A universality theorem for Voevodsky's algebraic
cobordism spectrum
I. Panin*y K. Pimenovy O. R"ondigsz
June 06, 2007x
Abstract
An algebraic version of a theorem due to Quillen is proved. More preci*
*sely,
for a ground field k we consider the motivic stable homotopy category SH (*
*k) of
P1-spectra, equipped with the symmetric monoidal structure described in [P*
*PR1 ].
The algebraic cobordism P1-spectrum MGL is considered as a commutative mo*
*noid
equipped with a canonical orientation thMGL 2 MGL 2,1(Th (O(-1))). For a c*
*ommu-
tative monoid E in the category SH(k) it is proved that assignment ' 7! '(*
*thMGL )
identifies the set of monoid homomorphisms ': MGL ! E in the motivic stabl*
*e ho-
motopy category SH(k) with the set of all orientations of E. The result wa*
*s stated
originally in a slightly different form by G. Vezzosi in [Ve].
1 Oriented commutative ring spectra
We refer to [PPR1 , Appendix] for the basic terminology, notation, construction*
*s, defini-
tions, results. For the convenience of the reader we recall the basic definitio*
*ns. Let S be a
Noetherian scheme of finite Krull dimension. One may think of S being the spect*
*rum of
a field or the integers. Let Sm=S be the category of smooth quasi-projective S-*
*schemes,
and let sSet be the category of simplicial sets. A motivic space over S is a fu*
*nctor
A: SmOp=S ! sSet
(see [PPR1 , A.1.1]). The category of motivic spaces over S is denoted M(S). Th*
*is defi-
nition of a motivic space is different from the one considered by Morel and Voe*
*vodsky in
[MV ] - they consider only those simplicial presheaves which are sheaves in th*
*e Nisnevich
topology on Sm=S. With our definition the Thomason-Trobaugh K-theory functor ob-
tained by using big vector bundles is a motivic space on the nose. It is not a *
*simplicial
Nisnevich sheaf. This is why we prefer to work with the above notion of "space".
________________________________
*Universit"at Bielefeld, SFB 701, Bielefeld, Germany
ySteklov Institute of Mathematics at St. Petersburg, Russia
zInstitut f"ur Mathematik, Universit"at Osnabr"uck, Osnabr"uck, Germany
xThe authors thank the SFB-701 at the Universit"at Bielefeld, the RTN-Networ*
*k HPRN-CT-2002-
00287, the RFFI-grant 03-01-00633a, and INTAS-03-51-3251 for their support.
1
We write Hcmo(S) for the pointed motivic homotopy category and SHcm(S) for t*
*he sta-
ble motivic homotopy category over S as constructed in [PPR1 , A.3.9, A.5.6]. B*
*y [PPR1 ,
A.3.11 resp. A.5.6] there are canonical equivalences to Ho(S) of [MV ] resp. S*
*H (S) of [V1 ].
Both Hcmo(S) and SHcmo(S) are equipped with closed symmetric monoidal structure*
*s such
that the P1-suspension spectrum functor is a strict symmetric monoidal functor
1P1:Hcmo(S) ! SHcm(S).
Here P1 is considered as a motivic space pointed by 1 2 P1. The symmetric monoi*
*dal
structure (^, IS = 1P1S+) on the homotopy category SHcm(S) is constructed on t*
*he model
category level by employing symmetric P1-spectra. It satisfies the properties r*
*equired by
Theorem 5.6 of Voevodsky congress talk [V1 ]. From now on we will usually omit*
* the
superscript (-)cm.
Every P1-spectrum E = (E0, E1, . .).represents a cohomology theory on the ca*
*tegory
of pointed motivic spaces. Namely, for a pointed motivic space (A, a) set
Ep,q(A, a) = Hom SHo(S)( 1P1(A, a), p,q(E))
and E*,*(A, a) = p,qEp,q(A, a). This definition extends to motivic spaces via *
*the functor
A 7! A+ which adds a disjoint basepoint. That is, for a non-pointed motivic sp*
*ace A
set Ep,q(A) = Ep,q(A+, +) and E*,*(A) = p,qEp,q(A). Recall that there is a ca*
*nonical
element in E2n,n(En), denoted as 1P1En(-n) ! E. It is represented by the cano*
*nical
map (*, . .,.*, En, En ^ P1, . .).! (E0, E1, . .,.En, . .).of P1-spectra.
Every X 2 Sm=S defines a representable motivic space constant in the simplic*
*ial
direction taking an S-smooth scheme U to Hom Sm=S(U, X). It is not possible in *
*general
to choose a basepoint for representable motivic spaces. So we regard S-smooth v*
*arieties
as motivic spaces (non-pointed) and set
Ep,q(X) = Ep,q(X+, +).
Given a P1-spectrum E we will reduce the double grading on the cohomology th*
*eory
E*,*to a grading. Namely, set Em = m=p-2qEp,qand E* = m Em . We often write
E*(k) for E*(Spec(k)) below.
To complete this section, note that for us a P1-ring spectrum is a monoid (E*
*, ~, e)
in (SH(S) , ^, IS). A commutative P1-ring spectrum is a commutative monoid (E, *
*~, e) in
(SH(S) , ^, 1). The cohomology theory E* defined by a P1-ring spectrum is a rin*
*g coho-
mology theory. The cohomology theory E* defined by a commutative P1-ring spectr*
*um
is a ring cohomology theory, however it is not necessary graded commutative. T*
*he co-
homology theory E* defined by an oriented commutative P1-ring spectrum is a gra*
*ded
commutative ring cohomology theory.
1.1 Oriented commutative ring spectra
Following Adams and Morel we define an orientation of a commutative P1-ring spe*
*ctrum.
However we prefer to use Thom classes instead of Chern classes. Consider the p*
*ointed
motivic space P1 = colimn 0Pn having base point g1: S = P0 ,! P1 .
2
The tautological "vector bundle" T(1) = OP1 (-1) is also known as the Hopf b*
*undle.
It has zero section z :P1 ,! T(1). The fiber over the point g1 2 P1 is A1. *
*For a
vector bundle V over a smooth S-scheme X, with zero section z :X ,! V , its Tho*
*m space
Th(V ) is the Nisnevich sheaf associated to the presheaf Y 7! V (Y )= V r z(X) *
*(Y ) on
the Nisnevich site Sm=S. In particular, Th (V ) is a pointed motivic space in *
*the sense
of [PPR1 , Defn. A.1.1]. It coincides with Voevodsky's Thom space [V1 , p. 42*
*2], since
Th(V ) already is a Nisnevich sheaf. The Thom space of the Hopf bundle is then *
*defined
as the colimit Th(T(1)) = colimn 0Th OPn(-1) . Abbreviate T = Th(A1S).
Let E be a commutative P1-ring spectrum. The unit gives rise to an element *
*1 2
E0,0(Spec(k)+). Applying the P1-suspension isomorphism to that element we get *
*an
element P1(1) 2 E2,1(P1, 1). The canonical covering of P1 defines motivic weak*
* equiv-
alences
__~_//_1 1 oo~__ 1 1
P1 P =A A =A r {0} = T
of pointed motivic spaces inducing isomorphisms E(P1, 1) E(A1=A1r {0}) ! E(T *
*).
Let T(1) be the image of P1(1) in E2,1(T ).
Definition 1.1.1. Let E be a commutative ring P1-spectrum. A Thom orientation *
*of
E is an element th 2 E2,1(Th (T(1)) such that its restriction to the Thom space*
* of the
fibre over the distinguished point coincides with the element T(1) 2 E2,1(T ).*
* A Chern
orientation of E is an element c 2 E2,1(P1 ) such that c|P1 = - P1(1). An orien*
*tation of
E is either a Thom orientation or a Chern orientation. One says that a Thom ori*
*entation
th of E coincides with a Chern orientation c of E provided that c = z*(th) or e*
*quivalently
the element th coincides with the one th(O(-1)) given by (2) below.
Remark 1.1.2. The element th should be regarded as the Thom class of the tautol*
*ogical
line bundle T(1) = O(-1) over P1 . The element c should be regarded as the Cher*
*n class
of the tautological line bundle T(1) = O(-1) over P1 .
Example 1.1.3. The following orientations given right below are relevant for ou*
*r work.
Here MGL denotes the P1-ring spectrum representing algebraic cobordism obtaine*
*d below
in Definition 2.1.1, and BGL denotes the P1-ring spectrum representing algebra*
*ic K-
theory constructed in [PPR1 , Theorem 2.2.1].
o Let u1 : 1P1(Th (T(1)))(-1) ! MGL be the canonical map of P1-spectra. *
*Set
thMGL = u1 2 MGL 2,1(Th (T(1))). Since thMGL |Th(1)= P1(1) in MGL 2,1(T*
*h (1)),
the class thMGL is an orientation of MGL .
o Set c = (-fi) [ ([O] - [O(1)]) 2 BGL 2,1(P1 ). The relation (11) from [PPR*
*1 ] shows
that the class c is an orientation of BGL .
2 Oriented ring spectra and infinite Grassmannians
Let (E, c) be an oriented commutative P1-ring spectrum. In this section we comp*
*ute the
E-cohomology of infinite Grassmannians and their products. The results are the *
*expected
ones - see Theorems 2.0.6and 2.0.7.
3
The oriented P1-ring spectrum (E, c) defines an oriented cohomology theory o*
*n SmOp
in the sense of [PS1 , Defn. 3.1] as follows. The restriction of the functor E*
**,*to the
category Sm=S is a ring cohomology theory. By [PS1 , Th. 3.35] it remains to co*
*nstruct a
Chern structure on E*,*|SmOp in the sense of [PS1 , Defn. 3.2]. Let Ho(k) be th*
*e homotopy
category of pointed motivic spaces over k. The functor isomorphism Hom Ho(k)(-,*
* P1 ) !
Pic(-) on the category Sm=S provided by [MV , Thm. 4.3.8] sends the class of t*
*he identity
map P1 ! P1 to the class of the tautological line bundle O(-1) over P1 . For *
*a line
bundle L over X 2 Sm=S let [L] be the class of L in the group Pic(X). Let fL :X*
* ! P1
be a morphism in H(k) corresponding to the class [L] under the functor isomorph*
*ism
above. For a line bundle L over X 2 Sm=S set c(L) = f*L(c) 2 E2,1(X). Clearly,
c(O(-1)) = c. The assignment L=X 7! c(L) is a Chern structure on E*,*|SmOp sin*
*ce
c|P1 = - P1(1) 2 E2,1(P1, 1). With that Chern structure E*,*|SmOp is an oriente*
*d ring
cohomology theory in the sense of [PS1 ]. In particular, (BGL , cK ) defines an*
* oriented ring
cohomology theory on SmOp.
Given this Chern structure, one obtains a theory of Thom classes V=X 7! th(V*
* ) 2
E2rank(V ),rank(VT)hX(V ) on the cohomology theory E*,*|SmOp=S in the sense of*
* [PS1 ,
Defn. 3.32] as follows. There is a unique theory of Chern classes V 7! ci(V ) 2*
* E2i,i(X)
such that for every line bundle L on X one has c1(L) = c(L). For a rank r vecto*
*r bundle
V over X consider the vector bundle W := 1 V and the associated projective ve*
*ctor
bundle P(W ) of lines in W . Set
~th(V ) = cr(p*(V ) OP(W)(1)) 2 E2r,r(P(W )). (1)
It follows from [PS1 , Cor. 3.18] that the support extension map
2r,r
E2r,rP(W )=(P(W ) r P(1)) ! E P(W )
is injective and ~th(E) 2 E2r,rP(W )=(P(W ) r P(1)) . Set
th(E) = j*(~th(E)) 2 E2r,rThX (V ) , (2)
where j :ThX (V ) ! P(W )=(P(W ) r P(1)) is the canonical motivic weak equivale*
*nce
of pointed motivic spaces induced by the open embedding V ,! P(W ). The assig*
*n-
ment V=X to th(V ) is a theory of Thom classes on E*,*|SmOp (see the proof of [*
*PS1 ,
Thm. 3.35]). Hence the Thom classes are natural, multiplicative and satisfy the*
* following
Thom isomorphism property.
Theorem 2.0.4. For a rank r vector bundle p: V ! X on X 2 Sm=S with zero section
z :X ,! V , the map
- [ th(V ): E*,*(X) ! E*+2r,*+rV=(V r z(X))
is an isomorphism of two-sided E*,*(X)-modules, where - [ th(V ) is written for*
* the
composition map - [ th(V ) O p*.
*
* __
Proof. See [PS1 , Defn. 3.32.(4)]. *
* |__|
4
Analogous to [V1 , p. 422] one obtains for vector bundles V ! X and W ! Y *
* in
Sm=S a canonical map of pointed motivic spaces Th(V ) ^ Th(W ) ! Th(V xS W ) wh*
*ich
is a motivic weak equivalence as defined in [PPR1 , Defn. 3.1.6]. In fact, the *
*canonical map
becomes an isomorphism after Nisnevich (even Zariski) sheafification. Taking Y *
*= S and
W = 1 the trivial line bundle yields a motivic weak equivalence Th(V ) ^ T ! T *
*h(V 1).
The canonical covering of P1 defines motivic weak equivalences
T = A1=A1 r {0} _~__//_P1=A1~ooP1_
and the arrow T = A1=A1 r {0} ! P1=P1 r {0} is an isomorphism. Hence one may
switch between T and P1 as desired.
Corollary 2.0.5. For W = V 1 consider the motivic weak equivalences
ffl: Th(V ) ^ P1 ! Th(V ) ^ P1=A1 Th(V ) ^ T ! Th(W )
of pointed motivic spaces over S. The diagram
P1
E*+2r,*+r(ThO(VO))__//_E*+2r+2,*+r+1(ThO(VO) ^ P1)
id|| |ffl*|
| T *+2r+2,*+r+1|
E*+2r,*+r(ThO(VO))____//_E OO (Th (W ))
-[th(V )|| |-[th(W)|
| |
E*,*(X) _______id_______//_E*,*(X)
commutes.
Let Gr(n, n+m) be the Grassmann scheme of n-dimensional linear subspaces of *
*An+mS.
The closed embedding An+m = An+m x {0} ,! An+m+1 defines a closed embedding
Gr(n, n + m) ,! Gr(n, n + m + 1). (3)
The tautological vector bundle is denoted T(n, n + m) ! Gr (n, n + m). The clo*
*sed
embedding (3) is covered by a map of vector bundles T(n, n + m) ,! T(n, n + m +
1). Let Gr(n) = colimm 0Gr (n, n + m), T(n) = colimm 0T(n, n + m) and Th (T(n))*
* =
colimm 0Th (T(n, n+m)). These colimits are taken in the category of motivic spa*
*ces over
S.
Theorem 2.0.6. Let E be an oriented P1-ring spectrum. Then
E*,*(Gr (n)) = E*,*(k)[[c1, c2, . .,.cn]]
is the formal power series ring, where ci:= ci(T(n)) 2 E2i,i(Gr (n)) denotes th*
*e i-th Chern
class of the tautological bundle T(n). The inclusion incn:Gr (n) ,! Gr(n + 1) *
*satisfies
inc*n(cm ) = cm for m < n + 1 and inc*n(cn+1) = 0.
5
Proof.The case n = 1 is well-known (see for instance [PS1 , Thm. 3.9]). For a *
*finite
dimensional vector space W and a positive integer m let F(m, W ) be the flag va*
*riety of
flags W1 W2 . . .Wm of linear subspaces of W such that the dimension of Wi *
*is i.
Let Ti(m, W ) be the tautological rank i vector bundle on F(m, W ).
Let V = A1 be an infinite dimensional vector bundle over S and set e = (1, *
*0, . .)..
Then Vn denotes the n-fold product of V , and eni2 Vn the vector (0, . .,.0, e,*
* 0, . .,.0)
having e precisely at the ith position. Let F (m) = colimW F(m, W ) and let Ti*
*(m) =
colimWTi(m, W ), where W runs over all finite-dimensional vector subspaces of V*
*n. Thus
we have a flag T1(m) T2(m) . . .Tm (m) of vector bundles over F (m). Set Li*
*(m) =
Ti(m)=Ti-1(m). It is a line bundle over F (m).
Consider the morphism pm :F (m) ! F (m - 1) which takes a flag W1 W2 . .*
* .
Wm to the flag W1 W2 . . .Wm-1. It is a projective vector bundle over F (m *
*- 1)
such that the line bundle Li(m) is its tautological line bundle. Thus there exi*
*sts a tower
of projective vector bundles F (m) ! F (m - 1) ! . .!.F (1) = P(Vn). The projec*
*tive
bundle theorem implies that
E*,*(F (n)) = E*,*(k)[[t1, t2, . .,.tn]]
(the formal power series in n variables), where ti= c(Li(n)) is the first Chern*
* class of the
line bundle Li(n) over F (n).
Consider the morphism q :F (n) ! Gr(n), which takes a flag W1 W2 . . .Wn*
* to
the space Wn. It can be decomposed as a tower of projective vector bundles. In *
*particular,
the pull-back map q*: E*,*(Gr (n)) ! E*,*(F (n)) is a monomorphism. It takes th*
*e class
cito the symmetric polynomial oei= t1t2. .t.i+ . .+.tn-i+1. .t.n-1tn. So the im*
*age of q*
contains E*,*(k)[[oe1, oe2, . .,.oen]]. It remains to check that the image of q*
** is contained in
E*,*(k)[[oe1, oe2, . .,.oen]]. To do that consider another variety.
Namely, let V 0be the n-dimensional subspace of Vn generated by the vectors *
*eni's. Let
lnibe the line generated by the vector eni. Let Vi0be a subspace of V 0generate*
*d by all
enj's with j i. So one has a flag V10 V20 . . .Vn0. We denote this flag F 0*
*. For each
vector subspace W in Vn containing V 0consider three algebraic subgroups of the*
* general
linear group GL W . Namely, set
PW = Stab(V 0), BW = Stab(F 0), TW = Stab(ln1, ln2, . .,.lnn).
The group TW stabilizes each line lni. Clearly, TW BW PW and Gr(n, W ) = GL*
* W=PW ,
F(n, W ) = GL W=BW Set M(n, W ) = GL W=TW . One has a tower of obvious morphis*
*ms
rW qW
M(n, W ) --! F(n, W ) --! Gr(n, W ).
Set M(n) = colimW M(n, W ), where W runs over all finite dimensional subspace W*
* of
Vn containing V 0. Now one has a tower of morphisms
q
M(n) r-!F (n) -!Gr (n).
The morphisms rW can be decomposed in a tower of affine bundles. Hence it indu*
*ces an
isomorphism on any cohomology theory. The same then holds for the morphism r and
E*,*(M(n)) = E*,*(k)[[t1, t2, . .,.tn]].
6
Permuting vectors eni's yields an inclusion n GL(V 0) of the symmetric group*
* n in
GL (V 0). The action of n by the conjugation on GL W normalizes the subgroups *
*TW and
PW . Thus n acts as on M(n) so on Gr(n) and the morphism q O r : M(n) ! Gr(n)
respects this action. Note that the action of n on Gr(n) is trivial and the ac*
*tion of n
on E*,*(M(n)) permutes the variable t1, t2, . .,.tn. Thus the image of (q O r)**
* is contained
in E*,*(k)[[oe1, oe2, . .,.oen]]. Whence the same holds for the image of q*. Th*
*e Theorem is
__
proven. |_*
*_|
The projection from the product Gr(m) x Gr(n), to the j-th factor is called *
*pj. For
every integer i 0 set c0i= p*1(ci(T(m))) and c00i= p*2(ci(T(n)))
Theorem 2.0.7. Suppose E is an oriented commutative P1-ring spectrum. There is *
*an
isomorphism
*,* 0 0 0 00 00 00
E*,*(Gr (m) x Gr(n)) = E (k)[[c1, c2, . .,.cm , c1, c2, . .,.cn]]
is the formal power series on the c0i's and c00j's. The inclusion im,n:G(m) x *
*Gr(n) ,!
G(m+1)xG(n+1) satisfies i*m,n(c0r) = c0rfor r < m+1, i*m,n(c0m+1) = 0, and i*m,*
*n(c00r) = c00r
for r < n + 1, i*m,n(c00n+1) = 0.
*
* __
Proof. Follows as in the proof of Theorem 2.0.6. *
* |__|
2.1 The symmetric ring spectrum representing algebraic cobor-
dism
To give a construction of the symmetric P1-ring spectrum MGL , recall the exter*
*nal prod-
uct of Thom spaces described in [V1 , p. 422]. For vector bundles V ! X and W !*
* Y
in Sm=S one obtains a canonical map of pointed motivic spaces Th (V ) ^ Th(W ) !
Th (V xS W ) which is a motivic weak equivalence as defined in [PPR1 , Defn. 3.*
*1.6]. In
fact, the canonical map becomes an isomorphism after Nisnevich (even Zariski) s*
*heafifi-
cation.
The algebraic cobordism spectrum appears naturally as a T -spectrum, not as *
*a P1-
spectrum. Hence we describe it as a symmetric T -ring spectrum and obtain a sym*
*metric
P1-ring spectrum (and in particular a P1-ring spectrum) by switching the suspen*
*sion
coordinate (see [PPR1 , A.6.9]). For m, n 0 let T(n, mn) ! Gr (n, mn) denote*
* the
tautological vector bundle over the Grassmann scheme of n-dimensional linear su*
*bspaces
of AmnS= AmSxS. .x.SAmS. Permuting the copies of AmSinduces a n-action on T(n,*
* mn)
and Gr(n, mn) such that the bundle projection is equivariant. The closed embed*
*ding
AmS = AmSx {0} ,! Am+1Sdefines a closed n-equivariant embedding Gr (n, mn) ,!
Gr (n, (m + 1)n). In particular, Gr(n, mn) is pointed by gn: S = Gr(n, n) ,! Gr*
*(n, mn).
The fiber of Gr(n, mn) over gn is AnS. Let Gr(n) be the colimit of the sequence
Gr(n, n) ,! Gr(n, 2n) ,! . .,.! Gr(n, mn) ,! . . .
7
in the category of pointed motivic spaces over S. The pullback diagram
T(n, mn)______//T(n, (m + 1)n)
| |
| |
fflffl| fflffl|
Gr (n, mn)____//_Gr(n, (m + 1)n)
induces a n-equivariant inclusion of Thom spaces
Th (T(n, mn)) ,! Th(T(n, (m + 1)n)).
Let MGL ndenote the colimit of the resulting sequence
MGL n = colimTh(T(n, mn)) (4)
m n
with the induced n-action. There is a closed embedding
Gr (n, mn) x Gr(p, mp) ,! Gr(n + p, m(n + p)) (5)
which sends the linear subspaces V ,! Amn and W ,! Amp to the product subspace
V x W ,! Amn x Amp = Am(n+p). In particular (gn, gp) maps to gn+p. The inclusio*
*n (5)
is covered by a map of tautological vector bundles and thus gives a canonical m*
*ap of
Thom spaces
Th (T(n, mn)) ^ Th(T(p, mp)) ! Th(T(n + p, m(n + p))) (6)
which is compatible with the colimit (4). Furthermore, the map (6) is nx p-equ*
*ivariant,
where the product acts on the target via the standard inclusion n x p n+p.*
* After
taking colimits, the result is a n x p-equivariant map
~n,p:MGL n^ MGL p! MGL n+p (7)
of pointed motivic spaces (see [V1 , p. 422]). The inclusion of the fiber Ap ov*
*er gp in T(p)
induces an inclusion Th (Ap) Th (T(p)) = MGL p. Precomposing it with the can*
*onical
p-equivariant map of pointed motivic spaces
Th (A1) ^ Th(A1) ^ . .^.Th(A1) ! Th(Ap)
defines a family of maps ep: ( 1TS+)p = T ^p! MGL p. Inserting it in the inclu*
*sion (7)
yields n x p-equivariant structure maps
MGL n ^ Th(A1) ^ Th(A1) ^ . .^.Th(A1) ! MGL n+p (8)
of the symmetric T -spectrum MGL . The family of n x p-equivariant maps (7) *
*form a
commutative, associative and unital multiplication on the symmetric T -spectrum*
* MGL
(see [J, Sect. 4.3]). Regarded as a T -spectrum it coincides with Voevodsky's *
*spectrum
MGL described in [V1 , 6.3].
8
__
Let T be the Nisnevich sheaf associated to the presheaf X 7! P1(X)=(P1 - {0}*
*)(X)
on the Nisnevich site Sm=S. The canonical covering of P1 supplies an isomorphism
~= __
T = Th(A1S)____//_T
of pointed motivic spaces. This isomorphism induces an isomorphism_MSS_ T(S) *
*~=
MSS __T(S) of the categories of symmetric T -spectra_and symmetric T -spectra. *
* In par-
ticular, MGL may be regarded as a symmetric T-spectrum by just changing the st*
*ructure
maps up to an isomorphism. Note that the isomorphism of categories respects bo*
*th
the symmetric_monoidal structure and the model structure. The canonical projec*
*tion
p: P1 ! T is a motivic weak equivalence, because A1 is contractible. It induces*
* a Quillen
equivalence
_p]_//_
MSS (S) = MSS P1(S)oo___MSS __T(S)
p*
when equipped with model structures as described in_[J] (see [PPR1 , A.6.9]). T*
*he right
adjoint p* is very simple: it sends a symmetric T -spectrum E to the symmetric*
* P1-
spectrum having terms p*(E) n = En and structure maps
En^p __structure map
En ^ P1__________//_E ^ T_________//_En+1.
In particular MGL := p*MGL is a symmetric P1-spectrum by just changing the s*
*tructure
maps. Since p* is a lax symmetric monoidal functor, MGL is a commutative monoi*
*d in a
canonical way. Finally, the identity is a left Quillen equivalence from the mod*
*el category
MSS cm(S) used in [PPR1 ] to Jardine's model structure by the proof of [PPR1 , *
*A.6.4]. Let
fl :Ho(MSS cm(S)) ! SH (S) denote the equivalence obtained by regarding a symm*
*etric
P1-spectrum just as a P1-spectrum.
Definition 2.1.1. Let (MGL , ~MGL , eMGL ) denote the commutative P1-ring spec*
*trum
which is the image fl(MGL ) of the commutative symmetric P1-ring spectrum MGL *
* in the
motivic stable homotopy category SH (S).
2.2 A universality theorem for the algebraic cobordism spec-
trum
The complex cobordism spectrum, equipped with its natural orientation, is a uni*
*versal
oriented ring cohomology theory by Quillen's universality theorem [Q ]. In thi*
*s section
we prove a motivic version of Quillen's universality theorem. The statement is *
*contained
already in [Ve]. Recall that the P1-ring spectrum MGL carries a canonical or*
*ientation
thMGL as defined in 1.1.3. It is the canonical map thMGL : 1P1(T h(O(-1)))(-1) *
*! MGL
of P1-spectra.
9
Theorem 2.2.1 (Universality Theorem). Let E be a commutative P1-ring spectrum a*
*nd
let S = Spec(k) for a field k. The assignment ' 7! '(thMGL ) 2 E2,1(Th (T(1))) *
*identifies
the set of monoid homomorphisms
': MGL ! E (9)
in the motivic stable homotopy category SH cm(S) with the set of orientations o*
*f E. The
inverse bijection sends an orientation th 2 E2,1(Th (T(1))) to the unique morph*
*ism
' 2 E0,0(MGL ) = Hom SH(S)(MGL , E)
such that u*i(') = th(T(i)) 2 E2i,i(Th (T(i))), where th(T(i)) is given by (2) *
*and
ui: 1P1(Th (T(i)))(-i) ! MGL is the canonical map of P1-spectra.
Proof.Let ': MGL ! E be a homomorphism of monoids in SH (S). The class th :=
'(thMGL ) is an orientation of E, because
'(th)|Th(1)= '(th|Th(1)) = '( P1(1)) = P1('(1)) = P1(1).
Now suppose thE 2 E2i,i(Th (O(-1))) is an orientation of E. We will construct a*
* monoid
homomorphism ': MGL ! E in SH(S) such that u*i(') = th(T(i)) and prove its uni*
*que-
ness. To do so, we compute E*,*(MGL ). By [PPR1 , Cor. 2.1.4], this group fits*
* into the
short exact sequence
0 ! lim-1E*+2i-1,*+i(Th (T(i))) ! E*,*(MGL ) ! lim-E*+2i,*+i(Th (T(i))) !*
* 0
where the connecting maps in the tower are given by the top line of the commuta*
*tive
diagram
-1P1
E*+2i-1,*+i(ThOi)ooE*+2i+1,*+i+1(Th_i^OP1)oo_E*+2i+1,*+i+1(ThOi+1)OOO
-[th(T(i))|| ffl*O(-[th(T(i)|1))| -[th(T(i+1))||
| | inc* |
E*,*(Gr (i))oo__id_____E*,*(Gr (i))oo__i_____E*,*(Gr (i + 1))
Here ffl: Th(V ) ^ P1 ! T h(V 1) is the canonical map. The pull-backs inc*ia*
*re all
surjective by Theorem 2.0.4. So we proved the following
Claim 2.2.2. The canonical map
E*,*(MGL ) ! lim-E*+2i,*+i(Th (T(i))) = E*,*(k)[[c1, c2, c3, . .].]
is an isomorphism of two-sided E*,*(k)-modules.
10
The family of elements th(T(i)) is an element in the lim--group, thus there *
*is a unique
element ' 2 E0,0(MGL ) with u*i(') = th(T(i)). We claim that ' is a monoid hom*
*omor-
phism. To check that it respects the multiplicative structure, consider the dia*
*gram
1P1(~i,j)(-i-j)
1P1(Th (T(i)))(-i) ^ 1P1(Th (T(j)))(-j)_______//_ 1P1(Th (T(i + j)))(-i - *
*j)
ui^uj|| ui+j||
fflffl| ~MGL fflffl|
MGL ^ MGL _________________________________//_MGL
'^'|| '||
fflffl| ~E fflffl|
E ^ E _____________________________________//_E.
Its enveloping square commutes in SH (S) by the chain of relations
' O ui+jO 1P1(~i,j)(-i -=j) ~*i,j(th(T(i + j))) = th(~*i,j(T(i + j))) = th(T(i*
*) x T(j))
= th(T(i)) x th(T(j)) = ~E(th(T(i)) ^ th(T(j)))
= ~E O ((' O ui) ^ (' O uj)).
To obtain the equality ~E O (' ^ ') = ' O ~MGL in SH(k) consider the short exac*
*t sequence
0 ! lim-1E*+4i-1,*+2i(Th (T(i)) ^ Th(T(i))) ! E*,*(MGL ^ MGL )
! lim-E*+4i,*+2i(T h(T(i)) ^ T h(T(i))) ! 0.
Note that since Th (T(i)) ^ Th(T(i)) ~= Th(T(i) x T(i)), there is a Thom isomor*
*phism
E*+4i-1,*+2i(Th (T(i) x T(i))) ~=E*-1,*(Gr (i) x Gr(i)) by Theorem 2.0.4. The l*
*im-1-group
is trivial because the connecting maps coincide with the pull-back maps
E*-1,*(Gr (i + 1) x Gr(i + 1)) ! E*-1,*(Gr (i) x Gr(i))
and these are surjective by Theorem 2.0.7. This implies the following
Claim 2.2.3. The canonical map
E*,*(MGL ^ MGL ) ! lim-E*+2i,*+i(Th (T(i)) ^ Th(T(i))) =
E*,*(k)[[c01, c001, c02, c002, . .].]
is an isomorphism of two-sided E*,*(k)-modules. Here c0iis the i-th Chern class*
* coming
from the first factor of Gr x Gr and c00iis the i-th Chern class coming from th*
*e second
factor.
Now the equality
' O ui+iO 1P1(~i,i)(-2i) = ~E O ((' O ui) ^ (' O ui))
shows that ~E O (' ^ ') = ' O ~MGL in SH (k).
11
To prove the Theorem it remains to check that the two assignments described *
*in
the Theorem are inverse to each other. An orientation th 2 E2,1(Th (O(-1))) in*
*duces
a morphism ' such that for each i one has ' O ui = th(Ti). And the new orienta*
*tion
th0:= '(thMGL ) coincides with the original one, due to the chain of relations
th0= '(thMGL ) = '(u1) = ' O u1 = th(T1) = th(O(-1)) = th.
On the other hand a monoid homomorphism ' defines an orientation th := '(thM*
*GL )
of E. The monoid homomorphism '0we obtain then satisfies u*i('0) = th(Ti) for e*
*very
i 0. To check that '0= ', recall that MGL is oriented, so we may use Claim 2*
*.2.2with
E = MGL to deduce an isomorphism
MGL *,*(MGL ) ! lim-MGL *+2i,*+i(Th (T(i))).
This isomorphism shows that the identity '0= ' will follow from the identities *
*u*i('0) =
u*i(') for every i 0. Since u*i('0) = th(Ti) it remains to check the relatio*
*n u*i(') =
th(Ti). It follows from the
Claim 2.2.4. There is an equality ui= thMGL (Ti) 2 MGL 2i,i(Th (T(i))).
In fact, u*i(') = ' O ui= '(ui) = '(thMGL (T(i))) = th(T(i)). The last equal*
*ity in this
chain of relations holds, because ' is a monoid homomorphism sending thMGL to t*
*h. It
remains to prove the Claim. We will do this in the case i = 2. The general case*
* can be
proved similarly. The commutative diagram
1P1(~1,1)(-2)
1P1Th(T(1))(-1) ^ 1P1Th(T(1))(-1)_________//_ 1P1Th(T(2))(-2)
u1^u1|| |u2|
fflffl| ~MGL fflffl|
MGL ^ MGL __________________________//_MGL
in SH (k) implies that
~*1,1(u2) = u1 x u1 2 MGL 4,2(Th (T(1)) ^ Th(T(1))) = MGL 4,2(Th (T(1) x T(*
*1))).
The equalities
~*1,1(thMGL (T(2)))= thMGL (~*1,1(T(2))) = thMGL (T(1) x T(1))
= thMGL (T(1)) x thMGL (T(1))
imply that it remains to prove the injectivity of the map ~*1,1. Consider the c*
*ommutative
diagram
~*1,1
MGL *,*(ThO(T(1)Ox T(1)))ooMGL_*,*(ThO(T(2)))O
Thom|~=| ~=Thom||
| *1,1 |
MGL *,*(Gr (1) x Gr(1))oo__MGL *,*(Gr (2))
12
where the vertical arrows are the Thom isomorphisms from Theorem 2.0.4and 1,1:
Gr(1) x Gr(1) ,! Gr (2) is the embedding described by equation (5). For an ori*
*ented
commutative ring P1-spectrum (E, th) one has E*,*(Gr (2)) = E*,*(k)[[c1, c2]] (*
*the formal
power series on c1, c2) by Theorem 2.0.6. From the other hand
E*,*(Gr (1) x Gr(1)) = E*,*(k)[[t1, t2]]
(the formal power series on t1, t2) by Theorem 2.0.7and the map *1,1takes c1 t*
*o t1 + t2
and c2 to t1t2. Whence *1,1is injective. The proofs of the Claim and of the Th*
*eorem are
__
completed. |_*
*_|
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