CHROMATIC MOTIVIC HOMOTOPY THEORY
Jens Hornbostel
November 26, 2003
Abstract
We construct a motivic version of the chromatic filtration and the chro*
*matic spectral
sequence. This should be used to study the stable A1-homotopy groups of th*
*e motivic sphere
spectrum. We also study different localization techniques both for classi*
*cal and motivic
spectra.
Introduction
The aim of this paper is to provide some tools which allow a better understandi*
*ng of the A1-
homotopy groups of the motivic sphere spectrum. To this purpose, we construct c*
*ertain chro-
matic localization functors L0nand N0nin Voevodsky's stable motivic homotopy ca*
*tegory which
(provided the chain complex 0 ! BP **(X) ! BP **(L00(N00(X))) ! BP **(L01(N01(X*
*))) ! ... is
exact for some smooth scheme X) leads to the motivic chromatic spectral sequence
En,s1= ExtsBP**(BP()BP**, BP **(L0n(N0n(X)))) ) Extn+sBP**(BP()BP**, BP **
**(X))
where BP **denotes a motivic version of the Brown-Peterson homology groups.
The most fundamental problem in classical homotopy theory is to compute the *
*homotopy
groups of spheres, which are the building blocks of any reasonable topological *
*space, that is
CW -complexes and in particular manifolds. Serre proved that the homotopy group*
*s of spheres
are finitely generated and finite except ßn(Sn) and ß4n-1(S2n), which are the d*
*irect sum of
Z and a finite abelian group. The Freudenthal suspension isomorphism says that *
*ßn+k(Sk) is
independent of k for k large enough, it is denoted by ßstn(S0) in this stable r*
*ange. The motivic
analogues of these two results are hitherto unknown.
A crucial idea in homotopy theory is that one should use other spectra to de*
*tect elements
of ßstn(S0). For instance, the unit map S0 ! BO from the sphere spectrum to the*
* spectrum
representing real topological K-theory induces isomorphisms ßstn(S0) ! KOn(pt) *
*for n = 0, 1, 2,
and a similar statement is expected (at least for n = 0, 1 see [Hor]) and parti*
*ally proved [Mo4 ]
in the motivic setting. A more sophisticated tool is a spectrum built from the *
*connected version
of BO (or BU) and called Im J. This spectrum is related to the EHP spectral seq*
*uence and
detects most of the 2-torsion [Mah ]. Also, there is a spectrum E(1) such that*
* the Bousfield
localization LE(1)(S0) of S0 with respect to E(1) detects the p-primary part of*
* Im J (compare
[Ra2 , Theorem 5.3.7]).
Another important tool is the Adams spectral sequence Er,q2= Extr,qA(H*(S0, *
*Z=p), Z=p) )
ßstq-r(S0)^pand more generally the Adams-Novikov spectral sequence
Er,q2= Extr,qi*(E^E)(E*(S0), E*(S0)) ) ßstq-r(LE(S0))
where E is a ring spectrum fulfilling the assumptions of [Ad , section III.15],*
* e. g., E might be
MU or BP .
1
Chromatic motivic homotopy theory *
* 2
There is another spectral sequence (see [MRW ]), called the chromatic spect*
*ral sequence, which
converges to the E2-term of the Adams-Novikov spectral sequence for E = BP and *
*hence is
extremly useful for computations. The related chromatic filtration is defined u*
*sing the spectrum
E(1) and higher chromatic spectra E(n), and it gives a beautiful decomposition *
*of ßst*(S0) into
vn-periodic elements.
The aim of this paper is to construct motivic analogues of the chromatic fil*
*tration and the
chromatic spectral sequence. As in topology, it should be useful both for const*
*ructing concrete
elements in the bigraded ring [S*, (Gm )^*]SH(k)of stable A1-homotopy groups of*
* the motivic
sphere spectrum as well as to get a better conceptual understanding of its gene*
*ral struture.
The self-contained appendices of this paper provide many facts concerning Bo*
*usfield and
Hirschhorn localization and related topics. In contrast, the previous part cont*
*ains many defini-
tons and constructions, but only a rather small number of new results. We belie*
*ve that at least
some of the questions we ask are both interesting and non-trivial, and deserve *
*to be studied by
the author and other mathematicians in forthcoming papers. Moreover, one might *
*try to relate
the first three slices of the chromatic filtration to motivic cohomology, algeb*
*raic K- and KO-
theory (or even an algebraic version of Im J constructed via algebraic Adams op*
*erations) and
some algebraic version of tmf, respectively. But as everybody knows, computatio*
*ns in SH(k)
are very hard, namely Morel only recently accomplished his computations of the *
*stable ß0 of a
point (see Theorem 1.1 below), and even [HM ] does not include the computation *
*of MGL** of
a point.
In section 1, we review what is know on [S*, (Gm )^*]SH(k)and motivic versio*
*ns of the
Adams-Novikov spectral sequence, mainly due to the work of F. Morel.
In section 2, we construct the motivic analogues of the spectra E(n), Morava*
* K-theory
spectra, the chromatic filtration and the chromatic spectral sequence. We estab*
*lish some basic
properties and list some more that we hope will hold. In particular, we establ*
*ish conditions
which imply that the E1 -term of the motivic chromatic spectral sequence and th*
*e E2-term of
the motivic Adams-Novikov spectral sequence for the motivic spectrum BP are is*
*omorphic.
In appendix A, we study the theory of Bousfield localization in cellular mod*
*el categories fol-
lowing Hirschhorn and describe how to apply it to A1-homotopy. This is crucial *
*for Definition
2.11. We observe (following Hirschhorn) that both the unstable and the stable A*
*1-homotopy
category of Morel and Voevodsky are obtained using Hirschhorn's techniques, hen*
*ce they are
cellular and can be further localized. Moreover, both the injective and project*
*ive model struc-
tures are monoidal in the sense of Hovey. These and other technical results in*
* the appendix
have their own interest and can be used for other applications than the chromat*
*ic filtration.
In appendix B, we show that Hirschhorn's localization techniques can be used*
* to recover
Bousfield localization in the categories of simplicial sets and spectra with re*
*spect to a given
homology theory. This allows us to apply all results of Hirschhorn's book when *
*dealing with
classical Bousfield localizations. We include this result in this paper as it m*
*otivates our Definition
3.8 of localization in the category of motivic spectra in appendix A.
I thank Paul Goerss for some discussions.
Chromatic motivic homotopy theory *
* 3
1 Recollections on the motivic Adams spectral sequence
We denote the stable A1homotopy category (sometimes also called the "stable mot*
*ivic homotopy
category") of Voevodsky [Vo] by SH(k). An object in this category is called a m*
*otivic spectrum
or a P1-spectrum. The following theorem is due to Morel [Mo4 ].
1.1 Theorem. For any perfect field k of characteristic different from 2, ther*
*e are natural
isomorphisms
[Sj, (Gm )^n]SH(k)= 0 8 j < 0
and
[S0, (Gm )^n]SH(k)~=KMWn (k).
Here [ , ]SH(k)means HomSH(k)( , ), and KMW* (k) is the Milnor-Witt K-th*
*eory of k.
It is the tensor algebra generated by the units of k in degree 1 and an element*
* j in degree -1
modulo certain relations. The Hurewicz map S0 ! HZ yields a map KMW* (k) ! KM*(*
*k) given
by mapping j to 0. See [Mo3 ], [Mo4 ] for more details.
The problem of computing [Sj, (Gm )^n]SH(k)for j > 0 is entirely open, and w*
*e hope that the
motivic chromatic spectral sequence we construct in the next section will lead *
*to computations
by proceeding similarly to ordinary topology (as sketched in the introduction).*
* That is, the
motivic chromatic spectral sequence should converge to the E2-term of the motiv*
*ic Adams-
Novikov spectral sequence we now describe.
The construction of the motivic version of the Adams-Novikov spectral sequen*
*ce for a given
motivic ring spectrum E in SH(k) is due to Morel [Mo1 ], [Mo2 ]. He then furthe*
*r studies the
case E = HZ =2 and shows that the spectral sequence described below convergence*
*s to a certain
completion of GW (k) when applied to the sphere spectrum. This led him to his c*
*onjecture on
[S0, S0]SH(k)which is now part of his Theorem 1.1 as we have GW (k) ~=KMW0 (k).*
* It seems
desirable to proceed with "finer" spectra (having a larger öm tivic Bousfield c*
*lass") than HZ =p.
We will be mainly interested in the cases E = MGL and E = BP . Of course, as *
*in topology
the spectral sequence for BP should be easier to compute as the one for MGL .
Following Morel [Mo2 , p. 10], the E2-term of the motivic Adams-Novikov spec*
*tral sequence
(which converges to something related to [X, Y ]SH(k)) is given by
Es,u2~=Exts[ **E,E]SH(k)(E**(Y ), s+uE**(X)),
provided that E ^ E is a projective locally finite E-module (see Definition 2.1*
*7). This property
will hold for E = MGL or BP , see Theorem 2.18. The E1 -term is much harder t*
*o identify,
one reason being the absence of a Serre finiteness theorem.
2 The chromatic constructions
2.1. Instead of considering the Adams-Novikov spectral sequence for MU, topol*
*ogist ofter
study only the situation localized at a given prime p. The localized spectrum *
*MU(p)then
decomposes into a wedge of Brown-Peterson spectra BP (see [BP ]), and BP corres*
*ponds to a
Chromatic motivic homotopy theory *
* 4
universal p-typical formal group law. Traditionally, the notation of BP and al*
*l objects built
from it does not reflect the once and for all fixed prime p.
Using the existence of certain elements in ß*(BP ), one then further constru*
*cts spectra E(n)
for all nonnegative integers n. These define Landweber exact cohomology theorie*
*s and are crucial
to define the chromatic filtration. Their Bousfield classes decompose into Mora*
*va K-theories
K(n) with coefficient ring M(n)* = Fp[vn, v-1n] where vn sits in degree 2(pn-1)*
*. See e. g. [Ra4 ]
for a more detailed survey and references.
2.2. In this section, we discuss the motivic analogues BP and E(n) of the spe*
*ctra BP and
E(n), and we show how they can be used to set up the motivic chromatic spectral*
* sequence
which conjecturally converges to the E2-term of the motivic Adams spectral sequ*
*ence. Two
motivic versions of connected Morava K-theories k(i) have been defined by Borgh*
*esi [Bor, pp.
402 and 411] along with computations of their motivic cohomology groups with fi*
*nite coefficients
[Bor, Theorem 12, Corollary 8]. We conjecture (see Conjecture 2.15) that Bousfi*
*eld localization
with respect to our motivic spectrum E(n) decomposes as Bousfield localization *
*with respect to
motivic non-connected Morava-K-theories K(i) as it does in topology (see e. g. *
*[Ra4 , Theorem
7.3.2 (d)]).
2.3. Given a simplicial presheaf, we use the same symbol for it and its P1-sus*
*pension spectrum
if no confusion may arise. We write pq for the functor Sp-q ^ (Gm )^q^ and =*
* 1,0. For
any P1-spectra E and X, we set Epq(X) := [ pqS0, E ^ X]SH(k)following [Vo, sect*
*ion 6].
Observe that Morel [Mo2 ] writes ~Epq(X) instead. The submodule E2n,n(X) of th*
*e bigraded
E**-module E**(X) is denoted by E2*,*(X), and we write ßpq(X) instead of S0pq(X*
*). We further
set E-p,-q= Epq.
2.4. Morel and Levine [LMo ] suggested a definition of algebraic cobordism **
* for objects
in Sm=k. Observe that we have a map *(k) ! MGL 2*,*of graded presheaves of a*
*belian
groups on Sm=k by [LMo ], and this map is an isomorphism after tensoring with Q*
* [HM ]. if k
is a subfield of C or if we had an isomorphism MGL 2*,*~=MU2*, then using the *
*composition
* ! MGL 2*,*! MU2* ! E(n)2*, and looking at Landweber's Exactness Theorem [La*
*],
one might expect that the groups E(n)2*,*could also be obtained by tensoring wi*
*th ß*E(n).
But in any case there is no motivic version of Brown's [Br] representability th*
*eorem yet that
assigns P1-spectra to a bigraded presheaf on Sm=k fulfilling some geometric pro*
*perties. Some
people believe that such a theorem can be deduced from the general Brown repres*
*entability
theorem of Neeman [Ne] in the context of triangulated categories. A naive motiv*
*ic version of
the chromatic resolution BP* ! M0 ! M1 ! ... can be constructed as in topology.*
* Any
~=
reasonable proof of the conjectured isomorphism MGL 2*,*! MU* should also imp*
*ly that
~=
BP 2*,*! BP*. Nevertheless, this will not imply that the naive motivic resoluti*
*on of BP will
consist of BP *,*(BP )-comodules as in topology [Ra2 , Lemma 5.1.6]. Also, it i*
*s not clear if the
motivic Mn will be isomorphic to the BP **(Ln(Nn(S0))) of Definition 2.12 (see *
*[Ra3 , Theorem
1] for the corresponding proof in topology).
2.5. Let now BP be the P1-spectrum defined in [HK ], [Ve] (a different constru*
*ction is suggested
in [Ya]) which is by construction a direct summand of the p-localisation MGL (*
*p)of MGL using
a certain idempotent e. Both MGL and BP are motivic ring spectra. Using [Hu *
*] or [Ja4], we
Chromatic motivic homotopy theory *
* 5
may indeed assume that they are strictly associative and not only up to homotop*
*y. Similarly,
all the spectra we construct in the sequel are strictly associative ring spectr*
*a, and we have maps
of ring spectra MGL ! MGL (p)e!BP .
2.6. Levine and Morel prove [LMo , Theorem 12.8] that there is an morphism of *
*graded rings
Z[x1, x2, x3, ...] ! * where xisits in degree 2i, and it is an isomorphism if *
*the characteristic of
the base field k is 0. The composition OE : Z[x1, x2, x3, ...] ! * ! MGL 2*,**
*! MGL 2*,*(p)maps
each xn to an element OE(xn) 2 [(P1)^n, BP ]SH(k). Let mn = OE(xn) if n is not *
*a power of p.
2.7 Definition. For all n 0, define vn recursively (as n increases) to be *
*the element
P n pi
in BP -2n,-n= [(P1)^n, BP ] given by vn := pln - i=1livn-i where li = e(mpi-1*
*) and in
particular l0 = 1.
2.8. Although this might not be the standard definition in topology, the theor*
*y of p-typical
P n pi
formal group laws and Araki's formula [Ra2 , A.2.2.2] pln = i=0livn-iimply th*
*at our Definition
2.7 is the correct one when carried out in ordinary topology. Conjecturally, th*
*ese vn are related
to the elements an [Bor, Theorem 10] as in topology, and our definition of conn*
*ected motivic
Morava K-theory should be equivalent to the one of Borghesi [Bor, p. 402]. An*
*yway, the
chromatic constructions below can be carried out starting with Borghesi's defin*
*ition just as
well.
2.9 Definition. For any ring spectrum F with multiplication ~ : F ^ F ! F, any*
* spectrum E
which is an F-module (e.g., E = F) and any element a 2 ß2s,sF, we set
~(a^id)
E=a := hocof((P1)^s^ E -! E)
where hocof denotes the homotopy cofiber, and
~(a^id)1 ^-s ~(a^id)1 ^-2s
a-1E := hocolim(E -! (P ) ^ E -! (P ) ^ E ... )
We define
E(n) := v-1nBP=(vn+1, vn+2, ...)
k(n) := BP =(v0, v1, ..., vn-1, vn+1, vn+2, ...)
K(n) := v-1nk(n).
2.10. The hocolim and hocof are carried out in Hu's category of motivic S-modu*
*les ([Hu ],
which is the A1-version of [EKMM ]). It follows that if F is a strict (associa*
*tive, unital, commu-
tative) ring spectrum, then so is a-1F. Moreover, if E is a F-module for some r*
*ing spectrum F,
then so are a-1E and E=a. The fact that it should be possible construct motivic*
* analogues of
the spectra E(n) and K(n) by killing and inverting appropriate elements is alre*
*ady mentioned in
[Hu , section 14]. Hu suggests to proceed by killing elements in MGL (p)instea*
*d of BP (compare
also [EKMM , V.4]). By [Hu , Proposition 7.2] the homotopy category of motivic*
* S-modules is
equivalent to Jardine's [Ja4] homotopy category of motivic symmetric spectra an*
*d hence [Ja4, p.
Chromatic motivic homotopy theory *
* 6
473 and Theorem 4.3.1] to the stable motivic homotopy category of Voevodsky [Vo*
*]. By abuse
of notation, we will use the same symbol for an object in each of these equival*
*ent categories.
Assuming that the definitions of algebraic cobordism of Voevodsky [Vo, secti*
*on 6.3] and
Levine-Morel [LMo ] coincide in bidegree 2*, *, it seems natural to ask whether*
* E(n)2*,*(X) ~=
MGL 2*,*(X) MU2* E(n)2*.
For the existence of Bousfield localization functors and their properties in*
* general as well as
the definition of LE for a P1-spectrum E, we refer the reader to the appendix. *
*We then obtain
the following.
2.11 Definition. For any motivic spectrum E , we denote by LE : SH(k) ! SH(k)*
* the
Bousfield localization functor of Theorem 3.1 and Definition 3.8.
Next, we can define the chromatic filtration.
2.12 Definition. For any object X of SH(k), we set Ln(X) := LE(n)(X). We furth*
*er set
N0(X) = X and inductively Nn+1(X) := hocof(Ln : Nn(X) ! Ln(Nn(X))). Denote the
induced map -1Nn(X) ! Nn-1(X) by ffn-1. We define the chromatic tower Chr(X) o*
*f X by
n-1ffn-1-(n-1) n-2ffn-2-(n-2) -1 ff0
... -nNn(X) -! Nn-1(X) -! Nn-2(X) ... N1 ! X.
This corresponds in fact to the complement of the chromatic tower as defined*
* in classical
topology, see Proposition 2.16 below.
One may ask if the following motivic version of the smash product theorem [R*
*a4 , Theorem
7.5.6] holds.
2.13 Question. For any P1-spectrum X, is there is a natural isomorphism X ^ Ln*
*(S0) '!
Ln(X) in SH(k)?
2.14 Lemma. If the answer to question 2.13 is positive, then the isomorphism *
*induces an
isomorphism X ^ Chr(S0) '!Chr(X) of towers in SH(k).
Proof. Trivial as the functor X^ on SH(k) is exact.
The following is of course inspired by [Ra4 , Theorem 7.3.2 (d)].
2.15 Question. Define L0n= LK(n)_..._K(1)_K(0). Is there a natural isomorphism*
* of Bousfield
localization functors Ln ! L0n?
Similar to Definition 2.12, we set N00(X) = X and inductively N0n+1(X) := ho*
*cof(L0n:
N0n(X) ! L0n(N0n(X))). As usual, this defines a filtration on Hom-sets by setti*
*ng
F s[Sk, X]SH(k):= Im([Sk, -sNs(X)]SH(k)! [Sk, X]SH(k)). Hence applying [S*, ]*
*SH(k)to the
chromatic tower of X, we get a filtered graded object and an associated spectra*
*l sequence.
2.16 Proposition. We have an objectwise exact triangle of towers
-*N0*(X) ! X ! L0*-1(X)
Chromatic motivic homotopy theory *
* 7
in SH(k). In particular, we have L0n(N0n(X)) ' hofib(N0n+1(X) ! N0n(X)) ' hofi*
*b( nL0n(X) !
nL0n-1(X). Moreover, there is a spectral sequence
Es,t1= ßt(L0s(N0s(X))) ) ßt-s(holim L0n(X)).
If Conjecture 2.15 holds, the same statements hold for Li and Ni.
Proof. The first part can be shown by proceeding essentially as in ordinary top*
*ology [Ra1 ,
Theorem 5.10], using that SH(k) is triangulated and that our localization funct*
*ors Ln are
idempotent. Observe in particular that Definition 3.8 implies that the motivic *
*analogue of Cnf
in the proof of [Ra1 , Theorem 5.3 a)] is well defined. The spectral sequence i*
*s standard, compare
also [Ra1 , Proposition 5.12].
The following definition is essentially taken from [Mo2 , page 9].
2.17 Definition. Given a motivicWring spectrum F, an F-module E is called free*
* if there is a
stable weak equivalence E ' ff nff,iffF that commutes with the F-action up to *
*homotopy. We
say that E is projective if E is a retract (as an F-module) of a free F-module.*
* If in the above
decomposition for any integer N there is only a finite number of ff such that n*
*ff N, we say
that the module is locally finite.
Following standard terminology in topology (see e. g. [Ra4 , Definition A.2.*
*9]), one might
say that F is flat if F ^ F is F-free.
The fact that MGL ^ MGL is locally finite free is due to Morel (personal c*
*ommunication,
February 2003).
2.18 Theorem. The motivic spectra MGL ^ MGL and BP ^ BP are locally finite f*
*ree over
MGL resp. BP .
Proof. Following Morel, the proof for MGL is similar to the one in topology,Ws*
*eeW[Ad , Lemma 4.5
and Lemma 11.1]. In particular, we have a stable weak equivalence g = ffgff: *
*ff nff,iffMGL'!
MGL ^ MGL where gffis given by the composition of nff,iffMGLfff^id-!MGL^ M*
*GL ^
MGL id^~-!MGL^MGL and the fffrun through a system of generators of the free*
* ß**(MGL )-
*
* (fff)(p)^id
module MGL **(MGL ). We still have a stable equivalence g(p): nff,iffMGL(p)*
* -!
MGL (p)^ MGL (p)^ MGL (p)id^~-!MGL(p)^ MGL (p)after localizing at (p). As *
*BP :=
hocolim(MGL (p)e!MGL (p)e!...) (see [Ve, Definition 4.3]), we see that nff,i*
*ffBP'!BP^BP
is also a stable weak equivalence.
2.19 Proposition. Assume that E ^ E is a locally finite projective E-modul*
*e. Then
(ß**(E), E**(E)) is a Hopf algebroid, and E**(X) is a left comodule over E**(E)*
* for any motivic
spectrum X.
Proof. Similar to topology, see e. g. [Ra4 , section B.3].
Chromatic motivic homotopy theory *
* 8
Composing the maps L0n(N0n(X)) ! N0n+1(X) and N0n+1(X) ! L0n+1(N0n+1(X)), we*
* get a
sequence of BP **(BP )-comodules
0 ! BP **(X) ! BP **(L00(N00(X))) ! BP **(L01(N01(X))) ! ... (*)
Observe that (*) is a chain complex by Proposition 2.16.
2.20 Theorem. Assume that the chain complex (*) is exact. Then there is a spec*
*tral sequence
En,s1= ExtsBP**(BP()BP**, BP **(L0n(N0n(X))))
converging to Extn+sBP**(BP()BP**, BP **(X)). It is called the motivic chromati*
*c spectral sequence
of X.
Proof. The proof is purely homological algebra, one may proceed exactly as in [*
*Ra2 , Proposition
5.1.8, Corollary A.1.2.12 and Theorem A.1.3.2].
Observe that this is not the standard description of the chromatic spectral *
*sequence when
carried out in classical topology, but is equivalent to it by the results of [R*
*a3 ].
In classical topology, computations of the E1-term can be reduced to the com*
*putation of
Ext-groups of Morava K-theories over the Morava stabilizer algebra (see [Ra4 , *
*section B.8]).
We do not know whether a similar statement holds in the motivic setting. For co*
*mputations of
the chromatic spectral sequence in topology, for instance the link with Im J me*
*ntioned in the
introduction, the reader may consult [Ra2 , section 5].
Of course, we would like to know if holim Ln(X) is isomorphic to X in SH(k) *
*(or at least if
the BP -homology groups are isomorphic) provided X has p-local homotopy groups *
*and satisfies
some finiteness conditions. In topology, this is called the chromatic convergen*
*ce theorem [Ra4 ,
Theorem 7.5.7] and is deduced from the smash product theorem (compare Question *
*2.13).
Even if the questions in this article were all answered, one is probably sti*
*ll quite far away
from establishing the analogues of the big theorems about nilpotence and period*
*icity (or at least
the precursors of Toda or Nishida, see e. g. [Ra4 , section 9.6]) in A1-homotop*
*y. Observe that
Morel's Theorem 1.1 implies that the stable algebraic Hopf map j : Gm ! S0 is n*
*ot nilpotent,
so the naive motivic analogue of Nishida's nilpotence theorem will not hold.
3 Appendix A: Bousfield localizations in A1-homotopy
In this appendix, we will first review some general results concerning (left) B*
*ousfield [Bo2]
localizations in cellular model categories [Hi1] and in associated categories o*
*f spectra [Hov2].
Then we will explain how these techniques may be applied to the unstable and st*
*able A1-
homotopy category, which fills the gap before Definition 2.11. I thank Lars Hes*
*selholt who was
the first to tell me about the work of Hirschhorn and Hovey, and I thank Dan Du*
*gger, Christian
Häsemeyer for discussions about earlier drafts of the appendix and Phil Hirschh*
*orn for providing
a detailed proof of Lemma 3.5 and allowing me to include it here.
We assume that the reader is familiar with the following definitions (see e.*
* g. [Hi1]) con-
cerning a given model category C: left proper, cofibrantly generated, cellular,*
* size, S-local and
Chromatic motivic homotopy theory *
* 9
S-acyclic objects with respect to a given set of morphisms S in C. For the def*
*inition of the
category Sp(C, T ) of T -spectra for a left Quillen endofunctor T : C ! C, see *
*e. g. [Hov2,
section 1], and for the definition of ä lmost finitely generated", see [Hov2, s*
*ection 4]. Adding
and forgetting base points yields adjoint functors (see e.g. [MV , p. 109]), so*
* everything in the
sequel about simplicial sets and presheaves also holds for pointed objects, and*
* it is this pointed
version we use when passing to spectra. When we write P1, we always mean the re*
*presented
simplicial presheaf (P1, 1) pointed at infinity, which is different from the si*
*mplicial presheaf
P1+represented by the variety P1 with an added disjoint basepoint.
In order to apply the techniques of Bousfield localization from [Hi1], we ne*
*ed to know that
our model structure is cellular and left proper. The stable injective model st*
*ructure of [Ja4,
Theorem 2.9] (which is the stable version of the model structure of Morel-Voevo*
*dsky [MV ],
which in turn is a localization of the unstable model structure of Jardine [Ja1*
*]) is left proper
and cellular, see Corollary 3.7 below.
The general results we will need are the following:
3.1 Theorem. Suppose S is a set of maps in a left proper cellular model catego*
*ry C. Then
there is a left proper cellular model structure on C where the weak equivalence*
*s are the S-local
equivalences and the cofibrations are the cofibrations of C. We denote this new*
* model category
by LS(C) and call it the öB usfield localization of C with respect to S". The S*
*-local objects are
precisely the fibrant objects of LS(C), and thus we also write LS for a fixed c*
*hoice of a S-fibrant
replacement functor. The functor LS is idempotent.
Proof. See [Hi1, Theorem 4.1.1].
3.2 Theorem. Suppose C is a left proper cellular model category, and T is a le*
*ft Quillen end-
ofunctor on C. Then the category Sp(C, T ) of T -spectra with the levelwise def*
*ined fibrations and
weak equivalences is a left proper cellular model category. Hence by Theorem 3.*
*1, its localization
with respect to the stable weak equivalences as defined in [Hov2, Definition 3.*
*3] is a left proper
cellular model category.
Proof. See [Hov2, Theorem A.9].
The localized model structure given by Theorem 3.2 is called the stable mode*
*l structure with
respect to Sp(C, T ). So the strategy is to start with a left proper cellular m*
*odel category and
then to construct other model categories from it using these two theorems. We w*
*ill often say
Bousfield localization when applying these two theorems. This terminology is ju*
*stified by the
results of Appendix B. Observe also the related paper [GJ ] which discusses Bou*
*sfield localizations
for sheaves of S1-spectra.
The following model structure is sometimes called the "Heller model structur*
*e" (compare
[He]).
3.3 Definition. For any small category C, we say that a map of simplicial pres*
*heaves on C
is a global injective cofibration (resp. global injective weak equvalence) if *
*it is a sectionwise
cofibration (resp. sectionwise weak equvalence) of simplicial sets.
Chromatic motivic homotopy theory *
*10
Defining the global injective fibrations via the lifting property, we obtain*
* the global injective
model structure if C is the big Nisnevich site; and the model structure of Jard*
*ine [Ja1] with the
same cofibrations and the weak equivalences being the stalkwise (for the Nisnev*
*ich topology)
weak equivalences of simplicial sets yields the local injective model structure.
3.4 Theorem. The global injective model structure on opP rShv(Sm=k)Nis is cel*
*lular and
left proper. The local injective model structure of Jardine on opP rShv(Sm=k)N*
*is is cellular
and left proper.
Proof. That both model structures are left proper is proved in [Ja2, Propositio*
*n 1.4]. The
cellularity result is due to Hirschhorn [Hi2, Proposition 5.6 and Theorem 6.1].*
* As he doesn't
give the complete proof and moreover [Hi2] is not published, we include a proof*
* here. Both
the global and the local injective model structure are cofibrantly generated us*
*ing the classes of
[Ja1, pp. 65 and 68] as generating (trivial) cofibrations. An alternative choic*
*e for the sets of
generating (trivial) cofibrations is given in [Hi2, section 4]. By [Hi1, Defini*
*tion 12.1.1], there are
three properties to check for cellularity. Property (iii) is clear as it holds *
*for sets and can be
checked sectionwise. Condition (ii) is also satisfied as any presheaf of sets i*
*s small with respect to
the whole category, and simplicial presheaves on a category C are nothing else *
*but presheaves of
sets on C x . Condition (i) (the domains and codomains of the generating cofib*
*rations have to
be compact in the sense of [Hi1, Definition 11.4.1]) holds for the same reason,*
* that is presheaves
of sets are compact with respect to everything when choosing a cardinal suffici*
*ently large with
respect to the site. The details are given in Lemma 3.5 below. Observe that l*
*eft properness
and cellularity of the local model structure would also follow from the corresp*
*onding properties
of the global model structure by applying Theorem 3.1 if you are willing to bel*
*ieve or verify
Hirschhorn's [Hi2, p. 10] claim that the local structure is obtained by localiz*
*ing with respect to
the set of S of [Ja1, p. 265].
3.5 Lemma. (P. S. Hirschhorn) Both the global and the local injective model st*
*ructure for
simplicial presheaves on a small category C fulfill condition (1) of [Hi1, Defi*
*nition 12.1.1].
Proof. We reproduce the argument that Hirschhorn (personal communication, June *
*2003) pro-
vided. Suppose we have some cofibrantly generated model category structure on a*
* category of
diagrams of simplicial sets. Let I be a set of generating cofibrations. We ne*
*ed to show that
the domains and codomains of I are compact relative to I. Each such domain or c*
*odomain is a
diagram of simplicial sets, and so you one take the cardinal of the sets of sim*
*plices that appear
(i. e., the union over all of the domains and codomains of the union over all *
*objects in the
indexing category of all the simplices in the simplicial sets at all of those o*
*bjects), and let fl be
the cardinal that's the successor of the cardinal of that union. Since fl is a *
*successor cardinal, it
is regular.
The next step is to prove that if X is any cell complex (i. e., anything bui*
*lt from the initial
object by taking a transfinite composition of pushouts of elements of I), then *
*every cell of X is
contained in a subcomplex of X of size less than fl. This is similar to [Hi1, P*
*roposition 10.7.6]
where it is proved that, for cell complexes of topological spaces, every cell o*
*f a cell complex
is contained in a finite subcomplex of the cell complex. More precisely, we mak*
*e induction on
the "presentation ordinalö f the cell (that is, if X is constructed by means o*
*f a ~-sequence, an
Chromatic motivic homotopy theory *
*11
induction on the ordinal ff such that the cell is attached at stage ff of the ~*
*-sequence). A cell
attached at stage 0 is a subcomplex all by itself (of size 1). If every cell at*
* stage ff is contained in
a subcomplex of size less than fl, then each cell that one attaches at stage ff*
*+1 has an attaching
map that hits fewer than fl many simplices, each of which is contained in a uni*
*que cell, each of
which is contained in a subcomplex of size smaller than fl, and so taking the u*
*nion of all of those
subcomplexes one still gets a subcomplex of size less than fl (since fl is a re*
*gular cardinal). At
limit ordinals there are no cells attached, so the induction is thus complete.
Now if W is a domain or codomain of an element of I, then W has fewer than f*
*l simplices
(counting all of the simplices at all of the objects in the indexing category),*
* and if one maps W
into a cell complex X, one hits fewer than fl simplices, each of which is part *
*of a cell that is
contained in a subcomplex of size less than fl, and the union of all of those i*
*s of size less than fl
(since fl is regular).
3.6. Observe that [MV , Lemma 2.2.8 and Proposition 2.2.9] implies that the A1*
*-local injective
model structure of Morel and Voevodsky extended to presheaves as in [Ja4, Theor*
*em 1.1] is
obtained precisely by applying Hirschhorn's Theorem 3.1 to the local injective *
*model structure
of Jardine and the set S of morphisms A1 x Y !prY for all smooth varieties Y . *
* Hence, the
Morel-Voevodsky model structure is also cellular and left proper. Passing to P1*
*-spectra, we see
by Theorem 3.11 below that the stable injective model structure obtained by app*
*lying Hovey's
Theorem 3.2 to opP rShv(Sm=k)Nis equipped with the A1-local injective model st*
*ructure of
Morel and Voevodsky is identical as a model category to the stable model struct*
*ure of Jardine
[Ja4, Theorem 2.9] and hence has a homotopy category equivalent to the one of V*
*oevodsky [Vo].
Hence by Theorem 3.2 we obtain the following:
3.7 Corollary. The stable model structure on motivic P1-spectra of [Ja4, Theo*
*rem 2.9] is
cellular and left proper.
Proof.
Now we are ready to define the Bousfield localization with respect to a P1-s*
*pectrum E.
Recall that by Theorem 3.1 we have a functor LS : SH(k) ! SH(k).
3.8 Definition. Given a motivic spectrum E = (E0, E1, ...), we define the Bous*
*field localization
LE with respect to E to be the Bousfield localization LS (see Theorem 3.1) of S*
*p( opP rShv(Sm=k)Nis, P1^)
with respect to a set S = S(E) of representatives of isomorphism classes of the*
* class consisiting
of stable projective cofibrations (i. e., those having the lifting property wit*
*h respect to level-
wise A1-local projective trivial fibrations) ' : C ! B such that id ^ ' : E ^ C*
* ! E ^ B is an
isomorphism in SH(k) and moreover that the size of B is at most fl (see [Hi1, D*
*efinition 4.5.3]).
The above definition is motivated by the results of appendix B, in particula*
*r Theorem 4.6.
Observe also that by definition of SH(k) (see [Vo, Definition 5.1], [Mo3 , Defi*
*nition 5.1.4]) the
condition that id ^ ' : E ^ C ! E ^ B is an isomorphism in SH(k) is stronger th*
*an than just
requiring that E**(') is an isomorphism.
We will now discuss an alternative model structure which also admits Bousfie*
*ld localizations.
This is not necessary for the chromatic constructions of this paper, but is rat*
*her included for
the sake of completeness and further reference.
Chromatic motivic homotopy theory *
*12
Given the category of simplicial presheaves on a small category C, we can de*
*fine the global
projective model structure by defining the weak equivalences and the fibrations*
* sectionwise (op-
posed to Definition 3.3 where we took the cofibrations instead). Hirschhorn pro*
*ves the following:
3.9 Theorem. If M is a cellular model category and C is a small category, the*
*n the category
of presheaves on C with values in M is a cellular model structure when defining*
* the fibrations
and weak equivalences as being the sectionwise fibrations and sectionwise weak *
*equivalences in
M.
Proof. See [Hi1, Proposition 12.1.5].
Applied to our situation, we may begin with the category of simplicial presh*
*eaves or sheaves
on a site (e. g., (Sm=k)Nis) with weak equivalences and fibrations defined sect*
*ionwise; and then
prove that each of the two steps passing to local weak equivalences and localiz*
*ing with respect
to A1k! Spec(k) as done in [MV ] is a Bousfield localization in the sense of Th*
*eorem 3.1. This
has been carried out by [Bl]:
3.10 Theorem. The categories of simplicial presheaves and sheaves on Sm=k admi*
*t simplicial
proper cellular model structures if we define the weak equivalences to be the A*
*1-equivalences as
defined in [MV , Definition 3.2.1], the cofibrations to be the maps having the *
*left lifting property
with respect to the sectionwise trivial fibrations and the fibrations being tho*
*se having the right
lifting property with respect to the cofibrations.
Proof. See [Bl, Theorem 3.1]. Blander first proves the Theorem for the stalkwis*
*e weak equiv-
alences [Bl, Theorem 2.1] which yields the local projective model structure. I*
*t is possible to
replace this part of his proof by [Hi1, Proposition 12.1.5] and then localizing*
* with respect to the
subset of morphisms P ! X of S0 as defined in [Hov2, section 4], see [Bl, Lemma*
* 4.2]. Then
Blander shows that Hirschhorn's Bousfield localization (Theorem 3.1) applies wh*
*en passing to
A1-equivalences.
This model structure on simplicial presheaves is called the A1-local project*
*ive model structure.
Recall that the one of [MV ] extended to presheaves as in [Ja4, Theorem 1.1] is*
* called the A1-
local injective model structure. The identity functors between these two model *
*structures form
obviously a Quillen equivalence (see also [Du , Proposition 8.1]).
Now assume that T is a compact (see [Ja4, p. 466]) simplicial presheaf, in p*
*articular that
the functor T preserves sequential colimits. Observe that if T ^ is a left Qui*
*llen endofunctor on
P rShv(Sm=k)Nisequipped with the A1-local projective model structure, then (as*
* explained
below) the underlying category of the stable projective model structure on Sp( *
*P rShv(Sm=k)Nis, T ^)
is identical to the underlying category of motivic spectra of Jardine's [Ja4, T*
*heorem 2.9] which
we will call the stable injective model structure. Let ø : P1cof! P1 be an A1-l*
*ocal projective
cofibrant replacement of P1. Then P1cof^ is a left Quillen endofunctor on opP *
*rShv(Sm=k)Nis
equipped with the A1-local projective model structure, see below.
Of course, now the question arises if the identity functor on opP rShv(Sm=k*
*)Nisand ø in-
duce a Quillen equivalence from the stable injective model structure on Sp( opP*
* rShv(Sm=k)Nis, P1^)
of Jardine to the stable projective model structure on Sp( opP rShv(Sm=k)Nis, P*
*1cof^).
Note that if T ^ is not a left Quillen functor, then Theorem 3.2 does not ap*
*ply, and we do
not get a stable projective model structure. One possible strategy of proving t*
*he equivalence of
Chromatic motivic homotopy theory *
*13
the projective and the injective stable model structure is suggested by the fol*
*lowing result due
to Hovey:
3.11 Theorem. Let C be a left proper, almost finitely generated model category*
* where sequential
colimits preserve finite products. Suppose T : C ! C is a left Quillen functor *
*whose right adjoint
U commutes with sequential colimits. Then the following holds:
(i) For any object A of Sp(C, T ) and 1 as in [Hov2, Definition 4.4], the m*
*ap A ! 1 (Afib)
is a stable equivalence.
(ii) A map f : A ! B is a stable weak equivalence if and only if 1 (ffib) i*
*s a level
equivalence.
Proof. See [Hov2, Theorem 4.12].
We will now discuss if the hypotheses are fulfilled in our case. The questio*
*n is if we want to
define our spectra with respect to P1 or P1cof. In the A1-local projective mode*
*l structure, the
object P1 might be not cofibrant, so the functor P1^ is not necessarily a left *
*Quillen functor
and we can't use Hovey's Theorem 3.11 to identify the weak equivalences. If P1*
*^ was a left
Quillen functor, then comparing (i) and (ii) of Theorem 3.11 with [Ja4, p. 470]*
*, we would see
that the identity yields a stable Quillen equivalence between the projective an*
*d the injective
stable model structures, both being defined by levelwise weak equivalences on t*
*he associated
infinite loop spaces.
Smashing with P1cofinstead will give us a left Quillen functor as desired. *
*This will be a
consequence of the following stronger result:
3.12 Theorem. The category P rShv(Sm=k)Nis is a monoidal model category in the*
* sense
of Hovey (see [Hov1, Definition 4.2.6] or [Hov2, Definition 6.2]) when equipped*
* with either the
A1-local injective or with the A1-local projective model structure.
Proof. The injective case follows as smashing with a given object preserves A1-*
*local weak equiv-
alences (see [MV , Lemma 3.2.13]) and sectionwise monomorphisms. For the projec*
*tive case, we
first observe that the category opP rShv(Sm=k)Niswith the global projective mo*
*del structure
given by [Hi1, Theorem 11.6.1] (i. e., fibrations and weak equivalences are def*
*ined sectionwise)
is monoidal. The condition on the unit object S0 follows as S0 is cofibrant. *
*This follows as
Spec(k) is a final object, so we can construct liftings starting with the secti*
*on of our given
trivial fibration on Spec(k).
The condition on pushouts can be replaced by an adjoint condition (see [Hov1*
*, Lemma 4.2.2])
which is fulfilled as global projective cofibrations are in particular sectionw*
*ise cofibrations and
the category of simplicial sets is a monoidal model category (see e. g. [Hov1, *
*Proposition 4.2.8]).
The A1-local projective model structure is then also a monoidal model category *
*by Lemma 3.13
below. The assumption of Lemma 3.13 is fulfilled as for any simplicial set K an*
*d any object V
of (Sm=k)Nis, the pointed simplicial presheaf K x V+ is cofibrant for the A1-lo*
*cal projective
model structure. The argument is precisely the same as for S0 = 0 x Spec(k)+ *
*as we may
restrict to the site (Sm=V )Nis when checking the lifting property. In particul*
*ar, the domains
@ n x V+ and codomains n x V+ of the generating cofibrations of opP rShv(Sm=k*
*)Nis as
given by [Hi1, Theorem 11.6.1] are cofibrant. This implies that A^ preserves co*
*fibrations (as C
Chromatic motivic homotopy theory *
*14
is monoidal), and from the fact that A^ preserves A1-local weak equivalences we*
* see that the
assumption of Lemma 3.13 is satisfied.
I thank M. Hovey for drawing my attention to the following result of him, wh*
*ich is essentially
the same as the end of the proof of [Hov2, Theorem 6.3].
3.13 Lemma. Let C be a cellular monoidal model category with a set I of genera*
*ting cofibra-
tions and S a set of morphisms in C. Assume that for any domain or codomain A *
*of I, the
functor A^ preserves local trivial cofibrations. Then LS(C) is also a monoidal *
*model category.
Proof. Because localization preserves cofibrations, the only thing we have to c*
*heck is that if
f : A ! B is a cofibration and g : C ! D is a local trivial cofibration, then t*
*he pushout product
f g is a local weak equivalence. We may assume that f is a map of I by [Hov2, *
*Corollary
4.2.5]. The assumption implies that idA ^ g and idB ^ g are local trivial cofib*
*rations. Hence
the pushout h : B ^ C ! P of idA ^ g along f ^ idC is a local trivial cofibrati*
*on. On the other
hand, the map idB ^ g is also a local trivial cofibration. Therefore, the map P*
* ! B ^ D, which
is f g, is also a local weak equivalence.
We now can apply the pushout condition of loc. cit. to the map * ! P1cofto s*
*ee that P1cof^
is a left Quillen functor. Observe that although P1+is cofibrant in the A1-loca*
*l projective model
structure, P1 pointed at infinity may not be.
3.14. Concerning the other assumptions of Theorem 3.11, we do not know that th*
*e functor
U = P1cofpreserves sequential colimits ( P1 does as P1 is compact, see [Ja4, L*
*emma 2.2]).
Left properness and cellularity follow Theorem 3.2 and Theorem 3.10. A proof of*
* the property
ä lmost finitely generated" is sketched in [Hov2, section 4]. Some details (see*
* page 84 of loc.
cit.) are not verified, but they follow immediately from [Bl, Lemma 4.2]. Seque*
*ntial colimits
preserve finite products because they do so for simplicial sets. So if Theorem *
*3.11 applies to
P1cof, the identity on Sp( opP rShv(Sm=k)Nis, P1cof^) yields a Quillen equivale*
*nce between the
stable projective and the stable injective model structure on this category.
As indicated above, the map ø : P1cof! P1 induces a functor ~øfrom P1-spectr*
*a to P1cof-
spectra by mapping the structure maps oen : P1 ^ En ! En+1 of the P1-spectrum E*
* to the
composition oen(ø^id). This functor ~øis a Quillen equivalence by [Ja4, Proposi*
*tion 2.13] provided
P1cofis also compact. Assuming this, the identity functor on P1cof-spectra yie*
*lds a Quillen
equivalence between the stable projective and the stable injective model struct*
*ure because it
does so for the unstable A1-local structures and hence also between the model s*
*tructures on
P1cof-spectra where fibrations and weak equivalences are defined levelwise. No*
*w observe that
this gives Quillen equivalence also between the stable projective model structu*
*re and the stable
injective model structure as the weak equivalences in both model structures coi*
*ncide (compare
Theorem 3.11 and [Ja4, p.470], and choose an injective fibrant replacement func*
*tor in Theorem
3.11).
3.15. Using Hovey's techniques [Hov2, section 5], it is possible to get rid of*
* the above com-
pactness condition. First, we may apply [Hov2, Theorem 5.7] to the identity on*
* the pair
( opP rShv(Sm=k)Nis, P1cof^) where opP rShv(Sm=k)Nisis equipped with the A1-lo*
*cal pro-
jective resp. with the A1-local injective model structure. This is a Quillen ma*
*p of pairs as defined
Chromatic motivic homotopy theory *
*15
in [Hov2, Definition 5.4] by Theorem 3.12, taking ø = id which then trivially f*
*ulfills the extra con-
dition of [Hov2, Theorem 5.7]. So we obtain a Quillen equivalence between the s*
*table projective
and the stable injective model structure on P1cof-spectra. Next, we apply [Hov2*
*, Theorem 5.7] to
the Quillen map of pairs (Id, ~ø) : ( opP rShv(Sm=k)Nis, P1cof^) ! ( opP rShv(S*
*m=k)Nis, P1^)
with ~øinduced by ø : P1cof! P1 and opP rShv(Sm=k)Nisequipped with the A1-loca*
*l injec-
tive model structure on both sides (so everything is cofibrant) to obtain a Qui*
*llen equivalence
between the stable injective model structure on P1cof-spectra and the stable in*
*jective model
structure on P1-spectra.
Composing these two Quillen equivalences, we obtain the following:
3.16 Theorem. Choose an A1-local projective cofibrant replacement ø : P1cof! P*
*1. Then the
identity functor on opP rShv(Sm=k)Nisand ø induce a Quillen equivalence from t*
*he the stable
projective model structure on Sp( opP rShv(Sm=k)Nis, P1cof^) of Hovey to the st*
*able injective
model structure on Sp( opP rShv(Sm=k)Nis, P1^) of Jardine. In particular, writi*
*ng SH0(k) for
the homotopy category of Sp( opP rShv(Sm=k)Nis, P1cof^), we have an equivalence*
* of categories
~ø: SH0(k) ! SH(k).
Proof.
3.17. Dan Dugger (personal communication, June 2003) has outlined a strategy h*
*ow to con-
struct an object P1cofthat is compact in the sense of Jardine (so in particular*
* U = P1cof
will commute with sequential colimits, and we obtain the "explicit" description*
* of stable weak
equivalences of Theorem 3.11 rather than just the abstract one of [Hov2, Defini*
*tion 3.3]). Both
Dugger and Hovey informed the author about the existence of some unpublished wo*
*rk of J.
Smith on combinatorial model categories (see e. g. [Du , Definition 6.2]). Acco*
*rding to them,
this should imply that Theorem 3.1 and Theorem 3.2 remain true after replacing *
*"cellular" by
öc mbinatorial".
4 Appendix B: Bousfield localization for classical spectra is a
Hirschhorn localization
Throughout this section, spectra means Bousfield-Friedlander spectra with the s*
*table model
structure of [BF ], and we denote the homotopy category of spectra by SH. The *
*purpose of
this section is to show that if given a homology category E* on simplicial sets*
* resp. spectra
represented by a spectrum E, it is possible to choose a set of morphisms S = S(*
*E) such that
applying Hirschhorn's abstract localization (Theorem 3.1), one obtains a model *
*structure on
simplicial sets resp. spectra whose weak equivalences are precisely the E*-iso*
*morphisms and
whose cofibrations are cofibrations of simplicial sets resp. spectra. Recall th*
*at a cofibration of
spectra is a map having the lifting property with respect to levelwise fibratio*
*ns. We call these
cofibrations stable cofibrations throughout this appendix.
The following localization theorem is due to Bousfield [Bo1]:
4.1 Theorem. There is a model struture on simplicial sets whose cofibrations a*
*re the monomor-
phisms and whose weak equivalences are the E*-isomorphisms.
Chromatic motivic homotopy theory *
*16
Proof. [Bo1, Theorem 10.2].
We will prove that this E-local model structure on simplicial sets is identi*
*cal to one that is
obtained using the set-up of Hirschhorn for a suitable set S(E) of morphisms.
4.2 Definition. We define the S(E)-local model structure on the category of si*
*mplicial sets
as the Hirschhorn localization of Theorem 3.1 with respect to a set S = S(E) of*
* representatives
of isomorphism classes of the class consisiting of cofibrations i : X ! Y such *
*that E*(i) is an
isomorphism, and moreover that the size of Y is at most fl (see [Hi1, Definitio*
*n 4.5.3]).
4.3 Theorem. The two model structures of Theorem 4.1 and of Definition 4.2 yie*
*ld identi-
cal model structures on the category of simplicial sets. In particular, a map *
*is an S(E)-local
equivalence if and only if it induces an E*-isomorphism.
Proof. The set JS of [Hi1, p. 81] is contained in S. This follows when applying*
* [Hi1, Definition
3.1.1 and Theorem 3.3.19] to C = S and N the category of simplicial sets with t*
*he E-local model
structure of Theorem 4.1. As inclusion of subcomplexes are precisely the monomo*
*rphisms in the
category of simplicial sets and we have fl c for the cardinals defined in [Bo*
*1, p. 146] and [Hi1,
Definition 4.5.3], we see that the set JS contains up to isomorphisms the set o*
*f cofibrations of
[Bo1, Lemma 11.3], and thus the claim follows.
In [Bo2, p. 261], Bousfield claims the existence of a model structure on spe*
*ctra whose weak
equivalences are precisely those maps that induce an isomorphism after applying*
* E*. His paper
already contains most of the necessary techniques to prove such a result. The *
*first complete
proof for the existence of this model structure seems to be due to Goerss and J*
*ardine (who prove
a much more general result in [GJ ]).
4.4 Theorem. There is a model struture on spectra whose cofibrations are the s*
*table cofibra-
tions and whose weak equivalences are the E*-isomorphisms.
Proof. Apply [GJ , Theorem 3.10 and Remark 3.12] to C = D = the trivial site an*
*d f the identity
map.
This Theorem 4.4 is phrased in [GJ ] using bispectra. See [Ja3] for the defi*
*nition of bispectra
an the diagional functor d from bispectra to spectra. To say that a map f : X !*
* Y induces
~= '
an isomorphism E*(f) : E*(X) ! E*(Y ) is equivalent to say that d(E ^ X) ! d(E *
*^ Y ) is an
isomorphism in SH.
4.5 Definition. We define the S(E)-local model structure on the category of sp*
*ectra as the
Hirschhorn localization of Theorem 3.1 with respect to a set S = S(E) of repres*
*entatives of
isomorphism classes of the class consisiting of stable cofibrations i : X ! Y s*
*uch that E*(i) is
an isomorphism, and moreover that the size of Y is at most fl (see [Hi1, Defini*
*tion 4.5.3]).
4.6 Theorem. The two model structures of Theorem 4.4 and of Definition 4.5 yie*
*ld identical
model structures on the category of spectra. In particular, a map is an S(E)-lo*
*cal equivalence if
and only if it induces an E*-isomorphism.
Chromatic motivic homotopy theory *
*17
In order to prove this, we will need a couple of lemmata. We say that a map*
* of spectra
X ! Y is an inclusion if Xn ! Yn is an inclusion of pointed simplicial sets. Re*
*call that any
stable cofibration is an inclusion, but the converse does not hold. By [GJ , Le*
*mma 3.1], if A ! B
is a cofibration of spectra and V ! B an inclusion, then the induced map V \ A *
*! V is also
a cofibration of spectra. This will be used in the following lemma, which is a *
*variant of [Bo1,
Lemma 1.12]. Let oe be the cardinal of [Bo1, p. 260]. Thus if ]X oe, the set *
*E*(X) has at
most oe elements.
4.7 Lemma. For any cofibration of spectra i : A ! B which is an E*-isomorphism*
*, there
exists an inclusion W ! B such that ]W oe, W 6 A and the induced cofibration*
* W \ A ! W
is an E*-isomorphism.
Proof. Proceed as in the proof of [Bo2, Lemma 1.12] to construct the desired W .
The next lemma is a spectrum version of [Bo1, Lemma 11.3]
4.8 Lemma. Let f : X ! Y be a map of spectra having the RLP with respect to *
*each
cofibration of spectra i : A ! B such that ]B oe and that E*(i) is an isomorp*
*hism. Then f has
the RLP with respect to each cofibration of spectra i : A ! B such that E*(i) i*
*s an isomorphism.
Proof. Applying Lemma 4.7, one can proceed by transfinite induction exactly as *
*in the proof
of [Bo1, Lemma 11.3] (observe that in our case A ! W [ W is a stable cofibratio*
*n of spectra
because W \ A ! W is).
Proof of Theorem 4.6. We have to analyze the set JS = JS(E) as defined in [H*
*i1, p. 81],
which is a set of generating trivial cofibrations for the model structure of De*
*finition 4.5. First,
observe that one can show similarly to the proof of Theorem 4.3 that JS is cont*
*ained in S.
The category of spectra is cofibrantly generated and even cellular (see Theorem*
* 3.2). The set
I of generating cofibrations is described in [Hov2, Definition 1.8]. The defini*
*tion of inclusions
of subcomplexes that appears in the definition of the set JS = JS(E) is thus gi*
*ven by [Hi1,
Definition 11.1.2] applied to this set I. A careful verification now shows that*
* JS(E) is contained
in the class of all stable cofibrations A ! B that are E*-isomorphisms, and it *
*contains up to
isomorphisms all of those with ]B oe as fl oe. Now Lemma 4.8 allows us to c*
*onclude that
both model structures have not only the same cofibrations, but also the same fi*
*brations and
hence the same weak equivalences.
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Jens Hornbostel, NWF I - Mathematik, Universität Regensburg, 93040 Regensbur*
*g, Ger-
many, jens.hornbostel@mathematik.uni-regensburg.de