Motivic Landweber exact theories and their effective covers

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by  Marc Levine

MOTIVIC LANDWEBER EXACT THEORIES AND THEIR
EFFECTIVE COVERS
5
MARCLEVINE
1
0
2 Abstract. Letkbeafieldofcharacteristiczeroandlet(F,R)beaLandweber
n exactformalgrouplaw. WeconsideraLandweberexactT-spectraE:=R⊗L
a MGLanditseffectivecoverf0E→E withrespecttoVoevodsky’sslicetower.
J The coefficient ring R0 of f0E is the subring of R consisting of elements of
R of non-positive degree; the power series F ∈ R[[u,v]] has coefficients in
1
R0 although (F,R0) is not necessarily Landweber exact. We show that the
] geometricpartX 7→f0E∗(X):=(f0E)2∗,∗(X)off0Eiscanonicallyisomorphic
G totheorientedcohomologytheoryX 7→R0⊗LΩ∗(X),whereΩ∗ isthetheory
of algebraic cobordism as defined in [12]. This recovers results of Dai-Levine
A
[2]asthespecialcaseofalgebraicK-theoryanditseffectivecover,connective
. algebraicK-theory.
h
t
a
m
[ Contents

2
Introduction 1
v
1. Oriented ring T-spectra 4
4
8 2. The slice spectral sequence 5
2 3. Oriented theories and their T-effective covers 7
0 4. Oriented duality theories 15
.
1 5. Algebraic cobordism and oriented duality theories 16
0 6. The comparison map 19
4
References 22
1
:
v
i
X
Introduction
r
a Let S be a fixed base-scheme, Sm/S the category of smooth quasi-projective
S-schemes,and SH(S) the motivic stable homotopycategoryof T-spectra. In this
paper we consider two types of cohomology theories which carry the designation
“oriented”. The first type are those bi-graded theories on Sm/S, X 7→ E∗,∗(X)
represented by a (weak) commutative ring T-spectrum E ∈ SH(S) with an orien-
tationc inthe reducedcohomologyE˜2,1(P∞). The secondarethe orientedtheories
in the sense of [12, definition 1.1.1], that is, contravariant functors X 7→ A∗(X)
from Sm/S to commutative graded rings, together with pushforward maps f for
∗
projective morphisms f : Y → X, satisfying a number of functorialities and com-
patibilities. We will refer to the first type as “motivic” theories, the second as
1991 Mathematics Subject Classification. Primary 14C25, 19E15; Secondary 19E08 14F42,
55P42.
Key words and phrases. Algebraiccobordism,orientedtheories,slicetower.
TheauthorthankstheHumboldtFoundationforsupportthroughtheHumboldtProfessorship.
1
2 MARCLEVINE
“geometric”. Assigning to a motivic theory E∗,∗ its geometric part X 7→ E2∗,∗(X)
gives the link between these two notions.
Among the orientedmotivic theories,the theory representedby Voevodsky’sal-
gebraiccobordismspectrumMGListheuniversalone(see[28]fortheconstruction
of MGL, [21] for its universality). For S = Speck, k a field of characteristic zero,
algebraic cobordism Ω∗ as defined in [12] is the universal geometric theory.
For S =Speck as above, the universal property of Ω∗ gives a canonical natural
transformationΩ∗ →MGL2∗,∗ofgeometrictheories. RelyingonresultsofHopkins-
Morel-Hoyois[5,6],thiswasshowntobeanisomorphismin[11]. Themainpurpose
of this paper is to extend this isomorphism property to other motivic theories and
their geometric parts.
For a formal group law (F,R), F(u,v) ∈ R[[u,v]], one can form the oriented
geometric theory X 7→ R⊗ Ω∗(X), where L is the Lazard ring and L → R the
L
classifying homomorphism for F. In the motivic setting, the situation is more
delicate, however, just as in the classical case, if (F,R) is a Landweber exact for-
mal group law, there is a corresponding oriented weak commutative ring spectrum
MGL(R) with MGL(R)∗,∗(X)∼=R⊗LMGL∗,∗(X) (see [19]).
One can go a bit farther by considering the effective cover f E → E of a T-
0
spectrumE ∈SH(S). Heref isthetruncationfunctorwithrespecttoVoevodsky’s
0
slice tower. f E is ananalogof the classical-1 connective coverof a spectrum, and
0
inheritsmanypropertiesfromE. Inparticular,forE anorientedweakcommutative
ringT-spectrum,the effectivecoverf E inheritsfromE acanonicalstructureofan
0
orientedweakcommutativeringT-spectrum;thecoefficientringR :=(f E)2∗,∗(S)
0 0
is just the degree ≤ 0 part of the coefficient ring R := E∗,∗(S) of E (at least for S
the spectrum ofa characteristiczerofield, see theorem3.9). The coefficients ofthe
group law F associated to E actually lie in R , but even if (F ,R) is Landweber
E 0 E
exact, it is usually the case that (F ,R ) is not.
E 0
For E the oriented spectrum MGL(R) associated to a Landweber exact formal
grouplaw, with effective coverE , our mainresult is the following: For k a charac-
0
teristic zero field, S =Speck, the canonical natural transformations
E2∗,∗(k)⊗L∗ Ω∗ →E2∗,∗, E02∗,∗(k)⊗L∗ Ω∗ →(f0E)2∗,∗
areisomorphismsofgeometricorientedcohomologytheoriesonSm/k. Weactually
prove a stronger result (see corollary 6.3) concerning the oriented Borel-Moore
homology theories on Sch/k defined by E and f E.
0
The case of the Landweber exact theory E∗,∗ follows immediately from the case
E = MGL, proved in [11], so our efforts are directed at the effective cover E .
0
The main idea for these results already appears in our treatment (with S. Dai)
of the case of algebraic K-theory and its effective cover [2]. We axiomatize the
situationviathe notionofageometrically Landweber exactmotivic orientedtheory
(see definition 3.7) and show that for such a theory E, the canonical map
E2∗,∗(k)⊗L∗ Ω∗(X)→E2∗,∗(X)
is an isomorphism for all X ∈ Sm/k (theorem 6.2). Our result on a Landweber
exact theory and its effective cover then follow once we show that these are both
geometrically Landweber exact (theorem 3.9).
The paper is organized as follows: We begin by recalling some of the basic no-
tions concerning oriented (weak) commutative ring spectra in the motivic stable
MOTIVIC LANDWEBER EXACT THEORIES AND THEIR EFFECTIVE COVERS 3
homotopy category in §1, where we also recall the main results on the univer-
sality of MGL. In §2 we recall basic facts about the slice tower in the motivic
stable homotopy category; we discuss as well some issues of convergence of the
slice spectral sequence. In §3 we introduce the effective cover of an oriented weak
ring T-spectrum and show that it too defines an oriented weak commutative ring
spectrum. We discuss Landweber exact theories and introduce the notion of a ge-
ometrically Landweber exact theory. We discuss oriented duality theories on the
category of smooth pairs over k in §4. This theory provides the link between geo-
metric theories and motivic theories via the operationoftaking the geometric part
of a motivic oriented theory. We describe the relation of oriented duality theories
with algebraic cobordism in §5 and put everything together and prove our main
results in the final section, §6.
I would like to thank the referee for a number of helpful comments and sugges-
tions.
Notation and conventions. WeletSpcandSpc denotethecategoriesofsimplicial
•
sets and pointed simplicial sets, respectively, with homotopy categories H and H .
•
Spt the categoryof spectra (for the usual suspension operator Σ:=(−)∧S1) and
SH the stable homotopy category.
We denote by G the pointed S-scheme (A1 − 0 ,1 ). Pn will denote the
m S S S ∗
pointed S-scheme Pn, with base-point [1,0,...,0]; we similarly give the limit P∞
S
the base-point [1,0,...].
Weuseabase-schemeS thatisseparated,regular,noetherianandoffiniteKrull
dimension; we will at a certain point specialize to the case S = Speck, k a field
of characteristic zero. Sch/S will denote the category of quasi-projective schemes
overS and Sm/S the full subcategoryof smooth,quasi-projectiveschemes overS.
Spc(S) and Spc (S) are respectively the categories of presheaves on Sm/S with
•
values in Spc and Spc . Spt (S) will denote the category of S1-spectra over S,
• S1
this being the category of presheavesof spectra on Sm/S. We let Spt (S) denote
T
the category of T-spectra in Spc (S), with T :=A1/A1\{0}.
•
ThecategoriesSpc(S),Spc (S), Spt (S)andSpt (S)allhavemotivicmodel
• S1 T
structures (the original source for the unstable theory is [15], see also [3, 4] for a
compact description. For the stable theory, we refer the reader to [7]), with ho-
motopy categories denoted H(S), H (S), SH (S) and SH(S), respectively. The
• S1
categoriesSH (S) and SH(S) are triangulatedtensor categorieswith translation
S1
given by S1-suspension; the tensor structure is constructed using symmetric spec-
trum versions of Spt (S) and Spt (S), with appropriate model structures, also
S1 T
discussed in [7]. The unit for the tensor structure in SH(S) is the motivic sphere
spectrum S :=Σ∞S .
S T +
We have suspension functors Σa,b : SH(S) → SH(S) defined for a ≥ b ≥ 0 by
Σa,b(E)=E ∧Sa−b∧G∧b, and extending to a,b∈Z using the invertibility of Σ1,0
m
andΣ1,1. WehaveaswelltheT-suspensionΣ (E):=E∧T,canonicallyisomorphic
T
to Σ2,1. In addition, we have infinite suspension functors Σ∞ : H (S) → SH(S),
T •
Σ∞ : H (S) → SH (S) and Σ∞ : SH (S) → SH(S) with respective right
s • S1 Gm S1
adjoints Ω∞ : SH(S) → H (S), Ω∞ : SH (S) → H (S) and Ω∞ : SH(S) →
T • S1 S1 • Gm
SH (S).
S1
We let L denote the Lazard ring, that is, the coefficient ring of the universal
rank one commutative formal group law F ∈ L[[u,v]]. We let L∗ denote L with
L
4 MARCLEVINE
the grading determined by dega =1−i−j if F (u,v)=u+v+ a uivj
ij L i,j≥1 ij
and let L denote L with the opposite grading L :=L−n.
∗ n P
1. Oriented ring T-spectra
We recall that a morphism f : E → F in a compactly generated triangulated
category T is a phantom map if for each compact object A in T, the induced map
f :Hom (A,E) →Hom (A,F) is zero; it is enough to check on compact objects
∗ T T
of the form X[n], where X is an element in a given set of compact generators and
n ∈ Z. The subset of phantom maps Hom (E,F) ⊂ Hom (E,F) is clearly a
T ph T
two-sidedideal,sowemayformthecategoryT/phwiththesameobjectsasT and
morphisms Hom (E,F):=Hom (E,F)/Hom (E,F) .
T/ph T T ph
We will be mainly interested in the case T = SH(S), in which the shifted
suspensionspectraΣnΣ∞X ,X ∈Sm/S,n∈Z,formasetofcompactgenerators.
T T +
Thus,twomapsf,g :E →F areequalmodulophantommapsifandonlyiff =g
∗ ∗
as maps of the associated bi-graded cohomology theories f ,g : E∗,∗ → F∗,∗ on
∗ ∗
Sm/S.
Definition 1.1. A commutative ring T-spectrum is a T-spectrum E ∈ Spt (S)
T
together with maps µ:E ∧E →E, 1:S →E such that (E,µ,1) is a commutative
S
monoid in SH(S). A weak commutative ring T-spectrum is a T-spectrum E ∈
Spt (S) together with maps µ : E ∧E → E, 1 : S → E such that (E,µ,1) is a
T S
commutative monoid in SH(S)/ph.
ForE,F (weak)commutativeringT-spectra,amorphismf :E →F inSH(S)is
a monoidmap (resp. weakmonoid map) if f is a map ofmonoid objects in SH(S)
(resp. in SH(S)/ph).
Definition 1.2. An orientation on a weak commutative ring T-spectrum (E,µ,1)
is an element c∈E2,1(P∞) such that the restriction c ∈E2,1(P1) corresponds to
∗ |P1 ∗
1∈E0,0(S) under the suspension isomorphism E0,0(S)∼=E2,1(T)∼=E2,1(P1).
∗
A pair (E,c) consisting of a weak commutative ring T-spectrum(E,µ,1) and an
orientationcisanorientedweak commutativeringT-spectrum. Wesaythataweak
commutative ring T-spectrum (E,µ,1) is orientable if there is an orientation c on
E. We sometimes omit the explicit mention of the orientation c.
A morphism f :(E,c)→(E′,c′) of orientedweak commutative ring T-spectra is
amorphismf :E →E′ inSH(S)suchthatf isaweakmonoidmapandf (c)=c′.
∗
Example 1.3. The algebraic cobordism spectrum MGL has been studied in [21].
MGL is the T-spectrum (MGL ,MGL ,...) with MGL the Thom space Th(E ),
0 1 n n
with E → BGL the universal n-plane bundle. MGL is a commutative ring
n n S
T-spectrum in SH(S) with an orientation c ∈ MGL2,1(P∞) given by noting
MGL ∗
that the diagram
P∞ oo E //Th(E )=MGL
1 1 1
induces an isomorphism P∞ ∼= MGL in H (S) and thereby a morphism c :
∗ 1 • MGL
Σ∞P∞ →Σ2,1MGL in SH(S).
T ∗
We recall the following result of Panin-Pimenov-R¨ondigs:
Theorem 1.4 ([21, theorem 1.1]). For E a commutative ring T-spectrum in SH(S),
sending a monoid morphism ϕ :MGL →E to ϕ(c ) gives a bijection of the set
MGL
of monoid maps ϕ with the set of orientations c ∈E2,1(P∞).
E ∗
MOTIVIC LANDWEBER EXACT THEORIES AND THEIR EFFECTIVE COVERS 5
Given an oriented commutative ring T-spectrum (E,c) in SH(S), we let
ϕ :MGL→E
E,c
denote the corresponding morphism of commutative ring T-spectra.
In fact, this result extends directly to the setting of oriented weak commutative
ringspectra,replacing“monoidmap” with“weakmonoidmap”. Indeed, the proof
in [21] reduces to proving certain identities in E∗,∗(MGL(n)) or E∗,∗(MGL(n)∧
MGL(n)). It is shown in [21] that the canonical map
E∗,∗(MGL(n))→limE∗,∗(Th(T(n,m)))
←
is an isomorphism, where T(n,m)→Grass(n,n+m) is the universal bundle, and
Th(−) is the Thom space. Similarly
E∗,∗(MGL(n)∧MGL(n))∼=limE∗,∗(Th(T(n,m))∧Th(T(n,m)))
←
TheirproofreliesonshowingthatthemapsonE-cohomologyintheinversesystem
are surjective, hence the proofs only require the knowledge of E-cohomology on
objects in Sm/S, and therefore the proofs work for oriented weak commutative
ring spectra without change.
In case S = Speck, k a field, Vezzosi’s proof of the universality of MGL [25,
theorem4.3]isalsobasedonobtainingidentitiesinE∗∗(MGL(n))orE∗∗(MGL(n)∧
MGL(n))andthuscanalsobeadaptedtothesettingoforientedweakcommutative
ring T-spectra.
Remark 1.5. Let E ∈ SH(S) be a weak commutative ring T-spectrum. Let t ∈
E
E1,1(G )betheelementcorrespondingtotheunit1∈E0,0(S)underthesuspension
m
isomorphism. By functoriality, t gives a map of pointed sets
E
t (X):O×(X)→E1,1(X).
E X
If E admits an orientation c ∈ E2,1(P∞), then t (X) is a group homomorphism.1
E ∗ E
UsingtheE∗,∗(S)-modulestructureonE∗,∗(X),t (X)extendstoamapofE∗,∗(S)-
E
modules
t (X):E2∗,∗(S)⊗ O×(X)→E2∗+1,∗+1(X).
E Z X
2. The slice spectral sequence
Voevodsky introduced in [26] his slice tower as a motivic analog to the classical
Postnikov tower for spectra. In addition to [26], we refer the reader to [27] and
[8, 9] for the basic facts concerning the slice tower, some of which we briefly recall
here for the reader’s convenience.
We consider the localizing subcategory ΣpSHeff(S) of SH(S) generated by
T
objects ΣnX , with X ∈ Sm/S and n ≥ p, with inclusion i : ΣpSHeff(S) →
T + p T
SH(S). TheobjectsΣnX arecompact,henceeachofthecategoriesΣpSHeff(S)
T + T
is compactly generated; the set of all ΣnX , n ∈ Z, X ∈ Sm/S similarly forms a
T +
set of compact generators for SH(S).
ThankstoresultsofNeeman[17]oncompactlygeneratedtriangulatedcategories,
the functor i admits a right adjoint r : SH(S) → ΣpSHeff(S). Defining f :=
p p T p
1Letting Sdenote the spherespectrum andwriting[a]:=tS(a), thisfollowsfromtheidentity
[ab]=[a]+[b]+H[a][b](H :S∧Gm→SthestableHopfmap)andthefactthatH goestozero
inanyorientedtheoryE. BoththesefactsareprovenbyMorelin[14,§6].
6 MARCLEVINE
i ◦ r : SH(S) → SH(S), one has canonical natural transformations f → id,
p p p
f →f . Applying these to a T-spectrum E ∈SH(S) yields the the slice tower
p+1 p
...→f E →f E →...→E.
p+1 p
The slice functor s : SH(S) → SH(S) is characterized up to unique isomor-
p
phism by natural transformations f → s →f [1] so that for each E ∈ SH(S),
p p p+1
the triangle
f E →f E →s E →f E[1]
p+1 p p p+1
is distinguished.
There are canonical isomorphisms for all a,b,p,
Σa,b◦f ∼=f ◦Σa,b; Σa,b◦s ∼=s ◦Σa,b,
p p+b p p+b
compatible with the defining distinguished trianglesfor the s . The canonicalmap
p
f E →E induces an isomorphism Ω∞ f E ∼=Ω∞ E for all n≤0.
n Gm n Gm
For Y ∈Sm/S, we have the associated slice spectral sequence
(2.1) Ep,q(Y;n)=(s E)p+q,n(Y)=⇒Ep+q,n(Y).
2 −q
We conclude this section with a convergence criterion for the spectral sequence
(2.1).
Lemma2.1. SupposethatS =Speck, k aperfectfield. TakeE ∈SH(S). Suppose
that there is a non-decreasing function f : Z → Z with lim f(n) = ∞, such
n→∞
that π E = 0 for a ≤ f(b). Then the for all Y, the spectral sequence (2.1) is
a+b,b
strongly convergent.2
Proof. We may rewrite the E -term in (2.1) as
2
Ep,q(Y;n)=Hom (Σ∞Y ,Σp+q,ns E).
2 SH(S) T + −q
The spectral sequence will be strongly convergent(for fixed n) if for each m ∈Z,
0
there is an integer q(m ) such that
0
Hom (Σ∞Y ,Σm,nf E)=0
SH(S) T + q
for all q ≥q(m ) and all m≥−m . Indeed, we have
0 0
Hom (Σ∞Y ,Σm,nf E)=Hom (Σ−mΣ∞Y ,Ω∞ Σ0,nf E)
SH(S) T + q SHS1(S) S1 S1 + Gm q
=π (Ω∞ Σ0,nf E(Y)).
−m Gm q
Thus, the above condition is just saying that, given an integer m , the spectrum
0
Ω∞ Σ0,nf E(Y) is m -connected for all q ≥q(m ), this being a standard criterion
Gm q 0 0
forthestrongconvergenceofthespectralsequenceassociatedtothetowerofspectra
...→Ω∞ Σ0,nf E(Y)→...→Ω∞ Σ0,nf E(Y)∼=Ω∞ Σ0,nE(Y).
Gm q Gm −n Gm
We proceed to exhibit the existence of such an integer q(m ).
0
We recallthe 2-variablePostnikovtower f defined in [8, §3], with f =f .
a,b b −∞,b
We also recall that
i) π f F =0 for m<a, b∈Z [8, lemma 4.4],
m+r,r a,b
ii) the canonical map ρ : f F → f F induces an isomorphism on π
a,b a,b b m+r,r
for all m≥a, r ≥b,
2As spectral sequence {Erpq} ⇒ Gp+q converges strongly to G∗ if for each n, the spectral
sequence filtration F∗Gn on Gn is finite and exhaustive, there is an r(n) such that for all p
and all r ≥ r(n), all differentials entering and leaving Erp,n−p are zero and the resulting maps
Erp,n−p→E∞p,n−p=GrpFGn areallisomorphisms.
MOTIVIC LANDWEBER EXACT THEORIES AND THEIR EFFECTIVE COVERS 7
iii) ρ : f F → f F is an isomorphism if ρ induces an isomorphism on
a,b a,b b
π for all m∈Z and r ≥b [8, lemma 4.6].
m+r,r
The universal property of f gives the isomorphism for all r ≥q+n
q+n
π f Σ0,nE ∼=π Σ0,nE ∼=π E.
m+r,r q+n m+r,r m+r,r−n
By assumption on E, π E =0 for m+n≤f(r−n), and thus
m+n+r−n,r−n
π f (Σ0,nE)=0
m+r,r q+n
for m≤f(r−n)−n as well (assuming r ≥q+n).
Now take q(m ) to be an integer such that f(q(m )) ≥ m +dimY +n, and
0 0 0
take q ≥ q(m ). As f is non-decreasing, we have f(r−n)−n ≥ f(q(m ))−n ≥
0 0
m +dimY for all r ≥q+n, hence
0
π f (Σ0,nE)=0
m+r,r q+n
for m≤m +dimY, r ≥q+n. But by (i)-(iii), this implies that the map
0
f (Σ0,nE)→f (Σ0,nE)
m0+dimY+1,q+n q+n
is an isomorphism, and thus by (i), π f (Σ0,nE)=0 for all a≤m +dimY,
a+b,b q+n 0
b∈Z. Translating back to Ω∞ Σ0,nf E via the isomorphisms
Gm q
π f (Σ0,nE)∼=π Σ0,nf E ∼=π Ω∞ Σ0,nf E
a+b,b q+n a+b,b q a+b,b Gm q
for b≥0, a∈Z, we have
π Ω∞ Σ0,nf E =0
a+b,b Gm q
for all a≤m +dimY, b≥0, q ≥q(m ). In particular, taking b=0,
0 0
π Ω∞ Σ0,nf E =0
a Gm q
for all a≤m +dimY, q ≥q(m ).
0 0
We now apply the local-global spectral sequence (with q ≥q(m ))
0
Ea,b =Ha(Y ,π Ω∞ Σ0,nf E)=⇒π (Ω∞ Σ0,nf E(Y)).
2 Nis −b Gm q −a−b Gm q
This sequence is convergentby [1, theorem8]since Y has finite Nisnevichcohomo-
logicaldimension≤dimY andisstronglyconvergentsinceπ Ω∞ Σ0,nf E =0for
−b Gm q
−b≤m +dimY. In particular, Ea,b =0 for a+b≥−m , hence Ω∞ Σ0,nf E(Y)
is m -co0nnected for q ≥q(m ) 2 0 Gm q (cid:3)
0 0
3. Oriented theories and their T-effective covers
3.1. The T-effective cover.
Proposition 3.1. 1. Let (E,µ,1) be a weak commutative ring T-spectrum. Then
f E has a unique structure of a weak commutative ring T-spectrum such that the
0
canonical map ρ:f E →E is a weak monoid morphism.
0
2. If(E,c)is an orientedweak commutativeringT-spectrum,then thereis aunique
element c ∈ (f E)2,1(P∞) with ρ (c ) = c, and c defines an orientation on the
0 0 ∗ ∗ 0 0
weak commutative ring T-spectrum f E.
0
3. We have t =ρ (t ) in E1,1(G ).
E ∗ f0E m
8 MARCLEVINE
Proof. Write E for f E. We first lift the multiplication. We note that SHeff(S)
0 0
is closed under smash product, as the generators are, so E ∧E is in SHeff(S).
0 0
Thus, the composition
ρ∧ρ µ
E ∧E −−→E ∧E −→E
0 0
admits a unique lifting µ :E ∧E →E making
0 0 0 0
E ∧E µ0 //E
0 0 0
ρ∧ρ ρ
(cid:15)(cid:15) (cid:15)(cid:15)
E ∧E // E
µ
commute.
We claim that µ is associative modulo phantom maps. For this, let
0
ϕ:E ∧E ∧E →E
0 0 0 0
denote the difference µ ◦(µ ∧id)−µ ◦(id∧µ ). Let A be a compact object
0 0 0 0
of SH(S) and let h : A → E ∧E ∧E be a morphism in SH(S). We apply [17,
0 0 0
theorem 4.3.3] for the subcategory SHeff(S) of SH(S), generated as a localizing
subcategory of SH(S) by the set of compact objects S := {Σ∞X ,X ∈ Sm/S},
T +
with cardinal β =ℵ . This tells us that there is a compact object B in SHeff(S),
0
and morphisms f :A→B, g :B →E ∧E ∧E , such that h=g◦f. But as B is
0 0 0
compact, the composition ρ◦ϕ◦g in SH(S) is zero, and since B is in SHeff(S),
the universal property of ρ : E → E implies that ϕ◦g = 0 in SHeff(S). Thus
0
ϕ◦h=ϕ◦g◦f =0 and hence µ is associative modulo phantom maps.
0
The unit map 1 : S → E factors uniquely through ρ, since S is in SHeff(S),
S S
giving the unit map 1 :S →E ; the identities
0 S 0
µ ◦(id ∧1 )=id =µ ◦(1 ∧id )
0 E0 0 E0 0 0 E0
inSH(S)/phandthe commutativityofµ modulo phantommaps followasforthe
0
proof of associativity, as does the uniqueness of µ modulo phantom maps.
0
For(2),recallthat∗=[1,0,...,0]inPn. LetAn(∗)⊂Pnbetheopensubscheme
X 6= 0. Then the quotient map Pn → Pn/An(∗) is an isomorphism in H (S),
0 ∗ •
giving the isomorphism P∞ ∼= lim Pn/An(∗). In particular, both Pn and P∞ are
∗ −→n ∗ ∗
in Σ1SHeff(S).
T
We have the isomorphism Σ f E ∼= f Σ E, and under this isomorphism Σ ρ :
T 0 1 T T
Σ f E →Σ E goes over to the universal map ρ :f Σ E →Σ E. Thus c:P∞ →
T 0 T 1 1 T T ∗
Σ E factorsuniquely throughΣ ρ viaa mapc :P∞ →Σ f E. The restrictionof
T T 0 ∗ T 0
c toP1 issimilarlytheunique lifting ofc ,whichshowsthatc ∈(f E)2,1(P1)
0 ∗ |P1∗ 0|P1∗ 0 ∗
corresponds to 1 ∈ (f E)0,0(S) under the suspension isomorphism. Thus c ∈
0 0 0
(f E)2,1(P∞) is an orientation for f E.
0 ∗ 0
For (3), this follows directly from (1) and the naturality of the map t with
E
respect to weak monoid morphisms. (cid:3)
3.2. Landweber exact theories. We recall the Landweber exactness conditions
for a formal group law (F,R): Let ϕ : L∗ → R be the classifying map for (F,R).
ChoosehomogeneouspolynomialgeneratorsforL∗ overZ, L∗ =Z[x ,x ,...],with
1 2
degx =−i. Thenforeachprimep,the sequencep,x ,...,x ,...isaregular
i p−1 pi−1
sequence on R.
MOTIVIC LANDWEBER EXACT THEORIES AND THEIR EFFECTIVE COVERS 9
For an oriented theory (E,c), E ∈SH(S), let R∗ be the coefficient ring, that is,
E
R∗ :=E2∗,∗(S). Let F (u,v)∈R∗[[u,v]] be the formal group law of the oriented
E E,c E
theory (E,c).
We let SH(S) ⊂ SH(S) be the full subcategory of Tate spectra, that is, the
T
localizing subcategory generated by the spheres Σa,bS , a,b ∈ Z. We recall from
S
[19, §4] some basic facts concerning Tate spectra: There is an exact projection
functor p : SH(S) → SH(S) right adjoint to the inclusion. For a morphism
T T
S → S′, the functor Lf∗ : SH(S′) → SH(S) maps SH(S′) to SH(S) and the
T T
restriction Lf∗ : SH(S′) → SH(S) has right adjoint p ◦Rf . Finally, p is a
T T T ∗ T
SH(S) -module map, that is, for E ∈SH(S) and F ∈SH(S), there is a natural
T T
isomorphism
(3.1) p (E ∧F)∼=E ∧p (F).
T T
We let SH(S) ⊂ SH(S) denote the thick subcategory of finite spectra, namely,
fin
the thick subcategory generated by the spectra ΣnΣ∞X , X ∈ Sm/S, n ∈ Z.
T T +
Similarly, we let SH(S) ⊂ SH(S) be the thick subcategory generated by
T,fin T
spheres Σa,bS , a,b∈Z.
S
Definition 3.2. Let S =Speck, k a field of characteristic zero. An oriented weak
commutative ring T-spectrum (E,c) with classifying map ϕ : MGL → E is said
E,c
to be Landweber exact if
1. TheclassifyingmapL∗ →R∗ fortheformalgrouplawF satisfiestheLandwe-
E E,c
ber exactness conditions.
2. For all finite spectra F ∈ SH(k)fin, the map RE∗ ⊗R∗MGL MGL∗,∗(F) → E∗,∗(F)
induced by ϕ and the product map E2n,n(k)⊗Ea,b(F) → Ea+2n,b+n(F) is an
E,c
isomorphism.
3. E is in the Tate subcategory SH(k) of SH(k).
T
Remark 3.3. Letkbeafieldofcharacteristiczero. Naumann-Spitzweck-Østvær[19,
theorem8.7]showthat,foreachLandweberexactL∗-algebraR∗,thebigradedfunc-
torfromfinitespectratobi-gradedalgebras,F 7→R∗⊗L∗MGL∗,∗(F),isrepresented
by an object MGL(R∗) in SH(k) with morphisms µ : MGL(R∗)∧MGL(R∗) →
T
MGL(R∗), 1 : S → MGL(R∗) defining a (oriented) weak commutative ring T-
k
spectrum. It does not seem to be known if one can give MGL(R∗) the structure of
an oriented commutative ring T-spectrum.
In fact, Naumann-Spitzweck-Østvær work in the setting of homology theories
on SH(S) rather than cohomology theories on finite spectra, and they prove their
results for S a regular, noetherian separated scheme of finite Krull dimension. In
case the base-scheme is Speck, k a field of characteristic zero, all finite spectra
in SH(k) are strongly dualizable [23, theorem 1.4], so one may easily pass from
homology theories on SH(k) to cohomology theories on finite spectra in SH(k).
In the case of a general base-scheme S, the classifying map L∗ → MGL2∗,∗(S)
might not be an isomorphism, and one would need to modify the definition of a
Landweberexactspectrumaccordingly. Naumann-Spitzweck-Østværsimplydefine
a Landweber exact spectrum to be one of the form MGL(R∗), rather than giving
anintrinsiccharacterization. IncaseS =Speck,k afieldofcharacteristiczero,the
construction of Naumann-Spitzweck-Østvær in fact gives us all Landweber exact
oriented weak commutative ring spectra, as the next two lemmas show.
10 MARCLEVINE
Lemma 3.4. Suppose S = Speck, k a field of characteristic zero. Suppose that
(E,c) is Landweber exact (in the sense of definition 3.2). Then the classifying map
ϕ :MGL →E and the product maps for F ∈SH(k), RE ⊗ E (F)→E (F),
E,c ∗ Z ∗,∗ ∗,∗
together induce an isomorphism of homology theories on SH(k)
RE ⊗ MGL (−)→E (−).
∗ RM∗GL ∗,∗ ∗,∗
Proof. BothfunctorsRE⊗ MGL (−)andE (−)arehomologicalfunctorson
∗ RM∗GL ∗,∗ ∗,∗
SH(k), compatible with coproducts, hence the full subcategory C of objects F for
whichthelemmaholdsisalocalizingsubcategoryofSH(k). AsSH(k)isgenerated
as a localizing category by SH(k) , this reduces us to the case F ∈SH(k) .
fin fin
Take F in SH(k) . Then F is strongly dualizable with FD ∈ SH(k) [23,
fin fin
theorem1.4],andwehaveG (F)∼=G−∗,−∗(FD)for allG ∈SH(k). The factthat
∗,∗
ϕ induces an isomorphism
E,c
RE∗ ⊗R∗MGL MGL∗,∗(FD)∼=E∗,∗(FD).
shows that SH(k) ⊂C. (cid:3)
fin
Lemma 3.5. Suppose S = Speck, k a field of characteristic zero. Let (E,c) be
a Landweber exact oriented weak commutative ring T-spectrum (in the sense of
definition 3.2). Then there is an isomorphism of oriented weak commutative ring
T-spectra ψ :MGL(R∗)→E.
E
Proof. Since k has characteristic zero, the results of Hopkins-Morel-Hoyois [5, 6]
show that the classifying map L → RMGL is an isomorphism. We may therefore
∗ ∗
replace RMGL with L in our definition of Landweber exactness and in lemma 3.4.
∗ ∗
Let f : Speck → SpecQ be the canonical morphism of schemes, giving the
//
adjoint functors Lf∗ : SH(Q) oo SH(k) : Rf . Consider the object Rf E ∈
∗ ∗
SH(Z)andtakeF ∈SH(Z) . WehaveacanonicalisomorphismRf (E∧Lf∗F)∼=
T ∗
Rf E ∧F (see e.g. [19, pg. 578, eqn. (26)]) and therefore by lemma 3.4 we have
∗
(Rf∗E)∗,∗(F)=E∗,∗(Lf∗F)∼=R∗E ⊗L∗ MGLk∗,∗(Lf∗F)
for all F in SH(Q) . That is, Rf E represents the homology theory F 7→RE ⊗
T ∗ ∗ L∗
MGL (Lf∗F) on SH(Q) ; using the isomorphism (3.1), the same holds for the
k∗,∗ T
Tate projection p Rf E. The spectrum MGL (R∗)∈SH(Q) similarly represents
T ∗ Q E
the homology theory F 7→RE ⊗ MGL (F).
∗ L∗ Q∗,∗
We have a canonical isomorphism MGLk ∼= Lf∗MGLQ; we consider this as an
identity. The spectrum MGL (R∗) is similary by construction canonically isomor-
k E
phictothepull-backLf∗MGL (R∗). ApplyingRf andtheTateprojectiontopair
Q E ∗
of canonical maps ϕMGL(R∗E) : MGLk → MGLk(RE∗), ϕE : MGLk → E and using
the appropriate adjoint properties gives us the commutative diagram in SH(Q)
(3.2) MGL //p Rf MGL // Rf MGL
ϕMGLQ(R∗E) (cid:15)(cid:15) Q pTRf∗ϕMGTL(R∗E)∗(cid:15)(cid:15) ◗◗k◗◗◗◗◗p◗T◗R◗f◗∗◗ϕ◗E(( ∗ ❑❑k❑❑❑R❑f❑∗❑ϕ❑❑E%%
MGL (R∗) // p Rf MGL (R∗) p Rf E //Rf E
Q E T ∗ k E T ∗ ∗