Motivic Landweber exactness Niko Naumann, Markus Spitzweck, Paul Arne Ostvaer Contents 1 Introduction 2 2 Preliminaries on algebraic stacks 6 3 Conventions 8 4 Homology and cohomology theories 9 5 Tate objects and flat Hopf algebroids 13 6 The stacks of topological and algebraic cobordism 14 6.1 The algebraic stack of MU . . . . . . . . . . . . . . . . . . . . . .* * . . . . 14 6.2 The algebraic stack of MGL . . . . . . . . . . . . . . . . . . . . .* * . . . . 14 6.3 Formal groups and stacks . . . . . . . . . . . . . . . . . . . . . . * *. . . . 20 6.4 A map of stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . 22 7 Landweber exact theories 23 8 Representability and base change 26 9 Operations and cooperations 30 10 A Chern character 36 1 1 Introduction The Landweber exact functor theorem combined with Brown representability provid* *es an almost unreasonably efficient toolkit for constructing homotopy types out of* * purely algebraic data. Among the many examples arising this way is the presheaf of el* *liptic homology theories on the moduli stack of elliptic curves. In this paper we inci* *te the use of such techniques in the algebro-geometric setting of motivic homotopy theory. In what follows we shall state some of the main results in the paper, commen* *t on the proofs and discuss some of the background and relation to previous works. Throu* *ghout we employ a stacky viewpoint of the subject which originates with formulations * *in stable homotopy theory pioneered by Morava and Hopkins. Let S be a regular noetherian * *base scheme of finite Krull dimension and SH (S) the corresponding motivic stable ho* *motopy category. A complex point Spec(C) ! S induces a functor SH (S) ! SH to the cla* *ssical stable homotopy category. Much of the work in this paper is guidelined by the p* *opular quest of hoisting results in SH to the more complicated motivic category. To set the stage, denote by MGL the algebraic cobordism spectrum introduced* * by Voevodsky [29]. By computation we show (MGL *, MGL *MGL ) is a flat Hopf algeb* *roid in Adams graded Abelian groups. (Our standard conventions concerning graded objec* *ts are detailed in Section 3. Recall that MGL * MGL 2*,*.) The useful fact that * *MGL gives rise to an algebraic stack [MGL *=MGL *MGL ] in the formulation introduced by* * the first author comes to bear. (This apparatus is reviewed in Section 2.) By comparing w* *ith the complex cobordism spectrum MU we deduce a 2-categorical commutative diagra* *m: Spec (MGL *)________//Spec(MU *) (1) | | | | fflffl| fflffl| [MGL *=MGL *MGL_]__//_[MU *=MU *MU ] The right hand part of the diagram is well-known: Milnor's computation of MU * * *and Quillen's identification of the canonical formal group law over MU * with the u* *niversal formal group law are early success stories in modern algebraic topology. As a G* *m-stack the lower right hand corner identifies with the moduli stack of strict graded f* *ormal groups. Our plan from the get-go was to prove (1) is cartesian and use that des* *cription of the algebraic cobordism part of the diagram to deduce motivic analogs of the* *orems in stable homotopy theory. It turns out this strategy works for general base sc* *hemes. 2 Recall that an MU *-module M* is Landweber exact if v(p)0, v(p)1, . .f.orms * *a regular sequence in M* for every prime p. Here v(p)0= p and the v(p)ifor i > 0 are inde* *composable elements of degree 2pi-2 in MU *with Chern numbers divisible by p. Using the ca* *rtesian diagram (1) we show the following result for Landweber exact motivic homology t* *heories, see Theorem 7.3 for a more precise statement. Theorem: Suppose A* is a Landweber exact graded MU *-algebra. Then MGL **(-) MU *A* is a bigraded ring homology theory on SH (S). Using the theorem we deduce that MGL **(-) MU *A* is a ring cohomology theory on the subcategory of strongly dualizable objects o* *f SH (S). In the case of the Laurent polynomial ring Z[fi, fi-1] on the Bott element, thi* *s observation is part of the proof in [26] of the motivic Conner-Floyd isomorphism ~= MGL **(-) MU *Z[fi, fi-1]_//_KGL** for the motivic spectrum KGL representing homotopy algebraic K-theory. Define the category of Tate objects SH (S)T as the smallest localizing trian* *gulated subcategory of the motivic stable homotopy category containing the set T of all* * mixed motivic spheres Sp,q Sp-qs^ Gqm. The Tate objects are precisely the cellular spectra in the terminology of [7]. * *Our choice of wording is deeply rooted in the theory of motives. Since the inclusion SH (S)T * * SH (S) preserves sums and SH (S) is compactly generated the inclusion acquires a right* * adjoint pSH (S),T:SH(S) ! SH (S)T called the Tate projection. When E is a Tate object a* *nd F a motivic spectrum there is thus an isomorphism E**(F) ~=E**(pSH (S),TF). As in topology, it follows that the E**-homology of F is determined by the E**-* *homology of mixed motivic spheres. This observation is a key input in showing (E*, E*E) * *is a flat 3 Hopf algebroid in Adams graded Abelian groups provided one - and hence both - o* *f the canonical maps E** ! E**E is flat and the canonical map E*E E* E** ! E**E is an isomorphism. Specializing to the example of algebraic cobordism allows us to fo* *rm the algebraic stack [MGL *=MGL *MGL ] and (1). Our motivic analog of Landweber's exact functor theorem takes the following * *form, see Theorem 8.6. Theorem: Suppose M* is an Adams graded Landweber exact MU *-module. Then there exists a motivic spectrum E in SH (S)T and a natural isomorphism E**(-) ~=MGL **(-) MU *M* of homology theories on SH (S). In addition, if M* is a graded MU *-algebra then E acquires a quasi-multipli* *cation which represents the ring structure on the corresponding Landweber exact theory. When the base scheme is the integers we use motivic Landweber exactness and * *the fact that SH (Z) is a Brown category, so that all homology theories are represe* *ntable, to conclude the proof of the motivic exact functor theorem. For a general base * *scheme we provide base change results which allow us to reduce to the case of the inte* *gers. The derived category of modules over MGL - relative to Z - turns also out to be a * *Brown category. This suffices to show the above remains valid when translated verbati* *m to the setting of highly structured MGL -modules. Recall MGL is a motivic symmetric * *spectrum and the monoid axiom introduced in [25] holds for the motivic stable structure * *according to [15, Proposition 4.19]. Thus the modules over MGL acquire a closed symmet* *ric monoidal model structure. Moreover, for every cofibrant replacement of MGL the* *re is an induced Quillen equivalence of modules. We wish to emphasize the close connection between our results and the classi* *cal Landweber exact functor theorem. In particular, if M* is concentrated in even d* *egrees there exists a commutative ring spectrum ETopin SH which represents the corresp* *onding topological Landweber exact theory. Although E and ETopare objects in wildly di* *fferent categories of spectra, it turns out there is an isomorphism E**E ~=E** ETop*ETop*ETop. The last section of the paper describes (co)operations and phantom maps betw* *een Landweber exact motivic spectra. We use a spectral sequence argument to show th* *at 4 every MGL -module E gives rise to a surjection Ep,q(M)____//_Homp,qMGL**(MGL **M, E**). (2) The kernel of (2) identifies with the Ext-term Ext1,(p-1,q)MGL**(MGL **M, E**). (3) Imposing the assumption that ETop*ETop be a projective ETop*-module implies the* * given Ext-term in (3) vanishes, and hence (2) is an isomorphism. The assumption on ET* *opholds for unitary topological K-theory KU and localizations of Johnson-Wilson theori* *es. By way of example we compute the KGL -cohomology of KGL . That is, with the comple* *ted tensor product, there is an isomorphism of KGL **-algebras ~= ** * KGL **KGL ____//_KGL bKU*KU KU. By [2] the group KU 1KU is trivial and KU 0KU is uncountable [2]. We also show * *that KGL does not support any nontrivial phantom maps. Adopting the proof to SH reprove* *s the analogous result for KU . These techniques can also be utilized to show there i* *s a Chern character in SH (S) from KGL to the periodized rational motivic Eilenberg-MacL* *ane spectrum PM Q representing rational motivic cohomology. In the arguments we use (semi)model structures on E1 -algebras, but these can be skipped when restricte* *d to a smooth base over a field on account of the isomorphism between MQ and the Landw* *eber theory for the additive formal group law over the rationals. Inspired by the results herein we make some rather speculative remarks conce* *rning future works. The all-important chromatic approach to stable homotopy theory ac* *quires deep interplays with the algebraic geometry of formal groups. Landweber exact a* *lgebras over Hopf algebroids represent a central theme in this endeavor, leading for ex* *ample to the bicomplete closed symmetric monoidal abelian category of BP *BP-comodule* *s. The techniques in this paper furnish a corresponding Landweber exact motivic Br* *own- Peterson spectrum MBP equivalent to the constructions in [14] and [28]. The * *object MBP *MBP and questions in motivic chromatic theory at large can be investigat* *ed along the lines of this paper. An exact analog of Bousfield's localization machinery * *in motivic stable homotopy theory was worked out in [24, Appendix A], cf. also [11] for a * *discussion of the chromatic viewpoint. In a separate paper we shall dispense with the regu* *larity assumption on S. The results in this paper remain valid for noetherian base sc* *hemes of finite Krull dimension. Since this generalization uses arguments which are o* *therwise independent of the present work, the details will appear elsewhere. 5 2 Preliminaries on algebraic stacks By a stack we shall mean a category fibered in groupoids over the site comprise* *d by the category of commutative rings endowed with the fpqc-topology. A stack X is alge* *braic if its diagonal is representable and affine, and there exists an affine scheme * *U together with a faithfully flat map U ! X, called a presentation of X. We refer to [10] * *and [20] for motivation and basic properties of these notions. Lemma 2.1: Suppose there are 2-commutative diagrams of algebraic stacks Z ____//_Z0 Y ____//_Y0 (4) | | | | | | ss| | fflffl|fflffl|fflffl| fflffl| X ____//_X0 X ____//_X0 where ss is faithfully flat. Then the left hand diagram in (4) is cartesian if * *and only if the naturally induced commutative diagram Z xX Y _____//Z0xX0Y0 (5) | | | | fflffl| fflffl| Y __________//Y0 is cartesian. Proof. The base change of the canonical 1-morphism c : Z ! Z0xX0X over X along * *ss identifies with the canonically induced 1-morphism Z xX Y _cx1_//(Z0xX0X) xX Y ~=Z0xX0Y ~=(Z0xX0Y0) xY0Y. This is an isomorphism provided (5) is cartesian; hence so is c x 1. By faithfu* *lly flatness * * __ of ss it follows that c is an isomorphism. The reverse implication holds trivia* *lly. |__| Corollary 2.2: Suppose X and Y algebraic stacks, U ! X and V ! Y presentations and there is a 2-commutative diagram: U ____//_V (6) | | | | fflffl|fflffl| X ____//_Y 6 Then (6) is cartesian if and only if one - and hence both - of the commutative * *diagrams (i = 1, 2) U xX U ____//_V xY V (7) pri|| |pri| fflffl| fflffl| U __________//V is cartesian. * * __ Proof. Follows from Lemma 2.1 since presentations are faithfully flat. * * |__| A presentation U ! X yields a Hopf algebroid or cogroupoid object in commuta* *tive rings ( (OU), (OUxXU )). Conversely, if (A, B) is a flat Hopf algebroid, deno* *te by [Spec(A)=Spec(B)] the associated algebraic stack. We note that by [20, Theorem * *8] there is an equivalence of 2-categories between flat Hopf algebroids and rigidified a* *lgebraic stacks. Let QcX denote the category of quasi-coherent OX-modules and A 2 QcX a monoi* *d, or quasi-coherent sheaf of OX-algebras. If X0 is a scheme and ss : X0 ! X faith* *fully flat, then A is equivalent to the datum of the OX0-algebra A(X0) ss*A combined with* * a descent datum with respect to X1 X0 xX X0 _____////_X0. When X0 = Spec(A) is* * affine, X1 = Spec( ) is affine, (A, ) a flat Hopf algebroid and A(X0) a -comodule alg* *ebra. There is an evident adjunction between left A-modules in Qc X and left A(X0)- modules in Qc X0: ss* : A - mod___//_A(X0)o-omod_: ss* Since ss* has an exact left adjoint ss* it preserves injectives and there are i* *somorphisms ExtnA(M, ss*N ) ~=ExtnA(X0)(ss*M, N ) (8) between Ext-groups in the categories of quasi-coherent left A- and A(X0)-module* *s. Now assume i : U ,! X is the inclusion of an open algebraic substack. Then * *[20, Propositions 20, 22] imply i* : QcU ,! QcX is an embedding of a thick subcatego* *ry; see also [20, section 3.4] for a discussion of the functoriality of Qc X with respe* *ct to X. For F, G 2 QcU the Yoneda description of Ext-groups gives isomorphisms ExtnA(A OX i*F, A OX i*G) ~=Extni*A(i*A OU F, i*A OU G). (9) We will need the following result. 7 Proposition 2.3: Suppose there is a 2-commutative diagram of algebraic stacks X0|>> ff___|_ __ |ss __ f | X ______|_____//_Y |BB | - | | BBB | --- | | BB | -- | |fX B!fflffl|!|fY""-- ssX| X ssY| | _==aaBB | |iX___ BiYBB| | __ BB | fflffl|.ffi__Oifflffl|"0BP U ____________//_U0 where X, Y , X0 are schemes, ss, ssX , ssY faithfully flat, and iX , iY (hence * *also i) open inclusions of algebraic substacks. If ss*YssY,*OY 2 QcY is projective then ExtnA(X0)(A(X0) OX0 ss*fY,*OY , A(X0) OX0 ff*OX ) ae ~= 0 n 1, Hom OY (ss*YssY,*OY , A(Y ) OY f*OXn)= 0. Proof. By (8) the group ExtnA(X0)(ss*(A OX fY,*OY ), A(X0) OX0 ff*OX ) is isomo* *rphic to ExtnA(A OX fY,*OY , ss*(ss*A OX0 ff*OX )), which the projection formula ident* *ifies with ExtnA(A OX iY,*ssY,*OY , A OX iY,*i*ssX,*OX ). By (9) the latter Ext-groups i* *s isomorphic to Extni*YA(i*YA OU0ssY,*OY , i*YA OU0i*ssX,*OX ). Replacing i*ssX,*OX by ssY* *,*f*OX and applying (8) gives an isomorphism to ExtnA(Y()ss*Y(i*YA OU0ssY,*OY ), A(Y ) O* *Y f*OX ) = ExtnA(Y()A(Y ) OY ss*YssY,*OY , A(Y ) OY f*OX ). Now A(Y ) OY ss*YssY,*OY is a * *projective left A(Y )-module by the assumption on ss*YssY,*OY . Hence the Ext-term vanishe* *s in every positive degree, while for n = 0, Hom A(Y )(A(Y ) OY ss*YssY,*OY , A(Y ) OY f*OX ) ~=Hom OY (ss*YssY,*OY , A(Y ) * *OY f*OX ). __ |_* *_| 3 Conventions The category of graded objects in an additive tensor category A refers to integ* *er-graded objects subject to the Koszul sign rule x y = (-1)|x||y|y x. However, A will of* *ten have 8 a supplementary graded structure. The category of Adams graded objects in A ref* *ers to integer-graded objects in A, but no sign rule for the tensor product is intr* *oduced as a consequence of the Adams grading. It is helpful to think of the Adams grading a* *s being even. We will deal with graded Abelian groups, Adams graded graded Abelian grou* *ps, or Z2-graded Abelian groups with a sign rule in the first but not in the second* * variable, and Adams graded Abelian groups. For an Adams graded graded Abelian group A**, we define Ai A2i,iand let A* denote the corresponding Adams graded Abelian gro* *up. The smash product induces a closed symmetric monoidal structure on SH (S). We denote the internal function spectrum from E to F by Hom__(E, F) and the tensor* * unit or sphere spectrum by 1. The dual of E is by definition E_ Hom__(E, 1). Note tha* *t E**with the usual indexing is an Adams graded graded Abelian group. Let Ei be short for* * E2i,i. When E is a ring spectrum, i.e. a commutative monoid in SH (S), we implicitly a* *ssume E**is a commutative monoid in Adams graded graded Abelian groups. This latter h* *olds true for orientable ring spectra [14, Proposition 2.16] in view of [19, Theorem* * 3.2.23]. It is convenient to view evenly graded MU *-modules to be Adams graded. In * *this case we will implicitly divide the grading by 2. 4 Homology and cohomology theories An object F of SH (S) is called finite (another term is compact) if Hom SH (S)(* *F, -) respects sums. Using the 5-lemma one shows the subcategory of finite objects SH* * (S)f of SH (S) is thick [12, Definition 1.4.3(a)]. For a set R of objects in SH (S)f* *let SH (S)R,f denote the smallest thick triangulated subcategory of SH (S)fcontaining R and S* *H (S)R the smallest localizing subcategory of SH (S) containing R [12, Definition 1.4.* *3(b)]. The examples we will deal with are the sets of mixed motivic spheres T , the se* *t of (isomorphism classes of) strongly dualizable objects D and the set SH (S)f. Remark 4.1: According to [7, Remark 7.4] SH (S)T SH (S) is the full subcatego* *ry of cellular motivic spectra introduced in loc. cit. Recall F 2 SH (S) is strongly dualizable if for every G 2 SH (S) the canonic* *al map F_ ^ G_____//Hom_(F, G) is an isomorphism. A strongly dualizable object is finite since 1 is finite. 9 Lemma 4.2: SH (S)D,fis the full subcategory of SH (S)f of strongly dualizable o* *bjects of SH (S). Proof. Since D is stable under cofiber sequences and retracts, every object of * *SH (S)D,f * * __ is strongly dualizable. * * |__| Lemma 4.3: SH (S)R,fis the full subcategory of compact objects of SH (S)R and t* *he latter is compactly generated. Proof. Note SH (S)R is compactly generated since SH (S) is so [21, Theorem 2.1,* * 2.1.1]. If (-)c indicates a full subcategory of compact objects [21, Theorem 2.1, 2.1.3* *] implies SH (S)cR= SH (S)R \ SH (S)c = SH (S)R \ SH (S)f. Hence it suffices to show SH (S)R \ SH (S)f= SH (S)R,f. The inclusion " " is ob* *vious and to prove " " let R0be the smallest set of objects closed under suspension, * *retract and cofiber sequences containing R. Then R0 SH (S)f and SH (S)R,f= SH (S)R0,f SH (S)f, SH (S)R = SH (S)R0. By applying [21, Theorem 2.1, 2.1.3] to R0it follows that SH (S)R \ SH (S)f= SH (S)R0 \ SH (S)f= R0 SH (S)R0,f= SH (S)R,f. __ |_* *_| Corollary 4.4: If R R0are as above, the inclusion SH (S)R SH (S)R0 has a ri* *ght adjoint pR,R0. Proof. Since SH (S)R is compactly generated and the inclusion preserves sums th* *e claim * * __ follows from [21, Theorem 4.1]. * * |__| Definition 4.5: The Tate projection is the functor pSH (S)f,T: SH (S)__//_SH(S)T . Lemma 4.6: In the situation of Corollary 4.4, the right adjoint pR0,Rpreserves * *sums. Proof. Using [21, Theorem 5.1] it suffices to show that SH (S)R SH (S)R0 pres* *erves compact objects. Hence by Lemma 4.3 we are done provided SH (S)R,f SH (S)R0,f. * * __ Clearly this holds since R R0. * * |__| 10 Lemma 4.7: Suppose R as above contains T . Then pR,T : SH (S)R____//_SH(S)T is an SH (S)T -module functor. Proof. Let ' : SH (S)T ! SH (S)R be the inclusion and F 2 SH (S)T , G 2 SH (S)R* * . Then the counit of the adjunction between ' and pR,T yields the canonical map '(F ^ pR,T(G)) ~='(F) ^ '(pR,T(G))__//_'(F) ^ G, that is adjoint to F ^ pR,T(G)____//_pR,T('(F) ^ G). (10) We claim (10) is an isomorphism for all F, G. In effect, the full subcategory o* *f SH (S)T generated by the objects F for that (10) is an isomorphism for all G 2 SH (S)R * *is easily seen to be localizing, and hence we may assume F = Sp,qfor p, q 2 Z. The sphere* * Sp,qis invertible, so SH (S)T (-, pR,T('(Sp,q) ^ G)) ~=SH (S)R ('(-), Sp,q^ G) is isom* *orphic to SH (S)R ('(-)^S-p,-q, G) ~=SH (S)T (-^S-p-q, pR,T(G)) ~=SH (S)T (-, Sp,q^pR,T(G* *)). * * __ This shows pR,T('(Sp,q) ^ G) and Sp,q^ pR,T(G) are isomorphic, as desired. * * |__| Remark 4.8: (i) For every G 2 SH (S) the counit pR,T(G) ! G, where ' is omit* *ted from the notation, is an ss**-isomorphism. Using pSH (S),Trather than the * *cellular functor introduced in [7] refines Proposition 7.3 of loc. cit. (ii)If E 2 SH (S)T and F 2 SH (S) then Ep,q(F) ~=Ep,q(pSH (S),T(F)) on account* * of the isomorphisms between SH (S)(Sp,q, E ^ F) and SH (S)T (Sp,q, pSH (S),T(E ^ F)) ~=SH (S)T (Sp,q, E ^ pSH (S),T(F)* *). In [7] it is argued that most spectra should be non-cellular. On the other* * hand, the E-homology of F agrees with the E-homology of some cellular spectrum. We n* *ote that many conspicuous motivic (co)homology theories are representable by c* *ellular spectra: Landweber exact theories, including algebraic cobordism and homo* *topy algebraic K-theory, and also motivic (co)homology over fields of character* *istic zero according to work of Hopkins and Morel. Definition 4.9: A homology theory on a triangulated subcategory T of SH (S) is a homological functor T ! Ab which sends sums to sums. Dually, a cohomology theo* *ry on T is a homological functor Top! Ab which sends sums to products. 11 Lemma 4.10: Suppose R D is closed under duals. Then every homology theory on SH (S)R,fextends uniquely to a homology theory on SH (S)R . Proof. In view of Lemma 4.3 we can apply [12, Corollary 2.3.11] which we refer * *to for a * * __ more detailed discussion. * * |__| Homology and cohomology theories on SH (S)D,fare interchangeable according to the categorical duality equivalence SH (S)opD,f~=SH(S)D,f. The same holds for e* *very R for which SH (S)R,fis contained in SH (S)D,fand closed under duality, e.g. SH (* *S)T ,f. We shall address the problem of representing homology theories on SH (S) in Sec* *tion 8. Cohomology theories are always defined on SH (S)f unless specified to the contr* *ary. Definition 4.11: Let T SH (S) be a triangulated subcategory closed under the * *smash product. A multiplicative or ring (co)homology theory on T, always understood * *to be commutative, is a (co)homology theory E on T together with maps Z ! E(S0,0) and E(F) E(G) ! E(F ^ G) which are natural in F, G 2 T. These maps are subject to* * the usual unitality, associativity and commutativity constraints [27, pg. 269]. Ring spectra in SH (S) give rise to ring homology and cohomology theories. W* *e shall use the following bigraded version of (co)homology theories. Definition 4.12: Let T SH (S) be a triangulated subcategory closed under shif* *ts by all mixed motivic spheres Sp,q. A bigraded homology theory on T is a homological fu* *nctor from T to Adams graded graded abelian groups taking sums to sums together with natural isomorphisms (X)p,q~= ( 1,0X)p+1,q and (X)p,q~= ( 0,1X)p,q+1 for all p and q such that the diagram (X)p,q________// ( 1,0X)p+1,q | | | | fflffl| fflffl| ( 0,1X)p,q+1____//_ ( 1,1X)p+1,q+1 commutes. Bigraded cohomology theories are defined likewise. 12 It is clear that a (co)homology theory on T is the same as a bigraded (co)ho* *mology theory on T. 5 Tate objects and flat Hopf algebroids As in stable homotopy theory, we wish to associate a flat Hopf algebroid with s* *uitable motivic ring spectra. By a Hopf algebroid we shall mean a cogroupoid object in* * the category of commutative rings over either Abelian groups, Adams graded Abelian * *groups or Adams graded graded Abelian groups. Throughout this section we assume E is a* * ring spectrum in SH (S)T . We call E**flat provided one - and hence both - of the ca* *nonical maps E**! E**E is flat, and similarly for E* and E* ! E*E. Lemma 5.1: (i) If E** is flat then for every motivic spectrum F the canonica* *l map E**E E**E**F_____//(E ^ E ^ F)** is an isomorphism. (ii)If E* is flat and the canonical map E*E E* E** ! E**E is an isomorphism, * *then for every motivic spectrum F the canonical map E*E E*E*F ____//_(E ^ E ^ F)* is an isomorphism. Proof. (i): By Lemma 4.7, replacing F by its Tate projection we may assume that* * F is a Tate object. The proof follows now along the same lines as in topology by fir* *st noting that the statement clearly holds when F is a mixed motivic sphere, and secondly* * that we are comparing homology theories on SH (S)T which respect sums. (ii): The t* *wo assumptions imply the assumption of (i), so there is an isomorphism E**E E**E**F ____//_(E ^ E ^ F)**. By the second assumption the left hand side identifies with (E*E E*E**) E**E**F ~=E*E E*E**F. * * __ Restricting to bidegrees which are multiples of (2, 1) yields the claimed isomo* *rphism. |__| 13 Corollary 5.2: (i)If E** is flat then (E**, E**E) is canonically a flat Hopf * *algebroid in Adams graded graded Abelian groups and for every F 2 SH (S) the module * *E**F is an (E**, E**E)-comodule. (ii)If E* is flat and the canonical map E*E E* E** ! E**E is an isomorphism, * *then (E*, E*E) is canonically a flat Hopf algebroid in Adams graded Abelian gro* *ups and for every F 2 SH (S) the modules E**F and E*F are (E*, E*E)-comodules. The second part of Corollary 5.2 is really a statement about Hopf algebroids: Lemma 5.3: Suppose (A**, **) is a flat Hopf algebroid in Adams graded graded A* *belian groups and the natural map * A* A** ! ** is an isomorphism. Then (A*, *) h* *as the natural structure of a flat Hopf algebroid in Adams graded Abelian groups, * *and for every comodule M** over (A**, **) the modules M** and M* are (A*, *)-comodule* *s. 6 The stacks of topological and algebraic cobordism 6.1 The algebraic stack of MU Denote by FG_the moduli stack of one-dimensional commutative formal groups [20]* *. It is algebraic and a presentation is given by the canonical map FGL ! FG_, where * *FGL is the moduli scheme of formal group laws. The stack FG_carries a canonical line b* *undle ! and [MU *=MU *MU ] is equivalent to the corresponding Gm-torsor FG_sover FG_. 6.2 The algebraic stack of MGL In this section we first study the (co)homology of finite Grassmannians over re* *gular noetherian base schemes of finite Krull dimension. Using this computational inp* *ut we relate the algebraic stacks of MU and MGL . A key result is the isomorphism MGL **MGL ~=MGL ** MU *MU *MU . If S is the spectrum of a field this can easily be extracted from [6, Theorem 5* *]. Since it is crucial for the following, we will give a rather detailed argument for the gene* *ralization. We recall the notion of oriented motivic ring spectra formulated by Morel [1* *8], cf. [14], [23] and [28]: If E is a motivic ring spectrum, the unit map 1 ! E yields a cl* *ass 14 1 2 E0,0(1) and hence by smashing with the projective line a class c1 2 E2,1(P1* *). An orientation on E is a class c1 2 E2,1(P1 ) that restricts to c1. Note that KGL * * and MGL are canonically oriented. For 0 d n define the ring Rn,d Z[x1, . .,.xn-d]=(sd+1, . .,.sn), (11) where si is given by 1X 1 + sntn (1 + x1t + x2t2 + . .+.xn-dtn-d)-1 inZ[x1, . .,.xn-d][[t]]* *x. n=1 By assigning weight i to xievery sk 2 Z[x1, . .,.xk] is homogeneous of degree k* *. In (11), sj = sj(x1, . .,.xn-d, 0, . .).by convention when d + 1 i n. We note that R* *n,dis a free Z-module of rank nd. For every sequence a_= (a1, . .,.ad) subject to the ineq* *ualities n - d a1 a2 . . .ad 0, we set: 0 1 xa1 xa1+1 . . .xa1+d-1 B x x . . .x C a_ detBB a2-1 a2 a2+d-2CC @ . . . . . .. . . . . .A xad-d+1 . . .. . . xad Here x0 1 and xi 0 for i < 0 or i > n - d. The Schur polynomials { a_} for* *m a basis for Rn,das a Z-module. Let ss : Rn+1,d+1! Rn,d+1be the unique surjective* * ring homomorphism where ss(xi) = xi for 1 i n - d - 1 and ss(xn-d) = 0. It is ea* *sy to see that ss( a_) = a_if a1 n - d - 1 and ss( a_) = 0 for a1 = n - d. Hence t* *he kernel of ss is the principal ideal generated by xn-d. That is, ker(ss) = xn-d . Rn+1,d+1. (12) Moreover, let ' : Rn,d! Rn+1,d+1be the unique monomorphism of abelian groups su* *ch that for every a_, '( a_) = a_0where a_0= (n - d, a_) (n - d, a1, . .,.ad). * *Clearly we get im (') = ker(ss). (13) Note that ' is a map of degree n - d. We will also need the unique ring homomor* *phism f : Rn+1,d+1! Rn,d= Rn+1,d+1=(sd+1) where f(xi) = xifor all 1 i n-d. Elemen* *tary matrix manipulations establish the equalities f( (a1,...,ad,0)) = (a1,...,ad) (14) 15 and '( (a1,...,ad)) = xn-d . (a1,...,ad,0). (15) Next we discuss some geometric constructions involving Grassmannians. For 0 d n, denote by Gr n-d(An)=Z the scheme parameterizing subvector bu* *ndles of rank n-d of the trivial rank n bundle such that the inclusion of the subbund* *le is locally split. Similarly, G(n, d)=Z denotes the scheme parameterizing locally free quot* *ients of rank d of the trivial bundle of rank n; clearly G(n, d) ~=Gr n-d(An). It is kno* *wn that G(n, d)=Z is smooth of relative dimension d(n - d) and if n ____//_ ____//_ 0_____//Kn,d___//OG(n,d) Qn,d 0 (16) is the universal short exact sequence of vector bundles on G(n, d), letting K0n* *,ddenote the dual of Kn,d, the tangent bundle is given by TG(n,d)=Z~=Qn,d K0n,d. (17) The map " n+1 i : G(n, d) ~=Gr n-d(An)O_//_Grn-d(A ) ~=G(n + 1, d + 1) classifying Kn,d OnG(n,d),! On+1G(n,d)is a closed immersion. From (17) it fol* *lows that the normal bundle N (i) of i identifies with Kn,d. Next consider the compositi* *on on G(n + 1, d + 1) ff : OnG(n+1,d+1)O/"/_On+1G(n+1,d+1)//_Qn+1,d+1 for the inclusion into the first n factors. The complement of the support of c* *oker(ff) is an open subscheme U G(n + 1, d + 1) and there is a map ss : U ! G(n, d + 1) classifying ff|U. It is easy to see that ss is an affine bundle of dimension d,* * and hence ss is a motivic weak equivalence. (18) An argument with geometric points reveals that U = G(n + 1, d + 1) \ i(G(n, d))* *. We summarize the above with a diagram: i" ` ss G(n, d)O___//G(n + 1, d +o1)o?U_____//_G(n, d + 1). (19) With these precursors out of the way we are ready to compute the (co)homology o* *f finite Grassmannians with respect to any oriented motivic ring spectrum. 16 For every 0 d n there is a unique morphism of E**-algebras 'n,d: E** Z R* *n,d! E**(G(n, d)) such that 'n,d(xi) = chi(Kn,d) for 1 i n - d. This follows fr* *om (16) and the standard calculus of Chern classes in E-cohomology. Note that 'n,dis bi* *graded if we assign degree (2i, i) to xi2 Rn,d. Proposition 6.1: For 0 d n the map of E**-algebras 'n,d: E** Z Rn,d____//E**(G(n, d)) is an isomorphism. Proof. First observe the result holds when d = 0 and d = n since then G(n, d) =* * S. By induction it suffices to show that if 'n,dand 'n,d+1are isomorphisms, then s* *o is 'n+1,d+1. To that end we contemplate the diagram: fi ** E*-2r,*-r(G(n,Od))_ff__//E**(G(nO+O1,Od + 1))_//_E (G(n,OdO+ 1)) (20) 'n,d(-2r,-r)~=|| 'n+1,d+1|| 'n,d+1~=|| | 1 ' | 1 ss | (E** Z Rn,d)(-2r, -r)______//_E** Z Rn+1,d+1_____//_E** Z Rn,d+1 Here r codim(i) = n - d and (-2r, -r) indicates a shift. The top row is part * *of the long exact sequence in E-cohomology associated with (19) using the Thom isomorp* *hism E*+2r,*+r(Th (N (i))) ~=E**(G(n, d)) and the fact that E**(U) ~=E**(G(n, d + 1)* *) by (18). The lower sequence is short exact by (13). Since Kn+1,d+1|U ~=ss*(Kn,d+1) OU * *we get fi('n+1,d+1(xi)) = fi(chi(Kn+1,d+1)) = chi(Kn+1,d+1|U) = ss*(chi(Kn,d+1)) = 'n,* *d+1(1 ss(xi)). Therefore, the right hand square in (20) commutes, fi is surjective an* *d the top row in (20) is short exact. Next we study the Gysin map ff. Since i*(Kn+1,d+1) = Kn,dthere is a cartesian square of projective bundles: 0 P(Kn,d O) __i_//_P(Kn+1,d+1 O) p|| || fflffl| fflffl| G(n, d)___i___//_G(n + 1, d + 1) By the induction hypothesis 'n,dis an isomorphism. Thus the projective bundle t* *heorem gives Xr E**(P(Kn,d O)) ~=(E** Z Rn,d)[x]=(xr+1+ (-1)i'n,d(xi)xr+1-i), i=1 17 where x ch1(OP(Kn,d O)(1)) 2 E2,1(P(Kn,d O)). Similarly, Xr E**(P(Kn+1,d+1 O)) ~=E**(G(n + 1, d + 1))[x0]=(x0r+1+ (-1)i'n+1,d+1(x0i)x0* *r+1-i), i=1 where x0 ch1(OP(Kn+1,d+1 O)(1)) and x0i= chi(Kn+1,d+1) 2 Rn+1,d+1. (We denot* *e the canonical generators of Rn+1,d+1by x0iin order to distinguish them from xi 2 Rn* *,d.) Recall the Thom class of Kn,d~=N (i) is constructed from Xr th chr(p*(Kn,d) OP(Kn,d O)(1)) = xr + (-1)i'n,d(xi)xr-i2 E2r,r(P(Kn,d * *O)). i=1 Using i0*(x0) = x and i*('n+1,d+1(x0i)) = 'n,d(xi) for 1 i r, we get that Xr "th x0r+ (-1)i'n+1,d+1(x0i)x0r-i2 E2r,r(P(Kn+1,d+1 O)) i=1 0* satisfies i (t"h) = th, and if z : G(n+1, d+1) ! P(Kn+1,d+1 O) denotes the zer* *o-section, then z*(t"h) = (-1)n-d'n+1,d+1(x0n-d) 2 E2(n-d),n-d(G(n + 1, d + 1)).(21) Moreover, since i*(Kn+1,d+1) = Kn,dwe conclude that E**(i) O 'n+1,d+1= 'n,dO (1 f). (22) By inspection of the construction of the Thom isomorphism it follows that ff O E**(i) equals multiplication byz*(t"h). (23) Now for every partition a_as above we compute (14) (22) ** ff O 'n,d( a_) = ff O 'n,dO (1 f)( (a_,0)) = ff O E (i) O 'n+1,d+1(* * (a_,0)) (23)* (21) n-d 0 = z (t"h) . 'n+1,d+1( (a_,0)) = 'n+1,d+1((-1) xn-d . (a_,0)) (15) n-d = (-1) . 'n+1,d+1((1 ')( a_)). This verifies that the left hand square in (20) commutes up to a sign. Hence, * *by the * *__ 5-lemma, 'n+1,d+1is an isomorphism. |* *__| 18 Since 1+G(n, d) 2 SH (S) is dualizable and E is oriented we see that for al* *l 0 d n the Kronecker product E**(G(n, d)) E**E**(G(n, d))__//_E** (24) is a perfect pairing of finite free E**-modules. Proposition 6.2: (i) E**(BGL d) = E**[[c1, . .,.cd]] where ci2 E2i,i(BGL d)* * is the ith Chern class of the tautological rank d vector bundle. (ii) a) E**(BGL ) = E**[[c1, c2, . .].] where ci is the ith Chern class of t* *he universal bundle. b) E**(BGL ) = E**[fi0, fi1, . .].=(fi0 = 1) as E**-algebras where fii * *2 E2i,i(BGL ) is the image of the dual of ci12 E2i,i(BGL 1). (iii)There are Thom isomorphisms E**-modules E**(BGL )____//_E**(MGL ) and of E**-algebras E**(MGL )___//_E**(BGL ). Proof. Parts (i) and (ii)a) are clear from the above. From (24) we conclude the* *re are canonical isomorphisms E**(BGL d)____//_HomE**(E**(BGL d), E**), E**(BGL d)____//HomE**,c(E**(BGL d), E**). The notation Hom E**,crefers to continuous E**-linear maps with respect to the * *inverse limit topology on E**(BGL d) and the discrete topology on E**. From this, the * *proofs of * * __ parts (ii)b) and (iii) carry over verbatim from topology. * * |__| Corollary 6.3: (i)The tuple (MGL **, MGL **MGL ) is a flat Hopf algebroid in* * Adams graded graded Abelian groups. For every motivic spectrum F the module MGL * ***F is an (MGL **, MGL **MGL )-comodule. (ii)By restriction of structure the tuple (MGL *, MGL *MGL ) is a flat Hopf a* *lgebroid in Adams graded Abelian groups. For every motivic spectrum F the modules MGL * ***F and MGL *F are (MGL *, MGL *MGL )-comodules. 19 Proof. (i): We note MGL is a Tate object by [7, Theorem 6.4], Remark 4.1 and M* *GL **is flat by Proposition 6.2(iii) with E = MGL . Hence the statement follows from Co* *rollary 5.2(i). (ii): The bidegrees of the generators fiiin Proposition 6.2 are multipl* *es of (2, 1). * * __ This implies the assumptions in Corollary 5.1(ii) hold, and the statement follo* *ws. |__| The flat Hopf algebroid (MGL *, MGL *MGL ) gives rise to the algebraic stack [MGL *=MGL *MGL ]. Although the grading is not required for the definition, it defines a Gm-action* * on the stack and we may therefore form the quotient stack [MGL *=MGL *MGL ]=Gm. * * For F 2 SH (S), let F(F) be the Gm-equivariant quasi-coherent sheaf on [MGL *=MGL * * *MGL ] associated with the comodule structure on MGL *F furnished by Corollary 6.3(ii* *). Denote by F=Gm(F) the descended quasi-coherent sheaf on [MGL *=MGL *MGL ]=Gm. Lemma 6.4: (i) MGL **MGL ~=MGL ** MU *MU *MU ~=MGL **[b0, b1, . .].=(b0 = * *1). (ii)Let x, x0 be the images of the orientation on MGL with respect to the two* * natural P maps MGL *! MGL *MGL . Then x0= i 0bixi+1(where b0 = 1). Proof. Here biis the image under the Thom isomorphism of fiiin Proposition 6.2.* * Part (i) follows by comparing the familiar computation of MU *MU with our computati* *on of MGL **MGL . For part (ii), the computations leading up to [1, Corollary 6.8] ca* *rry over __ unchanged. |_* *_| 6.3 Formal groups and stacks A graded formal group over an evenly graded ring A* or more generally over an a* *lgebraic Gm-stack is a group object in formal schemes over the base with a compatible Gm* *-action such that locally in the Zariski topology it looks like Spf(R*[[x]]), as a form* *al scheme with Gm-action, where x has weight -1. (Note that every algebraic Gm-stack can be co* *vered by affine Gm-stacks.) This is equivalent to demanding that x has weight 0 (or a* *ny other fixed weight) by looking at the base change R ! R[y, y-1], y of weight 1. A str* *ict graded formal group is a graded formal group together with a trivialization of the lin* *e bundle of invariant vector fields with the trivial line bundle of weight 1. The strict gr* *aded formal group associated with the formal group law over MU * inherits a coaction of MU * **MU compatible with the grading and the trivialization; thus it descends to a stric* *t graded 20 formal group over FG_s. As a stack, FG_sis the moduli stack of formal groups w* *ith a trivialization of the line bundle of invariant vector fields, while as a Gm-sta* *ck it is the moduli stack of strict graded formal groups. It follows that FG_(with trivial G* *m-action) is the moduli stack of graded formal groups. For a Gm-stack X the space of Gm-m* *aps to FG_is the space of maps from the stack quotient X=Gm to FG_. Hence a graded * *formal group is tantamount to a formal group over X=Gm. An orientable theory gives rise to a strict graded formal group over the coe* *fficients: Lemma 6.5: If E 2 SH (S) is an oriented ring spectrum satisfying the assumptions in Corollary 5.2(ii) then the corresponding strict graded formal group over E* * *inherits a compatible E*E-coaction and there is a descended strict graded formal group o* *ver the stack [E*=E*E]. In particular, the flat Hopf algebroid (MGL *, MGL *MGL ) acqu* *ires a well defined strict graded formal group, [MGL *=MGL *MGL ] a strict graded formal * *group and the quotient stack [MGL *=MGL *MGL ]=Gm a formal group. Proof. Functoriality of E*(F) in E and F ensures the formal group over E* inher* *its an E*E-coaction. For example, compatibility with the comultiplication of the forma* *l group amounts to commutativity of the diagram: (E ^ E)*(P1 )_________//_(E ^ E ^ E)*(P1 ) | | | | fflffl| fflffl| (E ^ E)*(P1 x P1 )___//_(E ^ E ^ E)*(P1 x P1 ) All maps respect gradings, so there is a graded formal group over the Hopf alge* *broid. Different orientations yield formal group laws which differ by a strict isomorp* *hism, so there is an enhanced strict graded formal group over the Hopf algebroid. It ind* *uces a strict graded formal group over the Gm-stack [MGL *=MGL *MGL ] and quotientin* *g out by * * __ the Gm-action yields a formal group over the quotient stack. * * |__| For oriented motivic ring spectra E, F denote by '(E, F) the strict isomorph* *ism of formal group laws over (E ^ F)* from the pushforward of the formal group law ov* *er E* to the one of the formal group law over F* given by the orientations on E ^ F i* *nduced by E and F. Lemma 6.6: Suppose E, F, G are oriented spectra and let p: (E ^ F)* ! (E ^ F ^ * *G)*, q :(F ^ G)* ! (E ^ F ^ G)* and r :(E ^ G)* ! (E ^ F ^ G)* denote the natural ma* *ps. Then r*'(E, G) = p*'(E, F) O q*'(F, G). 21 Corollary 6.7: If E 2 SH (S) is an oriented ring spectrum and satisfies the ass* *umptions in Corollary 5.2(i), there is a map of Hopf algebroids (MU *, MU *MU ) ! (E**, * *E**E) such that MU * ! E** classifies the formal group law on E** and MU *MU ! E**E the s* *trict isomorphism '(E, E). If E satisfies the assumptions in Corollary 5.2(ii) then t* *his map factors through a map of Hopf algebroids (MU *, MU *MU ) ! (E*, E*E). The induc* *ed map of stacks classifies the strict graded formal group on [E*=E*E]. 6.4 A map of stacks Corollary 6.7 and the orientation of MGL furnish a map of flat Hopf algebroids (MU *, MU *MU )____//(MGL *, MGL *MGL ) such that the induced map of Gm-stacks [MGL *=MGL *MGL ] ! FG_sclassifies the* * strict graded formal group on [MGL *=MGL *MGL ]. Thus there is a 2-commutative diagr* *am: Spec(MGL *)_______//_Spec(MU *) (25) | | | | fflffl| fflffl| [MGL *=MGL *MGL ]______//_FG_s Quotienting out by the Gm-action yields a map of stacks [MGL *=MGL *MGL ]=Gm * *! FG_ which classifies the formal group on [MGL *=MGL *MGL ]=Gm. Proposition 6.8: The diagram (25) is cartesian. Proof. Combine Corollary 2.2 and Lemma 6.4. Part (ii) of the lemma is needed to* * ensure that the left and right units of (MU *, MU *MU ) and (MGL *, MGL *MGL ) are su* *itably * *__ compatible. |* *__| Corollary 6.9: The diagram Spec(MGL *)_________//_Spec(MU *) (26) | | | | fflffl| fflffl| [MGL *=MGL *MGL ]=Gm_______//_FG_ is cartesian. 22 7 Landweber exact theories Recall the Lazard ring L is isomorphic to MU *. For a prime p we fix a regular * *sequence p = v(p)0, v(p)1, . .2.MU * where v(p)nhas degree 2(pn - 1) as explained in the introduction. An (ungraded* *) L- module M is Landweber exact if (v(p)0, v(p)1, . .).is a regular sequence on M f* *or every p. An Adams graded MU *-module M* is Landweber exact if the underlying ungraded module is Landweber exact as an L-module [13, Definition 2.6]. In stacks this t* *ranslates as follows: An L-module M gives rise to a quasi-coherent sheaf M~ on Spec(L) an* *d M is Landweber exact if and only if M~ is flat over FG_with respect to Spec(L) ! * *FG_, see [20, Proposition 7]. Lemma 7.1: Let M* be an Adams graded MU *-module and M~*the associated quasi- coherent sheaf on Spec(MU *). Then M* is Landweber exact if and only if M~*is f* *lat over FG_swith respect to Spec(MU *) ! FG_s. Proof. We need to prove the "only if" implication. Assume M* is Landweber exact* * so that M~ has a compatible Gm-action. Let q: Spec(MU *) ! [Spec(MU *)]=Gm be the quotient map and N~*the descended quasi-coherent sheaf of M~*on [Spec(MU *)=Gm]. There is a canonical map N~*! q*M~*, which is the inclusion of the weight zero * *part of the Gm-action. By assumption, M~*is flat over FG_, i.e. q*M~*is flat over FG_. * *Since N~* is a direct summand of q*M~*it is flat over FG_. Hence M~*is flat over FG_ssinc* *e there is a cartesian diagram: Spec(MU *)_______//FG_s | | | | fflffl| fflffl| [Spec(MU *)]=Gm ____//_FG_ __ |_* *_| Remark 7.2: Lemma 7.1 does not hold for (ungraded) L-modules: The map Spec(Z) ! FG_sclassifying the strict formal multiplicative group over the integers is not* * flat, whereas the corresponding L-module Z is Landweber exact. In the following statements we view Adams graded Abelian groups as Adams gra* *ded graded Abelian groups via the line Z(2, 1). For example an MU *-module structur* *e on an 23 Adams graded graded Abelian group M** is an MU *-module in this way. Thus MGL * ***F is an MU *-module for every motivic spectrum F. Theorem 7.3: Suppose A* is a Landweber exact MU *-algebra, i.e. there is a map * *of commutative algebras MU * ! A* in Adams graded Abelian groups such that A* view* *ed as an MU *-module is Landweber exact. Then the functor MGL **(-) MU *A* is a big* *raded ring homology theory on SH (S). Proof. By Corollary 6.8 there is a projection p from Spec(A*) xFG_s[MGL *=MGL *MGL ] ~=Spec(A*) xSpec(MU *)Spec(MGL *) to [MGL *=MGL *MGL ] such that MGL *F MU *A* ~= (Spec(A*) xFG_s[MGL *=MGL *MGL ], p*F(F)). (27) (This is an isomorphism of Adams graded Abelian groups, but we won't use that f* *act.) The assignment F 7! F(F) is a homological functor since F 7! MGL *F is a homolo* *gical functor, and p is flat since it is the pullback of Spec(A*) ! FG_swhich is flat* * by Lemma 7.1. Thus p* is exact. Taking global sections over an affine scheme is an exact* * functor. Therefore, F 7! (Spec(A*) xFG_s[MGL *=MGL *MGL ], p*F(F)) is a homological f* *unctor on SH (S), so that by (27) F 7! MGL *F MU *A* is a homological functor with value* *s in Adams graded Abelian groups. It follows that F 7! (MGL *F MU *A*)0, the degre* *e zero part in the Adams graded Abelian group, is a homological functor, and it commut* *es with sums. Hence it is a homology theory on SH (S). The associated bigraded hom* *ology theory is clearly the one formulated in the theorem. Finally, the ring structur* *e is induced * * __ by the ring structures on the homology theory represented by MGL and on A*. * * |__| We note the proof works using F=Gm(F) instead of F(F); this makes the refere* *nce to Lemma 7.1 superfluous since neglecting the grading does not affect the proof. Corollary 7.4: The functor MGL **(-) MU *A* is a ring cohomology theory on stro* *ngly dualizable motivic spectra. Proof. Applying the functor in Theorem 7.3 to the Spanier-Whitehead duals of st* *rongly dualizable motivic spectra yields the cohomology theory on display. Its ring st* *ructure * * __ is induced by the ring structure on A*. * * |__| 24 Proposition 7.5: The maps [MGL *=MGL *MGL ] ! FG_sand [MGL *=MGL *MGL ]=Gm ! FG_are affine. Proof. Use Proposition 6.8, Corollary 6.9 and the fact that being an affine mor* *phism * * __ can be tested after faithfully flat base change. * * |__| Remark 7.6: We can formulate the above reasoning in more sheaf theoretic terms: Namely, denoting by i: [MGL *=MGL *MGL ] ! FG_sthe canonical map, the Landweb* *er exact theory is given by taking sections of i*F(F) over Spec(A*) ! FG_s. It is * *a homology theory by Proposition 7.5 since Spec(A*) ! FG_sis flat. Next we give the versions of the above theorems for MU *-modules. Proposition 7.7: Suppose M* is an Adams graded Landweber exact MU *-module. Then MGL **(-) MU *M* is a homology theory on SH (S) and MGL **(-) MU *M* a cohomo* *logy theory on strongly dualizable spectra. Proof. The map i: [MGL *=MGL *MGL ] ! FG_s is affine according to Proposition* * 7.5. With p: Spec(MU *) ! FG_sthe canonical map, the first functor in the propositio* *n is given by F O___//_ (Spec(MU *), M* MU *p*i*F(F)), which is exact by assumption. * * __ The second statement is proven by taking Spanier-Whitehead duals. * * |__| A Landweber exact theory refers to a homology or cohomology theory construct* *ed as in Proposition 7.7. There are periodic versions of the previous results: Proposition 7.8: Suppose M is a Landweber exact L-module. Then MGL *(-) L M is a (2, 1)-periodic homology theory on SH (S) with values in ungraded Abelian * *groups. The same statement holds for cohomology of strongly dualizable objects. These a* *re ring theories if M is a commutative L-algebra. Next we formulate the corresponding results for (highly structured) MGL -mo* *dules. This viewpoint goes back to [16] and plays an important role in our treatment, * *cf. sec. 9. 25 Proposition 7.9: Suppose M* is a Landweber exact Adams graded MU *-module. Then F 7! F** MU *M* is a bigraded homology theory on the derived category DMGL of M* *GL - modules. Proof. The proof proceeds along a now familiar route. What follows reviews the * *main steps. We wish to construct a homological functor from DMGL to quasi-coherent s* *heaves on [MGL *=MGL *MGL ]. Our first claim is that for every F 2 DMGL the Adams g* *raded MGL *-module F* is an (MGL *, MGL *MGL )-comodule. As in Lemma 5.1, MGL **MGL MGL**F** ____//_(MGL ^ F)** is an isomorphism restricting to an isomorphism MGL *MGL MGL* F* ____//_(MGL ^ F)*. This is proven by observing it holds for "spheres" p,qMGL, both sides are homo* *logical functors and commute with sums. This establishes the required comodule structu* *re. Next, the proof of Proposition 7.7 using flatness of M* viewed as a quasi-coher* *ent sheaf on [MGL *=MGL *MGL ] shows the functor in question is a homology theory. The * *remaining * * __ parts are clear. * *|__| Remark 7.10: We shall leave the formulations of the cohomology, algebra and per* *iodic versions of Proposition 7.9 to the reader. 8 Representability and base change Here we deal with the question when a motivic (co)homology theory is representa* *ble. Let R be a subset of SH (S)f such that SH (S)R,fconsists of strongly dualizable* * objects and is closed under smash products and duals. First, recall the notions of unital algebraic stable homotopy categories and* * Brown categories from [12, Definition 1.1.4 and next paragraph]: A stable homotopy ca* *tegory is a triangulated category equipped with sums, a compatible closed tensor produ* *ct, a set G of strongly dualizable objects generating the triangulated category as a * *localizing subcategory, and such that every cohomological functor is representable. It is* * unital algebraic if the tensor unit is finite (thus the objects of G are finite) and a* * Brown category if homology functors and natural transformations between them are representable. 26 A map between objects in a stable homotopy category is phantom if the induced map between the corresponding cohomology functors on the full subcategory of fi* *nite objects is the zero map. In case the category is unital algebraic this holds if* * and only if the map between the induced homology theories is the zero map. Lemma 8.1: The category SH (S)R is a unital algebraic stable homotopy category.* * The set G can be chosen to be (representatives of) the objects of SH (S)R,f. Proof. The nontrivial part is to verify that every cohomological functor on SH * *(S)R is representable. This follows from the generalized Brown representability theore* *m [21]. __ |_* *_| Lemma 8.2: Suppose S can be covered by affines which are spectra of countable r* *ings. Then SH (S)R is a Brown category and the category of homology functors on SH (S* *)R is naturally equivalent to SH (S)R modulo phantom maps. Proof. The first part follows by combining [12, Theorem 4.1.5] and [29, Proposi* *tion 5.5] * * __ and the second part by the definition of a Brown category. * * |__| Suppose R, R0are as above and SH (S)R,f SH (S)R0,f. Then a cohomology theory on SH (S)R0,frepresented by F restricts to a cohomology theory on SH (S)R,frepr* *esented by pR0,R(F). For Landweber exact theories the following holds: Proposition 8.3: Suppose a Landweber exact homology theory restricted to SH (S)* *T ,f is represented by a Tate spectrum E. Then E represents the theory on SH (S). Proof. Let M* be a Landweber exact Adams graded MU *-module affording the homol* *ogy theory under consideration. By assumption there is an isomorphism on SH (S)T ,f E**(-) ~=MGL **(-) MU *M*. By Lemma 4.10 the isomorphism extends to SH (S)T . Since MGL is cellular, an a* *rgument as in Remark 4.8 shows that both sides of the isomorphism remain unchanged when * * __ replacing a motivic spectrum by its Tate projection. * * |__| Next we consider a map f :S0 ! S of base schemes. The derived functor Lf*, s* *ee [22, Proposition A.7.4], sends the class of compact generators p,q 1 X+ of SH * *(S) - X a smooth S-scheme - to compact objects of SH (S0). Hence [21, Theorem 5.1] implie* *s Rf* 27 preserves sums, and the same result shows Lf* preserves compact objects in gene* *ral. A modification of the proof of Lemma 4.7 shows Rf* is an SH (S)T -module functo* *r, i.e. there is an isomorphism Rf*(F0^ Lf*G) ~=Rf*(F0) ^ G (28) in SH (S), which is natural in F02 SH (S0), G 2 SH (S)T . Proposition 8.4: Suppose a Landweber exact homology theory over S determined by* * the Adams graded MU *-module M* is representable by E 2 SH (S)T . Then Lf*E 2 SH (S* *0)T represents the Landweber exact homology theory over S0 determined by M*. Proof. For an object F0 of SH (S0), adjointness, the assumption on E and (28) i* *mply (Lf*E)**(F0) = ss**(F0^ Lf*E) is isomorphic to ss**(Rf*(F0^ Lf*E)) ~=ss**(Rf*F0^ E) ~=ss**(MGL ^ Rf*F0) MU *M*. Again by adjointness and (28) there is an isomorphism with ss**(MGL S0^ F0) MU *M* = MGL S0,**F0 MU *M*. __ |_* *_| In the next lemma we show the pullback from Proposition 8.4 respects multipl* *icative structures. In general one cannot expect that ring structures on the homology t* *heory lift to commutative monoid structures on representing spectra. Instead we will * *consider quasi-multiplications on spectra, by which we mean maps E ^ E ! E rendering the relevant diagrams commutative up to phantom maps. Lemma 8.5: Suppose a Landweber exact homology theory afforded by the Adams grad* *ed MU *-algebra A* is represented by a Tate object E 2 SH (S)T with quasi-multipli* *cation m: E ^ E ! E. Then Lf*m: Lf*E ^ Lf*E ! Lf*E is a quasi-multiplication and represents the ring structure on the Landweber exact homology theory determined* * by A* over S0. Proof. Let OE: F1^ F2 ! F3 be a map in SH (S)T . Let F0ibe the base change of F* *ito S0. If F0, G02 SH (S0) there are isomorphisms F0i,**F0~= Fi,**Rf*F0employed in the * *proof of 28 Proposition 8.4, and likewise for G0. These isomorphisms are compatible with OE* * in the sense provided by the commutative diagram: F01,**F0 F02,**G0________//_F03,**(F0^ G0) OO OO || |~=| || | |~= F3,**(Rf*(F0^ G0)) | OO | | | | | | F1,**Rf*F0 F2,**Rf*G0____//F3,**(Rf*F0^ Rf*G0) Applying the above to the quasi-multiplication m implies Lf*m represents the ri* *ng structure on the Landweber theory over S0. Hence Lf*m is a quasi-multiplication* * since * * __ the commutative diagrams exist for the homology theories, i.e. up to phantom ma* *ps. |__| We are ready to prove the motivic analog of Landweber's exact functor theore* *m. Theorem 8.6: Suppose M* is an Adams graded Landweber exact MU *-module. Then there exists a Tate object E 2 SH (S)T and an isomorphism of homology theories * *on SH (S) E**(-) ~=MGL **(-) MU *M*. In addition, if M* is a graded MU *-algebra then E acquires a quasi-multiplicat* *ion which represents the ring structure on the Landweber exact theory. Proof. First, let S = Spec(Z). By Landweber exactness, see Proposition 7.7, the* * right hand side of the claimed isomorphism is a homology theory on SH (Z). Its restri* *ction to SH (Z)T ,fis represented by some E 2 SH (Z)T since SH (Z)T is a Brown catego* *ry by Lemma 8.2. We may conclude in this case using Proposition 8.3. The general c* *ase follows from Proposition 8.4 since Lf*(SH (Z)T ) SH (S)T for f : S ! Spec(Z). Now assume M* is a graded MU *-algebra. We claim that the representing spect* *rum E 2 SH (Z)T has a quasi-multiplication representing the ring structure on the L* *andweber theory: The corresponding ring cohomology theory on SH (Z)T ,fcan be extended t* *o ind- representable presheaves on SH (Z)T ,f. Evaluating E(F) E(G) ! E(F ^ G) with * *F = G the ind-representable presheaf given by E on idE idEgives a map (E^E)0(-) ! E0(* *-) of homology theories. Since SH (Z)T is a Brown category this map lifts to a map E^* *E ! E of spectra which is a quasi-multiplication since it represents the multiplicati* *on of the * * __ underlying homology theory. The general case follows from Lemma 8.5. * * |__| 29 Remark 8.7: A complex point Spec(C) ! S induces a sum preserving SH (S)T -module realization functor r :SH (S) ! SH to the stable homotopy category. By the pr* *oof of Proposition 8.4 it follows that the topological realization of a Landweber exac* *t theory is the corresponding topological Landweber exact theory, as one would expect. Proposition 8.8: Suppose M* is an Adams graded Landweber exact MU *-module. Then there exists an MGL -module E and an isomorphism of homology theories on DMGL (E ^MGL -)**~= (-)** MU *M*. In addition, if M* is a graded MU *-algebra then E acquires a quasi-multiplicat* *ion in DMGL which represents the ring structure on the Landweber exact theory. Proof. We indicate the proof. By Proposition 7.9 it suffices to show that the h* *omology theory given by the right hand side of the isomorphism is representable. When * *the base scheme is Spec(Z) we claim that DMGL,T is a Brown category. In effect, SH* * (S)f is countable [29, Proposition 5.5] and MGL is a countable direct homotopy limi* *t of finite spectra, so it follows that DMGL,T ,fis also countable. So by [12, Theo* *rem 4.1.5] DMGL,T is a Brown category. Thus there is a an object of DMGL,T representing t* *he Landweber exact theory over Spec(Z). Now let f :S ! Spec(Z) be the unique map and Lf*MGL:DMGLZ ! DMGLS the pullback functor between MGL -modules. It has a r* *ight adjoint RfMGL,*. As prior to Proposition 8.4, we conclude RfMGL,*preserves sums* * and is a DMGLZ,T-module functor. The proof of Proposition 8.4 shows Lf*MGLrepresent* *s the Landweber theory over S. By inferring the analog of Lemma 8.5 our claim about the quasi-multiplicatio* *n is * * __ proven along the lines of the corresponding statement in Theorem 8.6. * * |__| 9 Operations and cooperations Let A* be a Landweber exact Adams graded MU *-algebra and E a motivic spectrum with a quasi-multiplication which represents the corresponding Landweber exact * *theory. Denote by ETopthe ring spectrum representing the corresponding topological Land* *weber exact theory. Then ETop*~=A*, ETop is a commutative monoid in the stable homoto* *py category and there are no even degree nontrivial phantom maps between such topo* *logical spectra [13, Section 2.1]. 30 Proposition 9.1: In the above situation the following hold. (i)E**E ~=E** ETop*ETop*ETop. (ii)E satisfies the assumption of Corollary 5.2(ii). (iii)The flat Hopf algebroid (E**, E**E) is induced from (MGL **, MGL **MGL )* * via the map MGL **! MGL ** MU *A* ~=E**. Proof. The isomorphism E**F ~=MGL **F MU *A* can be recasted as E**F ~=MGL **F MGL* MGL * MU *ETop*~=MGL **F MGL* E* and E**F ~=MGL **F MGL**MGL ** MU *ETop*~=MGL **F MGL**E**. In particular, E**E ~=MGL **E MGL**E**~= E**MGL MGL**E** is isomorphic to (MGL **MGL MGL**E**) MGL**E**~= E** MGL**MGL **MGL MGL**E**. (29) Moreover, since MGL **MGL ~=MGL ** MU *MU *MU , ETop* MU *MGL **MGL MU *ETop*~=ETop* MU *MGL ** MU *MU *MU MU *ETop* is isomorphic to MGL ** MU *ETop*ETop~= MGL ** MU *ETop* ETop*ETop*ETop~= E** ETop*ETop*ETo* *p. This proves the first part of the proposition. In particular, E*E ~=E* ETop*ETop*ETop (30) and E**E ~=E** E*E*E. (31) We note that ETop*ETopis flat over ETop*by the topological analog of (29) (this* * equation shows Spec(ETop*ETop) = Spec(ETop*) xFG_sSpec(ETop*)). Hence by (30) E*E is fla* *t over E*. * * __ Together with (31) this is Part (ii) of the proposition. Part (iii) follows fro* *m (29). |__| Remark 9.2: Let ETopand FTopbe evenly graded topological Landweber exact spectr* *a, E and F the corresponding motivic spectra. Then E ^ F is Landweber exact correspo* *nding to the MU *-module (ETop^ FTop)* (with either MU *-module structure). 31 Theorem 9.3: (i) The map afforded by the Kronecker product KGL **KGL_____//HomKGL**(KGL **KGL, KGL**) is an isomorphism of KGL **-algebras. (ii)With the completed tensor product there is an isomorphism of KGL **-algebr* *as KGL**KGL ~= KGL**b KU*KU*KU Item (i) and the module part of (ii) generalize to KGL **(KGL ^i) for i > 1. Proof. Recall KU *KU is free over KU *[2] and KGL is the Landweber theory deter* *mined by the MU *-algebra MU *! Z[fi, fi-1] which classifies the multiplicative forma* *l group law x+y -fixy over Z[fi, fi-1] with |fi| = 2 [26, Theorem 1.2]. The corresponding t* *opological Landweber exact theory is KU by the Conner-Floyd theorem. Thus by Proposition * *9.1 (i) KGL **KGL is free over KGL **. Moreover, KGL has the structure of an E1 -* *motivic ring spectrum, see [9], [26], so the universal coefficient spectral sequence [7* *, Proposition 7.7] can be applied to the KGL -modules KGL ^ KGL and KGL ; it converges condit* *ionally [5], [17], and the abutment is Hom **KGL-mod(KGL ^ KGL , KGL) = Hom **SH(S)(KG* *L , KGL). But the spectral sequence degenerates since KGL **KGL is a free KGL **-module, * *hence (i) and (ii). The more general statement is proved along the same lines by noting the isom* *orphism ETop*((ETop)^i) ~=ETop*ETop ETop*. .E.Top*ETop*ETop, * * __ and similarly for the Adams graded and Adams graded graded motivic versions. * * |__| In stable homotopy theory there is a universal coefficient spectral sequence* * for every Landweber exact ring theory [13, Proposition 2.21]. It appears there is no dire* *ct motivic analog: While there is a reasonable notion of evenly generated motivic spectrum* * as in [13, Definition 2.10] and one can show that a motivic spectrum representing a L* *andweber exact theory is evenly generated as in [13, Proposition 2.12], this does not ha* *ve as strong consequences as in topology because the coefficient ring MGL *is not concentra* *ted in even degrees as MU *, but see Theorem 9.7 below. We aim to extend the above res* *ults on homotopy algebraic K-theory to more general Landweber exact motivic spectra. 32 Proposition 9.4: Suppose M is a Tate object and E an MGL -module. Then there is* * a trigraded conditionally convergent right half-plane cohomological spectral sequ* *ence Ea,(p,q)2= Exta,(p,q)MGL**(MGL **M, E**) ) Ea+p,qM. * * __ Proof. MGL ^ M is a cellular MGL -module so this follows from [7, Proposition * *7.10]. |__| The differentials in this spectral sequence go dr: Ea,(p,q)r_//_Ea+r,(p-r+1,q)r. Theorem 9.5: Suppose M* is a Landweber exact graded MU *-module concentrated in even degrees and M 2 SH (S)T represents the corresponding motivic cohomology th* *eory. Then for p, q 2 Z and N an MGL -module spectrum there is a short exact sequence __ss//_ p,q ____//_ 0____//_Ext1,(p-1,q)MGL**(MGL__**M,/N**)/_Np,qMHomMGL**(MGL **M, N**) 0. Proof. Let MTop be the topological spectrum associated with M*. Then MU *MTop i* *s a flat MU *-module of projective dimension at most one [13, Propositions 2.12 and* * 2.16]. Hence MGL **M = MGL ** MU *MU *MTopis a MGL **-module of projective dimension * *at most one and consequently the spectral sequence of Proposition 9.4 degenerates * *at its E2-page. This implies the derived lim1-term lim1E***rof the spectral sequence i* *s zero; * * __ hence it converges strongly. The assertion follows because Ep,**1= 0 for all p * *6= 0, 1. |__| Remark 9.6: (i) For p, q 2 Z, the group of phantom maps Ph p,q(M, N) Np,qM* * is ' p,q defined as {Sp,q^ M ! N | for allE 2 SH (S)T ,fandE ! S ^ M : ' = 0}.* * It is clear that Ph p,q(M, N) ker(ss). (ii)The following topological example due to Strickland shows a nontrivial Ext* *1-term. The canonical map KU (p)! KU p from p-local to p-complete unitary topologi* *cal K-theory yields a cofiber sequence KU(p)____//_KUp___//_Effi//_ KU (p). Here E is rational and thus Landweber exact. Thus ffi is a degree 1 map be* *tween even Landweber spectra. However, ffi is a nonzero phantom map. 33 Over fields embeddable into C the corresponding boundary map for the motiv* *ic Landweber spectra is likewise phantom and non-zero. Using the notion of he* *ights for Landweber exact algebras from [20, Section 5], observe that E has heig* *ht zero while KU (p)has height one, compare with the assumptions in Theorem 9.7 b* *elow. Now fix Landweber exact MU *-algebras E* and F* concentrated in even degrees* * and a 2-commutative diagram f Spec (F*)___________//Spec(E*) (32) HHH vvv HHH vvv fF HH$$HzzvfEvv X where X is the stack of formal groups and fF (resp. fE) the map classifying the* * formal group GF (resp. GE) canonically associated with the complex orientable cohomolo* *gy theory corresponding to F* (resp. E*). This entails an isomorphism f*GE ~= GF * *of formal groups over Spec(F*). Hence the height of F* is less or equal to the he* *ight of E*. Let ETop, FTop (resp. E, F 2 SH (S)T ) be the topological (resp. motivic) * *spectra representing the indicated Landweber exact cohomology theory. Theorem 9.7: With the notation above assume ETop*ETopis a projective ETop*-modu* *le. (i)The map from Theorem 9.5 ** ~ Top Top ss : F**E___//HomMGL**(MGL **E, F**) = Hom ETop*(E* E , F**) is an isomorphism. (ii)Under the isomorphism in (i), the bidegree (0, 0) maps S*,*^ E ! F which r* *espect the quasi-multiplication correspond bijectively to maps of ETop*-algebras Hom ETop*-alg(ETop*ETop, F**). Remark 9.8: (i) The assumptions in Theorem 9.7 hold when ETop = KU and for certain localizations of Johnson-Wilson theories according to [2] respecti* *vely [3]. Theorem 9.7 recovers Theorem 9.3 with no mention of an E1 -structure on KG* *L . (ii)The theorem applies to the quasi-multiplication (E^E ! E) 2 E00(E^E) and s* *hows that this is a commutative monoid structure which lifts uniquely the multi* *plication 34 on the homology theory. For example, there is a unique structure of commut* *ative monoid on KGL S 2 SH (S) representing the familiar multiplicative structur* *e of homotopy K-theory, see [22] for a detailed account and an independent proo* *f in case S = Spec(Z). f (iii)The composite map ff : E* ! F* ! MGL ** MU *F* = F** yields a canonical bijection between the sets Hom ETop*-alg(ETop*ETop, F**) and {(ff0, ')}, w* *here ff0:E* ! F** is a ring homomorphism and ': ff*GE ! ff0*GE a strict isomorphism of s* *trict formal groups. (iv)Taking F = E in Theorem 9.7 and using Remark 9.6(i) implies that Ph**(E, E* *) = 0. For example, there are no nontrivial phantom maps KGL ! KGL of any bidegr* *ee. Proof. (of Theorem 9.7): We shall apply Proposition 2.3 with X0 Spec(MU *), X Spec(F*), Y Spec(E*), fX fF and fY fF, ss : Spec(MU *) ! X the map classi* *fying the universal formal group, f as given by (32) and ff : X = Spec(F*) ! X0 = Spe* *c(MU *) corresponding to the MU *-algebra structure MU * ! F*. Now by [20, Theorem 26],* * fX (resp. fY ) factors as fX = iX O ssX (resp. fY = iY O ssY ) with ssX and ssY fa* *ithfully flat and iX and iY inclusions of open substacks. The map i in Proposition 2.3 is ind* *uced by f. Finally, MGL **is canonically an MU *MU -comodule algebra and the OX-algebr* *a A in Proposition 2.3 corresponds to MGL **, i.e. A(X0) = MGL **and ss*YssY,*OY 2 Qc* *Y to the projective ETop*-module ETop*ETop. Taking into account the isomorphisms A(X0) OX0 ss*fY,*OY ~=MGL ** MU *MU Top*ETop ~=MGL **E A(X0) OX0 ff*OX ~=MGL ** MU *FTop*~=F** ss*YssY,*OY ~=ETop*ETop A(Y ) OY f*OX ~=F** OY ~=ETop* we obtain from Proposition 2.3 ( 0 n 1, ExtnMGL**(MGL **E, F**) ~= Top Top Hom ETop*(E* E , F**)n = 0. * * __ Hence (i) follows from Theorem 9.5 and (ii) by unwinding the definitions. * * |__| 35 10 A Chern character In what follows we define a ring map from KGL to periodized rational motivic co* *homology which induces the Chern character (or regulator map) from K-theory to (higher) * *Chow groups in the case when the base is a smooth scheme over a field. Let MZ denote the integral motivic Eilenberg-MacLane ring spectrum introduce* *d by Voevodsky [29, x6.1], cf. [8, Example 3.4]. Next we recall the canonical orient* *ation on MZ, in particular the construction of a map P1+! K(Z(1), 2) = L((P1, 1)). Recall the space L(X) assigns to any U the group of proper relative cycles o* *n U xSX over U of relative dimension 0 which have universally integral coefficients. T* *he line bundle OPn(1) OP1(n) carries the section ln Tnxn0+ Tn-1xn-10x1 + . .+.T0xn1* *, [T0 : . .:.Tn] homogeneous coordinates on Pn, [x0 : x1] coordinates on P1. Its zero * *locus is a relative divisor of degree n on P1 which induces a map Pn ! L(P1). These m* *aps arrange to maps Pn ! L((P1, 1)) compatible with the inclusions Pn ! Pn+1 induci* *ng a map ': P1 ! K(Z(1), 2). Moreover the maps Pn ! L(P1) are additive for the addit* *ion Pn x Pm ! Pn+m induced by multiplication of the sections ln. Hence ' is a ma* *p of commutative monoids and it restricts to the canonical map P1 ! K(Z(1), 2). This establishes an orientation on MZ with additive formal group law. Let MQ be the rationalization of MZ. In order to apply the spectral sequenc* *e of Proposition 9.4 to MQ we equip it with an MGL -module structure. Note that both* * MZ and MQ have canonical E1 -structures. Thus MQ ^ MGL is also E1 . As an MQ-module it has the form MQ[b1, b2, . .].. For any generator bi we let 'i: 2i,iMQ ! MQ ^* * MGL be the corresponding map. Taking its adjoint provides a map ' from the free MQ- W E1 -algebra on i>0S2i,ito MQ ^ MGL . Since everything is rational the contrac* *tion of these cells in E1 -algebras is isomorphic to MQ. Hence we get a map MGL ! MQ in E1 -algebras. This provides us in particular with an MGL -module structure on * *MQ. Let PM Q be the periodized rational Eilenberg-MacLane spectrum considered as* * an MGL -module, and LQ the Landweber spectrum corresponding to the additive formal group law over Q. By Remark 9.8 LQ is a ring spectrum. We let PL Q be the perio* *dic version. Both LQ and PLQ have canonical structures of MGL -modules. Finally, le* *t PH Q be the periodized rational topological Eilenberg-MacLane spectrum. Recall the map _ :KU *! PH Q* sending the Bott element to the canonical elem* *ent in degree 2. The exponential map establishes an isomorphism from the additive f* *ormal group law over PH Q* to the pushforward of the multiplicative formal group law * *over 36 KU * with respect to _. By Theorem 9.7 and Remark 9.8(iii) there is an induced * *map of ring spectra C :KGL ! PLQ. Theorem 10.1: The rationalization CQ :KGL Q ____//_PLQ of the map C is an isomorphism. Proof. This follows directly from the fact that the rationalization of _ is an * *isomorphism. __ |_* *_| Theorem 9.5 shows there is a short exact sequence 0 ____//_Ext1,(p-1,q)MGL**(MGL **LQ,/MQ**)/_MQp,qLQ __ss//_Homp,q _______//_ MGL**(MGL **LQ, MQ**) 0. Now since MQ carries the additive formal group law there is a natural transform* *ation of homology theories LQ**(-) _____//MQ**(-). The methods of Theorem 9.7 apply likewise to E = LQ, F = MQ and it follows that* * the above transformation again lifts uniquely to a map of ring spectra ': LQ ____//_MQ which can be prolonged to a map PL Q ! PM Q (denoted by the same symbol). The composition ' O C : KGL___//_MQ is called the Chern character. By construction it is functorial in the base sch* *eme with respect to the natural map Lf*MQS ! MQS0for f :S0! S. It is easily seen that ov* *er fields the map C coincides with the usual Chern character from K-theory to high* *er Chow groups with respect to the identification of higher Chow groups and motivic coh* *omology in [30]. For smooth quasi-projective schemes over fields this is known to be an isomo* *rphism after rationalization [4] (a map E ! F between periodic spectra is an isomorphi* *sm if it induces isomorphisms E-i,0(X) ! F-i,0(X) for all smooth schemes X over S and i * * 0). 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Fakult"at f"ur Mathematik, Universit"at Regensburg, Germany. e-mail: niko.naumann@mathematik.uni-regensburg.de Fakult"at f"ur Mathematik, Universit"at Regensburg, Germany. e-mail: Markus.Spitzweck@mathematik.uni-regensburg.de Department of Mathematics, University of Oslo, Norway. e-mail: paularne@math.uio.no 40