It is easy to chec k that the map f ∪ h is indep endent (as a map in H • ( k )) of the choices made. Thus, w e hav e a well-defined element c D 1 ( L ) ∈ MGL 2 , 1 D ( X ), assuming that L is generated by g lobal sections. In genera l, w e use Jouanoulo u’s trick. T ake an a ffine space bundle p : Y → X with Y affine, and r e pla ce ( X , D , L ) with ( Y , p − 1 ( D ) , p ∗ ( L )). As Y is affine, p ∗ ( L ) COMP ARISON OF COBORDISM THEORIES 15 is g e nerated b y global sec tio ns, so we can apply the constr uctio n of the pre ceding paragr aph, noting that p ∗ : MGL ∗ , ∗ D ( X ) → MGL ∗ , ∗ p − 1 ( D ) ( Y ) is an isomorphism. Since the collection of such affine spa ce bundles forms a directed system, it is easy to chec k that the resulting class c D 1 ( L ) := ( p ∗ ) − 1 ( c p − 1 ( D ) 1 ( p ∗ ( L ))) is independent of the choice o f p : Y → X , completing the construction. Let SP 1 be the full sub categ ory o f the catego r y SP of s mo oth pairs consisting of ( X , D ) with D ⊂ X a pure co dimension one close d subset. Clear ly , ( X , D ) 7→ Pic ef f D ( X ) defines a functor from SP 1 to the category of monoids, and c D 1 : Pic ef f D ( X ) → MGL 2 , 1 D ( X ) is a natural transforma tio n. Similar ly , it is easy to see that the dia g ram (5.1) Pic ef f D ( X ) / / c D 1   Pic( X ) c 1   MGL 2 , 1 D ( X ) / / MGL 2 , 1 ( X ) commutes. Example 5.2 . T ake X = A 1 = Sp ec k [ x ], D = 0, ( L , s ) = ( O A 1 , x ). Note that A 1 / A 1 \ 0 ∼ = ( P 1 , (1 : 0)) in H • ( k ), so MGL 2 , 1 0 ( A 1 ) ∼ = MGL 2 , 1 ( P 1 , (1 : 0)) ∼ = MGL 0 , 0 ( k ) = Z . Then c 0 1 ( O A 1 , x ) = ± 1. One can verify this by a direct c omputation. Alterna- tively , we ca n use the natura lit y of c 0 1 , replacing ( A 1 , 0 , ( O A 1 , x )) with ( P 1 , (1 : 0) , ( O (1) , X 1 )); b y excis ion, the restriction map MGL 2 , 1 (1:0) ( P 1 ) → MGL 2 , 1 0 ( A 1 ) is a n isomorphism. Using the comm utativity of the diagram (5.1), we see that c (1:0) 1 ( O (1) , X 1 ) maps to c 1 ( O (1)) ∈ MGL 2 , 1 ( P 1 ). By the pro jective bundle form ula, MGL 2 , 1 ( P 1 ) = c 1 ( O (1)) · MGL 0 , 0 ( k ) ⊕ MGL 2 , 1 ( k ) = c 1 ( O (1)) · MGL 0 , 0 ( k ) , so c 1 ( O (1)) is a genera tor of MGL 2 , 1 ( P 1 ) = Z , as desired. If w e nor ma lize the isomorphism MGL 2 , 1 0 ( A 1 ) ∼ = MGL 2 , 1 ( P 1 ) ∼ = Z by using c 1 ( O (1)) as the distinguished genera tor, then c 0 1 ( O A 1 , x ) = +1 . W e now show that the Chern class with suppor ts co mputes the b oundary map in our lo c alization sequence. Lemma 5. 3. L et f : X → A 1 b e a dominant morphism, with X ∈ Sm /k irr e- ducible, let D = f − 1 (0) and U = X \ D . L et u ∈ Γ( U, O ∗ ) b e the r estriction of f . Then ∂ X,D ( ψ ( u )) = c D 1 ( O X , f ) ∈ MGL 2 , 1 D ( X ) . 16 MARC LEVINE Pr o of. Both sides o f the equations a re natural in ( X, f ), so it suffices to handle the universal case X = A 1 , f = id. W r iting A 1 = Sp ec k [ x ], u is just the canonical unit x on U = A 1 \ { 0 } . The b oundar y map ∂ : MGL 1 , 1 ( U ) → MGL 2 , 1 0 ( A 1 ) is induced by the map δ in the Pupp e se quence U → A 1 → A 1 ∪ U × 1 U × A 1 /U × 0 δ − → ( A 1 / { 0 , 1 } ) ∧ U + Noting that ψ ( u ) descends cano nically to an elemen t in reduced coho mology ¯ ψ ( u ) ∈ MGL 1 , 1 ( U, 1) = MGL 1 , 1 ( G m ) , we can use the p ointed v ersion of the Pupp e sequence G m → ( A 1 , 1) → A 1 ∪ U × 1 U × A 1 / ( U × 0 ∪ 1 × A 1 ) ¯ δ − → ( A 1 / { 0 , 1 } ) ∧ G m Then ¯ δ is an isomo rphism in H • ( k ), a nd b oth terms are isomorphic to ( P 1 , (1 : 0)). The map Σ ∞ T ( A 1 / { 0 , 1 } ) ∧ G m → S 2 , 1 ∧ MGL representing ψ ( u ) is b y definition the map induced by the canonical inclusion P 1 → P ∞ follow ed by the map P ∞ → MGL 1 induced by the zero- s ection P ∞ → O (1), and pre-comp osed with the canonica l iso morphism ( A 1 / { 0 , 1 } ) ∧ G m ∼ = ( P 1 , (1 : 0)) in H • . But we ha ve a lready seen that this gives us c 1 ( O (1)) ∈ MGL 2 , 1 ( P 1 ), which is the same as c 0 1 ∈ MGL 2 , 1 0 ( A 1 ).  Suppo se the D ⊂ X is the supp ort o f a strict normal cr o ssing divisor, that is, if D has irreducible comp onents D 1 , . . . , D m , then for eac h I ⊂ { 1 , . . . , m } , the int ersectio n D I := ∩ i ∈ I D i is smo oth and has co dimensio n | I | o n X ; we call the D I the str ata of D . F o r ( L, s ) ∈ Pic ef f D ( X ), let div( s ) = P i n i D i denote the usua l divisor, that is, n i is the o rder of v anishing of s along D i . W e have defined in [3, definition 3.1.5] the c ob or dism diviso r class div Ω ( s ) ∈ Ω dim X − 1 ( D ) . with the following pro p er ties: (1) Let i : D → X b e the inclus ion. Then i ∗ (div Ω ( s )) = c 1 ( L ) := ˜ c 1 ( L )(1 X ) ∈ Ω dim X − 1 ( X ) . (2) W rite D = D 1 ∪ . . . ∪ D r , with e a ch D i a smoo th codimension one closed subscheme of X , such that div( s ) can be written a s div ( s ) = P r i =1 n i D i . Let ξ i = ˜ c 1 ( O X ( D i )). Then fo r each non-empty I ⊂ { 1 , . . . , r } there a re universal pow er series G I ( u 1 , . . . , u r ) ∈ Ω ∗ ( k )[[ u 1 , . . . , u r ]] such tha t div Ω ( s ) = X I ι I ∗ [ G I ( ξ 1 , . . . ξ r )([ D I ])] , COMP ARISON OF COBORDISM THEORIES 17 where D I = ∩ i ∈ I D i , ι I : D I → D is the inclusion, and [ D I ] ∈ Ω ∗ ( D I ) is the class of id D I . R emark 5.4 . Let X , D , s, L b e as ab ov e. In [3], w e used the notation [div ( s ) → D ] ∈ Ω ∗ ( D ) for div Ω ( s ). Lemma 5.5. L et ( X, D ) b e in SP , such that D is a r e duc e d strict normal cr ossing divisor on X . L et f : X ′ → X b e a morphism in Sm /k . Supp ose that f is tr ansverse to the inclusion D I → X for e ach stra tum D I of D . L et ( H, E ) b e a bi- gr ade d oriente d duality the ory on Sc h k , ϑ H : Ω ∗ → H 2 ∗ , ∗ the natur al tra nsformation given by [2, prop os ition 4 .2] . L etting D ′ = f − 1 ( D ) , we have the m aps α X,D ◦ ϑ H ( D ) : Ω d X − 1 ( D ) → E 2 , 1 D ( X ) α X ′ ,D ′ ◦ ϑ H ( D ′ ) : Ω d X ′ − 1 ( D ′ ) → E 2 , 1 D ′ ( X ′ ) f ∗ : E 2 , 1 D ( X ) → E 2 , 1 D ′ ( X ′ ) Then for ( L, s ) ∈ Pic ef f D ( X ) , we have ( f ∗ ( L ) , f ∗ ( s )) ∈ Pic ef f D ′ ( X ′ ) and f ∗ ( α X,D ◦ ϑ H ( D )(div Ω ( s ))) = α X ′ ,D ′ ◦ ϑ H ( D ′ )(div Ω ( f ∗ ( s )))) . Pr o of. W rite D = D 1 ∪ . . . ∪ D r with each D i a smo oth co dimensio n o ne clo sed subscheme of X , such that div( s ) can b e written as div ( s ) = P r i =1 n i D i . Let D ′ i = f − 1 ( D i ). Then D ′ is a r educed strict norma l c rossing diviso r o n X ′ , D ′ = D ′ 1 ∪ . . . ∪ D ′ r , each D ′ i is a smoo th codimension one closed subscheme of X ′ (or is empt y), and div( f ∗ ( s )) = P i n i D ′ i . Thus, letting ξ i = ˜ c 1 ( O X ( D i )), ξ ′ i = ˜ c 1 ( O X ′ ( D ′ i )), s ′ = f ∗ ( s )), we have div Ω ( s ) = X I ι I ∗ [ G I ( ξ 1 , . . . ξ r )([ D I ])] (5.2) div Ω ( s ′ ) = X I ι I ∗ [ G I ( ξ ′ 1 , . . . ξ ′ r )([ D ′ I ])] . (5.3) The maps α X,D ◦ ϑ H ( D ) are na tural with resp e ct to pro jective push-forward and commute with the res pect first Chern cla ss op erators , hence α X,D ◦ ϑ H ( D )( ι I ∗ [ G I ( ξ 1 , . . . ξ r )([ D I ])]) = ι I ∗ [ G I ( ξ 1 , . . . ξ r )( α X,D I ◦ ϑ H ( D I )([ D I ]))] and similarly for X ′ , D ′ , s ′ . F ur thermore, since D I and D ′ I are smo oth, we hav e ι I ∗ [ G I ( ξ 1 , . . . ξ r )( α X,D I ◦ ϑ H ( D I )([ D I ]))] = ι I ∗ [ G I ( ξ 1 , . . . ξ r )( ϑ E ( D I )(1 Ω D I ))] and similar ly for D ′ I . Here 1 Ω D I ∈ Ω ∗ ( D I ) is the unit. Finally , since ϑ E is a natural transformatio n of oriented co homology theories on Sm /k , w e hav e ϑ E ( D I )(1 Ω D I ) = 1 E D I , and similarly for D ′ I . P utting these identities together gives α X,D ◦ ϑ H ( D )( ι I ∗ [ G I ( ξ 1 , . . . ξ r )([ D I ])]) = ι I ∗ [ G I ( ξ 1 , . . . ξ r )(1 D I )] (5.4) α X ′ ,D ′ ◦ ϑ H ( D ′ )( ι I ∗ [ G I ( ξ ′ 1 , . . . ξ ′ r )([ D ′ I ])]) = ι I ∗ [ G I ( ξ ′ 1 , . . . ξ ′ r )(1 D ′ I )] . (5.5) Let f I : D ′ I → D I be the restriction of f . Since the diagr am D ′ I ι ′ I   f I / / D I ι I   X ′ f / / X 18 MARC LEVINE is transverse, the dia gram E a,b ( D ′ I ) ι ′ I ∗   E a,b ( D I ) f ∗ I o o ι I ∗   E 2 | I | + a, | I | + b D ′ ( X ′ ) E 2 | I | + a, | I | + b D ( X ) f ∗ o o commutes (see [2, lemma 1.7]). Similarly , f ∗ I commutes with the res pective first Chern class op erator s, so f ∗ I ◦ G I ( ξ 1 , . . . ξ r ) = G I ( ξ ′ 1 , . . . ξ ′ r ) ◦ f ∗ I . Finally , f ∗ I : E ∗ , ∗ ( D I ) | toE ∗ , ∗ ( D ′ I ) is a ring homomorphism, so f ∗ I (1 D I ) = 1 D ′ I . T ogether with the identities (5.2), (5.3), (5.4), (5.5), this completes the pr o of.  Lemma 5. 6. T ake X ∈ Sm /k , D a strict n ormal cr ossing divisor on X and ( L, s ) ∈ Pic ef f D ( X ) . Then c D 1 ( L, s ) = α X,D ( ϑ MGL ′ ( D )(div Ω ( s )) , wher e α X,D : MGL 2 dim X − 2 , dim X − 1 ( D ) → MGL 2 , 1 D ( X ) is t he c omp arison isomor- phism. Pr o of. With the help of lemma 5.5, both sides ar e natura l with resp ect to mo r- phisms that are trans verse to all the s trata of D . Thus, we ma y use Jouanoulou’s trick to reduce to the case in which X is affine. W r ite D a s a union of irreducible comp onents D = D 1 ∪ . . . ∪ D m and write div( s ) = P i n i D i , n i ≥ 0 . Letting L i = O X ( D i ), we therefor e have L ∼ = L ⊗ n 1 1 ⊗ . . . ⊗ L ⊗ n m m In addition, there are sections s ( i ) of L i , and a unit u on X such that the divisor of s i (as a cycle on X ) is D i and with s = u · ( s (1) 1 ) n 1 ⊗ . . . ⊗ ( s ( m ) ) n m . As ( L, s ) ∼ = ( L, u − 1 s ) and div Ω ( s ) = div Ω ( u − 1 s ), we may assume u = 1. As X is affine, each L i is g e nerated by global sections, and we can find a set of generating sections of L i of the form s ( i ) = s ( i ) 0 , . . . , s ( i ) N . Thu s, we have a morphism g : X → m Y i =1 P N such tha t g ∗ ( p ∗ i ( O (1))) ∼ = L i and with g ∗ ( X ( i ) 0 ) = s ( i ) . Here p i : Q m i =1 P N → P N is the pro jection on the i th factor , X 0 , . . . , X N are the standa rd co o rdinates on P N , and X ( i ) j := p ∗ i ( X j ). Let H i ⊂ Q m i =1 P N denote the subscheme defined b y X ( i ) 0 = 0 , and H = H 1 ∪ . . . ∪ H m . Since g is transverse to a ll the strata of H , we may use lemma 5 .5 to reduce us COMP ARISON OF COBORDISM THEORIES 19 to the case X = Q m i =1 P N , D = H , s = Q m i =1 ( X ( i ) 0 ) n i and L = N m i =1 p ∗ i ( O (1)) ⊗ n i . In this case, the natural map MGL 2 ∗ , ∗ H ( m Y i =1 P N ) → MGL 2 ∗ , ∗ ( m Y i =1 P N ) is injectiv e (see [3, lemma 5.2 .11]), so it suffices to see that c MGL 1 ( m O i =1 p ∗ i ( O (1)) ⊗ n i ) = ϑ MGL ( i H ∗ (div Ω ( s ))) ∈ MGL 2 , 1 ( m Y i =1 P N ) . But b y [3, prop ositio n 3.1.9] c Ω 1 ( m O i =1 p ∗ i ( O (1)) ⊗ n i ) = i H ∗ (div Ω ( s )) ∈ Ω 1 ( m Y i =1 P N ) . Since ϑ MGL is compatible with the resp e ctive Chern cla s s maps, this co mpletes the pro of.  Prop ositio n 5.7. L et f : X → P 1 b e a domina nt morphism with X ∈ Sm /k . L et D 0 = f − 1 (0) , D ∞ = f − 1 ( ∞ ) , D = D 0 ∐ D ∞ , U = X \ D , u = f | U ∈ Γ( U, O ∗ ) . Write ∂ X,D ( u ) ∈ MGL 2 , 1 D ( X ) as a su m ∂ X,D ( u ) := ∂ 0 X,D ( u ) − ∂ ∞ X,D ( u ); ∂ 0 X,D ( u ) ∈ MGL 2 , 1 D 0 ( X ) , ∂ ∞ X,D ( u ) ∈ MGL 2 , 1 D ∞ ( X ) , ac c or ding to the c anonic al dir e ct sum de c omp osition MGL 2 , 1 D ( X ) = MGL 2 , 1 D 0 ( X ) ⊕ MGL 2 , 1 D ∞ ( X ) . L et f 0 = f ∗ ( X 1 ) ∈ Γ( X , f ∗ O (1 )) and f ∞ = f ∗ ( X 0 ) ∈ Γ( X, f ∗ O (1 )) . Supp ose that D is a st rict normal cr ossing divisor. Then ∂ 0 X,D ( ψ ( u )) = α X,D 0 ( ϑ MGL ′ ( D 0 )(div Ω ( f 0 )) ∂ ∞ X,D ( ψ ( u )) = α X,D ∞ ( ϑ MGL ′ ( D ∞ )(div Ω ( f ∞ )) . Pr o of. W e first verify the formula for ∂ 0 X,D ( ψ ( u )). By excision, we can replace X with X \ D ∞ , so w e ma y assume that f is a regular function on X , f : X → A 1 . The formula for ∂ 0 X,D ( ψ ( u )) then follows from lemma 5.3 and lemma 5.6. The for mu la for ∂ ∞ X,D ( ψ ( u )) follows fro m the for mula for ∂ 0 X,D ( ψ ( u )). Indeed, ψ and ∂ X \ D 0 ,D ∞ are group homomorphisms , so ∂ ∞ X,D ( ψ ( u )) = − ∂ X \ D 0 ,D ∞ ( ψ ( u )) = ∂ X \ D 0 ,D ∞ ( ψ ( u − 1 )) . How ever, if w e re place u with u − 1 , this switches the roles of D 0 and D ∞ , a nd of f 0 and f ∞ . Thus the case w e hav e already handled shows that ∂ X \ D 0 ,D ∞ ( ψ ( u − 1 )) = α X,D ∞ ( ϑ MGL ′ ( D ∞ )(div Ω ( f ∞ )) , which completes the pro o f.  20 MARC LEVINE 6. P r oof of the main theorem W e a re now r eady to prove theorem 3.1. W e pro ceed by inductio n on the ma ximal dimension of a n irreducible c ompo nent of X ; the case of dimension 0 has b een verified in lemma 4.3. So, write X a s a union of clo sed subsets X = X 1 ∪ . . . ∪ X r ∪ X ′ with X 1 , . . . , X r irreducible of dimension d and X ′ of dimension < d . Let X ( d ) = x 1 ∐ . . . ∐ x r be the dimension d generic po int s of X . Set Ω (1) ∗ ( X ) := lim − → W Ω ∗ ( W ) as W ⊂ X runs ov er the closed subsets containing no x i and set MGL (1) 2 ∗ , ∗ ( X ) := lim − → W MGL ′ 2 ∗ , ∗ ( W ) ov er the same system of W . T aking the limit of the resp ective lo calization sequences gives us the co mm utative diagra m with exact columns: ⊕ i MGL ′ 2 ∗ +1 , ∗ ( k ( x i )) ∂   Ω (1) ∗ ( X ) ϑ (1) ( X ) / / i ∗   MGL (1) 2 ∗ , ∗ ( X ) i ∗   Ω ∗ ( X ) ϑ ( X ) / / j ∗   MGL ′ 2 ∗ , ∗ ( X ) j ∗   ⊕ i Ω ∗ ( k ( x i )) ⊕ i ϑ ( x i ) / /   ⊕ i MGL ′ 2 ∗ , ∗ ( k ( x i ))   0 0 W e hav e alrea dy seen (lemma 4.3) that the maps ϑ ( x i ) a r e isomo rphisms; the map ϑ (1) ( X ) is an isomor phis m by our induction hypothesis. This alrea dy implies that ϑ ( X ) is surjectiv e. T o show that ϑ ( X ) is injective, let Z [ k ( x i ) × ] b e the free abelia n group on k ( x i ) × . By lemma 4.3, we hav e a surjection ψ i : Z [ k ( x i ) × ] ⊗ L ∗ → MGL ′ 2 ∗ +2 d − 1 , ∗ + d − 1 ( k ( x i )) . Thu s, it suffices to define for ea ch i a map div Ω i : Z [ k ( x i ) × ] ⊗ L ∗ → Ω (1) ∗ + d − 1 ( X ) COMP ARISON OF COBORDISM THEORIES 21 making the diagram (6.1) Z [ k ( x i ) × ] ⊗ L ∗ α − 1 k ( x i ) ◦ ψ i / / div Ω i   MGL ′ 2 ∗ +2 d − 1 , ∗ + d − 1 ( k ( x i )) ∂ i   Ω (1) ∗ + d − 1 ( X ) ϑ (1) ( X ) / / MGL (1) 2 ∗ +2 d − 2 , ∗ + d − 1 ( X ) commute, and with i ∗ ◦ div Ω i = 0 . Since, for a ∈ k ( x i ) × , α ∈ MGL ′ 2 d − 2 ,d − 1 ( k ( x i )) = MGL 1 , 1 ( k ( x i )), b ∈ L ∗ , w e hav e ψ i ( a ⊗ b ) = ψ i ( a ) ∪ b ; ∂ i ( α ∪ b ) = ∂ ( α ) ∪ b , it suffices to define div Ω i on k ( x i ) × so that the dia gram c ommu tes and with i ∗ ◦ div Ω i = 0 on k ( x i ); w e then simply e xtend by linear it y a nd by using the L ∗ -mo dule structure. T ake u ∈ k ( x i ) × . Fix a blow-up π : ˜ X i → X of X i so that (1) ˜ X i is smo oth. (2) u defines a morphism f : ˜ X i → P 1 . (3) D 0 := f − 1 (0) and D ∞ := f − 1 ( ∞ ) are strict normal crossing divisor s . W e use the notation of prop osition 5.7. Let f 0 = f ∗ ( X 1 ), f ∞ = f ∗ ( X 0 ) and define div Ω i ( u ) := π ∗ (div Ω ( f 0 ) − div Ω ( f ∞ )) ∈ Ω (1) d − 1 ( X ) . Since div Ω i is defined on Z [ k ( x i ) × ], we need no t c heck that div Ω i ( u ) is indep endent of the ch oice of ˜ X i . W e first show tha t i ∗ ◦ div Ω i ( u ) = 0 . Indeed, let D = D 0 ∪ D ∞ and let ˜ ι : D → ˜ X i be the inclusion. Then by [3, prop osition 3.1.9] ˜ i ∗ (div Ω ( f 0 )) = c 1 ( f ∗ ( O (1)) = ˜ i ∗ (div Ω ( f ∞ )) , so ˜ i ∗ (div Ω ( f 0 ) − div Ω ( f ∞ )) = 0. Since π ∗ ◦ ˜ i ∗ = i ∗ ◦ π ∗ it follows that i ∗ ◦ div Ω i ( u ) = 0, as desired. Finally , we chec k that the diagr am (6.1) commutes. The bounda ry map in the lo calization sequence for MGL ′ ∗ , ∗ is co mpatible with the b oundary map in the Gysin sequence for MGL ∗ , ∗ -theory with suppo rts, i.e., ∂ MGL ˜ X i ,D ◦ α ˜ X i \ D = α ˜ X i ,D ◦ ∂ MGL ′ ˜ X i ,D . Thu s, prop osition 5.7 gives us (6.2) ∂ MGL ′ ˜ X i ,D ( α − 1 U ( ψ ( u ))) = ϑ ( D )(div Ω ( f 0 ) − div Ω ( f ∞ )) . The bo undary map in the localiza tion seq uence for MGL ′ ∗ , ∗ is natural with re- sp ect to pro jective push-for w ard (this follows fr om [2, lemma 2.6]), i.e., ∂ i ◦ π ∗ = π ∗ ◦ ∂ ˜ X i ,D . 22 MARC LEVINE Thu s, applying π ∗ to (6.2) yields ∂ i ( α − 1 U ( ψ ( u ))) = π ∗ ( ∂ MGL ′ ˜ X i ,D ( α − 1 U ( ψ ( u ))) = π ∗ ( ϑ ( D )(div Ω ( f 0 ) − div Ω ( f ∞ ))) = ϑ (1) ( X )( π ∗ (div Ω ( f 0 ) − div Ω ( f ∞ ))) = ϑ (1) ( X )(div Ω i ( u )) , as desir ed. This v erifies the commutativit y of the diagra m (6.1) and completes the pro of of theorem 3.1. References [1] F ulton, William. Intersection theory . Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 2 . Springer-V erlag, Berl i n-New Y ork, 1984. xi+470. [2] Levine, M. Or ien ted cohomology , Borel-Mo ore homology and algebraic cobordism , preprint 2008. http://www.math.neu .edu/ ∼ levine/publ/Publ.h tml [3] Levine, M .; Morel, F. Algebraic Cobo rdism Springer 2007. [4] Morel, F. A n introduction to A 1 -homotop y theory . Contemp or ary developments in algebr aic K -the ory 357–441, ICTP Lect. Notes, XV, Ab dus Salam In t. Cent . Theoret. Phys., T ri este, 2004. [5] Morel, F. and V o evodsky , V., A 1 -homotop y theo ry of schemes , Inst. Hautes ´ Etudes Sci. Publ. Math. 90 (1999), 45–143. [6] Østvær, P .A. and R¨ ondigs, O. Motive s and mo dules ov er motivic cohomology . C. R. Math. Acad. Sci. Paris 342 (2006) , no. 10, 751–754. [7] Pan in, I. Orien ted cohomology theories of algebraic v arieties . Special issue in honor of Hyman Bass on his s ev en tieth birthday . Part II I. K - Theory 30 (2003), no. 3, 265–314. [8] I. Panin Push-forwards in o riented cohomology theories of algebraic v arieties II . Preprint 2003. http://www.math.uiuc.edu /K-theory/0619/ [9] Pan in, I., Pimenov, K. and R¨ ondigs, O. A univ ersality theorem for V oevod- sky’s algebraic cobordism s pectrum . Preprint (2007) . K -theory preprint arch ive. h ttp://www.math.uiuc.edu/K-theory/084 6/ [10] V . V o evodsky , A 1 -homotop y theory , Pr oceedings of the In ternational Congress of Mathe- maticians, V ol. I (Berli n, 1998). Do c. Math. 1998, Extra V ol. I, 579–604. Dep ar tment of Ma thema tics, Nor theastern Un iversity, Boston, MA 0 2115, USA E-mail addr ess : marc@neu.edu