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UNIVERSALITY OF K-THEORY JOSE´ LUISGONZA´LEZANDKALLEKARU 3 ABSTRACT. We prove thatgraded K-theoryis universalamong oriented Borel-Mooreho- 1 0 mologytheorieswithamultiplicativeperiodicformalgrouplaw. 2 n a J 6 CONTENTS 1 1. Introduction 1 ] G 2. Oriented Borel-Moore Homology Theories 3 A . 3. Universality ofK-theory 9 h t a AppendixA. Descentfor K-theory 11 m References 12 [ 1 v 5 1 1. INTRODUCTION 8 3 . In [1] and [2] Shouxin Dai established a useful universal property enjoyed by the Gro- 1 0 thendieck K-theory functor of coherent sheaves G , that was conjectured by Levine and 0 3 Morel in [6]. Using results of Levine and Morel in [6], one can interpret Dai’s theorem 1 v: as describing some sufficient conditions under which K-theory can be recovered via ex- i X tension of scalars from algebraic cobordism. The argument that Dai provides in [1] and r [2] yields the universality of K-theory for schemes that admit embeddings into smooth a schemesin the category Sch of finite type separated schemesover a fixed field k of char- k acteristic zero. We will build on Dai’s work to show that this universality result holds over allof Sch , and more generallyover anyl.c.i. closedadmissiblesubcategory ofSch as k k definedin §2. TheproperstatementofDai’suniversalitytheoreminvolvesthenotionoforientedBorel- Moore homology theories (see § 2). One can regard each of these theories as an additive functor with values on graded abelian groups, endowed with projective push-forwards, l.c.i. pull-backs and exterior products, which satisfy some natural axioms. Each oriented Borel-Moore homology theory has a notion of Chern classes, and it is proved in [6] that for each such theory there is an associated formal group law that governs the relation ThisresearchwasfundedbyNSERCDiscoveryandAcceleratorgrants. 1 2 JOSE´ LUISGONZA´LEZANDKALLEKARU between the first Chern class of the tensor product of two line bundles with the first Chern classesofeach of the factors. The simplest example of an oriented Borel-Moore homology theory is the Chow homology functor A∗ described in [3], which has an additive formal group law. In fact, in characteristic zero, A∗ is universal among oriented Borel-Moore homology theories for which the first Chern class operators are additive with respect to tensor products of line bundles [6, Theorem 7.1.4]. On the other extreme, Levine and Morel constructed in [6] an oriented Borel-Moore homology theory Ω∗ called algebraic cobordism that is an algebraic analogue of the theory of complex cobordism and whose first Chern class operators behave on tensor products of line bundles according to the universal formal group law (FL,L). Moreover, Levine and Morel have proved that in characteristic zero, algebraiccobordism istheuniversalorientedBorel-Moore homologytheoryonSch . The k Grothendieck K-theory functor of coherent sheaves G , when made into a graded theory 0 byG [β,β−1],providesanotherimportantexampleofanorientedBorel-Moore homology 0 theory. Theformalgrouplawofthistheoryistheuniversalmultiplicativeperiodicformal group law(see 2.1.2and2.3). Let H∗ be any oriented Borel-Moore homology theory with a multiplicative periodic formal group law defined on the full subcategory S of Sch consisting of the smooth k quasiprojective schemes in Sch . In [6] Levine and Morel proved that there exists one k and only one transformation G0[β,β−1] H∗ of oriented Borel-Moore homology theories on S. In other words, they establishe→d that G [β,β−1] is the universal multiplicative 0 periodic oriented Borel-Moore homology theory on the category of smooth quasiprojective schemes over a field of characteristic zero. This led them to ask in [6] whether this property still holds overall ofSch . k Dai’suniversalityresultcameclosetoanswerLevineandMorel’squestion. Onewayto stateDai’sresultisthatincharacteristiczero,overanyl.c.i. closedadmissiblesubcategory C of Sch consisting of schemes that admit embeddings in smooth schemes in Sch (e.g. k k quasiprojective schemes), the natural morphism Ω∗ ⊗L Z[β,β−1] G0[β,β−1] induced by the universality of algebraic cobordism is an isomorphism. In→particular, it follows from the universality of algebraic cobordism that K-theory G [β,β−1] is universal among 0 orientedBorel-Moore homologytheorieswithamultiplicativeperiodicformalgrouplaw on anysuch category C. The goal of this article is to extend Dai’s universality to answer positively Levine and Morel’s question. This is done in Theorem 3.5 by using Dai’s result along with a descent sequence for projective envelopes that holds both in algebraic cobordism and K- theory (see Theorems 3.3 and 3.4), obtaining that in characteristic zero the morphism Ω∗ ⊗L Z[β,β−1] G0[β,β−1] is an isomorphism of oriented Borel-Moore homology theories over all of→Sch . In particular, in characteristic zero the graded K-theory functor k UNIVERSALITYOFK-THEORY 3 G [β,β−1]istheuniversalmultiplicative periodicorientedBorel-Moore homology theory 0 on anyl.c.i. closed admissiblesubcategory C ofSch . k The descent sequence in K-theory that we need in the main argument was proved by Gillet in [4] as a corollary to a more general descent theorem for simplicial schemes. In the appendix we give a shorter proof of Gillet’s theorem using Dai’s universality result and the descentsequencefor algebraiccobordism. Acknowledgments. This article builds on the result of Shouxin Dai establishing the de- sired universality property of K-theory for schemes that admit embeddings on smooth algebraicschemes. 2. ORIENTED BOREL-MOORE HOMOLOGY THEORIES 2.0.1. Throughout this article all of our schemes will be defined over a fixed field k of characteristic zero. Wedenote by Sch the category of separated finite type schemesover k Speck. For any full subcategory C of Sch , we denote by C′ the subcategory of C with the k same objects but whose morphisms are the projective morphisms. Smooth morphisms are assumed to be quasiprojective, and by definition locally complete intersection (l.c.i.) morphisms are assumed to admit a global factorization as the composition of a regular embeddingfollowedbyasmoothmorphism. Wefollowtheconventionthatsmoothmor- phisms,andmoregenerallyl.c.i. morphisms, havearelativedimension. Iff : X Y isan l.c.i. morphism of relative dimension d (or relative codimension −d) and Y is irr→educible, then Xisaschemeofpure dimensionequaltodimY +d. Ab∗ willdenote thecategory of graded abeliangroups. 2.0.2. WecallafullsubcategoryC ofSch admissibleifitsatisfiesthefollowingconditions k 1. Speckand the emptyscheme are in C. 2. IfY Xisa smooth morphism in Sch with X ∈ C, then Y ∈ C. k 3. IfX→and Y are in C, then so istheirproduct X×Y. 4. IfXand Y are in C, then so istheirdisjoint union X Y. ` 2.0.3. We call a full subcategory C of Sch l.c.i.-closed admissible if it is admissible and it k satisfies the following conditions 1. IfY Xisan l.c.i. morphism in Sch with X ∈ C, then Y ∈ C. k 2. Ifi :→Z Xisa regular embeddingin C, then the blowup ofX alongZis inC. → 2.0.4. Letf : X Zandg : Y Zbemorphismsinanadmissiblesubcategory C ofSch . k Wesaythat f an→dg are transve→rse in C ifthey satisfy O 1. Tor Z(O ,O ) = 0for all j > 0. j Y X 2. The fiberproduct X× Y isin C. Z 4 JOSE´ LUISGONZA´LEZANDKALLEKARU 2.1. Oriented Borel-Moore Homology Theories. Definition 2.1 (Levine-Morel). Let C be an admissible subcategory of Sch . An oriented k Borel-Moore homologytheory(or OBMhomologytheory)H∗ on C isgiven by: (D1) An additive functor H∗ : C′ Ab∗, i.e., a functor H∗ : C′ Ab∗ such that for any finite family (X ,...,X )of scheme→sin C′, the morphism → 1 r r r MH∗(Xi) H∗(aXi) i=1 → i=1 induced bythe projective morphisms X ⊆ r X isan isomorphism. i ì=1 i (D ) For each l.c.i. morphism f : Y X in C of relative dimension d a homomorphism 2 ofgraded groups → ∗ f : H (X) H (Y). ∗ ∗+d → (D ) An element 1 ∈ H (Speck) and for each pair (X,Y) of schemes in C, a bilinear 3 0 graded pairing × : H (X)×H (Y) H (X×Y) ∗ ∗ ∗ → (α,β) 7 α×β, → calledtheexteriorproduct,whichisassociative,commutativeandadmits1asunitelement. Thesedata satisfy the following axioms: ∗ (BM ) One has Id = Id for any X ∈ C. Moreover, for any pair of composable l.c.i. 1 X H∗(X) morphismsf : Y Xandg : Z Y,ofpurerelativedimensionsdanderespectively,one has → → ∗ ∗ ∗ (f◦g) = g ◦f : H (X) H (Z). ∗ ∗+d+e → (BM ) For any projective morphism f : X Z and any l.c.i. morphism g : Y Z that 2 are transverse in C, ifone forms the fiber dia→gram → W g′ // X f′ f (cid:15)(cid:15) (cid:15)(cid:15) Y g // Z ′ ′ (note that in this casef isprojective andg isl.c.i.) then ∗ ′ ′∗ g f = f g . ∗ ∗ (BM ) Given projective morphisms f : X X′ and g : Y Y′ one has that for any 3 classesα ∈ H∗(X) andβ ∈ H∗(Y) → → ′ ′ (f×g) (α×β) = f (α)×g (β) ∈ H (X ×Y ). ∗ ∗ ∗ ∗ UNIVERSALITYOFK-THEORY 5 Givenl.c.i. morphismsf : X X′andg : Y Y′onehasthatforanyclassesα ∈ H∗(X′) and β ∈ H∗(Y′) → → ∗ ∗ ∗ (f×g) (α×β) = f (α)×g (β) ∈ H (X×Y). ∗ (PB) For any line bundle L Y in C with zero section s : Y L, define the operator c˜1(L) : H∗(Y) H∗−1(Y)byc˜1(L→) = s∗s∗. LetE bethesheafofsect→ionsofarankn+1vector bundle E on X→in C, with associated projective bundle q : P(E) = Proj (Sym∗ E) X O O X X and associated canonical quotientline bundle q∗E OP(E)(1). For i = 0,...,n, let → → ξ(i) : H (X) H (P(E)) ∗+i−n ∗ → be the composition of q∗ : H∗+i−n(X) H∗+i(P(E)) with c˜1(OP(E)(1))i : H∗+i(P(E)) H∗(P(E)). Thenthe homomorphism → → n n Xξ(i) : MH∗+i−n(X) H∗(P(E)) i=0 i=0 → isan isomorphism. (EH) Let E X be a vector bundle of rank r over X in C, and let p : V X be an E-torsor. Then→p∗ : H∗(X) H∗+r(V) is anisomorphism. → → (CD) For integers r,n > 0, let W = Pn × ··· × Pn (r factors), and let p : W Pn i be the i-th projection. Let x ,...,x be the standard homogeneous coordinates o→n Pn, 0 n let a ,...,a be non-negative integers, and let j : E W be the subscheme defined by 1 r Qri=1p∗i(xn)ai = 0. Suppose that Eisin C. Then j∗ : H∗→(E) H∗(W)isinjective. → 2.1.1. If H∗ is an OBM homology theory, for each projective morphism f the homomorphism H∗(f) is denoted f∗ and called the push-forward along f. For each l.c.i. morphism g ∗ the homomorphism g is called the l.c.i. pull-back along g. For any line bundle L Y the operator c˜1(L) : H∗(Y) H∗−1(Y) is called the first Chern class operator of L. A m→orphism ϑ : H∗ H∗′ of OBM hom→ology theories is a natural transformation ϑ : H∗ H∗′ of functors C′ A→b∗ which moreover commutes with the l.c.i. pull-backs and prese→rves the exterior pro→ducts. Theaxiomsgive H∗(Speck)acommutative, gradedringstructure, givetoeach H∗(X) the structure of H∗(Speck)-module, and imply that the operations f∗ and g∗ pre- serve the H∗(Speck)-module structure. For each smooth quasiprojective X ∈ Schk with structure morphism f : X Speck, the elementf∗1is denoted by1 . X → 2.1.2. Recallthata(commutative)formalgroup lawonacommutativeringRisapower seriesF (u,v) ∈ RJu,vKsatisfying R (a) F (u,0) = F (0,u) = u, R R (b) F (u,v) = F (v,u), R R (c) F (F (u,v),w) = F (u,F (v,w)). R R R R 6 JOSE´ LUISGONZA´LEZANDKALLEKARU Thus FR(u,v) = u+v+Xai,juivj, i,j>0 where a ∈ R satisfy a = a and some additional relations coming from property i,j i,j j,i (c). A formal group law of the form (F (u,v) = u + v,R) is called additive. Likewise, + a formal group law of the form (F×(u,v) = u + v − βuv,R), for some β ∈ R, is called multiplicative; and it is in addition called periodic if β ∈ R is invertible. There exists a universal formal group law (FL,L), and its coefficient ring L is called the Lazard ring. In other words, todefinea formal group law(F,R)on acommutative ringR isequivalentto give aringhomomorphism L R. TheLazardringcanbeconstructed asthequotientof the polynomial ring Z[A ] →by the relations imposed by the three axioms above. The i,j i,j>0 imagesofthevariablesA inthequotientringarethecoefficientsa oftheformalgroup i,j i,j lawFL. Thering Lis graded,with Ai,j havingdegree i+j−1. 2.1.3. Levine and Morel proved that any OBM homology theory H∗ on an l.c.i-closed admissible subcategory C of Sch has an associated formal group law for its first Chern k classes (see [6, Definition 2.2.1, Corollary 4.1.8, Definition 4.1.9, Proposition 5.2.6]). More precisely,thereexistsauniqueformalgrouplaw(FH,H∗(Speck))suchthatforanysmooth quasiprojective scheme Y in Sch and for any pairof line bundles L and Mon Y in C, one k has c˜ (L⊗M)(1 ) = F (c˜ (L),c˜ (M))(1 ). 1 Y H 1 1 Y 2.2. Example: Chow Theory. The Chow group functor A∗ is an OBM homology theory onSch . Theexistenceofprojective push-forwards,l.c.i. pull-backsandexteriorproducts k satisfyingtherequiredaxiomscanbefoundin[3]. Moreover, givenlinebundlesLandM over the samebase one hasthe formula c˜ (L⊗M) = c˜ (L)+c˜ (M). 1 1 1 Hence,theformalgroup lawofChowtheoryisadditivegivenby(F (u,v) = u+v,Z). In A fact,LevineandMorelshowin[6,Theorem7.1.4]thatincharacteristiczeroA∗isuniversal among OBM homology theories with an additive formal group law on any l.c.i.-closed admissible subcategory C of Sch . k 2.3. Example: K-Theory. For any scheme X in Sch , let G (X) be the Grothendieck K- k 0 group of coherent O -modules (see [3, Chapter 15]). The class of any coherent sheaf F X on X in the group G (X) is denoted by [F]. Following Levine and Morel [6], for each 0 X in Schk we denote by G0(X)[β,β−1] the group G0(X) ⊗Z Z[β,β−1], where Z[β,β−1] is the ring of Laurent polynomials in a variable β of degree 1. One can verify that the functor G [β,β−1]becomes anOBM homology theory with the projective push-forwards, 0 UNIVERSALITYOFK-THEORY 7 l.c.i. pull-backs and exterior products given as follows. Define the push-forward for a projective morphism f : X Y by → ∞ f∗([F]·βm) := X(−1)i[Rif∗(F)]·βm ∈ G0(Y)[β,β−1] i=0 for any coherent OX-module F and any m ∈ Z, where Rif∗ denotes the i-th right derived functoroff∗. Thissumisfinitesincefisprojective. Thepull-backalonganl.c.i. morphism g : X Y ofrelative dimension disdefinedby → ∞ ∞ g∗([F]·βm) := X(−1)i[Lig∗(F)]·βm+d = X(−1)i[TorOi Y(OX,F)]·βm+d ∈ G0(X)[β,β−1] i=0 i=0 for any coherent O -module F and any m ∈ Z, where L g∗ denotes the i-th left derived Y i ∗ functor of g . This sum is finite since g is an l.c.i. morphism. For any X and Y in Sch the k external product G (X)[β,β−1]×G (Y)[β,β−1] G (X×Y)[β,β−1] 0 0 0 → isgiven by ([F ]·βm1,[F ]·βm2) 7 [π∗(F )⊗ π∗(F )]·βm1+m2, 1 2 X 1 X×Y Y 2 → for any coherent O -module F and any coherent O -module F , where π : X×Y X X 1 Y 2 X and π : X×Y Y are the projections. Moreover, given line bundles L and M over→the X same baseone h→asthe formula c˜ (L⊗M) = c˜ (L)+c˜ (M)−β·c˜ (L)◦c˜ (M). 1 1 1 1 1 Hence,theformalgrouplawofK-theoryismultiplicativeandperiodicgivenby(F (u,v) = K u+v−βuv,Z[β,β−1]). 2.4. Example: Algebraic Cobordism. Algebraic cobordism theory Ω∗ was constructed byLevineandMorelin[6]. Wenowrecallthegeometricpresentationofthistheorygiven byLevine andPandharipande in[7]. For X in Sch , let M(X) be the set of isomorphism classes of projective morphisms k f : Y X for Y ∈ Sch smooth quasiprojective. This set is a monoid under disjoint union k of the→domains; let M+(X) be its group completion. The class of f : Y X in M+(X) is denoted by[f : Y X]. → → Adoublepointdegenerationisamorphismπ : Y P1,withY ∈ Sch smoothquasipro- k jectiveofpuredimension,suchthatY∞ = π−1( )is→asmoothdivisoronYandY0 = π−1(0) is a union A∪B of smooth divisors intersectin∞g transversely along D = A∩B. Let N A/D andN bethenormalbundlesofDinAandB,anddefineP(π) = P(N ⊕O )(notice B/D A/D D that[P(N ⊕O ) X] = [P(N ⊕O ) X]becauseN ⊗N =∼ O (A+B)| =∼ O ). A/D D B/D D A/D B/D Y D D → → 8 JOSE´ LUISGONZA´LEZANDKALLEKARU Let X ∈ Sch , and let Y ∈ Sch be smooth quasiprojective of pure dimension. Let k k p ,p be the two projections of X×P1. A double point relation is defined by a projective 1 2 morphism π : Y X×P1, such thatp ◦π : Y P1 isa double pointdegeneration. Let 2 → → [Y X], [A X], [B X], [P(p ◦π) X] ∞ 2 → → → → be the elementsofM+(X) obtained bycomposing with p . The double pointrelation is 1 [Y X]−[A X]−[B X]+[P(p ◦π) X] ∈ M+(X). ∞ 2 → → → → Let R(X) be the subgroup of M+(X) generated by all the double point relations. The cobordism group ofX isdefinedto be Ω (X) = M+(X)/R(X). ∗ The group M+(X) is graded so that [f : Y X] lies in degree dimY when Y has pure dimension. Since double point relations are→homogeneous, this grading gives a grading on Ω∗(X). Itis shown in [6]that the graded group Ω∗(Speck) isisomorphic toL. The existence of projective push-forwards, l.c.i. pull-backs and exterior products satis- fying the required axioms can be found in [6]. Moreover, one of the main results in [6] is that in characteristic zero, Ω∗ is the universal OBM homology theory on any l.c.i.-closed admissiblesubcategoryC ofSchk. Inotherwords,foranyOBMhomologytheoryH∗ onC, there is a unique natural transformation of OBM homology theories ϑ : Ω∗ H∗. Given line bundlesL and Moverthe same base one hasthe formula → c˜1(L⊗M) = FL(c˜1(L),c˜1(M)), and therefore the formal group lawof Ω∗ isthe universal formal group law(FL,L). 2.5. Universality. We summarize Theorem 1.4.6 from [1]. Let (F ,R) be a formal group R law corresponding to the ring homomorphism L R. We define an OBM homology theory ΩFR on Sch by → ∗ k ΩFR := Ω ⊗ R. ∗ ∗ L Moreprecisely,ΩF∗R(X) := Ω∗(X)⊗LRforeachX ∈ Schk,andtheprojectivepush-forwards, l.c.i. pull-backs and exterior products are induced by functoriality by those of Ω∗. From the universality of Ω∗, it follows that in characteristic zero ΩF∗R is universal among OBM homology theories with formal group law (F ,R) on any l.c.i. closed admissible subcate- R gory of Sch . k Theformal group laws(F(u,v) = u+v,Z)and(F(u,v) = u+v−βuv,Z[β,β−1])arere- spectively the universal additive and the universal multiplicative periodic formal group laws. ThehomomorphismsL ZandL Z[β,β−1]classifyingtheseformalgrouplaws allowustodefinetheOBMho→mology the→oriesΩ+∗ := Ω∗⊗LZandΩ×∗ := Ω∗⊗LZ[β,β−1] on any l.c.i. closed admissible subcategory C of Sch . It follows from the universality k of algebraic cobordism as an OBM homology theory [6, Theorem 7.1.3] that these are UNIVERSALITYOFK-THEORY 9 respectively the universal additive OBM homology theory on C and the universal multiplicative periodic OBM homology theory on C. As we mentioned earlier, Levine and Morel showed that the canonical natural transformation Ω+∗ A∗ is an isomorphism [6, Theorem 7.1.4]. The main goal of this article is to show in The→orem 3.5 that the canonical natural transformation Ω× G [β,β−1] isanisomorphism. ∗ 0 → 3. UNIVERSALITY OF K-THEORY In this section we prove the universality of K-theory. We start by stating a version of Dai’s theorem proved in [1] which will constitute the central part of our argument. The assertionsin(a)and(b)inthefollowingtheorem aremerelyrestatementsofLemma2.1.4 and Corollary 1.6.4proved byDai in [1]. Theorem 3.1 (Dai). Let k be a field of characteristic zero. The canonical natural transformation of OBM homology theories Ω∗ G0[β,β−1] descends to a natural transformation of OBM homologytheoriesΩ× G [β,β−1]→thatsatisfies: ∗ 0 → (a) For everyXinSch the homomorphismΩ×(X) G (X)[β,β−1] issurjective. k ∗ 0 (b) For every quasiprojective scheme X in Sch the h→omomorphism Ω×(X) G (X)[β,β−1] k ∗ 0 isan isomorphism. → Definition 3.2 (Envelopes). An envelope of a scheme X in Sch is a proper morphism π : k X˜ X such that for every subvariety V of X there is a subvariety V˜ of X˜ that is mapped bir→ationallyontoV byπ. Ifπ : X˜ Xisanenvelope,wesaythatitisaprojectiveenvelope if π is a projective morphism. Li→kewise, we say that the envelope π : X˜ X is birational if for some dense open subset U of X, π induces an isomorphism π| : π→−1(U) U. The composition of envelopesisagainan envelopeandamorphism obtainedbyba→se change from an envelope is again an envelope. The domain X˜ of an envelope is not required to be connected, then by Chow’s lemma and induction on dimension it follows easily that for every scheme X in Sch there exists a birational envelope π : X˜ X such that X˜ is k quasiprojective. → Gillet in [4] established a descent sequence for envelopes in Chow theory and in K- theory. Westate the K-theory version only: Theorem3.3(Gillet). Letπ : X˜ XbeaprojectiveenvelopeinSch andletp ,p : X˜× X˜ X˜ k 1 2 X be thetwo projections. Thenthe s→equence → G (X˜ × X˜) −p−1−∗−−p−2∗ G (X˜) −π∗ G (X) 0 0 X 0 0 → → → isexact. Exactness ofGillet’sdescentsequence isequivalentto exactness ofthe sequence: G (X˜ × X˜)[β,β−1] −p−1−∗−−p−2∗ G (X˜)[β,β−1] −π∗ G (X)[β,β−1] 0. 0 X 0 0 → → → 10 JOSE´ LUISGONZA´LEZANDKALLEKARU Theanalogous descentsequence for algebraiccobordism wasproved in [5]: Theorem3.4. Letπ : X˜ XbeaprojectiveenvelopeinSch and letp ,p : X˜ × X˜ X˜ bethe k 1 2 X twoprojections. Thenthe→sequence → Ω (X˜ × X˜) −p−1−∗−−p−2∗ Ω (X˜) −π∗ Ω (X) 0 ∗ X ∗ ∗ → → → isexact. Since tensor product isa right exact functor, we alsogetan exact sequence Ω×(X˜ × X˜) −p−1−∗−−p−2∗ Ω×(X˜) −π∗ Ω×(X) 0. ∗ X ∗ ∗ → → → Weare nowreadyto prove the universality theorem. Theorem3.5(UniversalityofK-theory). Letkbeafieldofcharacteristiczero. LetC beanl.c.i.- closed admissible subcategory of Sch . Then G [β,β−1] is the universal multiplicative periodic k 0 orientedBorel-Moore homology theoryon C. Proof. Algebraic cobordism Ω∗ is the universal OBM homology theory on C, hence there is a canonical natural transformation of OBM homology theories φ : Ω∗ G0[β,β−1] on C. Since the formal group law of G [β,β−1] is multiplicative and peri→odic, then φ 0 induces a natural transformation of OBM homology theories ψ : Ω× G [β,β−1]. From ∗ 0 the universality of Ω∗, we know that Ω×∗ is the universal multiplica→tive periodic OBM homology theory on C. Therefore, it is enough to show that ψ : Ω×(X) G (X)[β,β−1] X ∗ 0 isan isomorphism for each Xin C. → We choose an envelope π : X˜ X, such that X˜ is quasi-projective. Consider the commutative diagram with exact row→s: (3.1) Ω×(X˜ × X˜) p1∗−p2∗ // Ω×(X˜) π∗ // Ω×(X) // 0 ∗ X ∗ ∗ (cid:15)(cid:15) ψX˜×XX˜ (cid:15)(cid:15) ψX˜ (cid:15)(cid:15) ψX G (X˜ × X˜)[β,β−1] p1∗−p2∗ // G (X˜)[β,β−1] π∗ // G (X)[β,β−1] // 0. 0 X 0 0 The mapsψ and ψ are isomorphisms byDai’s theorem since both X˜ × X˜ and X˜ are X˜×XX˜ X˜ X quasiprojective. The five-lemma impliesnowthat ψ isalso anisomorphism. (cid:3) X Remark3.6. The content ofTheorem 3.5isthat overafield ofcharacteristic zero, K-theory admits a unique natural transformation of OBM homology theories ϑ : G0[β,β−1] H∗ to any OBM homology theory H∗ with a multiplicative periodic formal group law→over any l.c.i.-closed admissible subcategory C of Sch . In the proof, one establishes the iso- k morphism ofOBMhomology theories Ω ⊗ Z[β,β−1] =∼ G [β,β−1], ∗ L 0