MOTIVIC CELL STRUCTURES
DANIEL DUGGER AND DANIEL C. ISAKSEN
Abstract.An object in motivic homotopy theory is called cellular if it c*
*an be
built out of motivic spheres using homotopy colimit constructions. We ex*
*plore
some examples and consequences of cellularity. We explain why the algebr*
*aic
K-theory and algebraic cobordism spectra are both cellular, and prove so*
*me
Künneth theorems for cellular objects.
Contents
1. Introduction 1
2. Cellular objects 3
3. Basic results 8
4. Grassmannians and Stiefel varieties 10
5. Other examples of cellular varieties 12
6. Algebraic K-theory and algebraic cobordism 13
7. Cellularity and stable homotopy 15
8. Finite cell complexes and Künneth theorems 19
9. Stable homotopy groups of filtered colimits 22
References 26
1.Introduction
If M is a model category, and A is a set of objects in M, one can consider th*
*e class
of A-cellular objects_things that can be built from the homotopy types in A
by iterative homotopy colimit constructions. In the case of topological spaces *
*such
cellular classes have been studied by Farjoun [DF ] and others. Another place t*
*hese
ideas have appeared is in the work of Dwyer, Greenlees, and Iyengar [DGI ], who
imported them into homological algebra. This paper is concerned with cellularity
in motivic homotopy theory.
Recall that in the motivic context there is a bi-graded family of `spheres' S*
*p,q;
we will take this family as our set A. One gets a slightly different theory dep*
*ending
on whether one works unstably or stably. In this paper we develop the basic the*
*ory
concerning cellular objects in both contexts, and collect an assortment of resu*
*lts
which we've found useful in applications. Specifically:
(1)We describe a collection of techniques for showing that schemes are cellular,
and apply these to toric varieties, Grassmannians, Stiefel manifolds, and ce*
*rtain
quadrics.
(2)We show that the algebraic K-theory spectrum KGL and the motivic cobor-
dism spectrum MGL are stably cellular.
1
2 DANIEL DUGGER AND DANIEL C. ISAKSEN
(3)For cellular objects, the usual collection of tools for computing carries ov*
*er
from ordinary stable homotopy theory to motivic stable homotopy theory (see
Section 7). If E is a motivic ring spectrum then we use these ideas to const*
*ruct
a convergent, tri-graded, Künneth spectral sequence for E*,*(X x Y ) as long
as X or Y satisfies some kind of cellularity condition. See Theorems 8.6 and
8.12.
Our experience has been that this material is a good starting point for under-
standing some of the inner workings of motivic homotopy theory, and so we have
tried to make the paper readable to people who only know the basic definitions
from [MV ].
1.1. Notions related to cellularity. Algebraic geometers have worked with the
related notion of a scheme with an algebraic cell decomposition [F2, Ex. 1.9.1]*
*, and
its generalization to that of a linear variety (introduced by Totaro [T ]). A s*
*cheme
X has an algebraic cell decomposition if it has a filtration by closed subschem*
*es
X = Xn Xn-1 . . .X0 ;`such that each complement Xi+1- Xi is a
disjoint union of affines nijAnij. The linear varieties constitute the smalle*
*st class
which contains the affine spaces An and has the property that if Z ,! X is a cl*
*osed
inclusion and at least two of the varieties Z, X, and X - Z are in the class, t*
*hen
so is the third.
These notions are useful when studying cohomology theories which have a lo-
calization (or Gysin) sequence, because the cohomology of a linear variety can *
*be
understood inductively. In the language of motivic homotopy theory these are the
algebraically oriented cohomology theories, i.e., the ones that have Thom isomo*
*r-
phisms. This means that the cohomology of the Thom space of a bundle over Z
is isomorphic to the cohomology of Z (up to a shift). If Z ,! X is a closed in-
clusion of smooth schemes then there is a homotopy cofiber sequence of the form
X - Z ! X ! Th NX=Z [MV , Thm. 3.2.23], where Th NX=Z is the Thom space
of the normal bundle of Z in X. One gets a long exact sequence relating the
cohomology of X - Z, X, and Z.
Our class of stably cellular varieties is very close to the class of linear v*
*arieties.
For every linear variety which we've tried to prove is stably cellular, we've b*
*een able
to do so (and vice versa); however, proving that something is cellular is often*
* much
harder. This is true for the Grassmannians Grk(An), for instance. The Schubert
cells give an `algebraic cell decomposition' showing that the Grassmannian is l*
*inear,
but to show the variety is cellular it is not enough just to see the cells insi*
*de the
variety: one has to produce an `attaching map' showing explicitly how to build
up the variety via homotopy colimits. In the Schubert cell approach one runs in*
*to
some hairy problems in trying to make this work, which we have not been able to
resolve. Our proof that Grassmannians are cellular follows a completely differe*
*nt
strategy.
If one is only interested in cohomology theories with Thom isomorphism, then
perhaps there is no reason for studying cellular varieties as opposed to linear*
* ones.
But the notion of `cellular' seems more familiar and sensible to a topologist, *
*and
most of our techniques for understanding the classical stable homotopy category
depend in some way on things being built from cells. As those techniques get
imported into motivic homotopy theory, the notion of cellularity may become more
useful.
MOTIVIC CELL STRUCTURES 3
1.2. Non-cellular varieties. Folklore says most schemes cannot be cellular. This
should be a consequence of the theory of weights in the cohomology of algebraic
varieties [De ]. The spheres Sp,qonly have even weights in their cohomology, and
so it should be impossible to construct something with odd weights (like an ell*
*iptic
curve, for instance) from the spheres.
Unfortunately, to write down a careful proof that an elliptic curve is not ce*
*llular
seems to require surmounting some obstacles. One possibility is to work over C *
*and
use mixed Hodge theory, but this requires showing that the mixed Hodge structur*
*es
are well-defined invariants of the motivic stable homotopy category. This takes*
* at
least a little work, due to the presence of infinite objects (like infinite wed*
*ges of
schemes, etc.) in the stable homotopy category.
Another possibility is to work over a number field k, and to use the weights
coming from the Galois actions on l-adic cohomology (again see [De ]). Here, one
should show that there is a realization functor from the motivic stable homotopy
category to the derived category of Gal(~k=k)-modules. Proposition 9.6 shows th*
*at
if an elliptic curve E is cellular then it can actually be built from spheres u*
*sing a
finite number of extensions and retracts, and so the argument with weights shou*
*ld
work out. We will not pursue these ideas further in this paper.
2.Cellular objects
Let M be a pointed model category, and let A be a set of objects in M.
Definition 2.1. The class of A-cellular objects is the smallest class of object*
*s of
M such that
(1)every object of A is A-cellular;
(2)if X is weakly equivalent to an A-cellular object, then X is A-cellular;
(3)if D :I ! M is a diagram such that each Diis A-cellular, then so is hocolimD.
The idea is that the A-cellular objects are the ones that can, up to homotopy*
*, be
built out of objects in A. This definition is precisely the same as the usual n*
*otion
of cellularity for the category of pointed topological spaces [DF , Ex. 2.D.2.1*
*].
We recall some useful results about cellularity in general. These properties *
*apply
to all possible M and A. To start with, note that any contractible object of M
(i.e., an object weakly equivalent to the initial and terminal object *) is A-c*
*ellular
because it is the homotopy colimit of an empty diagram.
Lemma 2.2. If X ! Y ! Z is a homotopy cofiber sequence in M such that X
and Y are A-cellular, then so is Z.
Proof.The object Z is the homotopy pushout of the diagram * X ! Y .
Lemma 2.3. If X is A-cellular, then so is X.
Proof.Apply Lemma 2.2 to the homotopy cofiber sequence X ! * ! X.
Recall that a pointed model category is stable if the suspension and loops fu*
*nc-
tors and form a self-equivalence of the homotopy category. Throughout the
paper we will abuse notation and also write (resp., ) for a chosen derived
functor of suspension (resp., loops) on the model category level. For instance,*
* to
compute ~X we can factor X ! * as a cofibration followed by trivial fibration
X æ CX -i * and then take CX=X.
4 DANIEL DUGGER AND DANIEL C. ISAKSEN
Lemma 2.4. Suppose that M is a stable model category. Also suppose that for
every object A of A, A is weakly equivalent to an object of A. An object X in M
is A-cellular if and only if X is A-cellular.
Proof.Consider the class C of objects X of M such that X is cellular; we want
to show that C coincides with the class of A-cellular objects. By Lemma 2.3, C
is contained in the class of A-cellular objects. To show that C contains the cl*
*ass
of A-cellular objects, it suffices to check that C has the three properties lis*
*ted in
Definition 2.1.
Property (1) follows immediately from the hypothesis of the lemma, property (*
*2)
follows immediately from the fact that respects weak equivalences, and proper*
*ty
(3) follows from the fact that respects homotopy colimits in a stable model
category.
Lemma 2.5. Suppose that M is a stable model category. Also suppose that for
every object A of A, A is weakly equivalent to an object of A. If X ! Y ! Z is*
* a
homotopy cofiber sequence in M such that any two of X, Y , and Z are A-cellular,
then so is the third.
Proof.One case is Lemma 2.2. For the other two cases, first observe from Lemma
2.4 that Y is A-cellular whenever Y is (and similarly for Z). Now, the object
Y is the homotopy pushout of the diagram * Z ! X, and the object X is the
homotopy pushout of the diagram * Y ! Z.
2.6. The motivic setting. Let MV denote the Morel-Voevodsky model category
on simplicial presheaves over Smk, the site of smooth schemes of finite type ov*
*er
some fixed ground field k. In fact, there are at least two versions of this mo*
*del
category, the injective [MV ], [Ja, App. B] and projective [Bl]. The identity f*
*unctors
give a Quillen equivalence between these model categories (which have the same
class of weak equivalences), and this guarantees that it doesn't matter which m*
*odel
structure is considered for the purposes of cellularity. Thus, in each situatio*
*n we
can choose whichever structure is more convenient. Unless otherwise stated, we
will use the injective version because it is convenient to have all objects cof*
*ibrant.
Actually, we will work with the pointed category MV*, in which every object is
equipped with a map from Spec(k).
We recall the following two important kinds of weak equivalences in MV. First,
if {U, V } is a basic Nisnevich cover of X [MV , Defn. 3.1.3], then the map
U qUxXV V ! X
is an A1-weak equivalence. Second, if X is any object of Smk, then the map
X x A1 ! X is an A1-weak equivalence. In a certain sense, these two kinds of
maps generate all A1-weak equivalences_cf. [D , Sec. 8.1], especially the parag*
*raph
following the proof of (8.1). Every proof that an object is cellular must neces*
*sarily
come down to using these two facts.
Actually, in this paper we will never have to explicitly use the Nisnevich
topology_all our arguments only involve Zariski covers and facts from [MV ]. Al*
*so,
the A1-weak equivalence for two-fold covers given in the last paragraph implies*
* a
corresponding statement for Zariski covers of any size: the A1-homotopy type of*
* a
scheme can be recovered from a Zariski cover by an appropriate gluing construct*
*ion.
This is what we will mostly use (see Lemma 3.6 below for a precise statement).
MOTIVIC CELL STRUCTURES 5
Recall that S1,1= A1-0, with the point 1 as basepoint; and S1,0is the constant
simplicial presheaf whose sections are the simplicial set 1=@ 1. For p q 0,
one defines
Sp,q= (S1,0^ . .^.S1,0) ^ (S1,1^ . .^.S1,1)
where there are p - q copies of S1,0and q copies of S1,1.
Definition 2.7. Let A = {Sp,q| p q 0} be the set of spheres in MV*. An obje*
*ct
X in MV* is unstably cellular if X is A-cellular.
If X is a pointed (possibly non-smooth) scheme, then the statement "X is un-
stably cellular" is to be interpreted as saying that the object of MV* represen*
*ted
by X is unstably cellular.
2.8. Motivic spectra. We let Spectra(MV) denote the category of symmetric P1-
spectra. Starting with the injective model structure on MV*, we get an induced
model structure on Spectra(MV) from [Ho2 , Defn. 8.7]. This turns out to be ide*
*n-
tical to the one provided by [Ja, Thm. 4.15]. Note that there is a Quillen pair
1 :MV* Æ Spectra(MV): 1 . We may desuspend the objects 1 (Sp,q) in both
variables, giving spectra which we will denote Sp,qfor all p, q 2 Z.
Remark 2.9. One can also use Bousfield-Friedlander P1-spectra rather than sym-
metric spectra. The model structure is provided by [Ho2 , Defn. 3.3] or [Ja,
Thm. 2.9]. Since this model category is Quillen equivalent to that of symmet-
ric P1-spectra [Ho2 , Sec. 10], all our basic results hold in either one. The s*
*mash
product for symmetric spectra will be important in Sections 7 and 8, however.
Definition 2.10. Let B = {Sp,q| p, q 2 Z} be the class of all spheres in
Spectra(MV). An object X of Spectra(MV) is cellular if X is B-cellular. We say
that an object X in MV* is stably cellular if 1 X is cellular.
Again, the statement that a (possibly non-smooth) pointed scheme is `stably
cellular' is really an abbreviation for the same statement about the object of *
*MV*
that it represents.
Example 2.11. An - 0 is unstably cellular, because after choosing any basepoint,
the variety An-0 is weakly equivalent to the sphere S2n-1,n. This fact is claim*
*ed in
[MV , Ex. 3.2.20]. For the convenience of the reader, we give a detailed explan*
*ation.
For n = 1 this is the definition that S1,1equals A1 - 0. For n = 2, we cover
A2 - 0 by the open sets U = (A1 - 0) x A1 and V = A1 x (A1 - 0). Then
U \ V = (A1- 0) x (A1- 0), and we have an associated homotopy pushout square
(A1 - 0) x (A1 - 0)__//_(A1 - 0) x A1
| |
| |
fflffl| fflffl|
(A1 - 0) x A1_________//A2 - 0.
By projecting away the A1 factors, we find that A2 - 0 is weakly equivalent to *
*the
homotopy pushout of
ß1 ß2 1
A1 - 0oo___(A1 - 0) x (A1 - 0)___//(A - 0).
6 DANIEL DUGGER AND DANIEL C. ISAKSEN
In order to compute this homotopy pushout, we look at the diagram
*Ooo_____________*O______________//_*OOOO
| | |
| | |
| | |
A1 - 0oo___(A1 - 0) _ (A1 - 0)___//A1 - 0
| | |
| | |
fflffl| fflffl| fflffl|
A1 - 0oo___(A1 - 0) x (A1 - 0)___//A1 - 0
in which the two middle horizontal arrows collapse one summand to a point, and
compute the homotopy colimit in two ways. If we first compute the homotopy
colimits of the rows and then the homotopy colimit of this new diagram, we get *
*the
desired homotopy pushout. On the other hand, if we first compute the homotopy
colimits of the vertical columns, then we get
* - (A1 - 0) ^ (A1 - 0) -! *,
and then the homotopy colimit of this new diagram is ((A1 - 0) ^ (A1 - 0)).
Since we must get the same homotopy type no matter which way we go about
computing the homotopy colimit, it must be that the desired homotopy pushout is
[(A1 - 0) ^ (A1 - 0)], which is S3,2.
For arbitrary n one proceeds by induction, covering An - 0 by (An-1 - 0) x A1
and An x (A1 - 0) and using the same argument to see that An - 0 is weakly
equivalent to [(An-1 - 0) ^ (A1 - 0)].
Example 2.12. According to [MV , Cor. 3.2.18], there is a cofiber sequence
Pn-1 ! Pn ! S2n,n
in MV* after choosing any basepoint for Pn-1. This shows inductively that Pn is
stably cellular. We will show below that Pn is in fact unstably cellular.
For n 1, there is a canonical projection map An - 0 ! Pn-1 sending a point
v to the line spanned by v. Also, there is a natural inclusion Pn-1 ,! Pn coming
from the inclusion An-1 ,! An-1 A1 = An.
Proposition 2.13. For n 1, there is a homotopy cofiber sequence An - 0 !
Pn-1 ,! Pn in MV* after choosing any basepoint in An - 0. Thus each Pn is
unstably cellular.
Proof.We decompose An+1 into An A1. Let l be the line spanned by the vector
(0, 1) (where the notation is with respect to this decomposition), and let U =
Pn - {l}. There is an open embedding An ,! Pn which sends v to the line spanned
by (v, 1)_call this open subset V . Then U \ V is isomorphic to An - 0, and we
have a homotopy pushout square
An - 0 ____//_Pn - {l}
| |
| |
fflffl| fflffl|
An ________//_Pn.
Since An is contractible, this identifies Pn with the homotopy cofiber of An - *
*0 !
Pn - {l}.
MOTIVIC CELL STRUCTURES 7
Now, there is a projection map Pn - {l} ! Pn-1 induced by the obvious projec-
tion An-1 A1 ! An-1. This is a line bundle, and is therefore a weak equivalen*
*ce
(in fact, an A1-homotopy equivalence). The composite An-0 ,! Pn-{l} ! Pn-1 is
precisely our projection map An - 0 ! Pn-1. So the homotopy cofiber of An - 0 !
Pn - {l} is weakly equivalent to the homotopy cofiber of An - 0 ! Pn-1.
Remark 2.14. Note that the homotopy cofiber sequence A1-0 ! * ! P1 identifies
P1 with (A1 - 0) ' (S1,1) ' S2,1.
2.15. Basepoints. When working with unstable cellularity one has to be a little
careful about basepoints. Here is one case where the issue disappears:
Proposition 2.16. Suppose X is an object of MV and a, b: * ! X are two choices
of basepoint. If a and b are weakly homotopic in MV, then (X, a) is weakly equi*
*va-
lent to (X, b) in MV* (hence one is unstably cellular if and only if the other *
*is).
Proof.First, one readily reduces to the case where X is fibrant. Let X0 be the
pushout in MV of the diagram
X -a * -i0!A1
where i0 is the inclusion of {0}; the idea is that X0 is X with an `affine whis*
*ker'
attached. The homotopy H : A1 x A1 ! A1 that takes (s, t) to st can be used to
show that the map X0! X which collapses the A1 onto a is a weak equivalence in
MV. If we regard X0 as pointed via the map 1 ! A1, then the same map is a weak
equivalence (X0, 1) ! (X, a) in MV*.
Since a and b are weakly homotopic and X is fibrant, there is a map H :A1 ! X
such that Hi0 = a and Hi1 = b. This induces a map X0 ! X sending 1 to b,
which is readily checked to be a weak equivalence. So we have a zig-zag of weak
equivalences in MV* of the form (X, a) (X0, 1) ! (X, b).
The applicability of the above result is limited by the fact that in motivic *
*ho-
motopy theory Ho (*, X) is often bigger than one would expect. For instance
Ho (*, A1 - 0) = k*, which is extremely big if k is the complex numbers. For
A1- 0 the basepoint doesn't matter for another reason, namely because all choic*
*es
are equivalent up to automorphism.
It turns out that we will almost always work with stable cellularity, and in *
*this
context the basepoint can be ignored:
Proposition 2.17. If X 2 MV*, then X is stably cellular if and only if X+ is
stably cellular. As a consequence, if X 2 MV has two basepoints x0 and x1 then
(X, x0) is stably cellular if and only if (X, x1) is stably cellular.
Proof.The first statement follows from considering the cofiber sequence 1 S0,0!
1 (X+ ) ! 1 X and applying Lemma 2.5. The second statement is true because
either condition is equivalent to X+ being stably cellular.
Because of Proposition 2.17 we can now rephrase the definition of stable cell*
*u-
larity.
Definition 2.18. A (pointed or unpointed) object X of MV is stably cellular if
1 (X+ ) is stably cellular in Spectra(MV).
This is the definition that we will use from now on. It saves us the trouble *
*of
having to worry about basepoints.
8 DANIEL DUGGER AND DANIEL C. ISAKSEN
3.Basic results
The notions of unstable cellularity and stable cellularity are related by the*
* fol-
lowing lemma.
Lemma 3.1. If X is an unstably cellular object of MV*, then it is also stably
cellular.
Proof.The functor 1 is a left Quillen functor, so it respects weak equivalence*
*s and
homotopy colimits. Thus, it suffices to show that 1 Sp,qis stably cellular. Bu*
*t this
is isomorphic to the stable sphere Sp,q, which is stably cellular by definition.
We will now study the basic constructions that behave well with respect to
cellularity.
`
Lemma 3.2. If each Xi is a stably cellular object of MV, then so is X = iXi.
Proof.ThisWfollows from the simple calculation that 1 (X+ ) is isomorphic to
i 1 (Xi+).
The set of spheres is closed under smash product. This implies that smash
products preserve unstably cellular objects.
Lemma 3.3. If X and Y are unstably cellular objects, then so is X ^ Y .
Proof.Since every object in MV* is injective cofibrant, the functor X ^(-) is a*
* left
Quillen functor from the pointed category MV* to itself, so it respects homotopy
colimits and weak equivalences. Thus, it suffices to show that X ^ Sp,qis unsta*
*bly
cellular for every p and q.
But the functor (-)^Sp,qalso respects homotopy colimits and weak equivalences,
so it suffices to show that Ss,t^ Sp,qis unstably cellular. This is isomorphic*
* to
Ss+p,t+q, which is unstably cellular by definition.
Note, in particular, that if X is a pointed unstably cellular object, then so*
* is
p,qX.
Lemma 3.4. If X and Y are stably cellular objects of MV, then so are X ^ Y and
X x Y .
Proof.The proof for X ^ Y works just as in Lemma 3.3, using the facts that
1 (X^Y ) is weakly equivalent to 1 X^ 1 Y and that every suspension spectrum
is cofibrant.
For X x Y , there is an unstable cofiber sequence X _ Y ! X x Y ! X ^ Y , so
we also have a stable cofiber sequence
1 (X _ Y ) ! 1 (X x Y ) ! 1 (X ^ Y ).
We just showed that the third term is stably cellular. The first term is isomor*
*phic
to 1 X _ 1 Y , which is a homotopy colimit of stably cellular spectra and thus
also stably cellular. Hence, the second term is stably cellular as well.
Example 3.5. By Lemma 3.4, the torus (A1 - 0)k is stably cellular for every k.
We do not know if tori are unstably cellular, but we suspect not. Throughout the
rest of the paper we will find that products arise all over the place, which is*
* why
we end up working primarily with stable cellularity.
MOTIVIC CELL STRUCTURES 9
Lemma 3.6. If X is a scheme and U* ! X is a hypercover in MV in the sense of
[SGA4 , Defn. 7.3.1.2] (or [DHI , Defn. 4.2]) and each Un is stably cellular, t*
*hen X
is also stably cellular. If the hypercover is in MV* and each Un is unstably ce*
*llular,
so is X.
Proof.This follows simply from the fact that the map hocolimnUn ! X is a weak
equivalence in MV [DHI , Thm. 6.2]. Then hocolimn 1 (Un+) ! 1 (X+ ) is also
a weak equivalence.
Note that if X is a scheme and {Ui} a Zariski open cover of X, then the assoc*
*iated
~Cech complex is a hypercover in the above sense. It is not necessary that X be
smooth here.
Definition 3.7. A Zariski cover {Uff} of a scheme X is completely stably cel-
lular if each intersection Uff1...ffn= Uff1\ . .\.Uffnis stably cellular.
A similar definition can be made with the notion of unstably cellular, but we
will not bother with it.
Lemma 3.8. If a variety X has a Zariski cover which is completely stably cellul*
*ar,
then X is also stably cellular.
Proof.Let {Uff} be the cover of X. Consider the ~Cech complex`C* of this cover,
which is a simplicial diagram of varieties such that Cn is Uff0ff1...ffn. Now*
* C* is
obviously a hypercover of X in MV (cf. [DHI , 3.4, 4.2]). Each Cn is stably cel*
*lular
by Lemma 3.2, so Lemma 3.6 applies.
By an `algebraic fiber bundle with fiber F ' we mean a map E ! B which in the
Zariski topology is locally isomorphic to a projection U x F ! U.
Lemma 3.9. If p: E ! B is an algebraic fiber bundle with fiber F such that F
is stably cellular and B has a completely stably cellular cover that trivialize*
*s the
bundle, then E is also stably cellular.
Proof.Let {U0, . .,.Uk} be the completely stably cellular cover of B. Consider *
*the
cover {V0, . .V.k} of E, where Vi = p-1Ui. Each Vi is isomorphic to F x Ui, so *
*it
is stably cellular by Lemma 3.4. Moreover, the intersections Vi0...inare isomor*
*phic
to F x Ui0...in, so this cover of E is completely stably cellular. Lemma 3.8 fi*
*nishes
the proof.
Corollary 3.10. If p: E ! B is an algebraic vector bundle such that B has a
completely stably cellular cover that trivializes the bundle, then the Thom spa*
*ce
Th(p) is also stably cellular.
Proof.Let s: B ! E be the zero section of the vector bundle. From the definition
of a Thom space [MV , Defn. 3.2.16], we have an unstable cofiber sequence
E - s(B) ! E ! Th(p),
where s(B) is the zero section of the bundle. So we just have to show that the
first two terms in this sequence are stably cellular. First, E is weakly equiva*
*lent to
B (because E can be fiberwise linearly contracted onto its zero section), and B*
* is
stably cellular by Lemma 3.8.
Next, E - s(B) ! B is an algebraic fiber bundle with fiber An - 0. By Lemma
3.9, we know that E - s(B) is stably cellular provided that An - 0 is. Recall t*
*hat
An - 0 is weakly equivalent to S2n-1,n, so it is stably cellular.
10 DANIEL DUGGER AND DANIEL C. ISAKSEN
Lemma 3.11. If x is a closed point of a smooth variety X, then X is stably cell*
*ular
if and only if X - {x} is stably cellular.
Proof.The homotopy purity theorem [MV , Thm. 3.2.23] tells us that there is a
cofiber sequence in MV of the form
X - {x} ! X ! Th(p),
where p is the normal bundle of {x} in X. Now Th(p) is just An=(An - 0) ' S2n,n,
where n is the dimension of X. Thus we have a cofiber sequence
X - {x} ! X ! S2n,n,
in which the third term is stably cellular. Lemma 2.5 finishes the proof.
Remark 3.12. Suppose that Z ,! X is a closed inclusion of smooth schemes. The
homotopy purity theorem gives a cofiber sequence X - Z ! X ! Th NX=Z. It is
tempting to conclude that Th NX=Z is cellular if Z is cellular, and so X is sta*
*bly
cellular if both Z and X - Z are stably cellular. Unfortunately we haven't been
able to prove the first implication, only the weaker version in Corollary 3.10.*
* This
weakness is the main reason that proving a variety is cellular often feels like*
* more
work than it should be; in particular it is what causes trouble with Grassmanni*
*ans
in the next section.
4. Grassmannians and Stiefel varieties
Proposition 4.1. The variety GLn is stably cellular for every n 1.
Proof.The proof is by induction on n. Let l be the line spanned by (1, 0, . .,.*
*0) in
An. There is a fibre bundle GLn ! Pn-1 that takes A to the line A(l) (where we
have GLn acting on An from the left). The fiber over [1, 0 . .,.0] is the parab*
*olic
subgroup P consisting of all invertible n x n matrices (aij) such that aj1 = 0 *
*for
j 2. As a variety (but not as a group), P is isomorphic to (A1-0)xAn-1xGLn-1,
which is stably cellular by induction and Lemma 3.4. The usual cover of Pn by
affines An is a completely cellular trivializing cover for the bundle, so Lemma*
* 3.8
applies.
Recall that the Grassmannian Grk(An) is the variety of k-planes in An. Also, *
*the
Stiefel variety Vk(An) consists of all ordered sets of k linearly independent v*
*ectors
in An. These objects are connected by a fiber bundle Vk(An) ! Grk(An) that takes
a set of k linearly independent vectors to the k-plane that it spans. The fiber*
* of
this bundle is GLk.
Proposition 4.2. For all n k 0, the Stiefel variety Vk(An) is stably cellul*
*ar.
Proof.Consider the fiber bundle p : Vk(An) ! An - 0 that takes the first vector
in an ordered set of k linearly independent vectors. The fiber of this bundle *
*is
Ak-1x Vk-1(An-1), which we may assume by induction is stably cellular. Because
of Lemma 3.9, it suffices to find a completely stably cellular cover of An - 0 *
*that
trivializes the bundle.
For 1 i n, let Uibe the open set of An-0 consisting of all vectors (x1, .*
* .,.xn)
such that xi6= 0. The intersections of these open sets are of the form (A1 - 0)*
*k x
An-k, so they are stably cellular.
It remains to show that the bundle is trivial over Ui. Without loss of genera*
*lity,
it suffices to consider U1. We regard An-1 as the subset of An consisting of ve*
*ctors
MOTIVIC CELL STRUCTURES 11
whose first coordinate is zero. Let f :U1 x Vk-1(An-1) x Ak-1 ! p-1U1 be the
map
(x, v1, . .,.vk-1, t1, . .,.tk-1) 7! (x, t1x + v1, t2x + v2, . .,.tk-1x + v*
*k-1).
One readily checks that this is an isomorphism.
Let G be an algebraic group. By a principal G-bundle we mean an algebraic
fiber bundle ß :E ! B together with an action E xG ! E, such that ß(eg) = ß(e)
and the induced map E x G ! E xB E sending (e, g) ! (e, eg) is an isomorphism.
Proposition 4.3. If E ! B is a principal G-bundle where both E and G are stably
cellular, then so is B.
Proof.Let C* be the ~Cech complex of the bundle. This means that Cm is the fiber
product E xB E xB . .x.BE (m + 1 copies of E). Because fiber bundles are locally
split, C* is a hypercover of B (cf. [DHI , 3.4, 4.2]). Using Lemma 3.6, we just*
* need
to show that each Cm is stably cellular.
From the definition of a principal bundle, Cm is isomorphic to E x Gm , which
is stably cellular by Lemma 3.4.
Proposition 4.4. For all n k 0, the Grassmannian variety Grk(An) is stably
cellular.
Proof.Consider the fiber bundle Vk(An) ! Grk(An). This is a principal G-bundle
with G = GLk. Thus, Proposition 4.3 applies because of Propositions 4.1 and
4.2.
Remark 4.5. One might also try to prove that Grassmannians are cellular by using
the Schubert cell decomposition. There are various approaches to this, and as f*
*ar
as we know all of them run into unpleasant problems. One possibility, for insta*
*nce,
is to consider the standard open cover of Grk(An) by affines Ak(n-k)_these are
precisely the top-dimensional open Schubert cells. If the finite intersections*
* of
these opens are all cellular, then so is Grk(An) by Lemma 3.8. Unfortunately th*
*ese
finite intersections become complicated, and we have only managed to prove they
are cellular for Gr1(An) and Gr2(An). The general case remains an intriguing op*
*en
question.
Our proof that Grassmannians are cellular generalizes easily to the case of f*
*lag
varieties. Given integers 0 d1 < d2 < . .<.dk n, let Fl(d1, . .,.dk; n) den*
*ote
the variety of flags V1 . . .Vk An such that dimVi= di.
Proposition 4.6. The flag variety Fl(d1, . .,.dk; n) is stably cellular.
Proof.Write Fl= Fl(d1, . .,.dk; n). There is an algebraic fiber bundle Vdk(An) !
Fltaking an ordered set of k linearly independent vectors to the flag whose ith
space is spanned by the first di vectors. This is a principal G-bundle, where G*
* is
the subgroup of GLn consisting of matrices in block form
2 3
A11 A12 . . . A1,k+1
66 0 A22 . . . A1,k+1 7
64 .. . . . 77
. .. .. .. 5
0 0 . . .Ak+1,k+1
12 DANIEL DUGGER AND DANIEL C. ISAKSEN
where A11 2 GLd1, Aii2 GLdi-di-1, and Ak+1,k+12 GLn-dk. As a variety G is
isomorphic to GLd1x GLd2-d1x . .x.GLdk-dk-1x GLn-dkx AN for some number
N. So G is stably cellular, hence Flis stably cellular by Proposition 4.3.
5. Other examples of cellular varieties
5.1. Toric varieties. We will show that every toric variety is weakly equivalent
in MV to a homotopy colimit of copies of tori T m = (A1 - 0)m . Since the tori *
*are
stably cellular by Lemma 3.4, so are toric varieties. This result is almost tri*
*vial in
the smooth case, since all smooth affine toric varieties have the form Anx(A1-0*
*)m .
The singular case takes a tiny bit more work. Recall that a singular variety, w*
*hen
regarded as an element of MV, is really the presheaf that it represents.
For background definitions and results, see [F1]. Let N denote an n-dimension*
*al
lattice; it is isomorphic to Zn, but we work in a coordinate-free context. Let
V be the corresponding R-vector space N R. Recall that if oe is a strongly
convex rational polyhedral cone in V , then one gets a finitely-generated semig*
*roup
Soe= oe_ \ Hom (N, Z) Hom (N, R), where oe_ is the dual cone of oe. The affine
toric variety X(oe) is defined to be Speck[Soe].
Lemma 5.2. If oe generates V as an R-vector space, then the corresponding toric
variety X(oe) is simplicially A1-contractible.
Proof.We need to construct a homotopy H :X(oe) x A1 ! X(oe) such that
H0: X(oe) x 0 ! X(oe) is constant and H1: X(oe) x 1 ! X(oe) is the identity.
To do this, we first choose nonzero generators u1, . .,.ut of Soe. Then we can *
*write
X(oe) as Speck[Y1, . .,.Yt]=I where I is generated by all polynomials of the fo*
*rm
Y1a1Y2a2. .Y.att- Y1b1Y2b2. .Y.btt
such that a1u1 + a2u2 + . .+.atut = b1u1 + b2u2 + . .+.btut (see [F1, Exercise,
p.19]).
We claim that because oe generates V , there is a vector w in V such that >
0 for all i (here <-, -> is the pairing between V *and V ). Accepting this for *
*the
moment, define
H(y1, y2, . .,.yt, s) = (sy1, sy2, . .,.syt).
To see that this map is well-defined, suppose that u = aiui = biui and that
(y1, . .,.yt) satisfies the equation ya11ya22. .y.att- yb11yb22. .y.btt= 0. Then
(sy1)a1(sy2)a2. .(.syt)at-(sy1)b1(sy2)b2. .(.syt)bt
equals
sya11ya22. .y.att- syb11yb22. .y.btt,
which is still equal to zero. Note that H0 is the constant map with value (0, 0*
*, . .,.0),
and that H1 is the identity.
We have only left to produce the vector w. If we had = 0 for all v 2 *
*oe\N,
then the fact that oe generates V and is rational would imply that ui= 0. There*
*fore,
for each i there exists a wi2 oe \ N such that 6= 0. Since ui2 oe_, we*
* must
in fact have > 0. Let w = w1+ . .+.wt. Again using the fact that ui2 o*
*e_
and wj 2 oe, we know that 0 for i 6= j. Hence > 0 for all i.
Proposition 5.3. Let oe be a strongly convex polyhedral rational cone. Then X(o*
*e)
is A1-homotopic to (A1 - 0)m , where m is the codimension of R . oe in V .
MOTIVIC CELL STRUCTURES 13
Proof.Split N as N0 N00so that R . oe = V 0= N0 R and (R . oe) \ V 00= 0,
where V 00= N00 R (cf. [F1, p. 29]). Then oe equals oe0x 0 in V 0x V 00, where*
* oe0
is the same cone as oe except that it lies in V 0. Now X(oe) ~=X(oe0) x (A1 - 0*
*)m
by [F1, p. 5] and [F1, p. 19]. The above lemma shows that X(oe0) is simplicially
A1-contractible, so X(oe) is simplicially A1-homotopic to (A1 - 0)m .
Theorem 5.4. Every toric variety can be obtained via homotopy colimits from the
tori (A1 - 0)m , and hence is stably cellular.
Proof.Given a fan , the toric variety X( ) has an open cover consisting of aff*
*ine
toric varieties X(oe) for cones oe belonging to . Since X(oe) \ X(ø) is equal*
* to
X(oe \ ø), this cover of X( ) has the property that every intersection of piece*
*s of
the cover is A1-homotopic to a torus. The usual argument with ~Cech complexes
tells us that X( ) is a homotopy colimit of the objects X(oe).
5.5. Quadrics. If q(x1, . .,.xn) is a quadratic form then one can look at the a*
*ffine
quadric AQ(q) ,! An defined by q(x1, . .,.xn) = 0, as well as the corresponding
projective quadric Q(q) 2 Pn-1. Note that if q is nondegenerate then AQ(q) is
singular (but only at the origin) whereas Q(q) is nonsingular. The varieties Pn*
*-1 -
Q(q) play a central role in [DI].
Proposition 5.6. If q is hyperbolic and non-degenerate, then An-AQ(q), AQ(q)-
0, Q(q), and Pn-1 - Q(q) are all stably cellular.
Proof.If q is hyperbolic then n is even and after a change of basis we have
q(a1, b1, . .,.ak, bk) = a1b1 + . .+.akbk (where n = 2k). We will abbreviate AQ*
*(q)
as just AQ, etc. The map A2k-AQ ! Ak-0 given by (ai, bi) 7! (ai) is an algebraic
fiber bundle with fiber (A1 - 0) x Ak-1. If Ui,! Ak - 0 is the open subscheme of
vectors whose ith coordinate is nonzero, then {Ui} is a completely stably cellu*
*lar
trivializing cover for the bundle. So Lemma 3.8 tells us that A2k- AQ is stably
cellular.
We consider the closed subscheme AQ - 0 ,! A2k- 0. The homotopy purity
sequence has the form A2k-AQ ,! A2k-0 ! Th N where N is the normal bundle.
But the normal bundle is easily checked to be trivial, so Th N ' S2,1^ (AQ - 0)*
*+ .
Since we know A2k - AQ is stably cellular, the cofiber sequence shows us that
AQ - 0 is also stably cellular.
For Q, we consider the principal (A1 - 0)-bundle A1 - 0 ! AQ - 0 ! Q and
apply Proposition 4.3. For Pn-1-Q we apply the same proposition to the principal
bundle A1 - 0 ! A2k- AQ ! Pn-1 - Q.
6.Algebraic K-theory and algebraic cobordism
We show that the motivic spectra KGL and MGL, representing algebraic K-
theory and algebraic cobordism respectively, are stably cellular.
In this section it will be more convenient to work in the category of naive s*
*pectra
(a.k.a. Bousfield-Friedlander spectra). We use the model structure on this cate*
*gory,
induced by that on MV, that is described in [Ho2 , Defn. 3.3] and [Ja, Thm. 2.9]
(the two turn out to be equal).
Several times in the following proofs we will use the fact that in MV filtered
colimits are homotopy colimits. This is inherited from the corresponding proper*
*ty
of sSet, using the fact that homotopy colimits for simplicial presheaves can be
computed objectwise.
14 DANIEL DUGGER AND DANIEL C. ISAKSEN
First recall a standard idea from stable homotopy theory:
Lemma 6.1. Let {En, 2,1En ! En+1} be a motivic spectrum. Then E is weakly
equivalent to the homotopy colimit of the diagram
1 E0 ! -2,-1 1 E1 ! -4,-2 1 E2 ! . ...
Proof.One model for -2n,-n 1 En is given by the formulas ( -2n,-n 1 En)k =
2(k-n),k-nEn if k n and ( -2n,-n 1 En)k = Ek otherwise. Thus, for every k,
( -2n,-n 1 En)k = Ek for sufficiently large n. Since homotopy colimits of spect*
*ra
are computed degreewise in k, this shows that the kth term of the homotopy coli*
*mit
is Ek, as desired.
Theorem 6.2. The algebraic K-theory spectrum KGL is stably cellular.
Proof.Recall that KGLn equals the object Z x BGL of MV* [V1 , Ex. 2.8]. By
Lemma 6.1, it suffices to show that Z x BGL is stably cellular, or equivalently
by Lemma 3.2 that BGL is stably cellular. Now BGL is weakly equivalent to
colimkcolimnGrk(An) by [V1 , Ex. 2.8]. Proposition 4.4 finishes the proof becau*
*se
these colimits are filtered colimits and hence homotopy colimits.
Remark 6.3. In the above result one can avoid the use of Grassmannians by
observing that BGL is the homotopy colimit of the usual bar construction_i.e., *
*of
the simplicial diagram [n] 7! GLn. Here GL = colimkGLk, as usual; note that this
is a filtered colimit, hence a homotopy colimit. Proposition 4.1 shows that ea*
*ch
GLk is stably cellular, and so GL is as well. Then so is each GLn, hence BGL is
stably cellular.
Theorem 6.4. The algebraic cobordism spectrum MGL is stably cellular.
Proof.Let pn,k:En,k! Grk(An) be the tautological k-dimensional bundle, and
let E0n,kbe the complement of the zero section. By Lemma 6.1, we need to show
that each MGLk is stably cellular. From [V1 , Ex. 2.10], the object MGLk is
colimnTh(pn,k), which is equal to colimnEn,k=E0n,k. Since the colimit is a fil-
tered colimit, it is also a homotopy colimit. Therefore, we only have to show t*
*hat
En,k=E0n,kis stably cellular. By Lemma 2.5, this reduces to showing that En,kand
E0n,kare stably cellular. The first is weakly equivalent to Grk(An), so it is s*
*tably
cellular by Proposition 4.4. The following proposition shows that the second is*
* also
stably cellular.
Proposition 6.5. Let pn,k:En,k ! Gr k(An) be the tautological k-dimensional
bundle, and let E0n,kbe the complement of the zero section. The variety E0n,kis
stably cellular.
Proof.To simplify notation, let V = Vk(An), G = Grk(An), and E0 = E0n,k. The
map E0 xG V ! E0 is a principal bundle with group GLk. By Proposition 4.3,
we just need to show that E0 xG V is stably cellular. Now E0 xG V is the variety
of ordered sets of k linearly independent vectors together with a non-zero vect*
*or
in the span of these vectors, which is isomorphic to V x (Ak - 0). This variety*
* is
stably cellular by Proposition 4.2 and Lemma 3.4.
Remark 6.6. It is known, at least in characteristic 0, that the Lazard ring
Z[x1, x2, . .].sits inside of MGL2*,*as a retract (here xihas degree (2i, i)). *
*Hopkins
and Morel have announced a proof that these two rings are actually equal. In th*
*is
MOTIVIC CELL STRUCTURES 15
case the xi's form a regular sequence, and one can inductively start forming ho*
*mo-
topy cofibers of MGL-module spectra: MGL=(x1) is defined to be the homotopy
cofiber of 2,1MGL ! MGL, then MGL=(x1, x2) is the homotopy cofiber of a map
4,2MGL=(x1) ! MGL=(x1), etc. According to Hopkins and Morel the spectrum
MGL=(x1, x2, . .).is weakly equivalent to the motivic cohomology spectrum HZ
(just as happens in classical topology). From this of course it would follow th*
*at HZ
is cellular: we already know MGL is cellular, and HZ would be built inductively
from suspensions of MGL via homotopy cofibers.
7. Cellularity and stable homotopy
The material in this section is completely formal, but at the same time sur-
prisingly powerful. If E is a motivic spectrum, write ßp,qE for the set of maps
Ho (Sp,q, E) in the stable homotopy category. First we will prove that stable A*
*1-
weak equivalences between cellular objects are detected by ßp,q. Then we'll pro*
*duce
a pair of spectral sequences for computing ßp,qof smash products and function s*
*pec-
tra that are applicable only under certain cellularity assumptions.
Proposition 7.1. If E is cellular and ßp,qE = 0 for all p and q in Z, then E is
contractible.
Proof.We may as well assume that E is both cofibrant and fibrant. Consider the
class of all spectra A such that the pointed simplicial set Map (A~, E) is cont*
*ractible,
where A~! A is a cofibrant-replacement for A. This class is closed under weak
equivalences and homotopy colimits, and our assumptions imply that it contains
the spheres Sp,q. Therefore the class contains every cellular object; in partic*
*ular,
it contains E. But if Map (E, E) is contractible, then the identity map is null*
* in
the stable homotopy category (because E is cofibrant and fibrant), and this imp*
*lies
that E is contractible.
Corollary 7.2. Let E ! F be a map between cellular spectra, and assume it
induces isomorphisms on ßp,qfor all p and q in Z. Then the map is a weak equiv-
alence.
This corollary is proved in [H , Thm. 5.1.5], but we include a proof for comp*
*lete-
ness and because it's short.
Proof.Let K be the homotopy fiber of E ! F . Since we are in a stable category,
it is enough to prove that K is contractible. But our assumptions imply that K *
*is
cellular, and that ß*,*K = 0. So the proposition gives us K ' *.
Proposition 7.3. If E is any motivic spectrum, there is a zig-zag A ! ^E E
in which E ! ^Eis a weak equivalence, A ! ^Einduces isomorphisms on bi-graded
homotopy groups, and A is cellular.
The following proof is an adaptation of the usual construction in ordinary to*
*pol-
ogy of cellular approximations to any space.
Proof.First let E ! E^ be a fibrant-replacement. Consider all possible maps
f :Sp,q! E^as p and q range over all integers, and let C0 = fSp,q. There is
a canonical map C0 ! ^E.~
Factor this map as C0 æ ^C0! ^E, where the first map is a trivial cofibration
and the second is a fibration. Next consider all possible maps f :Sp,q! ^C0which
16 DANIEL DUGGER AND DANIEL C. ISAKSEN
become zero in ßp,q(E^). We get a map fSp,q! C^0, and let C1 be the mapping
cone. There exists a commutative triangle of the form
C^0____//_^E>>
| ~~~~
| ~~
fflffl|~~
C1.
~
We again factor C1 ! ^Eas C1 æ ^C1! ^E, and repeat the procedure to get C2.
Continuing, we get a sequence of cofibrations between fibrant objects
C^0æ ^C1æ ^C2! . . .
all with maps down to ^E. We let C denote the homotopy colimit, and note that
there is a natural map C ! ^E. The map C ! ^Eis surjective on homotopy groups
because of the way in which C^0was defined. To show that C ! ^Eis injective on
homotopy groups, we need a technical result (see Corollary 9.1) that tells us t*
*hat
ßp,qC ~= colimnßp,q^Cn. From this observation, injectivity follows immediately.
Finally, note that each ^Ciis cellular and therefore C is cellular.
Remark 7.4. The above proof actually shows that the full subcategory of
Ho (Spectra(MV)) consisting of the cellular spectra is the same as the smallest
full triangulated subcategory which contains the spheres and is closed under in*
*fi-
nite direct sums. It might seem like we also need to include mapping telescopes*
* in
this statement, but we get these for free_see [BN ].
Definition 7.5. Given any motivic spectrum E, let Cell(E) be the corresponding
cellular spectrum constructed in Proposition 7.3.
It is easy to see from the proof of Proposition 7.3 that Cellis a functor. I*
*t's
slightly inconvenient that we don't obtain a natural map Cell(E) ! E. However,
because E ! ^Eis a weak equivalence, we do obtain a natural weak homotopy class
Cell(E) ! E.
The following simple lemma will be important later in the proof of Theorem
8.12.
Lemma 7.6. The functor Cell takes homotopy cofiber sequences to homotopy
cofiber sequences.
Proof.Let A ! B ! C be a homotopy cofiber sequence, and let D denote the
homotopy cofiber of Cell(A) ! Cell(B). Then D is cellular, and the induced
homotopy class D ! C is an isomorphism on ß*,*by the five-lemma. Now the map
Cell(B) ! Cell(C) induces D ! Cell(C), and this map is also an isomorphism on
ß*,*. By Proposition 7.2, this last map is a weak equivalence.
If E is a motivic ring spectrum, one can talk about E-modules, smash products
over E (denoted ^E ), and function spectra FE (-, -). The definitions are forma*
*l,
given a symmetric monoidal model category of spectra (see [EKMM , Ch. III], f*
*or
example). As in [EKMM ] we will blur the distinction between these constructi*
*ons
and their derived versions_to the seasoned homotopy-theorist it will always be
clear which we mean (and it's almost always the derived one).
We will need the following basic tool from the algebra of ring spectra:
MOTIVIC CELL STRUCTURES 17
Proposition 7.7. Let E be a motivic ring spectrum, M be a right E-module, and
N be a left E-module. Assume E and M are cellular. Then there there is a strong*
*ly
convergent tri-graded spectral sequence of the form
E2a,(b,c)= Torß*,*Ea,(b,c)(ß*,*M, ß*,*N) ) ßa+b,c(M ^E N),
and a conditionally-convergent tri-graded spectral sequence of the form
Ea,(b,c)2= Exta,(b,c)ß*,*E(ß*,*M, ß*,*N) ) ßa+b,cFE (M, N).
Some kind of cellularity hypothesis is essential for this proposition in orde*
*r to
guarantee convergence.
The proof of this result is almost exactly the same as the one of [EKMM ,
Thm. IV.4.1]; we will record some consequences before spelling out exactly what
changes need to be made. In the notation, a is the homological grading on
the Tor and (b, c) is the internal grading coming from the bi-graded homotopy
groups. The differentials in the first spectral sequence have the form dr:Era,(*
*b,c)!
Era-r,(b+r-1,c). The edge homomorphism of the spectral sequence is the obvious
map
[ß*,*M ß*,*Eß*,*N](b,c)! ßb,c(M ^E N).
Similar remarks apply to the Ext spectral sequence.
Corollary 7.8. Let M be a cellular motivic spectrum, and let N0 ! N be an
arbitrary map. If N0 ! N induces isomorphisms on bi-graded homotopy groups,
then M*,*(N0) ! M*,*(N) and (N0)*,*(M) ! N*,*(M) are both isomorphisms.
Proof.We apply the spectral sequences of the theorem in the case where E is the
sphere spectrum.
Here is one immediate consequence of Proposition 7.3 and the above corollary:*
* if
X is any scheme, then there is a cellular spectrum A and a zig-zag of maps betw*
*een
A and 1 X+ which induce isomorphisms on KGL- and MGL- homology. We
don't know whether the same statement can be made for cohomology.
Proof of Proposition 7.7.We follow the method explained in [EKMM , IV.5]. Fir*
*st,
we set K-1 = M and inductively build a sequence of homotopy cofiber sequences
Ki! Fi! Ki-1with the property that Fi is a free right E-module and ß*,*Fi!
ß*,*Ki-1 is surjective. The resulting chain complex
. .!.ß*,*F2 ! ß*,*F1 ! ß*,*F0 ! ß*,*M
is a free resolution over ß*,*E.
We now have a tower of homotopy cofiber sequences of the form
(7.9) . ._.___//M____//_M____//_O1,0K0__//_O2,0K1___//_.O.O.OOOO
| | | |
| | | |
| | | |
. . . * F0 1,0F1 2,0F2 . . .
(the tower is trivial as it extends left). We apply (-)^E N to this tower, and *
*consider
the resulting homotopy spectral sequence. Note that for each fixed q we will ge*
*t a
homotopy spectral sequence for ß*,q(-), with only the first variable changing: *
*so
we really have a family of spectral sequences, one for each q. We are free to t*
*hink
of this as a `tri-graded' spectral sequence.
18 DANIEL DUGGER AND DANIEL C. ISAKSEN
Note that Fi is a wedge of various suspensions of E, indexed by a free set of
generators for ß*,*Fias a ß*,*E-module. So Fi^E N is a wedge of suspensions of N
indexed by the same set. Therefore ß*,*(Fi^E N) is a direct sum of copies of ß**
*,*N
(with the grading shifted appropriately), and the identification of the E2-term*
* as
Torfalls out immediately. This is all the same as the argument in [EKMM ].
The place where we have to be careful is in convergence. By [Bd , Thm. 6.1(b)]
we only have to show that colimß*,*( n,0Kn ^E N) = 0. Let K1 be the ho-
motopy colimit of the sequence M ! 1,0K0 ! 2,0K1 ! . ... From a
technical result proved at the end of the paper (see Corollary 9.1), we have
ßp,q(K1 ) = colimnßp,q n+1,0Kn. The tower was constructed in such a way that
each Ki ! 1,0Ki+1 induces the zero map on ß*,*; so ß*,*K1 = 0. In ordinary
topology this would tell us that K1 is contractible, and therefore that K1 ^E N
is contractible_and we would be done. However, our assumptions imply induc-
tively that all the Ki are cellular, and therefore so is K1 . By Proposition 7.*
*1 the
vanishing of ß*,*tells us that K1 is contractible.
The proof for the Ext case follows the same ideas. For conditional con-
vergence [Bd , Defn. 5.10], we need to show that limß*,*FE ( n,0Kn, N) and
lim1ß*,*FE ( n,0Kn, N) are both zero. This follows from the usual short exact
sequence for homotopy groups of the homotopy limit of a tower [BK , IX.3.1] and
the fact that holimFE ( n,0Kn, N) is weakly equivalent to FE (K1 , N), which is
contractible.
7.10. Cellularity for E-modules. If E is a motivic ring spectrum, we can con-
sider the model category of right E-modules [SS]. We define the E-cellular mod-
ules to be the smallest class which contains the modules Sp,q^ E and is closed
under weak equivalence and homotopy colimits.
Most of the results of the previous section carry over to E-cellularity. The *
*key
observation is that Ho E(Sp,q^ E, X) is isomorphic to ßp,q(X).
It follows as in Proposition 7.2 that an E-module map between E-cellular spec*
*tra
is a weak equivalence if it induces isomorphisms on ßp,qfor all p and q in Z. F*
*or any
E-module X, the construction of Proposition 7.3 gives us a zig-zag CellE(X) !
^X X of E-module maps in which X ! ^Xis a weak equivalence, CellE(X) ! ^X
induces isomorphisms on bi-graded homotopy groups, and CellE(X) is E-cellular.
As in Definition 7.5, we obtain natural weak homotopy classes CellE(X) ! X, but
not actual maps. We can also prove that CellEtakes homotopy cofiber sequences
of E-modules to homotopy cofiber sequences.
We will need the following improvement on Proposition 7.7.
Proposition 7.11. The spectral sequences of Proposition 7.7 have the indicated
convergence properties as long as M is E-cellular (but without any other assump-
tions on E and M).
Proof.The basic setup is the same as in the proof of Proposition 7.7, and for t*
*he
Tor-spectral sequence we again only need to show that colimnß*,*( n,0Kn^E N) =
0. Because M is E-cellular, so is each Kiand so is K1 . We know that ß*,*K1 = 0.
From the E-cellular version of Proposition 7.2, we conclude that K1 is contract*
*ible.
The Ext case is again similar; see the end of the proof of Proposition 7.7 fo*
*r an
outline of the differences.
MOTIVIC CELL STRUCTURES 19
Note that if X is a cellular spectrum then X ^E is E-cellular. This lets us a*
*pply
the spectral sequences in the case where M has the form X ^ E, but without any
assumptions on the spectrum E.
8.Finite cell complexes and Künneth theorems
Definition 8.1. The finite cell complexes are the smallest class of objects in
Spectra(MV) with the following properties:
(1)the class contains the spheres Sp,q;
(2)the class is closed under weak equivalence;
(3)if X ! Y ! Z is a homotopy cofiber sequence and two of the objects are in the
class, then so is the third.
The full subcategory of Ho (Spectra(MV)) consisting of the finite cell comple*
*xes
is the smallest full triangulated subcategory containing all the spheres Sp,q(w*
*hich
will necessarily be closed under finite direct sums, but not infinite ones).
Remark 8.2. It is worth mentioning that a finite homotopy colimit of finite cell
complexes need not be a finite cell complex. This happens even in ordinary topo*
*l-
ogy: for example, the homotopy co-invariants of Z=2 acting on a point is RP 1.
Remark 8.3. If a scheme X has a finite Zariski cover {Ui} such that each in-
tersection Ui1\ . .\.Uik is a finite cell complex, then so is X. Our arguments
from Section 4 therefore show that GLn and Vk(An) are finite cell complexes. The
argument from Proposition 4.4 does not show that Grassmannians are finite cell
complexes, however. It turns out that they are, but the proof is much more elab*
*o-
rate. We have omitted it because for us the linear spectra (see below) are almo*
*st
as good as finite cell complexes, and Grassmannians are obviously linear.
For any two motivic spectra A and B and any motivic ring spectrum E, there
is a natural map
(8.4) F (A, E) ^ F (B, E) ! F (A ^ B, E ^ E) ! F (A ^ B, E).
In particular, using the identification F (S0, E) ~=E one finds that F (X, E) i*
*s a
bimodule over the ring spectrum E, and that the above map factors as
F (A, E) ^ F (B, E) ! F (A, E) ^E F (B, E) jA,B-!F (A ^ B, E).
Proposition 8.5. If A (or B) is a finite cell complex, then the map jA,B induces
isomorphisms on all ßp,q(-).
Proof.Fix B, and consider the class of objects A such that jA,B induces iso-
morphisms on bi-graded homotopy groups. One easily checks that this class is
closed under weak equivalences, and has the two-out-of-three property for ho-
motopy cofiber sequences. To see that the class contains Sp,q, use the fact th*
*at
F (Sp,q, E) ' S-p,-q^ E as E-modules.
Theorem 8.6. Suppose that X and Y are two motivic spectra such that X is a
finite cell complex. Let E be a motivic ring spectrum. Then there exists a stro*
*ngly-
convergent tri-graded Künneth spectral sequence of the form
*,* *,* *,* a+b,c
TorEa,(b,c)(E X, E Y ) ) E (X ^ Y ).
The grading conventions are the same as in Proposition 7.7.
20 DANIEL DUGGER AND DANIEL C. ISAKSEN
Proof.First note that F (Sp,q, E) is E-cellular (being equivalent to S-p,-q^ E).
Using this together with the fact that X is a finite cell complex, it follows t*
*hat
F (X, E) is E-cellular. We now apply Proposition 7.11 with M = F (X, E) and
N = F (Y, E). The groups ß*,*(M ^E N) are identified with ß*,*F (X ^ Y, E) by
Proposition 8.5.
The necessity of some kind of finiteness hypothesis in the above result is we*
*ll
known in ordinary topology_see [A , Lect. 1]. The result is often applied when
X = 1 A+ and Y = 1 B+ , where A and B are schemes, in which case X ^ Y =
1 (A x B)+ .
Remark 8.7. Note that the higher Tor's vanish if E*,*(X) is free as a module ov*
*er
E*,*, in which case we obtain a Künneth isomorphism
E*,*(X) E*,*E*,*(Y ) ~=E*,*(X ^ Y ).
Remark 8.8. In [J], Joshua produced a similar Künneth spectral sequence. His
result was stated only for algebraic K-theory and motivic cohomology, and assum*
*ed
that X and Y were schemes (rather than arbitrary objects of MV). Also, his
spectral sequence was only bi-graded rather than tri-graded: this is essential*
*ly
because he was applying the results of [EKMM ] rather than reproving them in *
*the
bi-graded context, and so his motivic cohomology rings were graded by total deg*
*ree.
Our proof is essentially the same as Joshua's (which in turn is the same as
the modern proof in stable homotopy theory) although we were able to streamline
things by using the language of motivic spectra.
Joshua's result assumes that one of the schemes X and Y is linear, as opposed
to being a finite cell complex in our sense. If one assumes the ring spectrum E
satisfies a Thom isomorphism theorem, then one can make our result encompass
Joshua's by expanding the class of finite cell complexes so as to be closed und*
*er the
process of `taking Thom spaces':
Definition 8.9. The linear motivic spectra are the smallest class of objects in
Spectra(MV) with the following properties:
(1)the class contains the spheres Sp,q;
(2)the class is closed under weak equivalence;
(3)if X ! Y ! Z is a homotopy cofiber sequence and two of the objects are in the
class, then so is the third;
(4)if , :E ! X is an algebraic vector bundle over a smooth scheme X, then
1 Th , belongs to the class if and only if 1 X+ belongs to the class.
Remark 8.10. If Z ,! X is a closed inclusion of smooth schemes, recall that the*
*re
is a stable homotopy cofiber sequence X - Z ! X ! Th NX=Z [MV , 3.2.23]. It
follows that if two of the three objects 1 Z+ , 1 X+ , and 1 (X -Z)+ are lin*
*ear,
then so is the third.
Let E be a motivic ring spectrum, and let , ! X be an algebraic vector bundle*
* of
rank n over a smooth scheme X. We'll say that E satisfies Thom isomorphism
for , if there is a class u 2 E2n,n(Th ,) such that multiplication by u gives an
isomorphism E*,*(X) ! ~E*+2n,*+n(Th ,). To be more precise, note that we have
a map of motivic spaces
__,__- ! ___,_x_,__~ __,__^ , ' __,__^ X .
, - 0 (, - 0) x ,=, - 0 + , - 0 +
MOTIVIC CELL STRUCTURES 21
This map induces ff: F (Th , ^ X+ , E) ! F (Th ,, E). If we write u as a homoto*
*py
class S-2n,-n ! F (Th ,, E), we can consider the composite
(8.11)
S-2n,-n^F (X+ , E)__//_F (Th ,, E) ^ F (X+_,/E)/F (Th ,^X+f,fE)//_F (Th ,, E),
where the second map is the same as in (8.4). The claim is that this composite *
*is
an isomorphism on ß*,*.
We say that E satisfies Thom isomorphism if it does so for every algebraic
vector bundle over a smooth scheme. The spectra HZ, KGL, and MGL are all
known to satisfy Thom isomorphism. The reader may wish to compare the above
discussion to the notion of algebraically orientable spectrum from [HK ].
Theorem 8.12. Suppose E is a ring spectrum satisfying Thom isomorphism. If X
and Y are motivic spectra such that X is linear, then there is a strongly conve*
*rgent
Künneth spectral sequence as in Theorem 8.6.
In the following proof, we continue our sloppiness about distinguishing vario*
*us
functors and their derived versions. It should be clear from context that we al*
*most
always mean the derived version.
Proof.The proof requires a little more care than the similar things we've done *
*so
far. If Z is a pointed motivic space, we abbreviate F ( 1 Z, E) as just F (Z, E*
*).
Recall from Section 7.10 that CellEis a functorial E-cellular approximation. *
*Let
C denote the class of all motivic spectra A such that for all motivic spectra Y*
* , the
composite
ßp,q(CellE(F (A, E)) ^E F (Y, E)) ! ßp,q(F (A, E) ^E F (Y, E)) ! ßp,qF (A ^ Y, *
*E)
is an isomorphism for all p and q. The first map above makes sense because we
have a homotopy class CellE(F (A, E)) ! F (A, E) even though we don't have an
actual map.
The class C clearly is closed under weak equivalences and contains the spheres
Sp,q. The E-cellular version of Lemma 7.6 and the five-lemma show that it also *
*has
property (3) of Definition 8.9. To show that C contains every linear spectrum, *
*we
must check that if , is a vector bundle over a smooth scheme X, then Th , belon*
*gs
to C if and only if 1 X+ belongs to C.
However, from (8.11) we have the map u: F (X+ , E) ! S2n,n^F (Th ,, E) which
is an isomorphism on ß*,*. It follows that CellE(F (X+ , E)) is weakly equivale*
*nt to
S2n,n^ CellE(F (Th ,, E)). We now look at the diagram
ßp,q(CellE(F (X+ , E)) ^E F (Y,_E))________//ßp,qF (X+ ^ Y, E)
| |
| |
fflffl| fflffl|
ßp,q(S2n,n^ CellE(F (Th ,, E)) ^E F (Y,_E))//_ßp,q(S2n,n^ F (Th , ^ Y, E)).
Both vertical maps induce isomorphisms on ß*,*_the one on the left is a weak
equivalence, and for the one on the right this is the `generalized Thom isomorp*
*hism'
from Lemma 8.13 below. So the top horizontal map is a ß*,*-isomorphism if and
only if the bottom map is one. This is equivalent to what we needed to prove.
We have shown that for every linear spectrum X, the map
ßp,q(CellE(F (X, E)) ^E F (Y, E)) ! ßp,qF (X ^ Y, E)
22 DANIEL DUGGER AND DANIEL C. ISAKSEN
is an isomorphism. By Proposition 7.11 there is a strongly convergent spectral
sequence of the form
*,* *,* *,*
TorEa,(b,c)(ß*,*CellE(F (X, E)), E (Y )) ! E (X ^ Y ).
But CellE(F (X, E)) ! F (X, E) is an isomorphism on bi-graded homotopy groups,
so this completes the proof.
Lemma 8.13. Let E be a ring spectrum satisfying Thom isomorphism, and let ,
be a vector bundle of rank n over a smooth scheme X. Then for every motivic
spectrum Y , the map
jY :F (X+ ^ Y, E) ! S2n,n^ F (Th , ^ Y, E)
induces isomorphisms on ß*,*. In other words, multiplication by the Thom class
gives an isomorphism
E*,*(X+ ^ Y ) ~=E*+2n,*+n(Th , ^ Y ).
The construction of the map jY is analogous to the discussion preceding Theo-
rem 8.12.
Proof.Let C be the class of all spectra Y such that jY induces isomorphisms on *
*ß*,*.
Clearly the class is closed under weak equivalences, homotopy colimits, and Sp,*
*q-
suspensions. Since every motivic spectrum is a homotopy colimit of desuspensions
of suspension spectra (see Lemma 6.1), it suffices to show that suspension spec*
*tra
belong to C. For F 2 MV*, the homotopy cofiber sequence
S0,0! 1 F+ ! 1 F
tells us that we can just prove the lemma in the case Y = 1 F+ .
But every motivic space F is a homotopy colimit of smooth schemes [D , Sec. 2*
*.6],
so it actually suffices to show that 1 Z+ belongs to C for every pointed smooth
scheme Z. But in this case the statement is just the Thom isomorphism for the
bundle ß*,, where ß :X x Z ! X is the projection.
9.Stable homotopy groups of filtered colimits
In this section we prove the following basic result about the category of mot*
*ivic
spectra, needed several times in the course of the paper. The phrase `directed
system', as used in the present section, refers to sequences E0 ! E1 ! . .i.nde*
*xed
by the natural numbers.
Proposition 9.1. Let i 7! Ei be a directed system of motivic spectra. Then
colimißp,qEi! ßp,q(hocolimEi) is an isomorphism for all p and q in Z.
9.2. Preliminary remarks. Let T be a triangulated category with infinite direct
sums, as in [BN , Def. 1.2]. An object X 2 T is called compact if T(X, iEi) ~=
iT(X, Ei) for every countable collection of objects {Ei}. The full subcategory*
* of
T consisting of compact objects is readily seen to be a triangulated subcategory
(and therefore closed under finite sums).
Recall from [BN ] that if E0 ! E1 ! . .i.s a directed system in T then one can
form what we'll call the triangulated homotopy colimit thocolimEi, defined as
the cofiber of Ei1-sh-! Ei. If M is a stable model category and E0 ! E1 ! . . .
is a directed system in M, one can check that hocolimEi (as defined in the model
category sense) is isomorphic to the object thocolimEi in Ho (M).
MOTIVIC CELL STRUCTURES 23
Lemma 9.3. Let M be a stable model category. An object X 2 Ho (M) is com-
pact if and only if the natural map colimiHo(X, Ei) ! Ho (X, thocolimEi) is an
isomorphism, for every directed system {Ei} in M.
Proof.The `only if' part is immediate, and works for any triangulated category.
For the `if' part, note that if {Ei} is any collection of cofibrant objects the*
*n the
hocolim of
E0 ! E0 _ E1 ! E0 _ E1 _ E2 ! . . .
is isomorphic to Eiin Ho (M). This fact is easily proven using facts about mod*
*el
categories, but it is unclear to us if it works for a general triangulated cate*
*gory.
9.4. Compact objects in motivic spectra. Since the model categories of mo-
tivic symmetric spectra and naive spectra are Quillen equivalent [Ho2 , Sec. 10*
*], we
can prove our theorems in either setting. It is easier to work with naive spect*
*ra.
If W 2 MV* then 1 W denotes the usual suspension spectrum of W . If E is
a spectrum and n 0, we let 2n,nE denote the naive spectrum with [ 2n,nE]i=
En+i. Likewise, when n < 0 let 2n,nE denote the spectrum with [ 2n,nE]i= * for
i < -n and [ 2n,nE]i= Ei+n for i -n. Finally, we abbreviate 2n,n( 1 W ) to
(2n,n)+1W . Note that these objects are cofibrant spectra by [Ho2 , Prop. 1.14*
*].
Our aim is the following:
Theorem 9.5. If X is any pointed smooth scheme and n 2 Z, then (2n,n)+1X
is a compact object of the motivic stable homotopy category.
From this one readily deduces the result about ßp,qof directed hocolims:
Proof of Proposition 9.1.The Sp,q's are in the smallest triangulated subcategory
which contains (2n,n)+1X for every X and n. By Theorem 9.5 and the observation
that the compact objects form a triangulated subcategory, the Sp,q's are theref*
*ore
compact. Now use Lemma 9.3.
Recall from Definition 8.1 the notion of a finite cell complex. One consequen*
*ce of
Theorem 9.5 is that we never really need infinite homotopy colimits to build st*
*ably
cellular schemes.
Proposition 9.6. Let X be a smooth scheme, and suppose X is stably cellular.
Then 1 X+ is a retract, in Ho (Spectra(MV)), of a finite cell complex.
Proof.Let T be the full subcategory of Ho (Spectra(MV)) consisting of the cellu*
*lar
objects. Then the spheres Sp,qare a set of weak generators for T: that is, the
smallest full triangulated subcategory of T containing the Sp,q's and closed un*
*der
infinite direct sums is T itself (see Remark 7.4). A result of Neeman, recount*
*ed
in [K , 5.3], shows that any compact object in T is a retract of something that*
* can
be built from the generators via finitely many extensions. The fact that 1 X+ *
*is
compact therefore finishes the proof.
We now turn to the proof of Theorem 9.5. A fundamental difficulty with the
injective model structure on MV (cf. [MV ] or [Ja, App. B]) is that a directed *
*colimit
of fibrant objects need not be fibrant. The projective structure [Bl] doesn't h*
*ave
this problem, but it has other difficulties instead. We will stick with the inj*
*ective
structure, but to get around the problem with colimits we need a new definition.
Definition 9.7. An object F 2 MV is almost-fibrant if the following conditions
are satisfied:
24 DANIEL DUGGER AND DANIEL C. ISAKSEN
(1)It is flasque, in the sense of [Ja, Sec. 1.4];
(2)It is objectwise-fibrant;
(3)For every elementary Nisnevich cover {U, V ! X}, the natural map F (X) !
F (U) xF(UxXV )F (V ) is a weak equivalence.
(4)For every smooth scheme X, the map F (X) ! F (XxA1) is a weak equivalence.
Note that the flasque condition implies in particular that for any inclusion *
*of
schemes W ! X the map F (X) ! F (W ) is a Kan fibration. It follows that the
pullback F (U) xF(UxXV )F (V ) appearing in (3) is actually a homotopy pullback
as well.
The following two results follow immediately from the ideas of [Bl], [DHI ], *
*and
[Ja, Sec. 1.4], so we only sketch the proofs. The first one explains the name `*
*almost-
fibrant':
Proposition 9.8. (a)Every fibrant object of MV is almost-fibrant.
(b)If F and F 0are almost-fibrant, then a map F ! F 0is a weak equivalence in
MV if and only if it is an objectwise weak equivalence.
(c)If F is almost-fibrant and X is a smooth scheme then Ho (X, F ) = ß0F (X) =
ß0Map (X, F ).
Proof.Part (a) follows from the definitions. For part (b), observe that almost-
fibrant objects are fibrant in the A1-local projective model structure of [Bl].*
* A weak
equivalence between fibrant objects in this structure is necessarily an objectw*
*ise
weak equivalence, as this is a general property of Bousfield localizations.
For part (c), let F ! F 0be a fibrant-replacement in MV. Then Ho (X, F ) ~=
ß0Map (X, F 0) ~= ß0F 0(X), and by (b) the latter is isomorphic to ß0F (X) ~=
ß0Map (X, F ).
Now we record some nice properties of almost-fibrant objects. Recall that if
F 2 MV* then 2,1F is the simplicial presheaf whose value at X is the pullback *
*of
* ! F (X) F (X x P1).
Proposition 9.9. (a)A directed colimit of almost-fibrant objects is almost-fibr*
*ant.
(b)Suppose {Ei} and {Fi} are two directed systems of almost-fibrant objects, and
{Ei! Fi} is a levelwise weak equivalence of systems. Then colimEi! colimFi
is also a weak equivalence.
(c)If F 2 MV* is almost-fibrant, then 2,1F is almost-fibrant. If F ! F 0is a
weak equivalence between pointed almost-fibrant objects, then 2,1F ! 2,1F 0
is an objectwise weak equivalence.
(d)If i 7! Fi is a directed system in MV*, then the canonical map colimi 2,1Fi!
2,1(colimiFi) is an isomorphism.
(e)If F ! F 0is a weak equivalence between almost-fibrant objects of MV* and X
is a pointed smooth scheme, then Map MV*(X, F ) ! Map MV*(X, F 0) is a weak
equivalence of simplicial mapping spaces.
Proof.Part (a) easily follows from the definitions, together with the fact that*
* fil-
tered colimits preserve fibrations and weak equivalences of simplicial sets. Pa*
*rt (b)
follows from Proposition 9.8(b) and the observation that colimits preserve obje*
*ct-
wise weak equivalences.
For the first statement in part (c), the only hard step is to check the flasq*
*ue
condition; but this is done in [Ja, Cor. 1.9, Cor. 1.10]. The second statement
follows from the first statement and Proposition 9.8(b). Part (d) is immediate
MOTIVIC CELL STRUCTURES 25
from the definitions. For part (e), note that if X is a pointed smooth scheme
and F 2 MV* then the simplicial mapping space Map MV*(X, F ) is the pullback
of * ! Map MV(*, F ) Map MV(X, F ). When F is almost-fibrant the right-hand
map is a fibration. Using Proposition 9.8(b) and the right properness of sSet, *
*it
follows that Map MV*(X, F ) ! Map MV*(X, F 0) is a weak equivalence.
Recall that a motivic spectrum {Ei} is fibrant if each Ei is fibrant in MV and
the maps Ei! 2,1Ei+1are all weak equivalences.
Definition 9.10. A motivic spectrum {Ei} is almost-fibrant if each Eiis almost-
fibrant in MV and the maps Ei! 2,1Ei+1 are weak equivalences.
Proposition 9.11. (a)Every fibrant motivic spectrum is almost-fibrant.
(b)A directed colimit of almost-fibrant spectra is almost-fibrant.
(c)If E and E0 are almost-fibrant spectra, a map E ! E0 is a stable weak equiv-
alence if and only if for every n 0 the map En ! E0nis an objectwise weak
equivalence in MV.
(d)If i 7! Eiis a directed system of almost-fibrant motivic spectra, then the n*
*atural
map hocolimEi! colimEi is a weak equivalence.
Proof.Part (a) follows from Proposition 9.9(a). Part (b) is routine, using Prop*
*o-
sition 9.9(a-d). Part (c) is well-known for fibrant spectra, so it suffices to *
*produce
a fibrant-replacement E ! ^Ewhich is an objectwise weak equivalence in all lev-
els (as the same can be done for E0). This can be accomplished by considering
naive spectra with levelwise weak equivalences and taking a fibrant replacement
in this setting, and then referring to Proposition 9.9(c) to see that this give*
*s an
-spectrum.
For (d), it suffices to show that if {Ei} and {Fi} are directed systems of al*
*most-
fibrant objects and each Ei ! Fi is a stable weak equivalence, then colimEi !
colimFi is a stable weak equivalence. By part (c), the map of nth spaces [Ei]n !
[Fi]n is an objectwise weak equivalence in MV*. It follows that [colimiEi]n !
[colimiFi]n is still an objectwise weak equivalence in MV*, and this implies th*
*at
colimEi! colimFi is a stable weak equivalence.
Recall that naive spectra form a simplicial model category, with the simplici*
*al
action being the level-wise one inherited from MV*. We write Map (-, -) for the
simplicial mapping space, both for motivic spectra and in MV*.
Lemma 9.12. If E is an almost-fibrant spectrum then Ho ( (2n,n)+1X, E) is iso-
morphic to ß0Map ( (2n,n)+1X, E), for any pointed smooth scheme X.
Proof.Let E ! ^Ebe a fibrant-replacement. If n 0 then
Map ( (2n,n)+1X, E) ~=Map MV*(X ^ P1 ^ . .^.P1, E0) ~=Map MV*(X, 2n,nE0).
Here 2n,nE0 = 2,1 2,1. . .2,1E0, and both isomorphisms come from vari-
ous adjunctions. Note that the same chain of isomorphisms exists with E re-
placed by E^. By Proposition 9.9(c), 2n,nE0 ! 2n,n^E0is an objectwise weak
equivalence between almost-fibrant objects. So Proposition 9.9(e) implies that
Map MV*(X, 2n,nE0) ' Map MV*(X, 2n,n^E0) ~=Map ( (2n,n)+1X, ^E). The set
of components of the latter space is of course Ho ( (2n,n)+1X, E) (using the fa*
*ct
that (2n,n)+1X is cofibrant).
26 DANIEL DUGGER AND DANIEL C. ISAKSEN
The case n < 0 is similar, except one instead starts with the isomorphism
Map ( (2n,n)+1X, E) ~=Map MV*(X, E-n ).
Lemma 9.13. Let F0 ! F1 ! . .b.e a directed system in MV*, and let X be a
pointed smooth scheme. Then Map MV*(X, colimFi) ~=colimMapMV* (X, Fi).
Proof.Note that the result is obvious for unpointed mapping spaces, s*
*ince
Map MV(X, F ) ~= F (X). In the pointed case Map MV* (X, colimFi) equals the
pullback of the diagram * ! Map MV (*, colimFi) Map MV (X, colimFi), which
is the same as the colimit of the pullbacks of diagrams * ! Map MV (*, Fi)
Map MV(X, Fi).
Finally we can prove the theorem:
Proof of Theorem 9.5.By Lemma 9.3 we must show that if E0 ! E1 ! . . .
is a directed system of motivic spectra then colimiHo( (2n,n)+1X, Ei) !
Ho ( (2n,n)+1X, hocolimEi) is an isomorphism. By taking a functorial fibrant
replacement for each Ei, we can assume that the Ei's are fibrant.
Since each Eiis (in particular) almost-fibrant, it follows from Proposition 9*
*.11(d)
that hocolimiEi! colimiEi is a stable weak equivalence. Thus, we have
Ho ( (2n,n)+1X, hocolimiEi) = Ho ( (2n,n)+1X, colimiEi).
This in turn is isomorphic to ß0Map ( (2n,n)+1X, colimiEi) by Lemma 9.12 be-
cause the spectrum colimiEiis almost-fibrant. So, we are reduced to showing that
colimß0Map ( (2n,n)+1X, Ei) ! ß0Map ( (2n,n)+1X, colimEi)
is an isomorphism. The idea is to prove that the mapping spaces themselves are
isomorphic, using adjointness to reduce to mapping spaces in MV*.
When n < 0 the mapping space Map( (2n,n)+1X, colimEi) is equal to
Map MV*(X, colim[Ei]-n ). By Lemma 9.13 we can pull the colimit outside, and
then adjointness gives us colimMap ( (2n,n)+1X, Ei).
When n > 0 one has
Map ( (2n,n)+1X, colimEi) ~=Map MV*(X, 2n,n(colim[Ei]0)),
where 2n,n(-) is shorthand for 2,1. . .2,1(-). By Proposition 9.9(d) we can
commute the 2n,npast the colimit, and then Lemma 9.13 lets us take the col-
imit outside. Using adjointness again, we get colimMap ( (2n,n)+1X, Ei). This
completes the proof.
References
[SGA4] M. Artin, A. Grothendieck, and J. L. Verdier, Th'eorie des topos et coho*
*mologie 'etale
des sch'emas, Lecture Notes in Mathematics 269, Springer, 1972.
[A] J. F. Adams, Lectures on generalised cohomology, Lecture Notes in Mathem*
*atics 99,
Springer-Verlag, 1969.
[Bl] B. A. Blander, Local projective model structures on simplicial presheave*
*s, K-Theory 24
(2001), no. 3, 283-301.
[Bd] M. Boardman, Conditionally convergent spectral sequences, in Homotopy in*
*variant al-
gebraic structures (Baltimore, MD, 1998), 49-84, Contemp. Math. 239, Ame*
*r. Math.
Soc., Providence, RI, 1999.
[BK] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localiza*
*tions, Lecture
Notes in Mathematics 304, Springer-Verlag, 1972.
[BN] M. Bökstedt and A. Neeman, Homotopy limits in triangulated categories, C*
*ompositio
Math. 86 (1993), 209-234.
MOTIVIC CELL STRUCTURES 27
[De] P. Deligne, Poids dans la cohomologie des vari'et'es alg'ebriques, in Pr*
*oceedings of the
International Congress of Mathematicians (Vancouver, 1974), Vol. 1, pp. *
*79-85, Canad.
Math. Congress, 1975.
[DF] E. Dror Farjoun, Cellular spaces, null spaces and homotopy localization,*
* Lecture Notes
in Mathematics 1622, Springer-Verlag, 1996.
[D] D. Dugger, Universal homotopy theories, Adv. in Math. 164 (2001), 144-17*
*6.
[DHI] D. Dugger, S. Hollander, and D. C. Isaksen, Hypercovers and simplicial p*
*resheaves,
Math. Proc. Cambridge Philos. Soc., to appear.
[DI] D. Dugger and D. C. Isaksen, The Hopf condition for bilinear forms over *
*an arbitrary
field, preprint, 2003.
[DGI] W. G. Dwyer, J. P. C. Greenlees, and S. Iyengar, Duality in algebra and *
*topology,
preprint.
[DS] W. G. Dwyer and J. Spali'nski, Homotopy theories and model categories, i*
*n Handbook
of algebraic topology, 73-126, North-Holland, 1995.
[EKMM] A. D. Elmendorff, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules,*
* and algebras
in stable homotopy theory, Mathematical Surveys and Monographs vol. 47, *
*American
Mathematical Society, 1997.
[F1] W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studie*
*s, vol. 131,
Princeton University Press, 1993.
[F2] W. Fulton, Intersection theory, Springer-Verlag Belin Heidelberg, 1998.
[H] P. S. Hirschhorn, Model Categories and Their Localizations, Mathematical*
* Surveys and
Monographs vol. 99, Amer. Math. Soc., 2003.
[Ho1] M. Hovey, Model Categories, Mathematical Surveys and Monographs vol. 63,*
* Amer.
Math. Soc., 1999.
[Ho2] M. Hovey, Spectra and symmetric spectra in general model categories, J. *
*Pure Appl.
Algebra 165 (2001), no. 1, 63-127.
[HK] P. Hu and I. Kriz, Some remarks on Real and Algebraic cobordism, K-theor*
*y 22, no. 4
(2001), 335-366.
[Ja] J. F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000), 445-552.
[J] R. Joshua, Algebraic K-theory and higher chow groups of linear varieties*
*, Math. Proc.
Cambridge Philos. Soc. 130 (2001), no. 1, 37-60.
[K] B. Keller, Deriving DG categories, Ann. Sci. 'Ecole Norm. Sup. (4) 27 (1*
*994), 63-102.
[MV] F. Morel and V. Voevodsky, A1-homotopy theory, Inst. Hautes Etudes Sci. *
*Publ. Math.
90 (2001), 45-143.
[SS] S. Schwede and B. Shipley, Algebras and modules in monoidal model catego*
*ries, Proc.
London Math. Soc. 80 (2000), 491-511.
[T] B. Totaro, Chow groups, Chow cohomology, and linear varieties, preprint,*
* 1995.
[V1] V. Voevodsky, Voevodsky's Seattle lectures: K-theory and motivic cohomol*
*ogy, in Alge-
braic K-theory (Seattle, WA, 1997), 283-303, Proc. Sympos. Pure Math., v*
*ol. 67, Amer.
Math. Soc., Providence, RI, 1999.
[V2] V. Voevodsky, A1-homotopy theory, Proceedings of the International Congr*
*ess of Math-
ematicians, Vol. I (Berlin, 1998), Doc. Math. 1998, Extra Vol. I, 579-60*
*4.
Department of Mathematics, University of Oregon, Eugene, OR 97403
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556
E-mail address: ddugger@math.uoregon.edu
E-mail address: isaksen@math.wayne.edu