On Voevodsky's algebraic K -theory spectrum BGL I. Panin*y K. Pimenov* O. R"ondigsz April 16, 2007x Abstract Under a certain normalization assumption we prove that the P1-spectrum * *BGL of Voevodsky which represents algebraic K-theory is unique over Spec(Z). F* *ollow- ing an idea of Voevodsky, we equip the P1-spectrum BGL with the structure* * of a commutative ring P1-spectrum in the motivic stable homotopy category. Furt* *her- more, we prove that under a certain normalization assumption this ring str* *ucture is unique over Spec(Z). For an arbitrary Noetherian base scheme S we pull* * this structure back to get a distinguished monoidal structure on BGL . 1 Preliminaries We refer to the Appendix A for the basic terminology, notation, constructions, * *definitions, results concerning homotopy theory. For a scheme S we write M(S), Mo(S), Hcmo(* *S) and SH (S) for the category of motivic spaces, the category of pointed motivic * *spaces, the pointed motivic homotopy category and the stable motivic homotopy category * *over S. These categories are equipped with symmetric monoidal structures. In parti* *cular, a symmetric monoidal structure (^, I) is constructed on the motivic stable homo* *topy category and its basic properties are proved. This structure is used intensivel* *y over the present text. Let S be a regular scheme, and let K0(S) denote the Grothendieck group of ve* *ctor bundles over S. Morel and Voevodsky proved in [MV , Thm. 4.3.13] that the Thom* *ason- Trobaugh K-theory [TT ] is represented in the pointed motivic homotopy category* * Ho(S) by the spaceSZ x Gr pointed by (0, x0). Here Gr is the union of the finite Gra* *ssmann varieties 1n=0Gr(n, 2n), considered as motivic spaces. There is an element o* *1 - 1 2 K0(Z x Gr) which corresponds to the identity morphism id:Z x Gr ! Z x Gr. It fo* *llows that there exists a unique morphism ~ :(Z x Gr) ^ (Z x Gr) ! Z x Gr ________________________________ *Steklov Institute of Mathematics at St. Petersburg, Russia yUniversit"at Bielefeld, SFB 710, Bielefeld, Germany zInstitut f"ur Mathematik, Universit"at Osnabr"uck, Osnabr"uck, Germany xThe authors thank the SFB-701 at the Universit"at Bielefeld, the RTN-Networ* *k HPRN-CT-2002- 00287, the RFFI-grant 03-01-00633a, and INTAS-03-51-3251 for their support. 1 ~ in Hcmo(S) such that the composition (ZxGr )x(ZxGr ) ! (ZxGr )^(ZxGr ) -! ZxGr represents the element (o1 - 1) (o1 - 1) in K0 (Z x Gr) x (Z x Gr) (see Cor.* * B.1.2). Let e : S0 ! Z x Gr be the map which corresponds to the point (1, x0) 2 Z x Gr* *. The triple (Z x Gr, ~ , e ) (1) is a commutative monoid in Hcmo(S). Using this fact Voevodsky constructed in [V1 ] a P1-spectrum BGL = (K0, K1, K2, . .). with structure maps ei:Ki^ P1 ! Ki+1such that 1. there is a motivic weak equivalence w :Z x Gr ! K0, and for all i one has * *Ki= K0 and ei= e0, 2. the morphism _can//_ 1w^P1//_ 1 ei__//_ w-1_//_ Z x Gr x P1 (Z x Gr) ^ P Ki^ P Ki+1 Z x Gr in Ho(S) represents the element (o1 - 1) ([O(-1)] - [O]) 2 K0(Z x Gr x P* *1), 3. and the adjoint Ki! P1(Ki+1) of eiis a motivic weak equivalence. With this spectrum in hand given a smooth X over S we may identify K0(X) with BGL 2i,i(X) as follows K0(X) = Hom Ho(S)(X+, Z x Gr) = Hom Ho(S)(X+, Ki) = BGL 2i,i(X) (2) Our first aim is to recall Voevodsky construction to show that this spectrum* * is essen- tially unique. This has also been obtained in [R ]. Our second and more importa* *nt aim is to give a commutative monoidal structure to the P1-spectrum BGL which respe* *cts the naive multiplicative structure on the functor X 7! BGL 2*,*(X). To be more prec* *ise, we construct a product ~BGL : BGL ^ BGL ! BGL (3) in the stable motivic homotopy category SH(S) such that for any X 2 Sm=S the di* *agram 0 K0(X) x K0(X) _______________//K (X) ~=|| |~=| fflffl| ~BGL fflffl| BGL 2i,i(X) x BGL 2j,j(X)___//_BGL2(i+j),i+j(X) commutes. We show that there is a unique product ~BGL 2 Hom SH(Z)(BGL ^ BGL , * *BGL ) satisfying this property. This induces a product ~BGL 2 Hom SH(S)(BGL ^ BGL , * *BGL ) for an arbitrary regular scheme S by pull-back along the structural morphism S ! Sp* *ec(Z). As well, we show that the product is associative, commutative and unital. The r* *esulting multiplicative structure on the bigraded theory BGL *,*coincides with the Waldh* *ausen multiplicative structure on the Thomason-Trobaugh K-theory, as we prove in Prop* *osi- tion 1.2.2. 2 1.1 Voevodsky's K-theory spectrum BGL The basic definitions, constructions and model structures used in the text are * *given in the Appendix A. Let S be a Noetherian finite-dimensional scheme. A motivic sp* *ace over S is a simplicial presheaf on the site Sm=S of smooth quasi-projective S-s* *chemes. A pointed motivic space over S is a pointed simplicial presheaf on the site Sm=* *S. Let Sn = Sn,0denote the n-fold smash product of the constant simplicial presheaf 1* *=@ 1 with itself. We write Mo(S) for the category of pointed motivic spaces over S. * *A closed motivic model structure Mcmo(S) is constructed in A.3.9. The adjective "closed"* * refers to the fact that closed embeddings in Sm=S are forced to become cofibrations. The * *resulting homotopy category Hcmo(S) is called the motivic homotopy category A.3.9 of S. B* *y A.3.11 it is equivalent to the Morel-Voevodsky A1-homotopy category [MV ], and we may* * drop the superscript in Hcmo(S) for convenience. The closed motivic model structure * *has the properties that 1. for any closed S-point x0: S ,! X in a smooth S-scheme, the pointed motivi* *c space (X, x0) is cofibrant in Mcmo(S), 2. a morphism f :S ! S0of base schemes induces a left Quillen functor f* :Mcm* *o(S0) ! Mcmo(S), and 3. taking complex points is a left Quillen functor RC :Mcmo(C) ! Top o. Conditions 2 and 3 do not hold for the Morel-Voevodsky model structure, conditi* *on 1 fails for the so-called projective model structure [DRO ]. For a morphism f :A* * ! B of pointed motivic spaces we will write [f] for the class of f in Ho(S). We will consider P1 as a pointed space over S pointed by 1: S ,! P1. A P1-sp* *ectrum E over S consists of a sequence E0, E1, . .o.f pointed motivic spaces over S, t* *ogether with structure maps oen: En ^ P1 ! En+1. Let SH (S) denote the homotopy category of * *P1- spectra, as described in A.5. By Theorem A.5.6 it is canonically equivalent to * *the motivic stable homotopy category constructed in [V1 ] and [J]. As we will see below the* *re exists an essentially unique P1-spectrum BGL over S = Spec(Z) satisfying properties 1* * to 2 from Section 1. In the following, we will construct BGL in a slightly differen* *t way than Voevodsky did originally in [V1 ]. In order to achieve this, we begin with a de* *scription of the known monoidal structure on the Thomason-Trobaugh K-theory [TT ]. 1.2 A construction of BGL Let S be a regular scheme. For every S-scheme X consider the category Big(X) of* * big vector bundles over X (see for instance [FS ] for the definition and basic prop* *erties). The assignments X 7! Big(X) and (f :Y ! X) 7! f* :Big(X) ! Big(Y ) form a functor from schemes to the category of small categories. The reason is that there is a* *n equality (f O g)* = g* O f*, not just a unique natural isomorphism. In what follows we * *will always consider the Thomason-Trobaugh K-theory space of X as the space obtained* * by applying Waldhausen's So-construction [W ] applied to the category Big(X) rathe* *r than 3 to the category Vect(X) of usual vector bundles on X. This has the advantage t* *hat the assignment taking an S-scheme X to the Thomason-Trobaugh K-theory space of X becomes a functor on the category of S-schemes, and in particular a pointed mot* *ivic space over S. Let KTT be the pointed motivic space defined in A.3.4. It has the properties* * that it is fibrant in Mcmo(S) and that KTT(X) is naturally weakly equivalent to the Wal* *dhausen K-theory space associated to the category of big vector bundles on X. For X 2 S* *m=S and S regular, KTT(X) is thus a Kan simplicial set having the Thomason-Trobaugh K-t* *heory groups KTT*(X) as its homotopy groups. Note that since X is regular, these K-t* *heory groups coincide with Quillen's higher K-theory groups. It follows immediately f* *rom the adjunction isomorphism p TT TT Hom Ho(S)(Sp,0^ X+, KTT) ~=Hom Ho S , K (X) = Kp (X) (4) that KTT regarded as an object in the motivic homotopy category Ho(S) (see A.3.* *9) represents the Thomason-Trobaugh K-theory on Sm=S. Using a result of Morel and Voevodsky we describe now a commutative monoidal structure on KTT as follows. Let (Gr , x0) and ZxGr , (0, x0) be the pointed motivic spaces described i* *n Section 1. Lemma 1.2.1. The canonical map Hom Hcmo(S)(Gr , KTT) ! lim-HomHo(S)(Gr (n, 2n), KTT) = lim-KTT0 Gr(n, 2n) is an isomorphism. A similar statement holds for the pointed motivic spaces Z x* * Gr and (Z x Gr) x (Z x Gr). Proof.The pointed motivic spaces (Gr (n, 2n), x0) are closed cofibrant and the * *inclusions (Gr (n, 2n), x0) in-!(Gr (n + 1, 2n + 2), x0) are closed cofibrations. Thus the* * colimit Gr is a model for the homotopy colimit. Since we already know that KTT is weakly equi* *valent to the zeroth term in a fibrant P1-spectrum BGL , there is an adjunction isomor* *phism Hom Hcmo(S)(A, KTT) ~=Hom SH(S)( 1P1A, BGL ). It follows from Lemma A.5.10 that there is an exact sequence of the form 0 ! lim-1[S1,0^ Gr(n, 2n), KTT] ! [Gr , KTT] ! lim-[Gr (n, 2n), KTT] ! 0 It remains to check that the lim-1-group vanishes. To prove this, note that by * *(4) there is an identification lim-1[S1,0^ Gr(n, 2n), KTT] = lim-1KTT1(Gr (n, 2n)). Lemma B.1.4 implies that lim-1KTT1(Gr (n, 2n)) = lim-1KTT1(S) KTT0(S)KTT0 Gr(n* *, 2n) , and the connecting homomorphisms in this tower are given by the tensor products* * id i*n, where i*n:KTT0 Gr(n + 1, 2n + 2) ! KTT0 Gr(n, 2n) . It is known that i*nis sur* *jective * * __ for all n 0. Thus the lim-1-group vanishes. The other cases are handled simil* *arly. |__| 4 For an integer n let [-n, n] = {a 2 Z: - n a n}. From now on we will ide* *ntify KTT0(Z x Gr) with the group lim-KTT0 [-n, n] x Gr(n, 2n) . Following Lemma 1.2.* *1 we will identify Hom Hcmo(S)(Z x Gr, KTT) with KTT0(Z x Gr). Let on be the tautological vector bundle over Gr(n, 2n), and let m be the tr* *ivial vector bundle of rank m. Let o1 - 1 2 KTT0(Z x Gr) be the unique element such that for* * each integer m 0 its restriction to the subspace {m} x Gr(n, 2n) Z x Gr coincides with the element [on] - n + m 2 KTT0(Gr (n, 2n)). Let f :Z x Gr ! KTT* * be the unique morphism in Hcmo(S) representing o1 - 1. By [MV , Thm. 4.3.13] the morp* *hism f :Z x Gr ! KTT is an isomorphism in Hcmo(S). The space Z x Gr is closed cofibrant and the space KTT is closed motivic fib* *rant. Thus by Quillen's Theorem A.2.1 there is a motivic weak equivalence Z x Gr w-!K* *TT in Mcmo(S) such that [w] 2 Hom Ho(S)(Z x Gr, KTT) = KTT0(Z x Gr) coincides with* * the element o1 - 1 2 KTT0(Z x Gr). Note that w is unique up to a simplicial homotop* *y. Set KTT0 (ZxGr )x(ZxGr ) := lim-KTT0 ([-n, n]xGr (n, 2n))x([-n, n]xGr (n, 2* *n)) . Following 1.2.1, identify Hom Ho(S)(ZxGr )x(ZxGr ), KTT with KTT0 (ZxGr )x(ZxG* *r ) . The space ZxGr pointed by (0, x0) is a pointed S-cellular scheme (see B.1 for t* *he definition of a pointed S-cellular scheme). The element o1 - 1 vanishes at the S-point (0* *, x0). Corollary B.1.2 and Lemma 1.2.1 show that there exists a unique morphism ~TT 2 Hom Ho(S)(KTT ^ KTT, KTT) (5) such that the composite morphism in Ho(S) ~TT TT (Z x Gr) x (Z x Gr) ! (Z x Gr) ^ (Z x Gr) w^w--!KTT ^ KTT --! K (6) represents the class (o1 - 1) (o1 - 1) in KTT0((Z x Gr) x (Z x Gr)). In fact,* * w ^ w is a weak equivalence in Mcmo(S) by Theorem A.3.9. Now define eTT as the compo* *site e w TT morphism S0 -! Z x Gr -!K . in Mo(S). Clearly the triple TT TT TT K , ~ , [e ] (7) is a commutative monoidal structure on KTT in Ho(S). Proposition 1.2.2. The commutative monoidal structure (7) on KTT regarded as an object in Ho(S) coincides with the structure defined by Waldhausen in [W ]. A sketch of the proof will be given in Appendix B. Now we continue our expos* *ition. Lemma 1.2.3. There is a trivial cofibration i: Z x Gr ! K in Mcmo(S) and a triv* *ial fibration p: K ! KTT such that w = p O i. In particular, the motivic space K is* * cofibrant and fibrant in Mcmo(S). Proof.This follows from factoring w as a cofibration followed by a trivial fibr* *ation in the * *__ model category Mcmo(S). |* *__| 5 Choose and fix i and p as required by the Lemma. The morphisms [w], [i] and* * [p] in Hcmo(S) are isomorphisms. The structure (7) of a commutative monoid on KTT * *in Ho(S) induces via the isomorphism [w] (resp. [p]) the structure of a commutativ* *e monoid (Z x Gr, ~Gr, eGr) (resp. (K, ~~, ~e)) on the motivic space Z x Gr (resp. K) * *in Hcmo(S) such that [i] and [p] are monoid isomorphisms. The space K is closed motivic fi* *brant and closed cofibrant, thus the morphisms ~~and ~ein Hcmo(S) can be represented by c* *ertain morphisms in Mo(S) by Theorem A.2.1. We will do this as follows. Let ~: (Z x Gr) ^ (Z x Gr) ! K be a morphism in Mo(S) representing the morph* *ism [iO~Gr] in Hcmo(S). There exists such a ~ since the space (ZxGr )^(ZxGr ) is co* *fibrant and the space K is fibrant in Mcmo(S). There exists a unique (up to a homotopy) mor* *phism ~K :K ^ K ! K in Mo(S) such that [~K O (i ^ i)] = [~] in Hcmo(S). Here one uses* * that the smash product of cofibrant objects in Mcmo(S), and in particular K ^ K, is * *cofibrant e i K by A.3.9. Define eK in Mo(S) as the composition S0 -! Z x Gr -!K. The maps ~ a* *nd eK represent the classes ~~and ~erespectively. So (K, [~K ], [eK ]) (8) is a commutative monoid in Hcmo(S) which coincides with (K, ~~, ~e). Remark 1.2.4. The functors Hom Hcmo(S)(-, K) and Hom Hcmo(S)(-, KTT) are isomor* *phic via the isomorphism sending g to [p]Og. Under this identification the class [i]* * of i represents the element o1 - 1 2 KTT0(Z x Gr) since w = p O i and [w] 2 [Z x Gr, KTT] repre* *sents o1 - 1. The composition of the inclusion P1 = Gr (1, 2) ,! Gr and the map i is deno* *ted b: P1 ! K. This map represents the class [O(-1)]-[O] in the kernel of the homom* *orphism 1*: K0(P1) ! K0(k) induced by 1: Spec(k) ! P1. Consider the map of pointed motivic spaces given by the composition ~K ffl : K ^ P1 id^b--!K ^ K -! K. (9) Definition 1.2.5. Define BGL as the P1-spectrum of the form (K0, K1, K2, . .).* *with Ki = K for all i and with the structure maps ei:Ki^ P1 ! Ki+1 equal to the map ffl: K ^ P1 ! K. It will be proved in Section 1.5 that in the case S = Spec(Z) this spectrum * *is essentially unique. In the next section, we will construct a monoidal structure on BGL rega* *rded as an object in the stable homotopy category SH(S). In the case of S = Spec(Z) such a* * monoidal structure is unique. Pulling it back via the structural morphism S0! Spec(Z) we* * get a monoidal structure on BGL in SH(S0) for an arbitrary Noetherian finite-dimensi* *onal base scheme S0. To complete this section we prove that BGL is an P1-spectrum which represe* *nts the Thomason-Trobaugh K-theory on Sm=S. For X 2 Sm=S we abbreviate BGL (X+) as BGL (X), which forces us to write BGL (X, x0) for a pointed S-scheme (X, x0). Lemma 1.2.6. The spectrum BGL is an P1-spectrum. 6 Proof.For Y 2 Sm=S and a closed subscheme Z ,! Y write KnZ(Y ) for the n-th Thomason-Trobaugh K-group of perfect complexes on Y with support on Z. It may be obtained as the n-th homotopy group of the homotopy fiber of the map KTT(Y )* * ! KTT(X r Z). For each smooth X over S the map Kn(X) = [Sn,0^ X+, Ki] ! [Sn,0^ X+ ^ P1, Ki^ P1] ! [Sn,0^ X+ ^ P1, Ki+1] ~=KnXx{1}(X x P1) induced by the structure map eicoincides with the multiplication by the class [* *O(-1)]-[O] in K0{1}(P1). This multiplication is known to be an isomorphism for the Thomas* *on- Trobaugh K-groups, by the projective bundle theorem [TT , Thm. 4.1] for X xP1. * *Whence __ the Lemma. |__| Corollary 1.2.7. For each pointed motivic space A over S the adjunction map Hom Ho(S)(A, K0) ! Hom SH(S)( 1P1A, BGL ) is an isomorphism. In particular, for every smooth scheme X over S and each cl* *osed subscheme Z in X one has KTTZ,p(X) = BGL -p,0X=(X r Z) . The family of these isomorphisms form an isomorphism Ad : KTT* ! BGL -*,0of cohomology theories on the category SmOp=S in the sense of [PS1 ]. Moreover the adjunction map [A, Ki* *] ! [ 1P1(A)(-i), BGL ] is an isomorphism. In particular, for every smooth scheme X* * over S and each closed subscheme Z in X one has KTTZ,p(X) = BGL 2i-p,i(X=(X - Z)). mij K The family of pairings Ki^ Kj --! Ki+jwith mij= ~ defines a family of pairi* *ngs [ : BGL p,i(A) BGL q,j(B) ! BGL p+i,q+j(A ^ B) (10) for pointed motivic spaces A and B. We will refer to the latter as the naive p* *roduct structure on the functor BGL *,*on the category M*(S). With this naive product * *in hand one has the following Corollary 1.2.8. The isomorphism Ad: KTT* ! BGL -*,0of cohomology theories on SmOp=S is an isomorphism of ring cohomology theories in the sense of [PS1 ]. 1.3 The Bott element The aim of this Section is to construct an element fi 2 BGL 2,1(Spec(k)), to sh* *ow that it is invertible and to check that for any pointed motivic space A one has BGL *,0(A)[fi, fi-1] = BGL *,*(A) (the Laurent polynomials over BGL *,0(A)). We will use the naive product struct* *ure on BGL described just above Corollary 1.2.8. eK 2,1 Definition 1.3.1. Set fi := [S0 -! K = K1] 2 BGL (Spec(k)), where eK is the u* *nit of the monoid K (see (8)). 7 Lemma 1.3.2. Let b: P1 ,! K be the map described just above (9). It represents* * the element [O(-1)] - [O] in BGL 0,0(P1, 1) = Ker 1*: K0(P1) ! K0(k) . There is a relation fi [ [O(-1)] - [O] = P1(1) 2 BGL 2,1(P1, 1), (11) where P1 is the suspension isomorphism and 1 2 BGL 0,0(Spec(k)) is the unit. T* *here is another relation 2,1 1 fi [ [O(1)] - [O] = - P1(1) 2 BGL (P , 1). (12) Proof.The element P1(1) is represented by the map eK^id 1 id^b m01 S0 ^ P1 ---! K0 ^ P --! K0 ^ K1 --! K1, where m01is from Theorem 2.2.1. The element fi [ [O(-1)] - [O] is represented* * by the map eK^b m01 S0 ^ P1 ---! K1 ^ K0 --! K1. Since K0 = K1 = K one has (id ^ b) O (eK ^ id) = eK ^ b. This implies the relat* *ion (11). * * __ Relation (12) follows from the first one since [O(1)]-[O] = -[O(-1)]+[O] in K0(* *P1). |__| Lemma 1.3.3. Let u 2 BGL -2,-1(Spec(k)) be the unique element such that P1(u) = [O(1)] - [O] in BGL 0,0(P1, 1). Then fi [ u = 1. Proof.Consider the commutative diagram BGL 2,1(Spec(k))OO BGL 0,0(P1,_1)[___//BGL2,1(P1,O1)O id P1|| |P1| | | BGL 2,1(Spec(k)) BGL -2,-1(Spec(k))[//_BGL0,0(Spec(k)). * * __ Now the Lemma follows from the relation (12). * * |__| Definition 1.3.4. For P1-spectra E and F set Ealg(F ) = +1-1E2i,i(F ). Proposition 1.3.5. For every pointed motivic space A the map BGL *,0(A) BGL alg(Spec(k)) ! BGL *,*(A) (13) given by a b 7! a [ b is a ring isomorphism and BGL alg(Spec(k)) = Z[fi, fi-1* *] is the Laurent polynomial ring. One can rewrite this ring isomorphism as BGL *,0(A)[fi, fi-1] = BGL *,*(A) (14) [fi *+2,1 Proof.In fact, BGL *,0(A) -! BGL (A) is an isomorphism since fi is invertib* *le. Since * * __ BGL 0,0(Spec(k)) = K0(Spec(k)) = Z the map (13) is a ring isomorphism. * * |__| 8 Using the isomorphism Ad : KTT*! BGL -*,0of ring cohomology theories from Co* *rol- lary 1.2.8 we get the following statement. Corollary 1.3.6. For every X 2 Sm=S and every closed subscheme Z ,! X one has KTTZ,-*(X)[fi, fi-1] = BGL *,*Z(X). (15) The family of these isomorphisms form an isomorphism of the ring cohomology the* *ories on SmOp in the sense of [PS1 ]. As well, there is an isomorphism KTTZ,-*(X) = BGL *,*Z(X)=(fi + 1)BGL *,*Z(X). (16) The family of these isomorphisms form an isomorphism of the ring cohomology the* *ories on SmOp in the same sense. Remark 1.3.7. The identification (18) of BGL -1,0(X) with BGL 2i-1,i(X) coincid* *es with [fii 2i-1,i the Bott periodicity isomorphism BGL -1,0(X) --! BGL (X). 1.4 BGL *,*as an oriented ring cohomology theory The restriction of the functor BGL *,*to the category SmOp is a ring cohomology* * theory. In this section we promote it to an oriented ring cohomology theory in the sens* *e of [PS1 ]. For that it suffices to construct a Chern structure on BGL *,*|SmOp in the sens* *e of [PS1 ]. Let P1 be the motivic space pointed by 1 2 P1 ,! P1 . A functor isomorphism Hom Ho(S)(-, P1 ) ~=Pic(-) on the category Sm=S is constructed in [MV , Prop. * *4.3.8]. That functor isomorphism takes the homotopy class of the canonical map P1+! (P1* * , 1) to the class of the tautological line bundle O(-1) over P1 . Consider the eleme* *nt c = (-fi) [ ([O] - [O(1)]) 2 BGL 2,1(P1 , 1). For a line bundle L over X 2 Sm=S set c(L) = f*L(c) 2 BGL 2,1(X), where fL 2 Hom Ho(S)(X+, P1 ) is the morphism corresponding to the class [L] of L in the g* *roup Pic(X). Clearly, c(O(-1)) = c. The assignment L=X 7! c(L) is a Chern structur* *e on BGL *,*|SmOp since c|P1 = - P1(1) 2 BGL 2,1(P1) by (12). With that Chern struc* *ture, BGL *,*|SmOp is an oriented ring cohomology theory in the sense of [PS1 ]. If the Thomason-Trobaugh K-theory is equipped with the Chern structure given* * by L=X 7! [O] - [L_] 2 KTT0(X), then the isomorphism (16) of the ring cohomology t* *heories respects the orientations. 1.5 Uniqueness of BGL We prove in this Section that over S = Spec(Z) the spectrum BGL is essentially* * unique regarded as an object in the stable homotopy category SH(S). This has also been* * obtained in [R ]. By [MV , Thm. 4.3.13] and Corollary B.1.2 there exists a unique morph* *ism ffl: (Z x Gr) ^ P1 ! Z x Gr 9 in Ho(S) such that the composite morphism (Z x Gr) x P1 ! (Z x Gr) ^ P1 ffl-!Z * *x Gr represents the element (o1 - 1) [O(-1)] - [O] 2 K0 (Z x Gr) x P1 . Let Z x Gr i-!K be a motivic weak equivalence and let e: K ^ P1 ! K be a morphism in Mo(S) such* * that the diagram (Z x Gr) ^ P1_ffl//_Z x Gr i^id|| |i| fflffl| e fflffl| K ^ P1__________//K commutes in Hcmo(S). Let i0:Z x Gr ! K0be another motivic weak equivalence and * *let e0:K0^ P1 ! K0be a morphism in Mo(S) such that e0O (i0^ id) = i0O ffl in Hcmo(S* *). Definition 1.5.1. Define BGL as the P1-spectrum of the form (K0, K1, K2, . .).* *with Ki = K for all i, and with the structure maps ei:Ki^ P1 ! Ki+1 equal to the map ffl: K ^ P1 ! K. Define BGL 0in the same way using K0and e0. Proposition 1.5.2. There exists a unique morphism ` :BGL ! BGL 0 in SH (S) such that for every integer i 0 the diagram 1P1Ki(-i)__ui//_BGL | 1P1OEi(-i)|| |`| fflffl|u0 fflffl| 1P1K0i(-i)__i//_BGL0 commutes, where OEi = i0O i-1 2 [Ki, K0i]Ho(S)and ui, u0iare the canonical morp* *hisms. Similarly, there exists a unique morphism `0:BGL 0 ! BGL in SH (S) such that f* *or every integer i 0 the diagram u0i 1P1K0i(-i)___//_BGL0 | 1P1OE0i(-i)|| |`0 fflffl|u fflffl|| 1P1Ki(-i)___i//_BGL commutes, where `i= i O (i0)-1 2 [K0i, Ki]Ho(S). Proof.Consider the exact sequence 0 ! lim-1BGL2i-1,i(K0i) ! BGL 0,0(BGL 0) ! lim-BGL2i,i(K0i) ! 0 from Lemma A.5.10. The family of elements (uiO 1P1`0i(-i)) is an element of th* *e group lim-BGL2i,i(K0i). Thus there exists the required morphism `0. To prove its un* *iqueness, observe that the lim-1-group vanishes by Proposition 1.7.1. Whence BGL 0,0(BGL* * 0) = lim-BGL2i,i(K0i) and `0is indeed unique. By symmetry there also exists a unique* * morphism * * __ ` with the required property. * * |__| 10 Proposition 1.5.3. The morphism ` :BGL ! BGL 0is the inverse of `0:BGL 0 ! BGL in SH (S), and in particular an isomorphism. Proof.The composite morphism `0O ` :BGL ! BGL has the property that for every integer i 0 the diagram 1P1Ki(-i)__ui_//BGL | id|| |`0O`| fflffl|u fflffl| 1P1Ki(-i)___i_//BGL0 commutes. However, the identity morphism id:BGL ! BGL has the same property. * * __ Thus `0O ` = id. by the uniqueness in Proposition 1.5.2, and similarly ` O `0= * *id. |__| Remark 1.5.4. The isomorphisms ` and `0are monoid isomorphisms provided that BGL and BGL 0are equipped with the monoidal structures given by Theorem 2.2.1. 1.6 Preliminary computations I In this section we prepare for the next section, in which we show that certain * *lim-1-groups vanish. Let BGL be the P1-spectrum defined in 1.2.5. We will identify in this * *section the functors BGL 0,0and BGL 2i,ion the category Ho(S) as follows: BGL 0,0(A) ~=Hom Ho(S)(A, K0) = Hom Ho(S)(A, K0) ~=BGL 2i,i(A). (17) Similarly, BGL -1,0(A) ~=Hom Ho(S)(S1,0^ A, K0) = Hom Ho(S)(S1,0^ A, Ki) ~=BGL 2i-1,i(A)* *.(18) These identifications respect the naive product structure (10) on the functor B* *GL *,*. In particular, the following diagram commutes for every pointed motivic space A ov* *er S. BGL -1,0(S) BGL 0,0(A)__//_BGL-1,0(A) (19) ~=|| |~=| fflffl| fflffl| BGL -1,0(S) BGL 2i,i(A)__//_BGL2i-1,i(A) Lemma 1.6.1. Let S be a regular scheme. For every integer i the map BGL -1,0(S) BGL0,0(S)BGL2i,i(K) ! BGL 2i-1,i(K) induced by the naive product structure is an isomorphism. The same holds if K * *^ P1 replaces K. Proof.The commutativity of the diagram (19) shows that it suffices to consider * *the case i = 0. Furthermore we may replace the pointed motivic space K with ZxGr since t* *he map i: Z x Gr ! K = K is a weak equivalence. The functor isomorphism KTT*! BGL -*,0 11 is a ring cohomology isomorphism by Corollary 1.2.8. Thus it remains to check t* *hat the map KTT1(S) KTT0(S)KTT0(Gr ) ! KTT1(Gr ) is an isomorphism. This is well-known. See Lemma B.1.4) for a proof. The ass* *ertion * * __ concerning K ^ P1 is proved similarly using Lemma B.1.3 instead. * * |__| To state the next lemma, consider the scheme morphism f :S = Spec(Z) ! Spec(* *C), the pull-back functor f* :SH(Z) ! SH(C) described in Proposition A.7.4, and the* * topo- logical realization functor RC :SH (C) ! SH CP1 described in Section A.7. Set * *r = RC O f* :SH(S) ! SH CP1. The functor r will be called for short the realization* * functor below in this Section. Lemma 1.6.2. Let BU be the periodic complex K-theory CP1-spectrum with terms Zx* *BU . There is a termwise zigzag weak equivalence BU- E -! rBGL of CP1-spectra. * * __ Proof.This follows from A.7.3 and the fact that Grassmann varieties pull back. * * |__| Lemma 1.6.3. Let S = Spec(Z) and let r :SH(Z) ! SH CP1 be the topological real- ization functor. Then for every integer i the realization homomorphism BGL 2i,* *i(K) ! (rBGL )2i(rK) is an isomorphism. Proof.Clearly it suffices to prove the case i = 0. We may replace the pointed * *motivic space Ki with Z x Gr as in the proof of Lemma 1.6.1. It remains to check that * *the topological realization homomorphism BGL 0,0(Gr ) ! (rBGL )0(Gr ) is an isomor* *phism. Since Gr(n, 2n) is a smooth cellular S-scheme, Lemma 1.6.4 below implies tha* *t the map BGL 0,0(Gr (n, 2n)) ! (rBGL )0(Gr (n, 2n)) is an isomorphism for every n. To c* *onclude the statement for Gr = [Gr(n, 2n), use the short exact sequence from Lemma A.5.10. * *In the resulting diagram 0,0 0 ____//_lim-1BGL-1,0Gr(n, 2n)____//_BGL0,0(Gr_)__//_lim-BGL Gr(n, 2n) ____//_0 | | | | | | fflffl| fflffl| fflffl| 0 ___//_lim-1rBGL -1,0rGr (n,_2n)//_rBGL 0,0(rGr/)/_lim-rBGL 0,0rGr (n,_2n)/* */_0 the map on the right hand side is then an isomorphism. Furthermore one conclude* *s from [Sw , Thm. 16.32] that lim-1rBGL -1,0rGr (n, 2n) = lim-1K1toprGr (n, 2n) = 0* *. On the other hand lim-1BGL-1,0Gr (n, 2n) = 0, as we already mentioned in the proof of* * 1.2.1. * * __ The result follows. * *|__| Lemma 1.6.4. Let X 2 Sm=S, where S = Spec(Z), and let X0 X1 . . .Xn = X be a filtration by closed subsets such that for every integer i 0 the S-schem* *e Xi- Xi-1 is isomorphic to a disjoint union of several copies of the affine space AiS. T* *he map BGL 0,0(X) ! (rBGL )0(rX) is an isomorphism. 12 Proof.Consider the class R of P1-spectra E such that BGL 0,0(E) ! rBGL 0(rE) i* *s an iso- morphism. It contains S0,0because in this case we obtain the isomorphism BGL 0,* *0(S0,0) ~= K0(Z) ~=Z ~=K0top(S0 which identifies the class of an algebraic resp. complex t* *opological vector bundle over Spec(Z) resp. o with its rank. The Bott periodicity isomorph* *isms for BGL and rBGL which are compatible by A.7.3 imply that S2m,m 2 R for all m 2 Z. Finally, if E ! F ! G ! S1,0^ E is a distinguished triangle in SH (S) such that* * E and G are in R, then so is F . For i 0 write Ui:= X r Xi, so that Ui is an open subset of Ui-1. In partic* *ular we have Un = ; and U-1 = X. The closed subscheme Xir Xi-1= Xi\ Ui-1 ,! Ui-1 is isomorphic to a disjoint union mi copies of affine spaces Ai, and is in particu* *lar smooth over S. Furthermore the normal bundle is trivial. The homotopy purity theorem [* *MV , 3.2.29] supplies a distinguished triangle 1P1Ui+! 1P1Ui-1+! 1P1Ui-1=Ui ~=_mij=1S2(n-i),(n-i) of P1-spectra. Since R contains 1P1Un = o we obtain inductively that R conta* *ins 1P1U-1 = 1P1X+. __ |__| Lemma 1.6.5. Let B0U be the sub-spectrum of BU with the n-th term equal to the * *con- nected component BU of the topological space Z x BU containing the basepoint * *o. The inclusion B0U -!BU is a weak equivalence of CP1-spectra. Proof.One has to check that the inclusion induces an isomorphism on stable homo* *topy groups. This follows because the structure map (Z x BU) ^ CP1 ! Z x BU factors * *over __ {0} x BU. |__| Lemma 1.6.6. There exists a sub-spectrum BfU of the CP1-spectrum BU with the n-* *th term Gr b(n), 2b(n) such that the inclusion BfU -!BU is a stable equivalence. Proof.The sequence b(n) will be constructed such that b(n) 2n + 1. Set b(0) = 1. We may assume that the structure map e0: BU ^ CP1 ! BU is cellular. Since rGr (b(0), 2b(0))^CP1 is a finite cell complex, it lands in a Grassmannian rGr * *(b(1), 2b(1)) for some integer b(1) 2 . 1 + 1. Continuing this process produces the require* *d sequence of b(n)'s. The inclusions induce an isomorphism colimn 0Gr b(n), 2b(n) ~=Gr. To observe that the inclusion j :BfU -! BU is then a stable equivalence, rec* *all that the number of 2k-cells in Gr(n, m) is given by the number of partitions of k in* *to at most n each of which is m - n. In particular, the 2k-skeleton of BU coincides wi* *th the 2k-skeleton of rGr (k, 2k). To prove the surjectivity of ssi(j) choose an eleme* *nt ff 2 ssiBU. It is represented by a cellular map a: Si+2m ! BU for some m with i + 2m 0. W* *e may choose m such that m i. Thus a lands in rGr (b(m), 2b(m)) and gives rise to a* *n element in ssiBfU mapping to ff. To prove the injectivity of ssi(j), choose an element * *ff 2 ssiBfU such that ssi(j)(ff) = 0. We may represent ff by some map a: ssi+2m(j)Gr (b(m)* *, 2b(m)) for some m with i + 2m > 0 and m i. The composition Si+2m ! aGr (b(m), 2b(m)) ,! BU 13 is nullhomotopic since ssi+2mBU ~= ssiBU via the homomorphism induced by the st* *ructure map. The nullhomotopy may be chosen to be cellular and thus lands in Gr(b(m), 2* *b(m)). * * __ This completes the proof. * *|__| 1.7 Vanishing of certain groups I Consider the stable equivalence hocolimi 0 1P1Ki(-i) ~=BGL (see (21)) and the * *respect- ing short exact sequence 0 ! lim-1BGL2i-1,i(Ki) ! BGL 0,0(BGL ) ! lim-BGL2i,i(Ki) ! 0 We prove in this section the following result Proposition 1.7.1. Let S = Spec(Z), then lim-1BGL2i-1,i(Ki) = 0. Proof.The connecting homomorphism in the tower of groups for the lim-1-term is * *the composite map -1P1 2i+1,i+1 e*i 2i+1,i+1 BGL 2i-1,i(Ki) -- BGL (Ki^ P1) - BGL (Ki+1) where -1P1is the inverse to the P1-suspension isomorphism and e*iis the pull-b* *ack induced by the structure map ei. Set A = BGL -1,0(S) and consider the diagram -1P1 e*i BGL 2i-1,i(Ki)oo_____BGLO2i+1,i+1(Ki^OP1)oo_______BGLOO2i+1,i+1(Ki+1)OO | | | | | | | id -11 | id e* | A BGL 2i,i(Ki)oo_PA_ BGL 2(i+1),i+1(Ki^ P1)oo_iA_ BGL 2(i+1),i+1(Ki+1) where the vertical arrows are induced by the naive product structure on the fun* *ctor BGL *,*. Clearly it commutes. Since S is regular, the vertical arrows are isomo* *rphisms by Lemma 1.6.1. It follows that lim-1BGL2i-1,i(Ki) = lim-1(A BGL 2i,i(Ki)) where * *in the last tower of groups the connecting maps are id ( -1P1O e*i). It remains to prove t* *he following assertion. Claim. lim-1(A BGL 2i,i(Ki)) = 0 Since S = Spec(Z) one gets A = BGL -1,0(S) = KTT1(Z) = Z=2Z. Thus A BGL 2i,i(K* *i) = BGL 2i,i(Ki)=mBGL 2i,i(Ki) with m = 2 and the connecting maps in the tower are * *just the mod-m reduction of the maps -1P1O e*i. Now a chain of isomorphisms completes t* *he proof of the Claim. lim-1BGL2i,i(Ki)=m~=lim-1(rBGL )2i(rKi)=m ~=lim-1BU2i(rKi)=m ~=lim1BU2i(Z x BU )=m ~=lim1BU2i(Z x BU ; Z=m) - - ~=K1top(BU; Z=m) ~=K1top(B0U; Z=m) ~=lim1K2i Gr (b(i), 2b(i)); Z=m = 0 - top 14 The first isomorphism follows from Lemma 1.6.3. The second isomorphism is indu* *ced by the levelwise weak equivalence BU ' rBGL mentioned in Lemma 1.6.2. The thi* *rd isomorphism is induced by the image of the weak equivalence Ki' Z x Gr under to* *po- logical realization. The forth and fifth isomorphism hold since BU2i+1(Z x BU )* * = 0. The sixth isomorphism is induced by the stable equivalence B0U ' BU from Lemma 1.6.* *5, the seventh one is induced by the stable equivalence BfU ' B0U from Lemma 1.6.6. Th* *e last * * __ one holds since all groups in the tower are finite. * * |__| 2 Smash-product, pull-backs, topological realization In this section we construct a smash-product ^ of P1-spectra, check its basic p* *roperties, consider its behavior with respect to pull-back and realization functors. We fo* *llow here an idea of Voevodsky [V1 , Comments to Thm. 5.6] and use results of Jardine [J]. 2.1 The smash product Definition 2.1.1. Let V :SH (S) ! SH (S) and U :SH (S) ! SH(S) be the equival* *ence described in Theorem A.6.4. For a pair of P1-spectra E and F set E ^ F := U(V E ^ V F ) as in Remark A.6.5. Proposition 2.1.2. Let S be a Noetherian finite-dimensional base scheme. The sm* *ash- product of P1-spectra over S induces a closed symmetric monoidal structure (^, * *I) on the motivic stable homotopy category SH (S) having the properties required by Theor* *em 5.6 of Voevodsky congress talk [V1 ]: 1. There is a canonical isomorphism E ^ 1P1A ~=(Ei^ A, ei^ id) for every poi* *nted motivic space A and every P1-spectrum. 2. There is a canonical isomorphism ( Eff) ^ F ~= (Eff^ F ) for P1-spectra Ei* *, F . 3. Smashing with a P1-spectrum preserves distinguished triangles. To be more * *precise, f ffl 1 if E -! F ! cone(f) -! E[1] is a distinguished triangle and G is a P -spec* *trum, f ffl the sequence E ^ G -!F ^ G ! cone(f) ^ G -!E ^ G[1] is a distinguished tri* *angle, where the last morphism is the composition of ffl^idG with the canonical i* *somorphism E[1] ^ G ! (E ^ F )[1]. * *__ Proof.Follows from Remark A.6.5 and Theorem A.6.4. |* *__| Lemma 2.1.3. Let E = hocolimi 0Eiis a sequential homotopy colimit of P1-spectra* *. For every P1-spectrum F there is an exact sequence of abelian groups 0 ! lim-1F p-1,q(Ei) ! F p,q(E) ! lim-F p,q(Ei) ! 0. (20) 15 __ Proof.This is Lemma A.5.10. |_* *_| By Lemma A.5.9, any P1-spectrum E can be expressed as the homotopy colimit hocolim 1P1Ei(-i) ~=E. (21) Corollary 2.1.4. For two P1-spectra E and F there is a canonical short exact se* *quence 0 ! lim-1F p+2i-1,q+i(Ei) ! F p,q(E) ! lim-F p+2i,q+i(Ei) ! 0.(22) Corollary 2.1.5. For a pair of spectra E and F and each spectrum G one has a ca* *nonical exact sequence of the form 0 ! lim-1Gp+4i-1,q+2i(Ei^ Fi) ! Gp,q(E ^ F ) ! lim-Gp+4i,q+2i(Ei^ Ei) !(0.23) Proof.For a pair of spectra E and F one has a canonical isomorphism of the form hocolim( 1P1(Ei^ Fi)(-2i)) ~=E ^ F (24) * * __ as deduced in Lemma A.6.8. The result follows from Corollary 2.1.4. * * |__| 2.2 A monoidal structure on BGL For a P1-spectrum E and an integer i 0 ui: 1P1Ei(-i) ! E denotes the canonical map. The aim of this section is to prove the following mij K Theorem 2.2.1. Consider the family of pairings Ki^ Kj --! Ki+jwith mij= ~ . For S = Spec(Z) there is a unique morphism ~BGL :BGL ^ BGL -! BGL in the motivic s* *table homotopy category SH (S) such that for every i the diagram 1 (mii) 1 1P1Ki(-i) ^ 1P1Ki(-i)_______// P1K2i(-2i) ui^ui|| u2i|| fflffl| ~BGL fflffl| BGL ^ BGL _________________//BGL commutes. Let eBGL :I ! BGL in SH (S) be adjoint to the unit eK :S0,0! K. Then (BGL , ~BGL, eBGL) is a commutative monoid in SH (S). Proof.The morphism ~BGL we are looking for is an element of the group BGL 0,0(B* *GL ^ BGL ). This group fits in the exact sequence 0 ! lim-1BGL4i-1,2i(Ki^ Ki) ! BGL 0,0(BGL ^ BGL ) ! lim-BGL4i,2i(Ki^ Ki) ! 0 by Corollary 2.1.5. The family of elements {u2iO 1 (mii)} is an element of the* * lim-group. The lim-1group vanishes by Proposition 2.4.1 below, whence there exist a unique* * element 16 ~BGL whose image in the lim-group coincides with the element {u2iO 1 (mii)}. * *That morphism ~BGL is the required one. In fact, the identities u2iO 1 (mii) = ~BGL O (ui^ ui) hold by the very con* *struction of ~BGL. The operation ~BGL is associative because the group lim-1BGL8i-1,4i(Ki^ K* *i^ Ki) vanishes by Proposition 2.5.1). That ~BGL is commutative follows from the vanis* *hing of the group lim-1BGL4i-1,2i(Ki^ Ki) (see Proposition 2.4.1). The fact that eBGL i* *s a two- sided unit for the multiplication ~BGL follows by Proposition 1.7.1, which show* *s that the * * __ group lim-1BGL2i-1,i(Ki) vanishes. * * |__| Definition 2.2.2. Let S be a regular scheme, with structural morphism f :S ! Sp* *ec(Z). Let f* :SH(Z) ! SH (S) be the strict symmetric monoidal pull-back functor from * *A.7.4. Set f*(~BGL) * ~SBGL:= f*(BGL ) ^ f*(BGL ) can--!f*(BGL ^ BGL ) -----! f (BGL ) f*(eBGL) * eSBGL:= S0 can--!f*(S0) -----! f (BGL ) and BGL S = f*(BGL ). Then (BGL S , ~SBGL, eSBGL) is a commutative monoid in SH* * (S). We will sometimes refer to a monoid in SH (S) as a P1-ring spectrum. Corollary 2.2.3. The multiplicative structure on the functor BGL *,*induced by * *the pair- ing ~SBGLand the unit eSBGLcoincides with the naive product structure (10). * *__ Proof.Follows from Theorem 2.2.1. |* *__| Corollary 2.2.4. The functor isomorphism [X, K0] ! [ 1P1(X), BGL ] respects the* * multi- plicative structures on both sides. The isomorphism Ad: KTT*! BGL -*,0of cohomo* *logy theories on SmOp=S is an isomorphism of ring cohomology theories in the sense o* *f [PS1 ]. * *__ Proof.Follows from Theorem 2.2.1. |* *__| 2.3 Preliminary computations II We will identify in this section the functors BGL 0,0and BGL 2i,i, BGL -1,0and * *BGL 2i-1,i on the motivic unstable category Ho(S) as in Section 1.6. Lemma 2.3.1. Let S be a regular base scheme. For every integer i the map BGL -1,0(S) BGL0,0(S)BGL2i,i(Ki^ Ki) ! BGL 2i-1,i(Ki^ Ki) induced by the naive product structure is an isomorphism. The same holds if we * *replace Ki^ Ki by Ki^ Ki^ P1. Proof.Since diagram (19) commutes, it suffices to consider the case i = 0. Furt* *hermore we may replace the pointed motivic space Kiwith ZxGr since the map i: ZxGr ! Ki= K* * is 17 a motivic weak equivalence. The functor isomorphism KTT*! BGL -*,0is an isomorp* *hism of ring cohomology theories. Thus it remains to check that the map KTT1(S) KTT0(S)KTT0(Gr ^ Gr) ! KTT1(Gr ^ Gr) is an isomorphism. This holds by Lemma B.1.3. The case of Ki^ Ki^ P1 is proved * *by __ the same arguments. |_* *_| Lemma 2.3.2. Let S = Spec(Z) and let r :SH(S) ! SHCP1 be the topological realiz* *ation functor. Then for every integer i the homomorphism BGL 2i,i(K ^ K) ! (rBGL )2i* *r(K ^ K) ~=(rBGL )2i(rK ^ rK) is an isomorphism. Proof.Since the Bott periodicity isomorphisms in the algebraic and complex topo* *logical setting correspond by A.7.3, it suffices to consider the case i = 0. We may rep* *lace the pointed motivic space K with Z x Gr as in the proof of Lemma 1.6.1. It remains * *to check that the realization homomorphism BGL 0,0(Gr ^ Gr) ! (rBGL )0(r(Gr ^ Gr)) = (rBGL )0(r(Gr ) ^ r(Gr ))) is an isomorphism. By Example A.5.8 1P1Gr^ Gr is a retract of 1P1Grx Gr in SH* * (S), whence it suffices to consider the topological realization homomorphism for Gr * *x Gr. Since the latter is an increasing union of the cellular S-schemes Gr(n, 2n) x G* *r(m, 2m), * * __ the result follows with the help of Lemma 1.6.4 as in the proof of Lemma 1.6.3.* * |__| 2.4 Vanishing of certain groups II Consider the stable equivalence hocolim 1P1(K^K)(-2i) ~=BGL ^BGL displayed in * *(24) and the corresponding short exact sequence 0 ! lim-1BGL4i-1,2i(K ^ K) ! BGL 0,0(BGL ^ BGL ) ! lim-BGL4i,2i(K ^ K) ! 0 from Corollary 2.1.5. We prove in this section the following result Proposition 2.4.1. If S = Spec(Z) then lim-1BGL4i-1,2i(K ^ K) = 0. Proof.For a pointed motivic space A we abbreviate A ^ A as A^2. The connecting homomorphism in the tower of groups forming the lim-1-term is the composite map ( O )-1Otw 4(i+1)-1,2(i+1)1 ^2 (ffl^ffl)*4(i+1)-1,2(i+1)^2 BGL 4i-1,2i(K^K) ------- BGL (K^P ) --- BGL (K ) where is the P1-suspension isomorphism, twis induced by interchanging the two* * pointed motivic spaces in the middle of the four-fold smash product, and ffl: K ^ P1 ! * *K is the structure map of BGL . Set A = BGL -1,0(S) and write B for BGL . Consider the d* *iagram ( O )-1Otw (ffl^ffl)* B4i-1,2i(K^2)oo__________B4i+3,2(i+1)(KO^OP1)^2oo___________ B4i+3,2i+2(K^2) OO OO | | | | | | | id ( O )-1Otw | id (ffl^ffl)* | A B4i,2i(K^2)oo________A B4i+4,2i+2(K ^ P1)^2oo_________A B4i+4,2i+2(K^2) 18 where the vertical arrows are induced by the naive product structure on the fun* *ctor BGL *,*. Clearly it commutes. The vertical arrows are isomorphisms by Lemma 1.6* *.1. It follows that lim-1BGL4i-1,2i(K^2) = lim-1A BGL 4i,2i(K^2) where in the last * *tower of groups the connecting maps are id ( O )-1P1O tw) O (ffl ^ ffl)* . It remain* *s to prove the following claim. Claim. lim-1(A BGL 4i,2i(K^2)) = 0. Since A = BGL -1,0(S) = KTT1(S) = Z=2Z, there is an isomorphism A BGL 4i,2i(K* *^2) = BGL 4i,2i(K^2)=mBGL 4i,2i(K^2) with m = 2. The connecting map in the tower are* * just the mod-m reduction of the maps ( O )-1 O twO (ffl ^ ffl)*. Now a chain of is* *omorphisms completes the proof of the Claim. lim-1BGL4i,2i(K^2)=m~=lim-1(rBGL )4i(rK ^ rK)=m ~=lim-1BU4i(rK ^ rK)=m ~=lim1BU4i(Z x BU ) ^ (Z x BU ) =m - ~=lim1BU4i(Z x BU ) ^ (Z x BU ); Z=m - ~=K1top(BU ^ BU; Z=m) ~=K1top(B0U ^ B0U; Z=m) ~=lim1K4i Gr (b(i), 2b(i)) ^ Gr(b(i), 2b(i)); Z=m = 0 - top The first isomorphism follows from Lemma 2.3.2. The second isomorphism is indu* *ced by the levelwise weak equivalence BU ' rBGL mentioned in Lemma 1.6.2. The thi* *rd isomorphism is induced by the image of the weak equivalence Ki' Z x Gr under to* *po- logical realization. The forth and fifth isomorphism hold since BU4i+1(Z x BU )* * = 0. The sixth isomorphism is induced by the stable equivalence B0U ' BU from Lemma 1.6.* *5, the seventh one is induced by the stable equivalence BfU ' B0U from Lemma 1.6.6. Th* *e last * * __ one holds since all groups in the tower are finite. * * |__| 2.5 Vanishing of certain groups III Consider the stable equivalence hocolim 1P1(K ^ K ^ K)(-3i) ~=BGL ^ BGL ^ BGL from (24) and the induced short exact sequence 0 ! lim-1BGL8i-1,4i(K^3) ! BGL 0,0(BGL ^ BGL ^ BGL ) ! lim-BGL8i,4i(K^3) ! 0. Proposition 2.5.1. If S = Spec(Z) then lim-1BGL8i-1,4i(K ^ K ^ K) = 0. * * __ Proof.This is proved in the same way as Proposition 2.4.1. * * |__| 2.6 BGL as an oriented commutative P1-ring spectrum Following Adams and Morel, we define an orientation of a commutative P1-ringSsp* *ectrum. However we prefer to use a Thom class rather than a Chern class. Let P1 = Pn * *be the motivic space pointed by 1 2 P1 ,! P1 . O(-1) be the tautological line bundle o* *ver P1 . 19 It is also known as the Hopf bundle. If V ! X is a vector bundle over X 2 Sm=S,* * with zero section z :X ,! V , let ThX (V ) = V=(V r z(X)) be the Thom space of V , c* *onsidered as a pointed motivic space over S. For example ThX (AnX) ' S2n,n. Define ThP1 * *O(-1) as the obvious colimit of the Thom spaces ThPn O(-1) . Definition 2.6.1. Let E be a commutative P1-ring spectrum. An orientation of E* * is an element th 2 E2,1(Th P1(O(-1))) = E2,1P1(O(-1)) such that its restriction to* * the Thom space of the fibre over the distinguished point coincides with the element* * P1(1) 2 E2,1(Th (1)) = E2,1(P1, 1). Remark 2.6.2. Let th be an orientation of E. Set c := z*(th) 2 E2,1(P1 ). It is* * proved in [PY , Prop. 6.5.1] that c|P1 = - P1(1). The class th(O(-1)) 2 E2,1P1(O(-1)) * *given by (26) coincides with the element th (see [PS1 , Thm.3.5]). Thus another possible* * definition of an orientation of E is the following. Definition 2.6.3. Let E be a commutative ring P1-spectrum. An orientation of E* * is an element c 2 E2,1(P1 ) such that c|P1 = - P1(1) (of course the element c shou* *ld be regarded as the first Chern class of the Hopf bundle O(-1) on P1 ). Remark 2.6.4. Let c be an orientation of E. Consider th(O(-1)) 2 E2,1P1(O(-1)) * *given by (26) and set th = th(O(-1)). It is straightforward to check that th|Th(1)= * * P1(1). Thus th is an orientation of E. Clearly c = z*(th) 2 E2,1(P1 ), whence the two * *definitions of orientations of E are equivalent. 2,1 Example 2.6.5. Set cK = (-fi) [ [O] - [O(1)] 2 BGL (P1 ). The relation (12)* * shows that cK is an orientation of BGL . Consider th(O(-1)) 2 BGL 2,1P1(O(-1)) given * *by (26) and set thK = th(O(-1)). The class thK is the same orientation of BGL . The orientation of BGL described in Example 2.6.5 has the following propert* *y. The map (16) BGL *,*! KTT* which takes fi to -1 is an oriented morphism of oriented cohomology theories, * *provided that KTT*is oriented via the Chern structure L=X 7! [O] - [L-1] 2 K0(X). 2.7 BGL *,*as an oriented ring cohomology theory An oriented P1-ring spectrum (E, c) defines an oriented cohomology theory on Sm* *Op in the sense of [PS1 , Defn.3.1] as follows. The restriction of the functor E** *,*to the category SmOp is a ring cohomology theory. By [PS1 , Th.3.35] it remains to con* *struct a Chern structure on E*,*|SmOp in the sense of [PS1 , Defn.3.2]. The functor is* *omorphism Hom Ho(S)(-, P1 ) ! Pic(-) on the category Sm=S provided by [MV , Thm. 4.3.8] * *takes the class of the canonical map P1+ ! P1 to the class of the tautological line * *bundle O(-1) over P1 . Now for a line bundle L over X 2 Sm=S set c(L) = f*L(c) 2 E2,1(* *X), where the morphism fL :X+ ! P1 in Ho(S) corresponds to the class [L] of L in t* *he group Pic(X). Clearly, c(O(-1)) = c. The assignment L=X 7! c(L) is a Chern stru* *cture 20 on E*,*|SmOp since c|P1 = - P1(1) 2 E2,1(P1, 1). With that Chern structure E*,** *|SmOp is an oriented ring cohomology theory in the sense of [PS1 ]. In particular, (BGL * *, cK ) defines an oriented ring cohomology theory on SmOp. Given this Chern structure, one obtains a theory of Thom classes V=X 7! th(V* * ) 2 E2rank(V ),rank(VT)hX(V ) on E*,*|SmOp in the sense of [PS1 , Defn. 3.32] as f* *ollows. There is a unique theory of Chern classes V 7! ci(V ) 2 E2i,i(X) such that for every * *line bundle L on X one has c1(L) = c(L). Now for a rank r vector bundle V over X consider * *the vector bundle W := 1 V and the associated projective vector bundle P(W ) of l* *ines in W . Set ~th(V ) = cr(p*(V ) OP(W)(1)) 2 E2r,r(P(W )). (25) It follows from [PS1 , Cor. 3.18] that the support extension map 2r,r E2r,rP(W )=(P(W ) r P(1)) ! E P(W ) is injective and ~th(E) 2 E2r,rP(W )=(P(W ) r P(1)) . Set th(E) = j*(~th(E)) 2 E2r,rThX (V ) , (26) where j :ThX (V ) ! P(W )=(P(W ) r P(1)) is the canonical motivic weak equivale* *nce of pointed motivic spaces induced by the open embedding V ,! P(W ). The assignment* * V=X to th(V ) is a theory of Thom classes on E*,*|SmOp (see the proof of [PS1 , Thm* *. 3.35]). So the Thom classes are natural, multiplicative and satisfy the following Thom iso* *morphism property. Theorem 2.7.1. For a rank r vector bundle p: V ! X on X 2 Sm=S the map - [ th(V ): E*,*(X) ! E*+2r,*+rThX(V ) is an isomorphism of the two-sided E*,*(X)-modules, where - [ th(V ) is written* * for the composition map - [ th(V ) O p*. * * __ Proof.See [PS1 , Defn. 3.32.(4)]. * * |__| A Motivic homotopy theory The aim of this section is to present details on the model structures we use to* * perform homotopical calculations. Our reference on model structures is [Ho ]. For the c* *onvenience of the reader who is not familiar with model structures, we recall the basic fe* *atures and purposes of the theory below, after discussing categorical prerequisites. A.1 Categories of motivic spaces Let S be a Noetherian separated scheme of finite Krull dimension (base scheme f* *or short). The category of smooth quasi-projective S-schemes is denoted Sm=S. A smooth mor* *phism is always of finite type. In particular, Sm=S is equivalent to a small category. The category of compactly generated topological spaces is denoted Top , the * *category of simplicial sets is denotes sSet. The set of n-simplices in K is Kn. 21 Definition A.1.1. A motivic space over S is a functor A: Sm=Sop ! sSet. The cat* *egory of motivic spaces over S is denoted M(S). For X 2 Sm=S the motivic space sending Y 2 Sm=S to the discrete simplicial * *set Hom Sm=S(Y, X) is denoted X as well. More generally, any scheme X over S defin* *es a motivic space X over S. Any simplicial set K defines a constant motivic space * *K. A pointed motivic space is a pair (A, a0), where a0: S ! A. Usually the basepoint* * will be omitted from the notation. The resulting category is denoted Mo(S). Definition A.1.2. A morphism f :S ! S0 of base schemes defines the functor f*: Mo(S) ! Mo(S0) sending A to (Y ! S0) 7! A(S xS0Y ). Left Kan extension produces a left adjoi* *nt f* :Mo(S0) ! Mo(S) of f*. ` If A is`a motivic space, let A+ denote the pointed motivic space (A S, i),* * where i: S ! A S is the canonical inclusion. The category Mo(S) is closed symmetr* *ic monoidal, with smash product A ^ B defined by the sectionwise smash product A ^ B (X): = A(X) ^ B(X) (27) and with internal hom Hom__Mo(S)(A, B) defined by Hom__Mo(S)(A, B)(x: X ! S)n: = Hom Mo(S)(A ^ n+, x*x*B). (28) In particular, Mo(S) is also enriched over the category of pointed simplicial s* *ets, with enrichment sSeto(A, B): = Hom__Mo(S)(A, B)(S). The mapping cylinder of a map f * *:A ! B is the pushout of the diagram ~= `idA`f ` A ^ @ 1+ ____//_A A___//_A .B (29) |ae| || fflffl| fflffl| A ^ 1+________________//Cyl(f) The composition of the canonical maps A ,! Cyl(f) ! B is f. The pushout product of two maps f :A ! C and g :B ! D of motivic spaces over* * S is the map f t g :A ^ D [A^B C ^ B ! C ^ D induced by the commutative diagram A ^ B ____//_A ^ D (30) | | | | fflffl| fflffl| C ^ B ____//_C ^ D. The functor f* :Mo(S0) ! Mo(S) is strict symmetric monoidal, since so is the pu* *llback functor sending X 2 SmS0to S xS0X 2 Sm=S. This ends the categorical considerati* *ons. 22 A.2 Model categories The basic purpose of a model structure is to give a framework for the construct* *ion of a homotopy category. Suppose wC is a class of morphisms in a bicomplete C one wan* *ts to be invertible. Call them weak equivalences. One can define the homotopy "catego* *ry" of the pair (C, wC) to be the universal "functor" : C ! Ho (C, wC) such that ever* *y weak equivalence is mapped to an isomorphism. In general, this homotopy "category" m* *ay not be a category: it has hom-classes, but not necessarily hom-sets. If one require* *s the exis- tence of two auxiliary classes of morphisms fC (the fibrations) and cC (the cof* *ibrations), together with certain compatibility axioms, one does get a homotopy category Ho* *(C, wC) and an explicit description of the hom-sets in it. Theorem A.2.1 (Quillen). Let (wC, fC, cC) be a model structure on a bicomplete * *category C. Then the homotopy category : C ! Ho(C, wC) exists and is the identity on ob* *jects. The set of morphisms in Ho(C, wC) from A to B is the set of morphisms in C fr* *om A to B modulo a homotopy equivalence relation, provided that ; ! A is a cofibrati* *on and B ! * is a fibration. Here ; is the initial object and * is the terminal object in C. An object A * *resp. B as in Theorem A.2.1 is called cofibrant resp. fibrant. Every object A in the homotop* *y category is isomorphic to an object C, where C is both fibrant and cofibrant. A (co)fi* *bration which is also a weak equivalence is usually called a trivial or acyclic (co)fib* *ration. To describe the standard way to construct model structures on a bicomplete c* *ategory, one needs a definition. Definition A.2.2. Let f :A ! B and g :C ! D be morphisms in C. If every commuta- tive diagram A _____//C f|| g|| fflffl|fflffl| B _____//D admits a morphism h: B ! C such that the resulting diagram A _____//C>>" "" f||h"" g|| fflffl|fflffl|"" B _____//D commutes (a lift for short), then f has the left lifting property with respect * *to g, and g has the right lifting property with respect to f. Here is the standard way of constructing a model structure on a given bicomp* *lete category. Choose the class of weak equivalences such that it contains all iden* *tities, is closed under retracts and satisfies the two-out-of-three axiom. Pick a set I (t* *he generating cofibrations) and define a cofibration to be a morphism which is a retract of a* * transfinite composition of cobase changes of morphisms in I. Pick a set J (the generating * *acyclic 23 cofibrations) of weak equivalences which are also cofibrations and define the f* *ibrations to be those morphisms which have the right lifting property with respect to every * *morphism in J. Some technical conditions have to be fulfilled in order to conclude that * *this indeed is a model structure, which is then called cofibrantly generated. See [Ho , The* *orem 2.1.19]. Example A.2.3. In Top , let the weak equivalences be the weak homotopy equivale* *nces, and set I = {@Dn ,! Dn}n 0 J = {Dn x {0} ,! Dn x I}n 0. Then the fibrations are precisely the Serre fibrations, and the cofibrations ar* *e retracts of generalized cell complexes ("generalized" refers to the fact that cells do not * *have to be attached in order of dimension). In sSet, let the weak equivalences be those ma* *ps which map to (weak) homotopy equivalences under geometric realization. Set I = {@ n ,! n}n 0 J = { nj,! n}n 1,0 j n where njis the sub-simplicial set of @ n obtained by removing the j-th face. T* *hen the fibrations are precisely the Kan fibrations, and the cofibrations are the inclu* *sions. Example A.2.4. For the purpose of this paper, model structures on presheaf cate* *gories Fun(Cop, sSet) with values in simplicial sets are relevant. There is a canonica* *l one, due to Quillen, which is usually referred to as the projective model structure. It* * has as weak equivalences those morphisms f :A ! B such that f(c): A(c) ! B(c) is a weak equivalence for every c 2 ObC (the objectwise or sectionwise weak equivalences)* *. Set I = {Hom S(-, c) x (@ n ,! n)}n 0 J = {Hom S(-, c) x ( nj,! n)}n 1,0 j n so that by adjointness, the fibrations are precisely the sectionwise Kan fibrat* *ions. There is another one with the same weak equivalences, due to Heller [He ], such that the* * cofibrations are precisely the injective morphisms (whence the name injective model structur* *e).The description of J involves the cardinality of the set of morphisms in C and is n* *ot explicit. Neither is the characterization of the fibrations. The morphisms of model categories are called Quillen functors. A Quillen fu* *nctor of model categories M ! N is an adjoint pair (F, G): M ! N such that F preserves cofibrations and G preserves fibrations. This condition ensures that (F, G) in* *duces an adjoint pair on homotopy categories (LF, RG), where LF is the total left derive* *d functor of F . A Quillen functor is a Quillen equivalence if the total left derived is * *an equivalence. For example, geometric realization is a strict symmetric monoidal left Quillen * *equivalence | - |: sSet! Top , and similarly in the pointed setting. If a model category has a closed symmetric monoidal structure as well, one h* *as the following statement. Theorem A.2.5 (Quillen). Let C be a bicomplete category with a model structure.* * Sup- pose that (C, , I) is closed symmetric monoidal. Suppose further that these st* *ructures are compatible in the following sense: 24 o The pushout product of two cofibrations is a cofibration, and o the pushout product of an acyclic cofibration with a cofibration is an acy* *clic cofibra- tion. Then A - is a left Quillen functor for all cofibrant objects A 2 C. In partic* *ular, there is an induced (total derived) closed symmetric monoidal structure on Ho(C, wC). One abbreviates the hypotheses of Theorem A.2.5 by saying that C is a symmet* *ric monoidal model category. This ends our introduction to model category theory. A.3 Model structures for motivic spaces To equip Mo(S) with a model structure suitable for the various requirements (co* *mpati- bility with base change, taking complex points, finiteness conditions, having t* *he correct motivic homotopy category), we construct a preliminary model structure first. S* *tart with the following construction, which is a special case of the considerations in [I* *]. Choose any X 2 SmS and a finite set {ij:Zj # X}mj=1 of closed embeddings in Sm=S. Regarding ij as a monomorphism of motivic spaces,* * one may form the categorical union (not the categorical coproduct!) ?mj=1ij: [mj=1Z* *j ,! X. That is, [mj=1Zj is the coequalizer in the category of motivic spaces of the di* *agram ma a Zj ' ZjxX Zj0 (31) j=1 j,j0 Call the resulting monomorphism ?mj=1ij: [mj=1Zj ,! X acceptable. The closed em* *bedding ; # X is acceptable as well. Consider the set Ace of acceptable monomorphisms. * *Let IcS be the set of maps {i+ t (@ n ,! n)+}i2Ace,n 0 (32) and let JcSbe the set of maps {i+ t ( nj,! n)+}i2Ace,n 1,0 j n (33) Definition A.3.1. A map f :A ! B in Mo(S) is a schemewise weak equivalence if f :A(X) ! B(X) is a weak equivalence of simplicial sets for all X 2 Sm=S. It i* *s a closed schemewise fibration if f :A ! B has the right lifting property with res* *pect to JcS. It is a closed cofibration if it has the left lifting property with respec* *t to all acyclic closed schemewise fibrations (closed schemewise fibrations which are also schem* *ewise weak equivalences). Theorem A.3.2. The classes defined in A.3.1 are a closed symmetric monoidal mod* *el structure on Mo(S), denoted Mcso(S). A morphism f :S ! T of base schemes induce* *s a strict symmetric monoidal left Quillen functor f* :Mo(T ) ! Mo(S). 25 Proof.The existence of the model structure follows from [I]. The pushout produ* *ct ax- iom follows, because the pushout product of two acceptable monomorphism is agai* *n ac- ceptable. To conclude the last statement, it suffices to check that f* maps an* *y map in IcTresp. JcTto a closed cofibration resp. schemewise weak equivalence. In f* *act, if i = ?mj=1ij: [mj=1Zj ,! X is an acceptable monomorphism in SmT, then f*(i) is t* *he acceptable monomorphism obtained from the closed embeddings {T xS Zj # T xS X.} Because f* is strict symmetric monoidal and a left adjoint, it preserves the pu* *shout product. Hence f* even maps the set IcTto the set IcS, and likewise for JcT. Th* *e result * *__ follows. |* *__| The resulting homotopy category is equivalent - via the identity functor - t* *o the usual homotopy category of the diagram category Mo(S) (obtained via the project* *ive model structure), since the weak equivalences are just the objectwise ones. Th* *e model structure Mcso(S) has the advantage that for any S-point x0: S # X in a smooth * *S- scheme, the pair (X, x0) is closed cofibrant. Not all pointed motivic spaces a* *re closed cofibrant. Let (-)cs! IdMo(S)denote a cofibrant replacement functor, for exampl* *e the one obtained from applying the small object argument to IcS. That is, the map A* *cs! A is a natural closed schemewise fibration and a schemewise weak equivalence, and* * Acsis closed cofibrant. Dually, let IdMo(S) ! (-)cf denote the fibrant replacement f* *unctor obtained by applying the small object argument to JcS. The closed schemewise fi* *brations may be characterized explicitly. Lemma A.3.3. A map f :A ! B is a closed schemewise fibration if and only if the following two conditions hold. 1. f(X): A(X) ! B(X) is a Kan fibration for every X 2 Sm=S, and 2. for every finite set {Zj # X}mj=1of closed embeddings in Sm=S, the induced* * map A(X) ! B(X) xsSeto([mj=1Zj,B)sSeto([mj=1Zj, A) is a Kan fibration. * * __ Proof.Follows by adjointness from the definition. * * |__| To obtain a motivic model structure, one localizes Mcso(S) as follows. Recal* *l that an elementary distinguished square (or simply Nisnevich square) is a pullback diag* *ram V ---! Y ? ? ? ? y y p j U ---! X 26 in Sm=S, where j is an open embedding and p is an 'etale morphism inducing an i* *somor- phism Y - V ~= X - U of reduced closed subschemes. Say that a pointed motivic s* *pace C is closed motivic fibrant if it is closed schemewise fibrant, the map 1 pr C X xS AS -! X is a weak equivalence of simplicial sets for every X 2 Sm=S, the square C(V ) --- C(Y ) ? ? ? ? y y C(U) --- C(X) is a homotopy pullback square of simplicial sets for every Nisnevich square in * *Sm=S and C(;) is contractible. Example A.3.4. Let X 7! KTT (X) be the pointed motivic space sending X 2 Sm=S to the first term of the Waldhausen K-theory spectrum [W ] associated to the ca* *tegory of big vector bundles over X [FS ], with isomorphisms as weak equivalences. That i* *s, KTT (X) = Sing|wSo(Vectbig(X), iso)| Suppose now that S is regular. Then so is X, and KTT (X) has the same homotopy * *type as the first term of the Thomason-Trobaugh K-theory spectrum of X [TT , Cor. 3.* *9]. It follows that the projection induces a weak equivalene KTT (X) ! KTT (X xS A1S) * *[TT , Prop. 6.8]. By [TT , Thm. 10.8] the square KTT (X) _______//KTT (U) KTT(p)|| || fflffl| fflffl| KTT (Y )____//_KTT p-1(U) associated to a Nisnevich square is a homotopy pullback square. Hence KTT is f* *ibrant in the projective motivic model structure on Mo(S). However, if i: Z ,! X is a * *closed embedding in Sm=S, the map KTT (i): KTT (X) ! KTT (Z) is not necessarily a Kan * *fi- bration. In particular, KTT is not closed schemewise fibrant. Choose a closed c* *ofibration which is also a schemewise weak equivalence KTT ! KTT such that KTT is closed s* *cheme- wise fibrant. It follows immediately that KTT is closed motivic fibrant, and th* *at TT(X) has the homotopy type of the first term of the Thomason-Trobaugh K-theory spect* *rum of X. Definition A.3.5. A map f :A ! B is a motivic weak equivalence if the map sSet o(fcs, C): sSeto(Bcs, C) ! sSeto(Acs, C) is a weak equivalence of simplicial sets for every closed motivic fibrant C. It* * is a closed motivic fibration if it has the right lifting property with respect to all acyc* *lic closed cofi- brations (closed cofibrations which are also motivic equivalences). 27 Example A.3.6. Suppose that f :A ! B is a map in Mo(S) inducing weak equivalenc* *es x*f :x*A ! x*B of simplicial sets on all Nisnevich stalks x*: Mo(S) ! sSet. The* *n f is a motivic weak equivalence. If f :A ! B is an A1-homotopy equivalence (for exam* *ple, the projection of a vector bundle), then it is a motivic weak equivalence. Example A.3.7. The canonical covering of P1 shows that it is motivic weakly equ* *ivalent as a pointed motivic space to the suspension S1^ (A1- {0}, 1), where S1 = 1=@ * *1. Set S1,0:= S1 and S1,1:= (A1 - {0}, 1), and define 1,0^p-q 1,1^q Sp,q:= S ^ S for p q 0 To generalize the example of P1, one can show that if Pn-1 ,! Pn is a linear em* *bedding, then Pn=Pn-1 is motivic weakly equivalent to S2n,n. To prove that the classes from Definition A.3.5 are part of a model structur* *e, it is helpful to characterize the closed motivic fibrant objects via a lifting pro* *perty. Let JcmSbe the union of the set JcSfrom (33) and the set JmS of pushout products of* * maps (@ n ,! n)+ with maps of the form zero+ 1 X+ ________________//(AS xS X)+ (34) U+ [V+ Cylh+ ____//_CylU+ [V+ Cyl(h+) ! X+ * _____________________//;+ where h is the open embedding appearing on top of a Nisnevich square V --h-! Y ? ? ? ? y py U ---! X in Sm=S. Lemma A.3.8. A pointed motivic space C is closed motivic fibrant if and only if* * the map C ! * has the right lifting property with respect to the set JcmS. * * __ Proof.This follows from adjointness, the Yoneda lemma and the construction of J* *cmS. |__| Theorem A.3.9. The classes of motivic weak equivalences, closed motivic fibrati* *ons and closed cofibrations constitute a symmetric monoidal model structure on Mo(S), d* *enoted Mcmo(S). The resulting homotopy category is denoted Hcmo(S) and called the poin* *ted mo- tivic unstable homotopy category of S. A morphism f :S ! T of base schemes indu* *ces a strict symmetric monoidal left Quillen functor f* :Mcmo(T ) ! Mcmo(S). 28 Proof.The existence of the model structure follows by standard Bousfield locali* *zation techniques. Here are some details. The problem is that JcmSmight be to small in* * order to characterize all closed motivic fibrations. Let ~ be a regular cardinal strictl* *y bigger than the cardinality of the set of morphisms in Sm=S. A motivic space A is ~-bounded* * if the union a A(X)n n 0,X2Sm=S has cardinality ~. Let J~Sbe a set of isomorphism classes of acyclic monomorp* *hisms whose target is ~-bounded. One may show that given an acyclic monomorphism j :A* * ,! B and a ~-bounded subobject C B, there exists a ~-bounded subobject C0 B conta* *ining C such that j-1(C0) ,! C0 is an acyclic monomorphism. Via Zorn's lemma, one th* *en gets that a map f 2 Mo(S) has the right lifting property with respect to all ac* *yclic monomorphisms if (and only if) it has the right lifting property with respect t* *o the set J~S. Such a map is in particular a closed motivic fibration. Any given map f :A* * ! B can now be factored (via the small object argument) as an acyclic monomorphism j :A* * ,! C followed by a closed motivic fibration. Factoring j as a closed cofibration fol* *lowed by an acyclic closed schemewise fibration in the model structure of Theorem A.3.2 imp* *lies the existence of the model structure. To prove that the model structure is symmetric monoidal, it suffices - by th* *e corre- sponding statement A.3.2 for Mcso(S) - to check that the pushout product of a g* *enerating closed cofibration and an acyclic closed cofibration is again a motivic equival* *ence. How- ever, from the fact that the injective motivic model structure is symmetric mon* *oidal, one knows that motivic equivalences are closed under smashing with arbitrary motivi* *c spaces [DRO ]. The first sentence is now proven. Concerning the third sentence, Theorem A.3.2 already implies that f* preserv* *es closed cofibrations. To prove that f* is a left Quillen functor, it suffices (by Dugge* *r's lemma [D , Cor. A2]) to check that it maps the set JcmTto motivic weak equivalences in Mo(* *S). One * * __ may calculate that f*(JcmT) = JcmS, whence the statement. * * |__| The closed motivic model structure is cofibrantly generated. As remarked in * *the proof of Theorem A.3.9, the set JcmSis perhaps not big enough to yield a full set of * *generating trivial cofibrations. By localization theory and Lemma A.3.8, a map with closed* * motivic fibrant codomain is a closed motivic fibration if and only if it has the right * *lifting property with respect to JcmS. Note that the domains and codomains of the maps in JcmSar* *e closed cofibrant and finitely presentable. The following Lemma is an easy consequence. Lemma A.3.10. Motivic equivalences and closed motivic fibrations with closed mo* *tivic fibrant codomain are closed under filtered colimits. Proof.This follows, since the domains of the maps in IcSand JcmSare finitely pr* *esentable. * * __ See [DRO ] for the corresponding statement for the projective motivic model st* *ructure. |__| Let Mo(S) be the category of simplicial objects in the category of pointed N* *isnevich sheaves on Sm=S. The functor mapping a (pointed) presheaf to its associated (po* *inted) 29 Nisnevich sheaf determines by degreewise application a functor aNis:Mo(S) ! Mo(* *S). Let Mo(S) i-!Mo(S) be the inclusion functor, the right adjoint of aNis. Theorem A.3.11. The pair (aNis, i) is a Quillen equivalence to the Morel-Voevod* *sky model structure. The functor aNisis strict symmetric monoidal. In particular, t* *he total left derived functor of aNisis a strict symmetric monoidal equivalence Hcmo(S) ! Ho(S) to the unstable pointed A1-homotopy category from [MV ]. Proof.Recall that the cofibrations in the Morel-Voevodsky model structure are p* *recisely the monomorphisms. Since every closed cofibration is a monomorphism and Nisnev* *ich sheafification preserves these, aNispreserves cofibrations. The unit A ! i aNis* *(A) of the adjunction is an isomorphism on all Nisnevich stalks, hence a motivic weak equi* *valence by Example A.3.6 for every motivic space A. In particular, aNismaps schemewise * *weak equivalences as well as the maps in JmS described in (34) to weak equivalences.* * Let IdMo(S)! (-)fibbe the fibrant replacement functor in Mcmo(S) obtained from the * *small object argument applied to JcmS. Hence if f is a motivic weak equivalence, the* *n ffibis a schemewise weak equivalence. One concludes that aNispreserves all weak equiva* *lences, thus is a left Quillen functor. Since the unit A ! i aNis(A) is a motivic weak* * equivalence * * __ for every A, the functor aNisis a Quillen equivalence. * * |__| Note that a map f in Mo(S) is a motivic weak equivalence if and only if aNis* *(f) is a weak equivalence in the Morel-Voevodsky model structure on simplicial shea* *ves. Conversely, a map of simplicial sheaves is a weak equivalence if and only if it* * is a motivic weak equivalence when considered as a map of motivic spaces. Remark A.3.12. Starting with the injective model structure on simplicial preshe* *aves mentioned in Example A.2.4, there is a model structure Mimo(S) on the category * *of pointed motivic spaces with motivic weak equivalences as weak equivalences and monomorp* *hisms as cofibrations. It has the advantage that every object is cofibrant, but the d* *isadvantage that it does not behave well under base change [MV , ??] or geometric realizat* *ion (to be defined below). The identity functor is a left Quillen equivalence Id:Mcmo(S) !* * Mimo(S), since the homotopy categories coincide. Further, let Spco(S) be the category of pointed Nisnevich sheaves on Sm=S. R* *ecall the cosimplicial smooth scheme over S whose value at n is Xn nS= Spec OS[X0, . .,.Xn]=( Xi= 1) (35) i=0 The functor Spco(S) ! Mo(S) sending A to the simplicial object SingS(A)n = A(-x* * nS) has a left adjoint | - |S :Mo(S). It maps B to the coend Z |B|S = Bn x nS n2 in the category of pointed Nisnevich sheaves. The following statement is proved* * in [MV ]. 30 Theorem A.3.13 (Morel-Voevodsy). There is a model structure on the category Spc* *o(S) such that the pair (| - |S, SingS) is a Quillen equivalence to the Morel-Voevod* *sky model structure. The functor | - |S is strict symmetric monoidal. In particular, th* *e total left derived functor of | - |S is a strict symmetric monoidal equivalence from Voevo* *dsky's pointed homotopy category to the unstable pointed A1-homotopy category A.4 Topological realization In the case where the base scheme is the complex numbers, there is a topologica* *l realization functor RC :Mcmo(C) ! Top owhich is a strict symmetric monoidal left Quillen fu* *nctor. It is defined as follows. If X 2 SmC, the set X(C) of complex points is a topo* *logical space when equipped with the analytic topology. Call this topological space Xan* *. It is a smooth manifold, and in particular a compactly generated topological space. O* *ne may view X 7! Xan as a functor SmC ! Top . Note that if i: Z # X is a closed embedd* *ing in SmC, then the resulting map ianis the closed embedding of a smooth submanifold,* * and in particular a cofibration of compactly generated topological spaces. Every motiv* *ic space A is a canonical colimit ~ colim X x n =-!A Xx n!A and one defines RC(A): = colim Xanx | n| 2 Top . Xx n!A Observe that if A is pointed, then so is RC(A). Theorem A.4.1. The functor RC :Mcmo(C) ! Top ois a strict symmetric monoidal le* *ft Quillen functor. Proof.The right adjoint of RC maps the compactly generated topological space Z * *to the motivic space SingC(Z) which sends X 2 SmC to the simplicial set sSetTop(Xa* *n, Z). To conclude that RC is strict symmetric monoidal, it suffices to observe that t* *here is a canonical homeomorphism (X xY )an~= XanxY an, and that geometric realization is* * strict symmetric monoidal. Suppose now that i: [mj=1Zj ,! X is an acceptable monomorphism. One computes RC([mj=1Zj) as the coequalizer am a an Zanj' ZjxX Zj0 . j=1 j,j0 an in Top . Every map ZjxX Zj0 ! Zanjis a closed embedding of smooth submanifol* *ds of complex projective space. In particular, one may equip Zanjwith a cell complex * *structure an 0 m such that Zj xX Zj0 is a subcomplex for every j . Then RC([j=1Zj) is the un* *ion of these subcomplexes, and in particular again a subcomplex. It follows that RC* *(i) is a cofibration of topological spaces. Since RC is compatible with pushout products* *, it maps the generating closed cofibrations to cofibrations of topological spaces. 31 To conclude that RC preserves trivial cofibrations as well, it suffices by D* *ugger's lemma [D , Cor. A2] to check that RC maps every map in JmCto a weak homotopy equivale* *nce. In fact, since the domains and codomains of the maps @ m ,! m are cofibrant, it s* *uffices to check the latter for the maps in diagram (34). In the first case, one obtain* *s the map Xan ,! (A1Cx X)an~= R2x Xan, in the second case one obtains up to simplicial ho* *motopy equivalence the canonical map Uan [p-1(V )anY an! Xan for a Nisnevich square V ____//_Y | | | p| fflffl|fflffl| U ____//_X * * __ This is in fact a homeomorphism of topological spaces. The result follows. * * |__| Suppose now that R ,! C is a subring of the complex numbers. Let f :Spec(C) ! Spec(R) denote the resulting morphism of base schemes. The realization with res* *pect to R (or better f) is defined as the composition RR = RC O f* :Mo(R) ! Mo(C) ! Top o. (36) It is a strict symmetric monoidal Quillen functor. The most relevant case is R * *= Z. Example A.4.2. The topological realization of the finite Grassmannian Gr(m, n) * *(over any base with a complex point) is the complex Grassmannian with the usual topol* *ogy. Since RC commutes with filtered colimits, RC(Gr ) is the infinite complex Grass* *mannian, which in turn is the classifying space BU for the infinite unitary group. Becau* *se RC is a left Quillen functor, the topological realization of any closed cofibrant mot* *ivic space weakly equivalent to Gr (such as K) is homotopy equivalent to BU. A.5 Spectra Definition A.5.1. Let P1Sdenote the pointed projective line over S. The categor* *y MS (S) of P1-spectra over S has the following objects. A P1-spectrum E consists of a s* *equence (E0, E1, E2, . .).of pointed motivic spaces over S, together with structure map* *s oeEn:En ^ P1 ! En+1 for every n 0. A map of P1-spectra is a sequence of maps of pointed* * motivic spaces which is compatible with the structure maps. Example A.5.2. Any pointed motivic space B over S gives rise to a P1S-suspension spectrum 1P1B = (B, B ^ P1, B ^ P1 ^ P1, . .). having identities as structure maps. More generally, let FrnB denote the P1-sp* *ectrum having values ( B ^ P1^m m 0 (FrnB)n+m = * m < 0 and identities as structure maps, except for oeFrnBn-1. The functor B 7! FrnB i* *s left adjoint to the functor sending the P1-spectrum E to En. 32 Remark A.5.3. In Definition A.5.1, one may replace P1 by any pointed motivic sp* *ace A, giving the category MS A (S) of A-spectra over S. Essentially the only relevant* * example for us is when A is weakly equivalent to the pointed projective line P1. The Th* *om space T = A1=A1 - {0} of the trivial line bundle over S admits motivic weak equivalen* *ces ___~_//1 1 oo___ P1 P =A ~ T (37) The motivic space P1 itself is not always the ideal suspension coordinate. For * *example, the algebraic cobordism spectrum MGL naturally comes as a T -spectrum. In ord* *er to switch between T -spectra and P1-spectra, consider the following general constr* *uction. A map OE: A ! B induces a functor OE*: MS B(S) ! MS A (S) sending the B-spectrum (E0, E1, . .,.oeEn) to the A-spectrum *E E (E0, E1, . .). with structure maps oeOEn = oen O (En ^ OE) Its left adjoint OE] maps the A-spectrum (F0, F1, . .,.oeFn) to the B-spectrum i j F0, B ^ F0 [A^F0F1, B ^ B ^ F0 [A^F0F1 [A^F1F2 , . . . (38) having the canonical maps as structure maps. Note that for the purpose of const* *ructing a model structure on A-spectra over S, the pointed motivic space A has to be co* *fibrant in the model structure under consideration. The next goal is to construct a model structure on MS (S) having the motivic* * stable homotopy category as its homotopy category. Definition A.5.4. Let P1 = Hom__Mo(S)(P1S, -) denote the right adjoint of P1^-* *. For a P1-spectrum E with structure maps oeEn:En ^ P1 ! En+1, let !En:En ! P1En+1 den* *ote the adjoint structure map. A P1-spectrum E is closed stably fibrant if o En is closed motivic fibrant for every n 0, and o !En:En ! P1En+1 is a motivic weak equivalence for every n 0. Any P1-spectrum E admits a closed stably fibrant replacement. First replace * *E by a levelwise P1-spectrum E` as follows. Let E`0= Efib0for a fibrant replacement in* * Mcmo(S). Given En ! E`n, set i j fib E`n+1:= E`n^ P1 [En^P1 En+1 which yields a levelwise motivic weak equivalence E ! E` of P1-spectra. To cont* *inue, observe that the adjoint structure maps of any P1-spectrum F may be viewed as a* * natural transformation q :F ! Q(F ) where Q(F ) is the P1-spectrum with terms P1F1, P1F2, . .a.nd structure maps P1(!Fn+1): P1Fn+1 ! 2P1Fn+2. Define Q1 (E) as the colimit of the sequence E` ___q//_Q(E`)Q(q)//_Q2(E`)_//_. * *. . 33 Definition A.5.5. A map f :E ! F of P1-spectra is a stable equivalence if the m* *ap Q1 (f)n is a weak equivalence for every n 0. It is a closed stable fibration * *if fn: En ! Fn is a closed motivic fibration and the induced map En ! Fn xQ1 (E)nQ1 (F )n i* *s a motivic weak equivalence for every n 0. It is a closed cofibration if fn: En * *! Fn and Fn ^ P1 [En^P1 En+1 ! Fn+1 are closed cofibrations for every n 0. Theorem A.5.6. The classes from Definition A.5.5 are a model structure on the c* *ategory of P1-spectra, denoted MS cm(S). The identity functor on P1-spectra from Jardin* *e's stable model structure to MS cm(S) is a left Quillen equivalence. In particular, the* * homotopy category SH cm(S): = Ho MS cm(S) is equivalent to the motivic stable homotopy* * category SH(S). Proof.Recall that P1 is closed cofibrant. The existence of the model structure * *follows as in [J, Thm. 2.9]. Moreover, the stable equivalences coincide with the ones in [* *J], because so do the stabilization constructions and the unstable weak equivalences. Sinc* *e every closed cofibration of motivic spaces is in particular a monomorphism, IdMS(S) i* *s a left Quillen equivalence. Note that the closed cofibrations are generated by the set n n {Frm X x @ ,! X x )+ }m,n 0,X2Sm=S (39) * * __ One may also describe a set of generating acyclic cofibrations. * * |__| Remark A.5.7. Note that one may form the smash product of a motivic space A and a P1-spectrum E by setting (A ^ E)n: = A ^ En and oeA^En:= A ^ oeEn. If A is c* *losed cofibrant, A^- is a left Quillen functor by Theorem A.3.9. From Theorem A.5.6 o* *ne may deduce that P1 ^ -: MS (S) ! MS (S) is a left Quillen equivalence. Since { ' S2* *,1= S1 ^ (A1 - {0}, 1), also S1 ^ - is a left Quillen equivalence. In particular, * *SH cm(S) is triangulated, with the total left derived of S1 ^ - as the shift functor. The t* *riangles are those which are isomorphic to the image of E ,! F ! F=E E ^ 1 [E F ! S1 ^ E where E ,! F is an inclusion of P1-spectra. As well, one has sphere spectra Sp* *,q2 SHcm(S) for all integers p, q 2 Z. Example A.5.8. Since SH(S) is an additive category, the canonical map E _F ! E * *xF is a stable equivalence. In the special case of P1-suspension spectra, the cano* *nical map factors as 1P1A _ 1P1B ~= 1P1(A _ B) ! 1P1(A x B) ! 1P1A x 1P1B which shows that 1P1(AxB) contains 1P1(A^B) as a retract in SH(S). Thus it is* * even a direct summand. The latter can be deduced as follows. The (reduced) join (A, a0* *)*(B, b0) is defined as the pushout in the diagram A x B x @ 1 [ {a0} x {b0} x 1___//_A _ B | | | | fflffl| fflffl| A x B x 1 _____________//_A * B. 34 of pointed motivic spaces over S. Attaching A ^ ( 1, 0) and B ^ ( 1, 0) to A ** * B via A _ B produces a pointed motivic space C which is equipped with a sectionwise w* *eak equivalence C ! (A x B) ^ S1,0. Collapsing {a0} * B and A * {b0} inside C yiel* *ds a sectionwise weak equivalence C ! (A ^ B ^ S1,0) _ (A ^ S1,0) _ (B ^ S1,0). Sinc* *e S1,0is invertible in SH(S), one gets a splitting 1P1(A x B) ' 1P1(A ^ B) _ 1P1A * * _ 1P1B in SH (S). For a P1-spectrum E let TrnE denote the P1-spectrum with ( Em m n TrnE m = En ^ (P1)^m-n = FrnEn m m n and with the obvious structure maps. The structure maps of E determine maps Trn* *E ! Trn+1E such that E = colimnTrnE. Since the canonical map FrnEn ! TrnE adjoint to the identity id:En ! En is an identity in all levels n, it is in particula* *r a stable equivalence. The identity id:En ^ (P1)^n ! Fr0En)n leads by adjointness to the* * map ~= 1 ^n Frn(En) ^ (P1)^n___//_FrnEn ^ (P ) ____//_Fr0En (40) and hence to a map FrnEn ! nP1Fr0En . Since the map (40) is an isomorphism in all levels n, it is a stable equivalence. Because P1 is a Quillen equivalenc* *e, the map FrnEn ! nP1(Fr0En)fib is a stable equivalence as well if En is closed cofibran* *t. In fact, the condition on En can be removed since Frnpreserves all weak equivalences. Th* *is leads to the following statement. Lemma A.5.9. Any P1-spectrum E is the colimit of a natural sequence Tr0E ____//_Tr1E___//Tr2E___//_. . . (41) of P1-spectra in which the n-th term is naturally stably equivalent to nP1( 1P* *1En)f . One may use the description in Lemma A.5.9 for computations as follows. Say * *that a P1-spectrum F is finite if it is stably equivalent to a P1-spectrum F 0such tha* *t * ! E0is obtained by attaching finitely many cells from the set (39). Lemma A.5.10. Let D(0) ! D(1) ! D(2) ! . . .be a sequence of P1-spectra, with colimit D(1). 1. Suppose that F is a finite P1-spectrum. The canonical map colimHom SH(S)F, D(i) ! Hom SH(S)F, D(1) i 0 is an isomorphism. 2. For any P1-spectrum E there is a canonical short exact sequence 1,0 0 ! lim1S ^ D(i), E ! D(1), E ! limD(i), E ! 0 (42) i 0 i 0 of abelian groups, where [-, -] denotes Hom SH(S)(-, -). 35 Proof.Observe first that stable equivalences and closed stable fibrations are d* *etected by the functor Q1 which is defined in A.5.5 as a sequential colimit. Lemma A.3.10 * *implies that stable equivalences and closed stable fibrations with closed stably fibran* *t codomain are closed under filtered colimits. Thus by Theorem A.2.1 one may compute Hom SH(S)(F, colimD(i)) ~=Hom MS (S)(F, colimD(i)fib)= ' i 0 i 0 for any cofibrant P1-spectrum F , where ' denotes the equivalence relation "sim* *plicial homotopy". This implies statement 1 because Hom MS (S)(F, -) commutes with fil* *tered colimits if * ! F is obtained by attaching finitely many cells. To prove the second statement, let C be the coequalizer of the diagram W f W 1 i 0D(i)____//_g//_i 0+^ D(i) where f resp. g is defined on the i-th summand D(i) as D(i) = 1+ ^ D(i) ,! 1+^* * D(i) resp. D(i) ! D(i + 1) = 0+ ^ D(i + 1) ,! 1+^ D(i + 1). The canonical map C ! colimi 0D(i) induced by the 1+^ D(i) ! D(i) ! colimi 0D(i) is a weak equivalen* *ce. In the stable homotopy category, which is additive, one may take the difference* * of f and g, and thus describe colimi 0D(i) via the distinguished triangle W f-g W W 1,0 i 0D(i)____//_i 0D(i)____//_colimiD0(i)__//_i 0S ^ D(i). (43) Applying [-, E]: = Hom SH(S)(-, E) to the triangle (43) produces a long exact s* *equence f*-g*Q Q . .o.o____ i 0 D(i), Eoo_____ D(1), E oo_____ i 0 S1,0^ D(i), Eoo_____. . . which may be split into the short exact sequence 1 0oo___limi 0D(i), E]oo__ D(1), E oo___limi 0S1,0^ D(i), Eoo__0_ __ |__| A.6 Symmetric spectra There seems to be no reasonable (i.e. symmetric monoidal) smash product for P1-* *spectra inducing a decent symmetric monoidal smash product on SH (S). This will be solv* *ed as in [HSS ] and [J]. Definition A.6.1. A symmetric P1-spectrum E over S consists of a sequence (E0, * *E1, . .). of pointed motivic spaces over S, together with group actions ( n)+ ^ En ! En a* *nd structure maps oeEn:En ^ P1 ! En+1 for all n 0. Iterations of these structure* * maps are required to be as equivariant as they can, using the permutation action of n o* *n (P1)^n. A map of motivic symmetric P1-spectra is a sequence of maps of pointed motivic * *spaces which is compatible with all the structure (group actions and structure maps). * * Call the resulting category MSS (S). 36 Example A.6.2. Analogous to Example A.5.2, the n-th shifted suspension spectrum FrnA of a pointed motivic space A has as values ( +m+n^ mx{1}A ^ (P1)^m m 0 (FrnB)m+n = * m < 0 where the m-th fold smash product (P1)^m carries the natural permutation action. Every symmetric P1-spectrum determines a P1-spectrum by forgetting the symme* *tric group actions. Call the resulting functor u: MSS (S) ! MS (S). It has a left ad* *joint v, which is characterized uniquely up to unique isomorphism by the fact that v FrnA = FrnA. (44) The smash product E ^ F of two symmetric P1-spectra E and F is constructed as follows. Set (E ^ F )n as the coequalizer of the diagram ` + 1 ___oeEr^Fs//_` + r+1+s=n n ^ rx 1x sEr^ P ^ Fs__________//_r+s=n n ^ rx sEr^ Fs (45) oeEr^(FsOtwist) where the coequalizer is taken in the category of pointed n-motivic spaces. Th* *e structure map oeE^Fnis induced by the structure maps oeF0, . .,.oeFnof F . One may provid* *e natural coherence isomorphisms for associativity, commutativity and unitality, where th* *e unit is IS = (S+, P1, P1^P1, . .,.(P1)^n, . .).with the obvious permutation action and * *identities as structure maps. We proceed with the homotopy theory of symmetric P1-spectra, as in [J]. Definition A.6.3. A map OE: E ! F of motivic symmetric P1-spectra is a levelwise acyclic fibration if OEn: En ! Fn is an acyclic closed motivic fibration of poi* *nted motivic spaces over S for all n 0. A map OE: E ! F of motivic symmetric P1-spectra i* *s a closed cofibration if it has the left lifting property with respect to all leve* *lwise acyclic fibrations. There exists a cofibrant replacement functor (-)cof! IdMSS(S). A * *levelwise fibrant motivic symmetric P1-spectrum E is closed stably fibrant if the adjoint* * En ! MX ({, En+1) of the structure map is a weak equivalence for every n 0. A map * *OE: E ! F is a stable equivalence if the map sSetMSSX(OEcof, G): sSetMSSX (F cof, G) ! sSetMSSX (Ecof, G) is a weak equivalence of pointed motivic spaces for all closed stably fibrant m* *otivic sym- metric P1-spectra G. The closed stable fibrations are then defined by the righ* *t lifting property. Theorem A.6.4 (Jardine). The classes of stable equivalences, closed cofibration* *s and closed stable fibrations from Definition A.6.3 constitute a symmetric monoidal * *model structure on MSS (S). The forgetful functor u: MSS cm(S) ! MS cm(S) is a right* * Quillen equivalence. 37 Proof.The proof of the first statement follows as in [J, Thm. 4.15, Prop. 4.19]* *. Note that the stable equivalences in MSS cm(S) and in Jardine's model structure MSS* * Jar(S) coincide, since the unstable weak equivalences do so by Remark A.3.12. In parti* *cular, the identity Id:MSS cm(S) ! MSS Jar(S) is a left Quillen equivalence. The closed * *cofibra- tions in MSS cm(S) are generated by the inclusions n n {Frm X x @ ,! X x )+ }m,n 0,X2Sm=S (46) By formula (44), the left adjoint v of u sends the generating cofibrations to t* *he generating cofibrations. It follows that v is a left Quillen functor. Since v :MS Jar(S) !* * MSS Jar(S) is * * __ a Quillen equivalence by [J, Thm. 4.31], so is the functor v :MS cm(S) ! MSS c* *m(S). |__| Remark A.6.5. In particular, the category Ho (MS cm(S)) inherits a closed symm* *etric monoidal product ^ by setting E ^ F := Ru Lv(E) ^ Lv(F ) In other words, if E and F are closed cofibrant P1-spectra, their smash product* * in SH(S) is given by the P1-spectrum u (v(E) ^ v(F ))fib. The unit is Ru Lv(I) ~=I the * *sphere P1-spectrum. Notation A.6.6. For P1-spectra E and F over S define the E-cohomology and the E- homology of F as Ep,q(F )= Hom SH(S)(F, Sp,q^ E) (47) Ep,q(F )= Hom SH(S)(Sp,q, F ^ E). (48) In the special case F = 1P1A, where A is a pointed motivic space over S one wr* *ites Ep,q(A) and Ep,q(A) instead. Sometimes we abbreviate S2n,n^ E by E(n). Remark A.6.7. Since (v, u) is a Quillen adjoint pair of stable model categories* * the total derived pair respects in particular the triangulated structures. Note that sinc* *e u preserves all colimits, both Lv and Ru preserve arbitrary coproducts. Lemma A.6.8. Let E and F be P1-spectra. Then E ^ F 2 SH(S) may be obtained as t* *he sequential colimit of a sequence whose n-th term is stably equivalent to 2nP1(* * 1P1En^Fn)fib. Proof.We may assume that both E and F are closed cofibrant. By Lemma A.5.9 E and F can be expressed as sequential colimits of their truncations. Since v preserv* *es colimits, v(E) is the sequential colimit of the diagram v(Tr0E) = v(Fr0E0) = Fr0E0 ! v(Tr1E) ! . . . and similarly for F . The stable equivalence FrnEn ! TrnE of cofibrant P1-spect* *ra induces a stable equivalence v(FrnEn) = FrnEn ! v(TrnE). Since smashing with a symmetr* *ic P1-spectrum preserves colimits, one has v(Trm E) ^ colimv(TrnF ) ~=colim v(Trm E) ^ v(TrnF ) n n 38 for every n. It follows that v(E) ^ v(F ) is the filtered colimit of the diagr* *am sending (m, n) to v(Trm E) ^ v(TrnF ). Since the diagonal is a final subcategory in N x* * N, there is a canonical isomorphism colimnv(TrnE) ^ v(TrnF ) ~=v(E) ^ v(F ). Theorem A.6.4 * *says that MSS cm(S) is symmetric monoidal, thus the canonical map Fr2n(En ^ Fn) ~=FrnEn ^n En ! v(TrnE) ^ v(TrnF ) (49) is a stable equivalence. There is a canonical map Fr2n(En^Fn) ! 2nP1Fr0(En^Fn)* * which is adjoint to the unit En ^ Fn ! 2nP1(P1)^2n^ (En ^ Fn). As in the case of P1-* *spectra, the map fib Fr2n(En ^ Fn) ! 2nP1Fr0(En ^ Fn) ! 2nP1Fr0(En ^ Fn) is a stable equivalence. It follows that v(E) ^ v(F ) is the colimit of a sequ* *ence whose fib n-th term is stably equivalent to 2nP1Fr0(En ^ Fn) . Hence it also follows * *that a fibrant replacement of v(E) ^ v(F ) may be obtained as the colimit of a sequenc* *e of closed stably fibrant symmetric P1-spectra whose n-th term is stably equivalent to 2n* *P1Fr0(En^ fib Fn) . Since the forgetful functor u preserves colimits, stable equivalences o* *f closed stably fibrant symmetric P1-spectra and P1, the P1-spectrum u v(E) ^ v(F ) is the co* *limit of * * fib a sequence of P1-spectra whose n-th term is stably equivalent to 2nP1u Fr0(En* *^Fn) . The map fib Fr0(En ^ Fn) ! u vFr0(En ^ Fn) * * __ is a stable equivalence because (v, u) is a Quillen equivalence A.6.4, whence t* *he result. |__| As in the case of non-symmetric spectra, one may change the suspension coord* *i- nate A.5.3. If A is a pointed motivic space over S, let MSS A(S) denote the ca* *tegory of symmetric A-spectra over S. Lemma A.6.9. A map A ! B in Mo(S) induces a strict symmetric monoidal functor MSS A (S) ! MSS B(S) having a right adjoint. If the map is a motivic weak equi* *valence of closed cofibrant pointed motivic spaces, this pair is a Quillen equivalence. * * __ Proof.This is quite formal. For a proof consider [Ho2 , Thm. 9.4]. * * |__| Because the change of suspension coordinate functors are lax symmetric monoi* *dal, they preserve (commutative) monoid objects, that is, (commutative) symmetric ri* *ng spec- tra. A.7 Stable topological realization Let Sp = Sp(Top , CP1) be the category of CP1-spectra (in Top ). An object in S* *p is thus a sequence of pointed compactly generated topological spaces E0, E1, . .w.ith s* *tructure maps En ^ CP1 ! En+1. The model structure on Sp is obtained as follows: Cofibra* *tions are generated by {FrTopm(|@ n ,! n|+)}m,n 0 39 so that every CP1-spectrum E has a cofibrant replacement Ecof! E mapping to E v* *ia a levelwise acyclic Serre fibration. For any CP1-spectrum E and any n 2 Z let ssnE := colim ssn+2mEm ! ssn+2m+2Em ^ CP1 ! ssn+2(m+1)E1+m ! . . . 2m+n 0,m 0 be the n-th stable homotopy group of E. Note that homotopy groups of non-degene* *rately based compactly generated topological spaces commute with filtered colimits. I* *f E is cofibrant, En is in particular non-degenerately based for all n 0. A map f :* *E ! F of CP1-spectra is a stable equivalence if the induced map ssnf :ssnEcof! ssnF c* *ofis an isomorphism for all n 2 Z. It is a stable fibration if it has the right lifting* * property with respect to all stable acyclic cofibrations. Similarly, one may form the category Sp = Sp (Top , CP1) of symmetric CP1-s* *pectra in Top . Cofibrations are generated by {FrTop,m(|@ n ,! n|+)}m,n 0 and a symmetric CP1-spectrum is stably fibrant if its underlying CP1-spectrum i* *s stably fibrant. A map f :E ! F of symmetric CP1-spectra is a stable equivalence if the* * induced map sSetSp (fcof, G) of simplicial sets of maps is an isomorphism for all stabl* *y fibrant symmetric CP1-spectra G. It is a stable fibration if it has the right lifting p* *roperty with respect to all stable acyclic cofibrations. Theorem A.7.1. Stable equivalences, stable fibrations and cofibrations form (sy* *mmetric monoidal) model structures on the categories of (symmetric) CP1-spectra in Top * *. The functor forgetting the symmetric group actions is a right Quillen equivalence. * *There is a zig-zag of strict symmetric monoidal left Quillen functors connecting Sp (Top ,* * CP1) and Sp (Top , S1). In particular, the homotopy category of (symmetric) CP1-spectra * *is equiv- alent as a closed symmetric monoidal and triangulated category to the stable ho* *motopy category. Proof.The statement about the model structures follows as in [HSS ] if one repl* *aces S1 by CP1 and simplicial sets by compactly generated topological spaces. The same hol* *ds for the statement about the functor forgetting the symmetric group actions. To cons* *truct the zig-zag, consider the corresponding stable model structure on the category of s* *ymmetric S1-spectra in the category Sp (Top , CP1), which is isomorphic as a symmetric m* *onoidal model category to the category of symmetric CP1-spectra in the category Sp (Top* * , S1) of topological symmetric S1-spectra. The suspension spectrum functors give a zi* *g-zag 1 1 1 1 1 Sp (Top , CP1) ! Sp Sp (Top , CP ), S ~=Sp Sp (Top , S )CP Sp (Top , S ) (50) of strict symmetric monoidal left Quillen functors. Since CP1^ - is a left Quil* *len equiva- lence on the left hand side in the zig-zag (50) and S1 ^ S1 ~=CP1, S1 ^ - is a * *left Quillen equivalence on the left hand side as well. By [Ho2 , Theorem 9.1], the arrow po* *inting to the right in the zig-zag (50) is a Quillen equivalence. A similar argument work* *s for the * * __ arrow on the right hand side, which completes the proof. * * |__| 40 Given a P1-spectrum E over C, one gets a CP1-spectrum RC(E) = (RCE0, RCE1, .* * .). with structure maps RC(En) ^ P1 ~=RC(En ^ P1) ! RC(En+1). The right adjoint for* * the resulting functor RC :MS (C) ! Sp is also obtained by a levelwise application o* *f SingC. The same works for symmetric P1-spectra over C. Theorem A.7.2. The functors RC :MS (C) ! Sp and RC :MSS (C) ! Sp are left Quillen functors, the latter being strict symmetric monoidal. Proof.Since the diagrams E7!En MSS (C)_u__//_MS(C)____//_Mo(C) SingC|| SingC|| |SingC| fflffl|u fflffl|E7!Efflffl|n Sp ________//_Sp_____//_Topo commute, RC preserves the generating cofibrations by A.4.1. Then Dugger's Lemma* * [D , Cor. A.2] implies that RC is a left Quillen functor, because SingCpreserves wea* *k equiva- lences and fibrations between fibrant objects. The fact that RC :MSS (C) ! Sp * * is strict * * __ symmetric monoidal follows from the definition of the smash product (45). * * |__| Example A.7.3. Let BGL be the P1-spectrum over C constructed in 1.2. Its n-th * *term is a closed cofibrant pointed motivic space K weakly equivalent to Z x Gr. Thus* * the n-th term of RC(BGL ) is weakly equivalent to BU. To show that the CP1-spectrum RC(B* *GL ) is the one representing complex K-theory, it suffices to check that the structu* *re map K ^ P1 ! K realizes to the structure map BU ^ CP1 ! BU of complex K-theory. Consider the diagram ~= alg Hom Ho(C)(Z x Gr, Z x Gr)___________//_K0 (Z x Gr) | | | | fflffl| ~ fflffl| Hom Ho(Topo)RC(Z x Gr), RC(Z x Gr)__=__//Ktop0RC(Z x Gr) where the vertical map on the left hand side is induced by RC and the vertical * *map on the right hand side is induced by the passage from algebraic to topological com* *plex vector bundles. The latter is a ring homomorphism, and in particular preserves the uni* *t, which is the image of the identity map under the horizontal isomorphisms. By natural* *ity, it follows that the diagram ~= alg Hom Ho(C)(A, Z x Gr)___________//K0 (Z x Gr) (51) | | | | fflffl| ~ fflffl| Hom Ho(Topo)RC(A), RC(Z x Gr) __=_//_Ktop0RC(Z x Gr) commutes for every pointed motivic space A over C. In particular, the structur* *e map of BGL which corresponds to (o1 - 1) ([O(-1)] - [O]) maps to the structure m* *ap of the complex K-theory spectrum, since it corresponds to the same bundle, viewed * *as a topological bundle. 41 Proposition A.7.4. A morphism f :S ! S0 of base schemes induces a strict symmet* *ric monoidal left Quillen functor f* :MSS (S0) ! MSS (S) such that f*(E) n = f*(En). Proof.The structure maps of f*(E) are defined via the canonical map f*(oeEn) * * f*(E)n ^ P1S0~=f*(En) ^ f*(P1S) ~=f*(En ^ P1S)_//_f (En+1) = f (E)n+1. It follows that f* has the functor f* as right adjoint, where f*(E))n = f*(En)* * and f*oeEn oef*En= f*En ^ P1S0_//_f*En ^ f*f*P1S0~=f*En ^ f*P1S//_f*(En ^ P1S)_//_f*En+1. Theorem A.3.9 and Dugger's lemma imply that f* preserves cofibrations and f* pr* *eserves fibrations. Since f* :Mo(S0) ! Mo(S) is strict symmetric monoidal and preserve* *s all colimits, then so is f* :MSS (S0) ! MSS (S) by the definition of the smash pro* *duct (45). __ |__| In particular, any complex point f :Spec(C) ! S of a base scheme S induces a* * strict symmetric monoidal left Quillen functor MSS (S) ! MSS (C) ! Sp (Top , CP1) (52) to the category of topological CP1-spectra. B Products in K-theory One of the aim of the Appendix is to prove Proposition 1.2.2. For that we need * *Proposi- tion B.1.1 and Corollary B.1.2. B.1 Cellular schemes Suppose that S is a regular base scheme. Recall that an S-cellular scheme is an* * S-scheme X equipped with a filtration X0 X1 . . .Xn = X by closed subsets such that * *for every integer i 0 the S-scheme XirXi-1is a disjoint union of several copies o* *f the affine space AiS. We do not assume that X is connected. A pointed S-cellular scheme * *is an S-cellular scheme equipped with a closed S-point x: S ,! X such that s(X) is co* *ntained in one of the open cells (a cell which is an open subscheme of X). The example* *s we are interested in are Grassmannians, projective lines and their products. We n* *eed the following proposition which will be proved below. Its proof uses only cup-prod* *ucts of the form Ki K0 ! Ki. The latter are easily defined without knowing the Waldhau* *sen products since the tensor product with a vector bundle is an exact functor. 42 Proposition B.1.1. Let (X, x) and (Y, y) be pointed smooth S-schemes. Then the* * se- quence 0 ! KTT0(X ^ Y ) ! KTT0(X x Y ) ! KTT0(X _ Y ) ! 0 is short exact. * * __ Proof.This follows from the isomorphism KTT0~=BGL 0,0and Example A.5.8. * * |__| Corollary B.1.2. Under the assumptions of Proposition B.1.1 let a 2 KTT0(X) and* * b 2 KTT0(Y ) be such that x*(a) = 0 = y*(b) in KTT0(S). Then the element a b 2 KTT0* *(X xY ) belongs to the subgroup KTT0(X ^ Y ). Furthermore the map KTT0(X=x(S)) KTT0(S)KTT0(Y=y(S)) ! KTT0(X ^ Y ) is an isomorphism. Proof.In fact, clearly the pull-back map KTT0(X _ Y ) ! KTT0(X) KTT0(Y ) is i* *njective. Since a b vanishes on x(S) x Y and on X x x(S) it follows that a b vanishes* * on X _ Y . Whence a b 2 KTT0(X ^ Y ). We skip a proof of the last assertion of the Corol* *lary since * * __ we do not need it. * *|__| With the Corollary B.1.2 in hand we are ready to prove Proposition 1.2.2. Fo* *r that we use below the notation of Section 1.2. Consider the diagram KTT(X) ^ KTT(X) = 1s(W1(X)) ^ 1s(W1(X)) m-! 2s(W2(X)) -adKTT(X) (53) with the Waldhausen morphisms m and the adjunction weak equivalence ad. The dia* *gram defines a morphism ~W TT KTT ^ KTT --! K (54) in Hcmo(S) which is the Waldhausen multiplication on KTT. We claim that the morphism ~TT given by (5) coincides with the Waldhausen mu* *lti- plication ~W on KTT. In fact, the Waldhausen multiplication ~W has the proper* *ty that for any smooth X over S the pairing KTT0(X) x KTT0(X) ! KTT0(X) coincides with the one given by the tensor product of vector bundles. Taking X = Z x Gr and u* *sing Corollary B.1.2 we see that the composite morphism ~W TT (Z x Gr) ^ (Z x Gr) w^w--!KTT ^ KTT --! K represents the class (o1 -1) (o1 -1) 2 KTT0((ZxGr )^(ZxGr )). However there exi* *sts a unique class in [KTT ^ KTT, KTT]Ho(S)with this property since w ^ w is an iso* *morphism in Hcmo(S). Whence ~W = ~TT. Similarly the known unit S0 ! KTT on the Thomason- Trobaugh K-theory coincides in Hcmo(S) with eTT defined above. Whence the mono* *idal structure (7) on the space KTT coincides with the Waldhausen one. Proposition 1* *.2.2 is proved. The following lemma is useful also. 43 Lemma B.1.3. Let (X, x) and (Y, y) be pointed smooth S-cellular schemes. Then * *the map KTTi(S) KTT0(S)KTT0(X ^ Y ) ! KTTi(X ^ Y ) is an isomorphism and KTT0(X ^ Y ) is a projective KTT0(S)-module. Proof.Consider the commutative diagram fi Ki(X ^OYO)______ff____//_Ki(XOxOY_)__________//_Ki(XO_OY ) ffl|| |ae| |`| | fl | ffi | Ki(S) K0(X ^ Y )____//_Ki(S) K0(X x Y_)__//_Ki(S) K0(X _ Y ) in which Kiis written for KTTiand the tensor product is taken over K0(S). The sequence fi 0 ! Ki(X ^ Y ) ff-!Ki(X x Y ) -!Ki(X _ Y ) ! 0 is short exact as was checked in the proof of Proposition B.1.1. In particular * *it is short exact for i = 0. Now the sequence fl ffi 0 ! Ki(S) K0(X ^ Y ) -!Ki(S) K0(X x Y ) -!Ki(S) K0(X _ Y ) ! 0 is short exact since K0(X _ Y ) is a projective K0(S)-module. The arrows ae and ` are isomorphisms by Lemmas B.1.4 and B.1.6 below respect* *ively. Whence ffl is an isomorphism as well. Finally K0(X ^ Y ) is a projective K0(S)* *-module since the sequence 0 ! K0(X ^ Y ) ! K0(X x Y ) ! K0(X _ Y ) ! 0 is short exact * *and * * __ K0(X x Y ) and K0(X _ Y ) are projective K0(S)-modules. * * |__| Lemma B.1.4. Let X be a smooth S-cellular scheme. Then the map KTTr(S) KTT0(S)KTT0(X) ! KTTr(X) is an isomorphism and KTT0(X) is a free KTT0(S)-module of rank equal to the num* *ber of cells. Proof.The Lemma easily follows from a slightly different claim. Claim B.1.5. Under the assumption of the Lemma the map of Quillen's K-groups Kr(S) K0(S)K00(Xj) ! K0r(Xj) is an isomorphism and K00(Xj) is a free K0(S)-module of the expected rank. We prove the Claim induction by j. If j = 0, then X = X0 is a disjoint union* * of several copies of the base scheme S. So the Claim is obvious in this case. Assume the* * Claim holds for all j d and prove it for j = d + 1. For that consider the subset Xd* *+1- Xd. It is a disjoint union of of several copies Uiof the affine space Ad+1S. Each of t* *he copy is an 44 open subset of Xd+1. Thus X1d:= Xd [ U1 is a closed subset of Xd+1. Consider th* *e long localization sequence of Quillen's K0-groups [Qu ] j* 0 . .!.K0r(Xd) i*-!K0r(X1d) -! Kr(U1) ! . . . where i* is the push-forward map and j* is the pull-back map. Let p : X1 ! S be* * the structure map. Then j* O p* is an isomorphism since S is regular. Thus j* is su* *rjective. So i* is injective and the sequence splits in short exact sequences (split ones) j* 0 0 ! K0r(Xd) i*-!K0r(X1d) -! Kr(U1) ! 0. Taking r = 0 and using the inductive assumption we see that K00(X1d) is a free * *K0(S)- module of the expected rank. Taking once again r = 0 and tensoring this sequenc* *e with Kr(S) over K0(S) we get an exact sequence of the form 0 ! Kr(S) K0(S)K00(Xd) -!Kr(S) K0(S)K00(X1d) -!Kr(S) K0(S)K00(U1) ! 0. The pairings of the form Kr(S) K0(S)K00(V ) ! K0r(V ) define a morphism of the * *last short exact sequence to the previous one. Using the inductive assumption we see that * *the pairing Kr(S) K0(S)K00(X1d) ! K0r(X1d) is an isomorphism. Repeating this procedure se* *veral times we get that the pairing Kr(S) K0(S)K00(Xd+1) ! K0r(Xd+1) is an isomorphi* *sm. * * __ The induction runs, whence the Claim and the Lemma. * *|__| Lemma B.1.6. Let (X, x) and (Y, y) be pointed smooth S-cellular schemes. Then * *the map KTTi(S) KTT0(S)KTT0(X _ Y ) ! KTTi(X _ Y ) is an isomorphism and KTT0(X _ Y ) is a projective KTT0(S)-module. 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