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).
By Mayer-Vietoris this holds in general for smooth schemes over fields.
37
Corollary 10.2: For smooth schemes over fields the map
': LQ ____//_MQ
is an isomorphism.
Corollary 10.3: For smooth schemes over fields
MQ**(-)
is the universal oriented homology theory with rational coefficients and additi*
*ve formal
group law.
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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