Birational Motivic Homotopy Theories and the Slice Filtration
aa r X i v : . [ m a t h . K T ] N ov BIRATIONAL MOTIVIC HOMOTOPY THEORIES AND THESLICE FILTRATION
PABLO PELAEZ
Abstract.
We show that there is an equivalence of categories between theorthogonal components for the slice filtration and the birational motivic stablehomotopy categories which are constructed in this paper. Relying on thisequivalence, we are able to describe the slices for projective spaces (including P ∞ ), Thom spaces and blow ups. Definitions and Notation
Our main result, theorem 3.6, shows that there is an equivalence of categoriesbetween the orthogonal components for the slice filtration (see definition 1.1) andthe weakly birational motivic stable homotopy categories which are constructedin this paper (see definition 2.9). Relying on this equivalence; we are able to de-scribe over an arbitrary base scheme (see theorems 4.2, 4.4 and 4.6) the slices forprojective spaces (including P ∞ ), Thom spaces and blow ups. We also constructthe birational motivic stable homotopy categories (see definition 2.4), which are anatural generalization of the weakly birational motivic stable homotopy categories,and show (see proposition 2.12) that there exists a Quillen equivalence betweenthem when the base scheme is a perfect field. Our approach was inspired by thework of Kahn-Sujatha [1] on birational motives, where the existence of a connectionbetween the layers of the slice filtration and birational invariants is explicitly sug-gested. Furthermore, this approach allows to obtain analogues for the slice filtrationin the unstable setting (see remark 3.8).In this paper X will denote a Noetherian separated base scheme of finite Krulldimension, Sch X the category of schemes of finite type over X and Sm X the fullsubcategory of Sch X consisting of smooth schemes over X regarded as a site withthe Nisnevich topology. All the maps between schemes will be considered over thebase X . Given Y ∈ Sch X , all the closed subsets Z of Y will be considered as closedsubschemes with the reduced structure.Let M be the category of pointed simplicial presheaves in Sm X equipped withthe motivic Quillen model structure [14] constructed by Morel-Voevodsky [8, p. 86Thm. 3.2], taking the affine line A X as interval. Given a map f : Y → W in Sm X , we will abuse notation and denote by f the induced map f : Y + → W + in M between the corresponding pointed simplicial presheaves represented by Y and W respectively.We define T in M to be the pointed simplicial presheaf represented by S ∧ G m ,where G m is the multiplicative group A X − { } pointed by 1, and S denotes the Mathematics Subject Classification.
Primary 14F42.
Key words and phrases.
Birational Invariants, Motivic Homotopy Theory, Motivic SpectralSequence, Slice Filtration. simplicial circle. Given an arbitrary integer r ≥ S r (respectively G rm ) will denotethe iterated smash product S ∧ · · · ∧ S (respectively G m ∧ · · · ∧ G m ) with r -factors; S = G m will be by definition equal to the pointed simplicial presheaf X + represented by the base scheme X .Let Spt ( M ) denote Jardine’s category of symmetric T -spectra on M equippedwith the motivic model structure defined in [6, Thm. 4.15] and let SH denote itshomotopy category, which is triangulated. We will follow Jardine’s notation [6, p.506-507] where F n denotes the left adjoint to the n -evaluation functor Spt ( M ) ev n / / M ( X m ) m ≥ ✤ / / X n Notice that F ( A ) is just the usual infinite suspension spectrum Σ ∞ T A .For every integer q ∈ Z , we consider the following family of symmetric T -spectra C q eff = { F n ( S r ∧ G sm ∧ U + ) | n, r, s ≥ s − n ≥ q ; U ∈ Sm X } (1.1)where U + denotes the simplicial presheaf represented by U with a disjoint basepoint. Let Σ qT SH eff denote the smallest full triangulated subcategory of SH whichcontains C q eff and is closed under arbitrary coproducts. Voevodsky [16] defines theslice filtration in SH to be the following family of triangulated subcategories · · · ⊆ Σ q +1 T SH eff ⊆ Σ qT SH eff ⊆ Σ q − T SH eff ⊆ · · · It follows from the work of Neeman [9], [10] that the inclusion i q : Σ qT SH eff → SH has a right adjoint r q : SH → Σ qT SH eff , and that the following functors f q : SH → SH s Let E ∈ Spt ( M ) be a symmetric T -spectrum. We will say that E is n -orthogonal , if for all K ∈ Σ nT SH eff Hom SH ( K, E ) = 0 Let SH ⊥ ( n ) denote the full subcategory of SH consisting of the n -orthogonal objects. IRATIONAL MOTIVIC HOMOTOPY THEORIES AND THE SLICE FILTRATION 3 The slice filtration admits an alternative definition in terms of (left and right)Bousfield localization of Spt ( M ) [11, 12]. The Bousfield localizations are con-structed following Hirschhorn’s approach [2]. In order to be able to apply Hirschhorn’stechniques, it is necessary to know that Spt ( M ) is cellular [2, Def. 12.1.1] and proper [2, Def. 13.1.1]. Theorem 1.2. The Quillen model category Spt ( M ) is:(1) cellular (see [5] , [3, Cor. 1.6] or [12, Thm. 2.7.4] ).(2) proper (see [6, Thm. 4.15] ). For details and definitions about Bousfield localization we refer the reader toHirschhorn’s book [2]. Let us just mention the following theorem of Hirschhorn,which guarantees the existence of left and right Bousfield localizations. Theorem 1.3 (see [2, Thms. 4.1.1 and 5.1.1]) . Let A be a Quillen model categorywhich is cellular and proper. Let L be a set of maps in A and let K be a set ofobjects in A . Then:(1) The left Bousfield localization of A with respect to L exists.(2) The right Bousfield localization of A with respect to the class of K -colocalequivalences exists. Now, we can describe the slice filtration in terms of suitable Bousfield localiza-tions of Spt ( M ). Theorem 1.4 (see [12]) . (1) Let R C q eff Spt ( M ) be the right Bousfield localiza-tion of Spt ( M ) with respect to the set of objects C q eff (see Eqn. (1.1) ).Then its homotopy category R C q eff SH is triangulated and naturally equiva-lent to Σ qT SH eff . Moreover, the functor f q is canonically isomorphic to thefollowing composition of triangulated functors: SH R / / R C q eff SH C q / / SH where R is a fibrant replacement functor in Spt ( M ) , and C q a cofibrantreplacement functor in R C q eff Spt ( M ) .(2) Let L Then its homotopy category S q SH is triangulated and the identity functor id : R C q eff Spt ( M ) → S q Spt ( M ) is a left Quillen functor. Moreover, the functor s q is canonically isomorphicto the following composition of triangulated functors: SH R / / R C q eff SH C q / / S q SH W q +1 / / R C q eff SH C q / / SH Proof. (1) and (3) follow directly from [12, Thms. 3.3.9, 3.3.25, 3.3.50, 3.3.68]. Onthe other hand, (2) follows from proposition 3.2.27(3) together with theorem 3.3.26;proposition 3.3.30 and theorem 3.3.45 in [12] (cid:3) Birational and Weakly Birational Cohomology Theories In this section, we construct the birational and weakly birational motivic stablehomotopy categories. These are defined as left Bousfield localizations of Spt ( M )with respect to maps which are induced by open immersions with a numerical condi-tion in the codimension of the closed complement (which is assumed to be smoothin the weakly birational case). The existence of the left Bousfield localizationsconsidered in this section follows immediately from theorems 1.2 and 1.3. Lemma 2.1. Let a, a ′ , b, b ′ , p, p ′ ≥ be integers such that a − p = a ′ − p ′ and b − p = b ′ − p ′ . Assume that p ≥ p ′ , then for every Y ∈ Sm X , there is a weakequivalence in Spt ( M ) , which is natural with respect to Yg a,bp,p ′ ( Y ) : F p ( S a ∧ G bm ∧ Y + ) → F p ′ ( S a ′ ∧ G b ′ m ∧ Y + ) Proof. We have the following adjunction (see [12, Def. 2.6.8])( F p , ev p , ϕ ) : M → Spt ( M )Using this adjunction, we define g a,bp,p ′ ( Y ) as adjoint to the identity map: S a ∧ G bm ∧ Y + id −→ ev p ( F p ′ ( S a ′ ∧ G b ′ m ∧ Y + )) ∼ = S p − p ′ ∧ G p − p ′ m ∧ S a ′ ∧ G b ′ m ∧ Y + ∼ = S a ∧ G bm ∧ Y + Thus, it is clear that g a,bp,p ′ ( Y ) is natural in Y , and it follows from [12, Prop. 2.4.26]that it is a weak equivalence in Spt ( M ). (cid:3) Definition 2.2 (see [13, section 7.5]) . Let Y ∈ Sch X , and Z a closed subscheme of Y . The codimension of Z in Y , codim Y Z is the infimum (over the generic points z i of Z ) of the dimensions of the local rings O Y,z i . Since X is Noetherian of finite Krull dimension and Y is of finite type over X , codim Y Z is always finite. Definition 2.3. We fix an arbitrary integer n ≥ , and consider the following setof open immersions which have a closed complement of codimension at least n + 1 B n = { ι U,Y : U → Y open immersion | Y ∈ Sm X ; Y irreducible ; ( codim Y Y \ U ) ≥ n + 1 } The letter B stands for birational. IRATIONAL MOTIVIC HOMOTOPY THEORIES AND THE SLICE FILTRATION 5 Now we consider the left Bousfield localization of Spt ( M ) with respect to asuitable set of maps induced by the families of open immersions B n describedabove. Definition 2.4. Let n ∈ Z be an arbitrary integer.(1) Let Spt ( B n M ) denote the left Bousfield localization of Spt ( M ) with respectto the set of maps sB n = { F p ( G bm ∧ ι U,Y ) : b, p, r ≥ , b − p ≥ n − r ; ι U,Y ∈ B r } . (2) Let b ( n ) denote its fibrant replacement functor and SH ( B n ) its associatedhomotopy category.For n = 0 we will call SH ( B n ) the codimension n + 1-birational motivic stablehomotopy category, and for n = 0 we will call it the birational motivic stablehomotopy category . Lemma 2.5. Let n ∈ Z be an arbitrary integer. Then for every a ≥ , the maps S a ∧ sB n = { F p ( S a ∧ G bm ∧ ι U,Y ) : b, p, r ≥ , b − p ≥ n − r ; ι U,Y ∈ B r } are weak equivalences in Spt ( B n M ) .Proof. Let F p ( G bm ∧ ι U,Y ) ∈ sB n with ι U,Y ∈ B r . Both F p ( G bm ∧ U + ) and F p ( G bm ∧ Y + ) are cofibrant in Spt ( M ) (see [12, Props. 2.4.17, 2.6.18 and Thm. 2.6.30]) andhence also in Spt ( B n M ). By construction, F p ( G bm ∧ ι U,Y ) is a weak equivalence in Spt ( B n M ); and [2, Thm. 4.1.1.(4)] implies that Spt ( B n M ) is a simplicial modelcategory. Thus, it follows from Ken Brown’s lemma (see [4, lemma 1.1.12]) that F p ( S a ∧ G bm ∧ ι U,Y ) is also a weak equivalence in Spt ( B n M ) for every a ≥ (cid:3) Proposition 2.6. Let E be an arbitrary symmetric T -spectrum. Then E is fibrantin Spt ( B n M ) if and only if the following conditions hold:(1) E is fibrant in Spt ( M ) .(2) For every a, b, p, r ≥ such that b − p ≥ n − r ; and every ι U,Y ∈ B r , theinduced map Hom SH ( F p ( S a ∧ G bm ∧ Y + ) , E ) ∼ = ι ∗ U,Y / / Hom SH ( F p ( S a ∧ G bm ∧ U + ) , E ) is an isomorphism.Proof. ( ⇒ ): Since the identity functor id : Spt ( M ) → Spt ( B n M )is a left Quillen functor, the conclusion follows from the derived adjunction( Q, b ( n ) , ϕ ) : SH → SH ( B n )together with lemma 2.5.( ⇐ ): Assume that E satisfies (1) and (2). Let ω , η denote the base pointsof the pointed simplicial sets Map ∗ ( F p ( G bm ∧ Y + ) , E ) and Map ∗ ( F p ( G bm ∧ U + ) , E )respectively. Since F p ( G bm ∧ Y + ) and F p ( G bm ∧ U + ) are always cofibrant, by [2, Def.3.1.4(1)(a) and Thm. 4.1.1(2)] it is enough to show that every map in sB n inducesa weak equivalence of simplicial sets:Map ∗ ( F p ( G bm ∧ Y + ) , E ) ι ∗ U,Y / / Map ∗ ( F p ( G bm ∧ U + ) , E ) PABLO PELAEZ Since Spt ( M ) is a pointed simplicial model category, we observe that lemma 6.1.2in [4] and remark 2.4.3(2) in [12] imply that the following diagram is commutativefor a ≥ π a,ω Map ∗ ( F p ( G bm ∧ Y + ) , E ) ι ∗ U,Y + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ ∼ = (cid:15) (cid:15) π a,η Map ∗ ( F p ( G bm ∧ U + ) , E ) ∼ = (cid:15) (cid:15) Hom SH ( F p ( S a ∧ G bm ∧ Y + ) , E ) ι ∗ U,Y + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ Hom SH ( F p ( S a ∧ G bm ∧ U + ) , E )by hypothesis, the bottom row is an isomorphism, hence the top row is also anisomorphism. This implies that for every map in sB n , the induced mapMap ∗ ( F p ( G bm ∧ Y + ) , E ) ι ∗ U,Y / / Map ∗ ( F p ( G bm ∧ U + ) , E )is a weak equivalence when it is restricted to the path component of Map ∗ ( F p ( G bm ∧ Y + ) , E ) containing ω . This holds in particular forMap ∗ ( F p +1 ( G b +1 m ∧ Y + ) , E ) ι ∗ U,Y / / Map ∗ ( F p +1 ( G b +1 m ∧ U + ) , E )Therefore, the following map is a weak equivalence of pointed simplicial sets, sincetaking S -loops kills the path components that do not contain the base pointMap ∗ ( S , Map ∗ ( F p +1 ( G b +1 m ∧ Y + ) , E )) (cid:15) (cid:15) Map ∗ ( S , Map ∗ ( F p +1 ( G b +1 m ∧ U + ) , E ))Now, since Spt ( M ) is a simplicial model category we deduce that the rows inthe following commutative diagram are isomorphismsMap ∗ ( S , Map ∗ ( F p +1 ( G b +1 m ∧ Y + ) , E )) ∼ = + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ ι ∗ U,Y (cid:15) (cid:15) Map ∗ ( F p +1 ( S ∧ G b +1 m ∧ Y + ) , E ) ι ∗ U,Y (cid:15) (cid:15) Map ∗ ( S , Map ∗ ( F p +1 ( G b +1 m ∧ U + ) , E )) ∼ = + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ Map ∗ ( F p +1 ( S ∧ G b +1 m ∧ U + ) , E ) IRATIONAL MOTIVIC HOMOTOPY THEORIES AND THE SLICE FILTRATION 7 Thus, by the three out of two property for weak equivalences, we conclude thatMap ∗ ( F p +1 ( S ∧ G b +1 m ∧ Y + ) , E ) ι ∗ U,Y / / Map ∗ ( F p +1 ( S ∧ G b +1 m ∧ U + ) , E )is also a weak equivalence of pointed simplicial sets. Finally, lemma 2.1 implies thatthe following diagram is commutative and the vertical arrows are weak equivalencesin Spt ( M )Map ∗ ( F p +1 ( S ∧ G b +1 m ∧ Y + ) , E ) ι ∗ U,Y / / Map ∗ ( F p +1 ( S ∧ G b +1 m ∧ U + ) , E )Map ∗ ( F p ( G bm ∧ Y + ) , E ) ι ∗ U,Y / / g ,b +1 p +1 ,p ( Y ) ∗ O O Map ∗ ( F p ( G bm ∧ U + ) , E ) g ,b +1 p +1 ,p ( U ) ∗ O O Thus, we conclude by the two out of three property for weak equivalences that thebottom arrow is also a weak equivalence in Spt ( M ). (cid:3) Proposition 2.7. The homotopy category SH ( B n ) is a compactly generated trian-gulated category in the sense of Neeman [9, Def. 1.7] .Proof. We will prove first that SH ( B n ) is a triangulated category. For this, it isenough to show that the smash product with the simplicial circle induces a Quillenequivalence (see [14, sections I.2, I.3])( − ∧ S , Ω S − , ϕ ) : Spt ( B n M ) → Spt ( B n M )It follows from [2, Thm. 4.1.1.(4)] that this adjunction is a Quillen adjunction,and the same argument as in [12, Cor. 3.2.38] (replacing [12, Prop. 3.2.32] withproposition 2.6) allows us to conclude that it is a Quillen equivalence.Finally, since SH is a compactly generated triangulated category (see [12, Prop.3.1.5]) and the identity functor is a left Quillen functor id : Spt ( M ) → Spt ( B n M )it follows from the derived adjunction( Q, b ( n ) , ϕ ) : SH → SH ( B n )that SH ( B n ) is also compactly generated, having exactly the same set of generatorsas SH . (cid:3) Definition 2.8. We fix an arbitrary integer n ≥ , and consider the following setof open immersions with smooth closed complement of codimension at least n + 1 W B n = { ι U,Y : U → Y open immersion | Y, Z = Y \ U ∈ Sm X ; Y irreducible ; ( codim Y Z ) ≥ n + 1 } Notice that every map in W B n is also in B n , but the converse doesn’t hold.The reason to consider maps ι U,Y in W B n is that if the closed complement issmooth, then the Morel-Voevodsky homotopy purity theorem (see [8, Thm. 2.23])characterizes the homotopy cofibre of ι U,Y in terms of the Thom space of the normalbundle for the closed immersion Y \ U → Y . Definition 2.9. Let n ∈ Z be an arbitrary integer. PABLO PELAEZ (1) Let Spt ( W B n M ) denote the left Bousfield localization of Spt ( M ) with re-spect to the set of maps sW B n = { F p ( G bm ∧ ι U,Y ) : b, p, r ≥ , b − p ≥ n − r ; ι U,Y ∈ W B r } . (2) Let wb ( n ) denote its fibrant replacement functor and SH ( W B n ) its associ-ated homotopy category.For n = 0 we will call SH ( W B n ) the codimension n + 1-weakly birational motivicstable homotopy category , and for n = 0 we will call it the weakly birationalmotivic stable homotopy category . Proposition 2.10. Let E be an arbitrary symmetric T -spectrum. Then E is fibrantin Spt ( W B n M ) if and only if the following conditions hold:(1) E is fibrant in Spt ( M ) .(2) For every a, b, p, r ≥ such that b − p ≥ n − r ; and every ι U,Y ∈ W B r , theinduced map Hom SH ( F p ( S a ∧ G bm ∧ Y + ) , E ) ∼ = ι ∗ U,Y / / Hom SH ( F p ( S a ∧ G bm ∧ U + ) , E ) is an isomorphism.Proof. The proof is exactly the same as in proposition 2.6. (cid:3) Proposition 2.11. The homotopy category SH ( W B n ) is a compactly generatedtriangulated category in the sense of Neeman.Proof. The proof is exactly the same as in proposition 2.7. (cid:3) Proposition 2.12. Assume that the base scheme X = Spec k , with k a perfectfield, then the Quillen adjunction: ( id, id, ϕ ) : Spt ( W B n M ) → Spt ( B n M ) is a Quillen equivalence.Proof. Consider the following commutative diagram Spt ( M ) id w w ♣♣♣♣♣♣♣♣♣♣♣ id & & ▼▼▼▼▼▼▼▼▼▼ Spt ( W B n M ) id / / ❴❴❴❴❴❴❴❴❴ Spt ( B n M )where the solid arrows are left Quillen functors. Clearly, W B r ⊆ B r for every r ≥ sW B n ⊆ sB n , and we conclude that every sW B n -local equivalence is a sB n -local equivalence. Therefore, the universal property of left Bousfield localizationsimplies that the horizontal arrow is also a left Quillen functor.The universal property for left Bousfield localizations also implies that it isenough to show that all the maps in sB n = { F p ( G bm ∧ ι U,Y ) : b, p, r ≥ , b − p ≥ n − r ; ι U,Y ∈ B r } become weak equivalences in Spt ( W B n M ). Given F p ( G bm ∧ ι U,Y ) ∈ sB n with ι U,Y ∈ B r , we proceed by induction on the dimension of Z = Y \ U . If dim Z = 0,then Z ∈ Sm X since k is a perfect field (and we are considering Z with the reducedscheme structure), hence F p ( G bm ∧ ι U,Y ) ∈ sW B n and then a weak equivalence in Spt ( W B n M ). IRATIONAL MOTIVIC HOMOTOPY THEORIES AND THE SLICE FILTRATION 9 If dim Z > 0, then we consider the singular locus Z s of Z over X . We havethat dim Z s < dim Z since k is a perfect field. Therefore, by induction on thedimension F p ( G bm ∧ ι V,Y ) is a weak equivalence in Spt ( W B n M ), where V = Y \ Z s .On the other hand, F p ( G bm ∧ ι U,V ) is also a weak equivalence in Spt ( W B n M ) since ι U,V is also in B r and its closed complement V \ U = Z \ Z s is smooth over X , byconstruction of Z s .But F p ( G bm ∧ ι U,Y ) = F p ( G bm ∧ ι V,Y ) ◦ F p ( G bm ∧ ι U,V ), so by the two out ofthree property for weak equivalences we conclude that F p ( G bm ∧ ι U,Y ) is a weakequivalence in Spt ( W B n M ). (cid:3) A Characterization of the Slices This section contains our main results. We give a characterization of the slicesin terms of effectivity and birational conditions (in the sense of definition 3.1), andwe also show that there is an equivalence between the notion of orthogonality (seedefinition 1.1) and weak birationality (see definition 3.1). Definition 3.1. Let E ∈ Spt ( M ) be a symmetric T -spectrum and n ∈ Z .(1) We will say that E is n +1-birational (respectively weakly n +1-birational ),if E is fibrant in Spt ( B n M ) (respectively Spt ( W B n M ) ). If n = 0 , we willsimply say that E is birational (respectively weakly birational ).(2) We will say that E is an n -slice if E is isomorphic in SH to s n ( E ′ ) forsome symmetric T -spectrum E ′ . Definition 3.2. (1) Let ι U,Y be an open immersion in Sm X . Let Y /U denotethe pushout of the following diagram in M (i.e. the homotopy cofibre of ι U,Y in M ) U + ι U,Y / / (cid:15) (cid:15) Y + (cid:15) (cid:15) X / / Y /U (2) Given a vector bundle π : V → Y with Y ∈ Sm X , let T h ( V ) denote theThom space of V , i.e. V / ( V \ σ ( Y )) , where σ : Y → V denotes the zerosection of V . Lemma 3.3. Let ι U,Y ∈ W B r for some r ≥ , and let a, b, p ≥ be arbitraryintegers such that b − p ≥ n − r . Then F p ( S a ∧ G bm ∧ Y /U ) ∈ Σ n +1 T SH eff Proof. Since Σ n +1 T SH eff is a triangulated category, it is enough to consider the case a = 0. It is also clear that it suffices to show that F ( Y /U ) ∈ Σ r +1 T SH eff .Now, it follows from the Morel-Voevodsky homotopy purity theorem (see [8,Thm. 2.23]) that there is an isomorphism in SH F ( Y /U ) → F ( T h ( N ))where Th(N) is the Thom space of the normal bundle N of the (smooth) comple-ment Z of U in Y : e : Y \ U = Z → Y But, ι U,Y ∈ W B r ; so e is a regular embedding of codimension c at least r + 1, hence N is a vector bundle of rank at least r + 1. Therefore, if N is a trivial vector bundlewe conclude from [8, Prop. 2.17(2)] that F ( T h ( N )) ∼ = F ( S c ∧ G cm ∧ Z + ) ∈ Σ cT SH eff ⊆ Σ r +1 T SH eff Finally, we conclude in the general case by choosing a Zariski cover of Z whichtrivializes N and using the Mayer-Vietoris property for Zariski covers. (cid:3) Lemma 3.4. Let U ∈ Sm X . Consider the open immersion in Sm X m U : A U \ U → A U given by the complement of the zero section. Then m U ∈ W B , and there exists aweak equivalence in Spt ( M ) between its homotopy cofibre in M , A U / ( A U \ U ) and S ∧ G m ∧ U + t U : A U / ( A U \ U ) → S ∧ G m ∧ U + Proof. Since the zero section i : U → A U is a closed embedding of codimension1 between smooth schemes over X , it follows from the definition of W B that m U ∈ W B . Finally, [8, Prop. 2.17(2)] implies the existence of the weak equivalence t U . (cid:3) Proposition 3.5. Let E ∈ Spt ( M ) be a symmetric T -spectrum and n ∈ Z . Con-sider the following conditions:(1) E is fibrant in L Spt ( M )is a left Quillen functor, therefore the following F p ( S a ∧ G bm ∧ U + ) F p ( S a ∧ G bm ∧ ι U,Y ) / / F p ( S a ∧ G bm ∧ Y + ) / / F p ( S a ∧ G bm ∧ Y /U )is a cofibre sequence in Spt ( M ). However, SH is a triangulated category and lemma2.1 implies that F p +1 ( S a ∧ G b +1 m ∧ Y /U ) ∼ = Ω S ◦ R ◦ F p ( S a ∧ G bm ∧ Y /U )are isomorphic in SH , where R denotes a fibrant replacement functor in Spt ( M ).Hence it suffices to show thatHom SH ( F p +1 ( S a ∧ G b +1 m ∧ Y /U ) , E ) = Hom SH ( F p ( S a ∧ G bm ∧ Y /U ) , E ) = 0But this follows from lemma 3.3 together with [12, Prop. 3.3.30], since we areassuming that E is fibrant in L 1, and 0 − p + ( b − ≥ n (i.e. b − p ≥ n + 1); then F p ( G b − m ∧ m U ) ∈ sW B n , i.e. a weak equivalence in Spt ( W B n M ).Since SH ( W B n ) is a triangulated category, id : Spt ( M ) → Spt ( W B n M ) isa left Quillen functor, and F p ( G b − m ∧ ( A U / ( A U \ U + ))) is the homotopy cofibre of F p ( G b − m ∧ m U ); we deduce that E being n + 1-weakly birational implies thatHom SH ( F p ( G b − m ∧ ( A U / ( A U \ U + ))) , E ) = 0Finally, it follows from lemma 3.4 that the following groups are isomorphic0 = Hom SH ( F p ( G b − m ∧ ( A U / ( A U \ U + ))) , E ) ∼ = Hom SH ( F p ( S ∧ G bm ∧ U + ) , E )(2) ⇔ (3): This follows directly from proposition 2.12. (cid:3) Theorem 3.6. The Quillen adjunction ( id, id, ϕ ) : Spt ( W B n M ) → L Spt ( W B n M ) and L Let E be fibrant in Spt ( M ) . Then E is an n -slice (see definition3.1(2)) if and only if the following conditions hold: S1: E is n -effective, i.e. E ∈ Σ nT SH eff . S2: E is n + 1 -weakly birational. In addition, if the base scheme X = Spec k , with k a perfect field, then E is an n -slice if and only if the following conditions hold: GSS1: E is n -effective, i.e. E ∈ Σ nT SH eff . GSS2: E is n + 1 -birational.Proof. Assume that E is an n -slice. Then theorems 1.4(1) and 1.4(3) imply that E is n -effective and fibrant in L Remark 3.8. Notice that theorem 3.6 implies that it is possible to construct theslice filtration directly from the Quillen model categories Spt ( W B n M ) described indefinition 2.9 without making any reference to the effective categories Σ qT SH eff .One of the interesting consequences of this fact is that it is possible to obtain ana-logues of the slice filtration in the unstable setting, since the suspension with respectto T or S does not play an essential role in the construction of Spt ( W B n M ) , i.e.we could consider the left Bousfield localization of the motivic unstable homotopycategory M with respect to the maps in definition 2.8. We will study the details ofthis construction in a future work. Some Computations In this section we use the characterization of the slices obtained in theorem 3.7to describe the slices of projective spaces, Thom spaces and blow ups.To simplify the notation, given a simplicial presheaf K ∈ M or a map f ∈M ; let s j ( K ), s j ( f ) (respectively s Let g : E → F be a map in SH such that s Theorem 4.2. Let Y ∈ Sm X . Then for any integer j ≤ n , the diagram 4.1 inducesthe following isomorphisms in SH s j ( P n ( Y ) + ) ∼ = / / s j ( P n +1 ( Y ) + ) ∼ = / / · · · ∼ = / / s j ( P ∞ ( Y ) + ) Proof. Let k > n , and consider the closed embedding induced by the diagram (4.1) λ kn : P n ( Y ) → P k ( Y ). It is possible to choose a linear embedding P k − n − ( Y ) → P k ( Y ) such that its open complement U k,n contains P n ( Y ) and has the structureof a vector bundle over P n ( Y ), with zero section σ kn : U k,n v kn / / (cid:15) (cid:15) P k ( Y ) P k − n − ( Y ) o o P n ( Y ) σ kn O O λ kn G G By homotopy invariance s Assume that the base scheme X = Spec k , with k a perfect field.Then, in the following diagram all the symmetric T -spectra are isomorphic to H Z : H Z ∼ = / / s ( P ( k ) + ) ∼ = / / s ( P ( k ) + ) ∼ = / / · · ·· · · ∼ = / / s ( P n ( k ) + ) ∼ = / / · · · ∼ = / / s ( P ∞ ( k ) + ) Proof. This follows immediately from theorem 4.2 together with the computationof Levine [7, Thm. 10.5.1] and Voevodsky [17] for the zero slice of the spherespectrum. (cid:3) Theorem 4.4. Let ι U,Y ∈ W B n , π : V → Y a vector bundle of rank r togetherwith a trivialization t : π − ( U ) → A rU of its restriction to U . Then for every integer j ≤ n , there exists an isomorphism in SH (see definition 3.2(2)) s j ( T h ( V )) ∼ = S r ∧ G rm ∧ s j − r ( Y + ) Proof. Let Z ∈ Sm X be the closed complement of ι U,Y . Consider the followingdiagram in Sm X , where all the squares are cartesian π − ( Z ) ∩ ( V \ σ ( Y )) / / (cid:15) (cid:15) V \ σ ( Y ) (cid:15) (cid:15) π − ( U ) ∩ ( V \ σ ( Y )) β o o (cid:15) (cid:15) π − ( Z ) / / (cid:15) (cid:15) V π (cid:15) (cid:15) π − ( U ) α o o (cid:15) (cid:15) Z / / Y U ι U,Y o o and let γ : T h ( π − ( U )) → T h ( V ) be the induced map between the correspondingThom spaces. We observe that α, β also belong to W B n ; thus, if j ≤ n we concludethat F ( ι U,Y ) , F ( α ) , F ( β ) are all weak equivalences in Spt ( W B j M ). Therefore,theorems 1.4(2) and 3.6 imply that if j ≤ n + 1, then s Assume that the base scheme X = Spec k , with k a perfect field.Let ι U,Y ∈ B n , π : V → Y a vector bundle of rank r together with a trivialization t : π − ( U ) → A rU of its restriction to U . Then for every integer j ≤ n , there existsan isomorphism in SH s j ( T h ( V )) ∼ = S r ∧ G rm ∧ s j − r ( Y + ) Proof. Proposition 2.12 implies that F ( ι U,Y ) is a weak equivalence in Spt ( W B j M )for j ≤ n . Hence, the result follows using exactly the same argument as in theorem4.4. (cid:3) Given a closed embedding Z → Y of smooth schemes over X , let B ℓ Z Y denotethe blowup of Y with center in Z . Theorem 4.6. Let ι U,Y ∈ W B n , and j ∈ Z an arbitrary integer. Consider thefollowing cartesian square in Sm X (4.2) D q (cid:15) (cid:15) d / / B ℓ Z Y p (cid:15) (cid:15) U u o o Z i / / Y U ι U,Y o o and let q j , d j , p j , i j denote s j ( q ) , s j ( d ) , s j ( p ) , s j ( i ) respectively. Then the cartesiansquare (4.2) induces the following distinguished triangle in SH (4.3) s j ( D + ) ( − djqj ) / / s j ( B ℓ Z Y + ) ⊕ s j ( Z + ) ( p j ,i j ) / / s j ( Y + ) If j ≤ n , then s j ( ι U,Y ) is an isomorphism in SH , and the following distinguishedtriangles in SH split s j ( D + ) ( − djqj ) / / s j ( B ℓ Z Y + ) ⊕ s j ( Z + ) o o ( p j ,i j ) / / s j ( Y + )( rj ) o o (4.4) s j ( Y + ) r j / / s j ( B ℓ Z Y + ) p j o o / / s j ( T h ( O D (1))) o o (4.5) where r j = s j ( u ) ◦ ( s j ( ι U,Y )) − , and O D (1) denotes the canonical line bundle of theprojective bundle q : D → Z . Proof. It follows from [8, Prop. 2.29 and Rmk. 2.30] that the following square ishomotopy cocartesian in M S ∧ D + id ∧ q (cid:15) (cid:15) id ∧ d / / S ∧ B ℓ Z Y + id ∧ p (cid:15) (cid:15) S ∧ Z + id ∧ i / / S ∧ Y + Thus, we deduce that the following diagram is a distinguished triangle in SH F ( D + ) ( − F d ) F q ) ) / / F ( B ℓ Z Y + ) ⊕ F ( Z + ) ( F ( p ) ,F ( i )) / / F ( Y + )Since the slices s j are triangulated functors, it follows that diagram (4.3) is adistinguished triangle in SH .Now, we prove that s j ( ι U,Y ) is an isomorphism for j ≤ n . By lemma 4.1, itsuffices to show that s Spt ( W B j M ) for j ≤ n .Thus, r j is well defined for j ≤ n , and the following diagram shows that it givesa splitting for the distinguished triangle (4.4)(4.6) s j ( U + ) s j ( u ) / / s j ( B ℓ Z Y + ) p j (cid:15) (cid:15) s j ( U + ) s j ( ι U,Y ) / / s j ( Y + )Finally, since the normal bundle of the closed embedding d : D → B ℓ Z Y is givenby O D (1), we deduce from the Morel-Voevodsky homotopy purity theorem (see [8,Thm. 2.23]) that the following diagram is a distinguished triangle in SH s j ( U + ) s j ( u ) / / s j ( B ℓ Z Y + ) / / s j ( T h ( O D (1)))Combining this distinguished triangle with diagram (4.6) above, we conclude thatdiagram (4.5) is a split distinguished triangle in SH for j ≤ n . (cid:3) Acknowledgements The author would like to thank Marc Levine for bringing to his attention theconnection between slices and birational cohomology theories, as well as for all hisstimulating comments and questions, and also thank Chuck Weibel for his interestin this work and several suggestions which helped to improve the exposition. References Model categories and their localizations , volume 99 of Mathematical Surveysand Monographs . American Mathematical Society, Providence, RI, 2003.[3] J. Hornbostel. Localizations in motivic homotopy theory. Math. Proc. Cambridge Philos.Soc. , 140(1):95–114, 2006.[4] M. Hovey. Model categories , volume 63 of Mathematical Surveys and Monographs . AmericanMathematical Society, Providence, RI, 1999. IRATIONAL MOTIVIC HOMOTOPY THEORIES AND THE SLICE FILTRATION 17 [5] M. Hovey. Spectra and symmetric spectra in general model categories. J. Pure Appl. Algebra ,165(1):63–127, 2001.[6] J. F. Jardine. Motivic symmetric spectra. Doc. 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