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 ! 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Available at http://www.math.uiuc.edu/K-theory/0627 Jens Hornbostel, NWF I - Mathematik, Universität Regensburg, 93040 Regensbur* *g, Ger- many, jens.hornbostel@mathematik.uni-regensburg.de