On the relation of Voevodsky's algebraic cobordism to Quillen's K -theory I. Panin*y K. Pimenovy O. R"ondigsz June 6, 2007x Abstract Quillen's algebraic K-theory is reconstructed via Voevodsky's al- gebraic cobordism. More precisely, for a ground field k the algebraic cobordism P1-spectrum MGL of Voevodsky is considered as a commu- tative P1-ring spectrum. Setting MGL i= 2q-p=iMGL p,qwe regard the bigraded theory MGL p,qas just a graded theory. There is a unique ring morphism OE: MGL 0(k) ! Z which sends the class [X]MGL of a smooth projective k-variety X to the Euler characteristic O(X, OX ) of the structure sheaf OX . Our main result states that there is a canon- ical grade preserving isomorphism of ring cohomology theories on the category SmOp=k ': MGL *(X, U) MGL 0(k)Z ! KTT-*(X, U) = K0-*(X - U), in the sense of [PS1 ], where KTT*is the Thomason-Trobaugh K-theory and K0*is Quillen's K0-theory. In particular, the left hand side is a ring cohomology theory. Moreover both theories are oriented in the sense of [PS1 ] and ' respects the orientations. The result is an algebraic version of a theorem due to Conner and Floyd. That theorem reconstructs complex K-theory via complex cobordism [CF ]. ______________________________ *Universit"at Bielefeld, SFB 701, Bielefeld, Germany ySteklov Institute of Mathematics at St. Petersburg, Russia zInstitut f"ur Mathematik, Universit"at Osnabr"uck, Osnabr"uck, Germany xThe authors thank the SFB-701 at the Universit"at Bielefeld, the RTN-Networ* *k HPRN- CT-2002-00287, the RFFI-grant 03-01-00633a, and INTAS-03-51-3251 for their supp* *ort. 1 1 A motivic version of a theorem by Conner and Floyd Our main result relates Voevodsky's algebraic cobordism theory MGL *,*to Quillen's K0-theory. We refer to [PPR1 , Appendix] for the basic terminology, notation, constructions, definitions, results. Let S be a Noetherian separated finite-dimensional scheme S. One may think of S being the spectrum of a field or the integers. A motivic space over S is a functor A :SmOp=S ! sSet (see [PPR1 , Appendix]). The category of motivic spaces over S is denoted M(S). This definition of a motivic space is different from the one consid- ered by Morel and Voevodsky in [MV ] - they consider only those simplicial presheaves which are sheaves in the Nisnevich topology on Sm=S. With our definition the Thomason-Trobaugh K-theory functor obtained by using big vector bundles is a motivic space on the nose. It is not a simplicial Nisnevich sheaf. This is why we prefer to work with the above notion of "space". We write Hcmo(S) for the pointed motivic homotopy category and SH cm(S) for the stable motivic homotopy category over S as constructed in [PPR1 , A.3.9, A.5.6]. By [PPR1 , A.3.11 resp. A.5.6] there are canonical equiva- lences to Ho(S) of [MV ] resp. SH (S) of [V1 ]. Both Hcmo(S) and SH cm(S) are equipped with closed symmetric monoidal structures such that the P1- suspension spectrum functor is a strict symmetric monoidal functor 1P1:Hcmo(S) ! SH cm(S). Here P1 is considered as a motivic space pointed by 1 2 P1. The symmetric monoidal structure (^, IS = 1P1S+ ) on the homotopy category SH cm(S) is constructed on the model category level by employing the category MSS (S) of symmetric P1-spectra. This symmetric monoidal category satisfies the properties required by Theorem 5.6 of Voevodsky congress talk [V1 ]. From now on we will usually omit the superscript (-)cm. Every P1-spectrum E represents a cohomology theory on the category of pointed motivic spaces. Namely, for a pointed motivic space (A, a) set Ep,q(A, a) = HomHcmo(S)( 1P1(A, a), p,q(E)) and E*,*(A, a) = p,qEp,q(A, a). This definition extends to motivic spaces via the functor A 7! A+ which adds a disjoint basepoint. That is, for a non-pointed motivic space A set Ep,q(A) = Ep,q(A+ , +) and E*,*(A) = p,qEp,q(A). Every X 2 Sm=S defines a representable motivic space constant in the simplicial direction taking an S-smooth scheme U to Hom Sm=S(U, X). It is 2 not possible in general to choose a basepoint for representable motivic spaces. So we regard S-smooth varieties as motivic spaces (non-pointed) and set Ep,q(X) = Ep,q(X+ , +). Given a P1-spectrum E we will reduce the double grading on the co- homology theory E*,* to a grading. Namely, set Em = m=p-2q Ep,q and E* = m Em . We often write E*(k) for E*(Spec (k)) below. A P1-ring spectrum is a monoid (E, ~, e) in (SH(S) , ^, IS). A commuta- tive P1-ring spectrum is a commutative monoid (E, ~, e) in (SH(S) , ^, 1). The cohomology theory E* defined by a P1-ring spectrum is a ring coho- mology theory. The cohomology theory E* defined by a commutative P1-ring spectrum is a ring cohomology theory, however it is not necessarily graded commutative. The cohomology theory E* defined by an oriented commuta- tive P1-ring spectrum (to be defined below) is a graded commutative ring cohomology theory. Occasionally a P1-ring spectrum (E, ~, e) might have a model (E0, ~0, e0) which is a symmetric P1-ring spectrum, that is, a symmetric P1-spectrum E0 equipped with a strict multiplication ~0:E0^ E0 ! E0 which is strictly associative and strictly unital for the unit e0: 1P1(S+ ) ! E0. This is the case for the algebraic cobordism P1-ring spectrum MGL , as described below. Such a model for the algebraic K-theory P1-ring spectrum BGL is currently not known to us. For the rest of the paper let k be a field and S = Spec(k). Usually S will be replaced by k in the notation. We work in this text with the algebraic cobordism P1-spectrum MGL and the algebraic K-theory P1-spectrum BGL as described in [PPR1 , Defn. 1.2.4] and [PPR2 , Sect. 2.1] respectively. The spectrum MGL is a commutative ring P1-spectrum by that construction. The spectrum BGL is equipped with a structure of a commutative P1-ring spec- trum as explained in [PPR1 , Thm. 2.1.1]. Let KTT* be Thomason-Trobaugh K-theory functor [TT ]. There is a canonical isomorphism Ad :KTT-*! BGL *,0 of ring cohomology theories on the category SmOp=S in the sense of [PS1 ]. An invertible Bott element fi 2 BGL 2,1(Spec (k)) is constructed in [PPR1 , Section 1.3]. For every pointed motivic space A the morphism BGL *,0(A) BGL 0(Spec (k)) ! BGL *,*(A) (1) given by a b 7! a[b is a ring isomorphism by [PPR1 , Sect. 1.3]. Furthermore BGL 0(Spec (k)) = Z[fi, fi-1 ] is the ring of Laurent polynomials on the Bott 3 element fi. To say the same in a different way, BGL *,0(A)[fi, fi-1 ] ~=BGL *,*(A). (2) The special case A = X=(X r Z) where X is a smooth k-variety and Z X is a closed subset implies the following result [PPR1 , Cor. 1.3.6]. Corollary 1.0.1. Let X be a smooth k-scheme, Z a closed subset of X and U = X r Z its open complement. Then there are isomorphisms KTT-*,Z(X)[fi, fi-1~]=BGL *,*(X=U) = BGL *(X=U) (3) KTT-*,Z(X) ~= BGL *,*(X=U)=(fi + 1)BGL *,*(X=U) (4) of ring cohomology theories on SmOp=k in the sense of [PS1 ]. We refer to [PPR2 ] for a construction of the commutative P1-ring spec- trum MGL . For the purposes of the present preprint we will need to know only two properties of that P1-spectrum, which we refer to as Quillen uni- versality and BGL -cellularity (see Subsection 2.1 below). 1.1 Oriented commutative ring spectra Following Adams and Morel we define an orientation of a commutative P1- ring spectrum. However we prefer to use Thom classes instead of Chern classes. Consider the pointed motivic space P1 = colimn 0 Pn having base point g1: S = P0 ,! P1 . The tautological "vector bundle" T(1) = OP1 (-1) is also known as the Hopf bundle. It has zero section z :P1 ,! T(1). The fiber over the point g1 2 P1 is A1. For a vector bundle V over a smooth S-scheme X, with zero section z :X ,! V , let the Thom space Th (V ) of V be the Nisnevich sheaf associated to the presheaf Y 7! V (Y )= V r z(X) (Y ) on the Nisnevich site Sm=S. In particular Th (V ) is a pointed motivic space in the sense of [PPR1 , Defn. A.1.1]. It coincides with Voevodsky's Thom space [V1 , p. 422], since Th (V ) is already a Nisnevich sheaf. The Thom space of the Hopf bundle is then defined as the colimit Th (T(1)) = colimn 0 Th OPn (-1) . Abbreviate T = Th (A1S). Let E be a commutative P1-ring spectrum. The unit gives rise to an ele- ment 1 2 E0,0(Spec (k)+ ). Applying the P1-suspension isomorphism to that element we get an element P1(1) 2 E2,1(P1, 1). The canonical covering of P1 defines motivic weak equivalences __~__//1 1 o~o__ 1 1 P1 P =A A =A r {0} = T of pointed motivic spaces inducing isomorphisms E(P1, 1) E(A1=A1 r {0}) ! E(T ) . Let T (1) be the image of P1(1) in E2,1(T ). 4 Definition 1.1.1. Let E be a commutative ring P1-spectrum. A Thom orientation of E is an element th 2 E2,1(Th (T(1)) such that its restriction to the Thom space of the fibre over the distinguished point coincides with the element T (1) 2 E2,1(T ). A Chern orientation of E is an element c 2 E2,1(P1 ) such that c|P1 = - P1(1). An orientation of E is either a Thom orientation or a Chern orientation. One says that a Thom orientation th of E coincides with a Chern orientation c of E provided that c = z*(th) or equivalently the element th coincides with the one th(O(-1)) given by (6) below. Remark 1.1.2. The element th should be regarded as the Thom class of the tautological line bundle T(1) = O(-1) over P1 . The element c should be regarded as the Chern class of the tautological line bundle T(1) = O(-1) over P1 . Example 1.1.3. The following orientations given right below are relevant for our work. Here MGL denotes the P1-ring spectrum representing algebraic cobordism obtained in [PPR2 , Defn 2.1.1] and BGL denotes the P1-ring spectrum representing algebraic K-theory constructed in [PPR1 , Theorem 2.2.1]. o Let u1 : 1P1(Th (T(1)))(-1) ! MGL be the canonical map of P1- spectra. Set thMGL = u1 2 MGL 2,1(Th (T(1))). Since thMGL |Th(1) = P1(1) in MGL 2,1(Th (1)), the class thMGL is an orientation of MGL . o Set c = (-fi) [ ([O] - [O(1)]) 2 BGL 2,1(P1 ). The relation (11) from [PPR1 ] shows that the class c is an orientation of BGL . 2 Oriented cohomology theories Let (E, c) be an oriented commutative P1-ring spectrum. In this section we compute the E-cohomology of infinite Grassmannians. The results are the expected ones - see Theorem 2.0.6. The oriented P1-ring spectrum (E, c) defines an oriented cohomology the- ory on SmOp in the sense of [PS1 , Defn. 3.1] as follows. The restriction of the functor E*,* to the category Sm=S is a ring cohomology theory. By [PS1 , Th. 3.35] it remains to construct a Chern structure on E*,*|SmOp in the sense of [PS1 , Defn.3.2]. Let Ho(k) be the homotopy category of pointed motivic spaces over k. The functor isomorphism Hom Ho(k)(-, P1 ) ! Pic(-) on the category Sm=S provided by [MV , Thm. 4.3.8] sends the class of the identity map P1 ! P1 to the class of the tautological line bundle O(-1) over P1 . For a line bundle L over X 2 Sm=S let [L] be the class 5 of L in the group Pic(X). Let fL :X ! P1 be the morphism in Ho(k) corresponding to the class [L] under the functor isomorphism above. For a line bundle L over X 2 Sm=S set c(L) = f*L(c) 2 E2,1(X). Clearly, c(O(-1)) = c. The assignment L=X 7! c(L) is a Chern structure on E*,*|SmOp since c|P1 = - P1(1) 2 E2,1(P1, 1). With that Chern structure E*,*|SmOp is an oriented ring cohomology theory in the sense of [PS1 ]. In particular, (BGL , cK ) defines an oriented ring cohomology theory on SmOp. Given this Chern structure, one obtains a theory of Thom classes V=X 7! th(V ) 2 E2rank(V ),rank(VT)hX(V ) on the cohomology theory E*,*|SmOp=S in the sense of [PS1 , Defn. 3.32] as follows. There is a unique theory of Chern classes V 7! ci(V ) 2 E2i,i(X) such that for every line bundle L on X one has c1(L) = c(L). For a rank r vector bundle V over X consider the vector bundle W := 1 V and the associated projective vector bundle P(W ) of lines in W . Set ~th(V ) = cr(p*(V ) OP(W) (1)) 2 E2r,r(P(W )). (5) It follows from [PS1 , Cor. 3.18] that the support extension map 2r,r E2r,rP(W )=(P(W ) r P(1)) ! E P(W ) is injective and ~th(E) 2 E2r,rP(W )=(P(W ) r P(1)) . Set th(E) = j*(~th(E)) 2 E2r,rTh X (V ) , (6) where j :Th X(V ) ! P(W )=(P(W ) r P(1)) is the canonical motivic weak equivalence of pointed motivic spaces induced by the open embedding V ,! P(W ). The assignment V=X to th(V ) is a theory of Thom classes on E*,*|SmOp (see the proof of [PS1 , Thm. 3.35]). So the Thom classes are nat- ural, multiplicative and satisfy the following Thom isomorphism property. Theorem 2.0.4. For a rank r vector bundle p :V ! X on X 2 Sm=S with zero section z :X ,! V , the map - [ th(V ) :E*,*(X) ! E*+2r,*+rV=(V r z(X)) is an isomorphism of two-sided E*,*(X)-modules, where - [ th(V ) is written for the composition map - [ th(V ) O p*. __ Proof. See [PS1 , Defn. 3.32.(4)]. |__| Analogous to [V1 , p. 422] one obtains for vector bundles V ! X and W ! Y in Sm=S a canonical map of pointed motivic spaces Th (V ) ^ Th (W ) ! Th (V xS W ) which is a motivic weak equivalence as defined 6 in [PPR1 , Defn. 3.1.6]. In fact, the canonical map becomes an isomorphism after Nisnevich (even Zariski) sheafification. Taking Y = S and W = 1 the trivial line bundle yields a motivic weak equivalence Th (V )^T ! T h(V 1). The motivic weak equivalences T = A1=A1 r {0} __~__//P1=A1 o~o__P1 introduced above imply that one may switch between T and P1 as desired. Corollary 2.0.5. For W = V 1 consider the motivic weak equivalences ffl :Th (V ) ^ P1 ! Th (V ) ^ P1=A1 Th (V ) ^ T ! Th (W ) of pointed motivic spaces. The diagram P1 E*+2r,*+r(ThO(VO))____//E*+2r+2,*+r+1(ThO(VO) ^ P1) id|| ffl*|| | T *+2r+2,*+r+1| E*+2r,*+r(ThO(VO))______//E OO (Th (W )) -[th(V )|| -[th(W)|| | | E*,*(X) ________id________//_E*,*(X) commutes. Let T(n) ! Gr (n) be the tautological vector bundle of n-dimensional linear subspaces. Theorem 2.0.6. Let ci = ci(T(n)) 2 E2i,i(Gr (n)) be the i-th Chern class of the tautological bundle T(n). Then E*,*(Gr (n)) = E*,*(k)[[c1, c2, . .,.cn]] is the formal power series on the ci's. The inclusion i :Gr (n) ,! Gr (n + 1) satisfies i*(cm ) = cm for m < n + 1 and i*(cn+1) = 0. 2.1 A general result The main result of this Section is Theorem 2.1.4. The complex cobordism spectrum, equipped with its natural orientation, is a universal oriented ring cohomology theory by Quillen's universality theorem [Qu1 ]. A motivic ver- sion of this universality theorem is proved in [PPR2 ] (see [Ve ] for the orig* *inal statement). We consider MGL with the commutative monoid structure de- scribed in [PPR2 , Defn 2.1.1] and with the orientation thMGL described in 1.1.3. 7 Definition 2.1.1 (Universality Property). Let (U, u) be an oriented com- mutative ring P1-spectrum over a field k. We say that (U, u) is Quillen uni- versal if for every commutative ring P1-spectrum E over k the assignment ' 7! '(u) 2 U2,1(Th (T(1))) identifies the set of monoid homomorphisms ' :U ! E (7) in the motivic stable homotopy category SH cm(S) with the set of orientations of E. Remark 2.1.2. The Universality Theorem ([Ve ] or [PPR2 ]) implies that the P1-spectrum MGL , equipped with its canonical orientation, is Quillen universal. If E is a commutative P1-ring spectrum over k and A is a pointed motivic space over k, E*(A) is an E0(k)-module in a natural way. A monoid homo- morphism OE :E1 ! E2 induces an E0(k)-module homomorphism E*1(A) ! E*2(A). In particular, if (U, u) is a Quillen universal oriented commutative ring P1-spectrum over k and (E, th) is an oriented commutative ring P1- spectrum over k, the monoid homomorphism ' :U ! E (8) in SH (k) induces the homomorphism '~A: U*(A) U0(k)E0(k) ! E*(A) (9) and in particular the homomorphism '~0A:U0(A) U0(k)E0(k) ! E0(A). (10) Both are natural in A. Since this moment choose (BGL , thK) for (E, th) (see Example 1.1.3). Set U~*(X, Y ) = U*(X, Y ) U0(k)BGL 0(k), ~U0(X, Y ) = U0(X, Y ) U0(k)BGL 0(k). Definition 2.1.3 (Weakly BGL-Cellular). Let (U, u) be a Quillen universal P1-ring spectrum, and let O~E0A:U0(A) U0(k)BGL 0(k) ! BGL 0(A) be the homomorphism induced by the orientation thK on BGL (see Example 1.1.3). Then (U, u) is is called weakly BGL -cellular if there exists an integer N such that the map ~'0Unis an isomorphism for n N. A pointed motivic space A is called small if the covariant functor Hom SH(S)( 1P1A, -) on SH (S) commutes with arbitrary coproducts. 8 Theorem 2.1.4. Let (U, u) be a Quillen universal oriented commutative P1- ring spectrum over a field k. Suppose (U, u) is weakly BGL -cellular. Then the homomorphism ~'Ais an isomorphism for all small pointed motivic spaces A. Proof. The proof consists of several steps. Let Our first aim is to prove that the homomorphisms ~'0Aare isomorphisms, where A is a small pointed motivic space. First we construct a section of the natural transformation '0,0:U0,0! BGL 0,0 of functors on the category of small pointed motivic spaces. To do this we begin with recalling that for every oriented commutative P1-ring spectrum (E, th) the ring cohomology theory E*,*|SmOp is an oriented cohomology the- ory on the category SmOp (see Section 2). Let FE,thbe the induced commu- tative formal group law over the ring E0(k). Let be the complex cobordism ring and let lE,th: ! E0(k) be the unique ring homomorphism sending the universal formal group F to FE,th. Set [Pn]E = lE,th([CPn]), (11) where [CPn] is the class of the complex projective space CPn in . Although the class [Pn]E depends on the orientation class th, we use the notation [Pn]E instead. If (E0, th0) is another oriented commutative P1-ring spectrum and _ :E ! E0 is a monoid homomorphism in the category SH cm(S) which preserves orientation classes, then it sends the formal group law FE,th to FE0,th0. In particular _([Pn]E ) = [Pn]E0. Applying this observation to the monoid homomorphism ' one obtains '([P1]U ) = [P1]BGL . To compute [P1]BGL recall that the coefficient at XY in the formal group law F coincides with the class -[CP1] in . The formal group law FBGL coincides with X + Y + fi-1 XY , since cBGL (L) = ([1] - [L_])(-fi). Thus the equality [P1]BGL = -fi-1 holds. We are ready to construct a section. Set Gr = colimn 0 Gr (n) and consider the map s : 1P1(Z x Gr) ! U (12) in the stable homotopy category category SH (S) given by the element cU1(1 - o1_) [ [P1]U 2 U0,0(Z x Gr). 9 Claim 2.1.5. One has '(cU1(1 - o1_) [ [P1]U ) = o1 - 1 2 BGL 0,0(Z x Gr). In fact, '(cU1(1 - o1_) [ [P1]U ) = cBGL1(1 - o1_) [ [P1]BGL = (1 - o1 ) [ fi [ (-fi-1 ) = o1 - 1 Claim 2.1.5 shows that the composition ' O s : 1P1(Z x Gr) ! BGL coincides with the adjoint of the motivic weak equivalence i :Z x Gr ! K = K0 from [PPR1 , Lemma 1.2.2]. Thus for every pointed motivic space A the map sA :BGL 0,0(A) = [A, K0] = [A, Z x Gr] ! [ 1P1(A), U] = U0,0(A) is a section of the map '0,0A:U0,0(A) ! BGL 0,0(A). Moreover, the section sA is natural_in A. * 0 __* * Let U (A) = U(A) U0(k)BGL (k) and recall that ~OEA:U (A) ! BGL (A) is the induced BGL 0(k)-module homomorphism. To extend the section s to __0 __0 0 a section ~s0:BGL 0 ! U of the natural transformation ~'0:U ! BGL of functors on pointed motivic spaces. note that BGL 0= BGL 0,0[fi, fi-1 ] for the Bott element fi 2 BGL 2,1(k) (see (2)). Thus for every pointed motivic space A, every element ff 2 BGL 0(A) can be presented in a unique way in the form a [ fii with a 2 BGL 0,0(A). Define __0 ~s0A:BGL 0 ! U (13) __0 0,0 by ~s0A(a [ fii) = sA (a) fii 2 U (A), where a 2 BGL (A). It is immediate that s0Ais natural in A.The computation '~0A(~s0(a [ fii)) = ~'0A(s(a) fii) = '(s(a)) [ fii = a [ fii proves the following Claim 2.1.6. The map ~s0Ais a section of ~'0A. 10 If for a pointed motivic space A the map ~'0Ais an isomorphism, then ~s0A is an isomorphism inverse to ~'0A. In particular, one has ~s0AO ~'0A= id. The homomorphism ~'0Ais an isomorphism for the pointed motivic spaces Un with n N, since U is weakly BGL -cellular. The class [un] 2 U2n,n(Un, *) of the canonical morphism un : 1P1Un(-n) ! U then satisfies the following relation: __0 (~s0UnO '0Un)([un]) = [un] 1 2 U (Un). (14) Now we are ready to check that ~'0Ais an isomorphism for all small pointed motivic spaces. Recall that for a small pointed motivic space A there is a canonical isomorphism of the form U2i,i(A) = colimn[ 2n,n(A), Ui+n]Ho(S) (15) where 2n,n = nP1. This isomorphism implies that for every element a 2 U2i,i(A) there exists an integer n 0 such that 2n,n(a) = f*([un]) for an appropriate map f : 2n,n(A) ! Ui+n in the homotopy category Ho(S). Here 2n,n(a) is the n-fold P1-suspension of a. The surjectivity of ~'0Ais clear, since ~s0Ais its section. It remains to ch* *eck __2i,i __0 the injectivity of '~0A. Take a homogeneous element ff 2 U (A) U (A) such that ~'0A(ff) = 0. It has the form ff = a fim for a homogeneous element a 2 U0(A). Since the element fi is invertible in BGL *,*(k), one concludes '0A(a) = 0. Choose an integer n 0 such that 2n,n(a) = f*([un]). The map ' of P1-spectra respects the suspension isomorphisms. Thus ' 2n,nA( 2n,n(a)) = 2n,n('A (a)) = 0 and (~s0 2n,nAO ' 2n,nA)( 2n,n(a)) = 0 too. The chain of __0 relations in U ( 2n,nA) given by 0 2n,n 0 * 0 = ~s 2n,nAO ' 2n,nA (a) = ~s 2n,nAO ' 2n,nA f ([un]) 0 * * = f* (~sUn+iO 'Un+i)([un]) = f ([un] 1) = f ([un]) 1 = 2n,n(a) 1 implies that 2n,n(a 1) = 2n,n(a) 1 = 0. Because the n-fold suspension map __ 0 __0 2n,n 2n,n:U (A) ! U ( A) __0 __0 is an isomorphism, a 1 = 0 in U (A) = U (A). This proves the injectivity and hence the bijectivity of ~'0Afor cofibrations of all small motivic spaces. To prove that ~'A is an isomorphism for all small motivic spaces we will use the fact that '~A respects the P1-suspension isomorphisms. For every 11 integer i 2 Z choose an integer n 0 with n i. Then for a pointed motivic space A one may form the suspension G^nm^ Sn-is^ A = Sn,n^ Sn-i,0^ A in the category of pointed motivic spaces, which supplies the commutative diagram 2n,n i i,0 0 BGL Oi(A)__~=_//BGLO(S2n,n^OA)Ooo~=_BGL (Sn,n^OSn-i,0^OA) '~iA|| ~'iS2n,n^A|| ~=|~'0Sn,n^Sn-i,0^A| | 2n,n | i,0 | Ui(A) ___~=__//_Ui(S2n,n^ A)oo_~=___U0(Sn,n^ Sn-i,0^ A) with the suspension isomorphisms 2n,n= nP1and i,0. The map ~'0Bis an isomorphism for B a small pointed motivic space, hence so is ~'iA. We proved __ that the map ~'A is an isomorphism for A being small. |__| 2.2 The BGL -cellularity of algebraic cobordism Theorem 2.2.1. The oriented commutative ring P1-spectrum (MGL , thMGL ) from Example 1.1.3 is weakly BGL -cellular. Proof. We will prove that the homomorphism ~'0Ais an isomorphism for A being one of the pointed motivic spaces Spec(k)+ , P1+, Gr(n)+ and Th (Tn) = MGL n , *. The map '~0kis an isomorphism, since_it_is the identity map. The case * 1 ______* MGL n = 1 of Theorem 2.0.6 implies that MGL (P ) = MGL (k)[[c ]], whence ______0 1 ______0 MGL MGL (P ) = MGL (k)[[c ]] (the formal power series on the first Chern class cMGL of the tautological line bundle O(-1)). The same holds for BGL . Namely BGL 0(P1 ) = BGL 0(k)[[cBGL ]]. By its definition the morphism ' takes the orientation class thMGL to the orientation class thK and so it preserves the first Chern class. Whence the map '~0P1 coincides with the map of formal power series induced by the isomorphism ~'0kof coefficients rings. Hence ~'0P1is an isomorphism as well. Consider now A = Gr (n)+ . By Theorem 2.0.6 its MGL -cohomology ring is the ring of formal power series on the Chern classes of the tautological bundle Tn over the coefficient ring MGL *,*(k). The same holds for the BGL - cohomology ring. As observed above, the map ' preserves the first Chern class, thus it preserves all Chern classes. Thus ~'0Gr(n)is an isomorphism as well. 12 Now consider A = Th (Tn). The morphism ' respects Thom classes (see (5) and (6)). The vertical arrows in the commutative diagram ______0 ~'0Th(Tn),*) 0 MGL ((ThO(Tn),O*))__________//_BGL(ThO(Tn))O ____thomMGL|| thomBGL| | ______0| ~'0Gr(n) 0 | MGL (Gr (n))_______________//BGL (Gr (n)) are isomorphisms induced by the the Thom isomorphism 2.0.4. The map ~'0Gr(n)is an isomorphism by the preceding case, whence ~'0Th(Tn),*)is an iso- morphism too. __ |__| 2.3 The main result Let k be a field and S = Spec(k). By Theorem [PPR2 , Theorem 2.2.1] and Example 1.1.3 there exists a unique monoid homomorphism ' :MGL ! BGL (16) in SH (S) such that '(thMGL ) = thK . It induces a natural transformation ______* * 0 * '~A: MGL (A) : = MGL (A) MGL 0(k)BGL (k) ! BGL (A). (17) Theorem 2.3.1. The homomorphism ~'A:MGL *(A) MGL 0(k)BGL 0(k) ! BGL *(A) is an isomorphism for all small pointed motivic spaces. Proof. In fact, (MGL , thMGL ) is Quillen universal by [Ve ] and [PPR2 ], and weakly BGL -cellular by Theorem 2.2.1. Theorem 2.1.4 completes the proof. __ |__| Remark 2.3.2. There is an unpublished result due to Morel and Hopkins, which states that there is a canonical isomorphism of the form MGL *,*(X) L Z[fi, fi-1 ] ! BGL *,*(X) where L denotes the Lazard ring carrying the universal formal group law. If the canonical homomorphism L ! MGL 0(k) is an isomorphism, Theorem 2.3.1 implies their result. 13 Let X be a smooth k-scheme and Z X a closed subset, with open complement U X. Consider the small pointed motivic space X=U and take the quotients of both sides of the isomorphism (17) modulo the principal ideal generated by the element 1 (fi + 1). Corollary 1.0.1 then implies that the natural transformation '~~X=U:MGL *(X=U) MGL 0(k)Z ! KTT-*,Z(X) (18) is an isomorphism, where KTT*,Z(X) are the Thomason-Trobaugh K-groups with supports. This family of isomorphisms shows that the functor (X, X r Z) 7! MGL *(X=(X r Z)) MGL 0(k)Z is a ring cohomology theory in the sense of [PS1 ]. This implies the first part of our main result. Theorem 2.3.3 (Main Theorem). Let X 2 Smk and Z X be a closed subset. o The family of isomorphisms '~~X=(X-Z):MGL *(X=U) MGL 0(k)Z ! KTT-*,Z(X) (19) form an isomorphism ~~' of ring cohomology theories on SmOp=k. o The natural isomorphism '~~respects orientations provided that MGL * and KTT-*are considered as oriented cohomology theories in the sense of [PS1 ] with orientations given by the Thom class thMGL 1 from 1.1.3 and the Chern structure L=X 7! [O] - [L-1]. In particular, the composition a7!a 1 0 b c7!'(b).c MGL 0(k)____________//MGL (k) Z ____________//Z sends the class [X] 2 MGL 0(X) of a smooth projective k-variety X to the Euler characteristic O(X, OX ) of the structure sheaf OX . Proof. The first part is already proven. To prove the second part, consider the orientations thMGL and thK from 1.1.3. Note that by the very definition of ' it sends thMGL to thK . Thus it respects the Chern structures on MGL * and BGL *described in Section 2. The quotient map BGL *! KTT-*takes the Bott element fi to (-1). Thus it takes the Chern structure on BGL *to the Chern structure on KTT-*given by L=X 7! [O] - [L-1] 2 K0(X). This shows that ~~' :MGL *(-) MGL 0(k)Z* ! KTT-*respects the orientations described in the Theorem 2.3.3. 14 Let f 7! fMGL resp. f 7! fK be the integrations on MGL *resp. KTT-*given by these Chern structures via Theorem [PS3 , Thm. 4.1.4]. By Theorem [PS2 , Thm. 1.1.10] the composition MGL * ! BGL * ! KTT-*respects the integrations on MGL * and KTT-*since it preserves the Chern structures. In particular, given a smooth projective S-scheme f :X ! Spec(k), the diagram '~~ MGL 0(X) MGL 0(k)Z0(X) ____//_KTT0(X) fMGL|| fK|| fflffl| '~~ fflffl| MGL 0(X) MGL 0(k)Z0(k) ______//KTT0(k) commutes where fMGL and fK are the push-forward maps for MGL * and KTT-*respectively. The integration f 7! fK on KTT-*respecting the Chern structure L 7! [O] - [L-1] coincides with the one given by the higher direct images by Theorem [PS2 , Thm. 1.1.11]. The last one sends the class [V ] 2 K0(X) of a vector bundle V over a smooth projective variety X to the Euler characteristic O(X, V) of the sheaf V of sections of V . Recall that for an oriented cohomology theory A with a Chern structure L 7! c(L) and for a smooth projective variety f :X ! Spec (k) its class [X]A 2 Aeven(Spec (k)) is defined as fA (1), where fA :A(X) ! A(Spec (k)) is the push-forward morphism respecting the Chern structure (see [PS3 , Thm. 4.1.4]). The notation fA is misleading, since it depends on the Chern structure. 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