Birational Motivic Homotopy Theories

and the Slice Filtration

Pablo Pelaez

Received: May 21, 2012 Revised: October 5, 2012

Communicated by Lars Hesselholt

Abstract. We show that there is an equivalence of categories between the orthogonal components for the slice filtration and the birational motivic stable homotopy categories which are constructed in this paper. Relying on this equivalence, we are able to describe the slices for projective spaces (including ℙ^{∞}), Thom spaces and blow ups.

2000 Mathematics Subject Classification: 14F42

Keywords and Phrases: Birational Invariants, Motivic Homotopy Theory, Motivic Spectral Sequence, Slice Filtration.

1 Definitions and Notation

Our main result, theorem 3.6, shows that there is an equivalence of categories between the orthogonal components for the slice filtration (see definition 1.1) and the weakly birational motivic stable homotopy categories which are constructed in this paper (see definition 2.9). Relying on this equivalence; we are able to describe over an arbitrary base scheme (see theorems 4.2, 4.4 and 4.6) the slices for projective spaces (including ℙ^{∞}), Thom spaces and blow ups. We also construct the birational motivic stable homotopy categories (see definition 2.4), which are a natural generalization of the weakly birational motivic stable homotopy categories, and show (see proposition 2.12) that there exists a Quillen equivalence between them when the base scheme is a perfect field. Our approach was inspired by the work of Kahn-Sujatha [1] on birational motives, where the existence of a connection between the layers of the slice filtration and birational invariants is explicitly suggested. Furthermore, this approach allows to obtain analogues for the slice filtration in the unstable setting (see remark 3.8). In this paper X will denote a Noetherian separated base scheme of finite Krull dimension, Sch_{X} the category of schemes of finite type over X and Sm_{X} the full subcategory of Sch_{X} consisting of smooth schemes over X regarded as a site with the Nisnevich topology. All the maps between schemes will be considered over the base X. Given Y ∈ Sch_{X}, all the closed subsets Z of Y will be considered as closed subschemes with the reduced structure. Let M be the category of pointed simplicial presheaves in Sm_{X} equipped with the motivic Quillen model structure [14] constructed by Morel-Voevodsky [8, p. 86 Thm. 3.2], taking the affine line 𝔸_{X}^{1} 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^{1}∧𝔾_{m}, where 𝔾_{m} is the multiplicative group 𝔸_{X}^{1} − {0} pointed by 1, and S^{1} denotes the simplicial circle. Given an arbitrary integer r ≥ 1, S^{r} (respectively 𝔾_{m}^{r}) will denote the iterated smash product S^{1}∧⋯∧S^{1} (respectively 𝔾_{m}∧⋯∧𝔾_{m}) with r-factors; S^{0} = 𝔾_{m}^{0} 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 equipped with the motivic model structure defined in [6, Thm. 4.15] and let SH denote its homotopy 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

\xymatrix{Spt(M) \ar[r]^{ev_{n}} & M \\ (X^{m})_{m≥0} \ar@{|->}[r] & X^{n}}

Notice that F_{0}(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_{eff}^{q} = {F_{n}(S^{r} ∧ 𝔾_{m}^{s} ∧ U_{+}) | n, r, s ≥ 0; s − n ≥ q; U ∈ Sm_{X}}  (1.1)

where U_{+} denotes the simplicial presheaf represented by U with a disjoint base point. Let Σ_{T}^{q} SH^{eff} denote the smallest full triangulated subcategory of SH which contains C_{eff}^{q} and is closed under arbitrary coproducts. Voevodsky [16] defines the slice filtration in SH to be the following family of triangulated subcategories

⋯ ⊆ Σ_{T}^{q+1}SH^{eff} ⊆ Σ_{T}^{q} SH^{eff} ⊆ Σ_{T}^{q−1}SH^{eff} ⊆ ⋯

It follows from the work of Neeman [9], [10] that the inclusion

i_{q} : Σ_{T}^{q} SH^{eff} → SH

has a right adjoint r_{q} : SH → Σ_{T}^{q} SH^{eff} , and that the following functors

f_{q} :SH → SH
s_{<q} :SH → SH
s_{q} :SH → SH

are triangulated, where f_{q} is defined as the composition i_{q} ∘ r_{q}; and s_{<q}, s_{q} are characterized by the fact that for every E ∈ SH, we have distinguished triangles in SH:

\xymatrix{f_{q}E \ar[r]^{θ_{q}^{E}} & E \ar[r]^{π_{<q}^{E}} & s_{<q}E \ar[r] & S^{1} ∧ f_{q}E \\ f_{q+1}E \ar[r]^{ρ_{q}^{E}} & f_{q}E \ar[r]^{π_{q}^{E}} & s_{q}E \ar[r] & S^{1} ∧ f_{q+1}E}

We will refer to f_{q}E as the (q − 1)-connective cover of E, to s_{<q}E as the q-orthogonal component of E, and to s_{q}E as the q-slice of E. It follows directly from the definition that s_{<q+1}E, s_{q}E satisfy that for every symmetric T -spectrum K in Σ_{T}^{q+1}SH^{eff} :

Hom_{SH}(K, s_{<q+1}E) = Hom_{SH}(K, s_{q}E) = 0

Definition 1.1. Let E ∈ Spt(M) be a symmetric T -spectrum. We will say that E is n-orthogonal, if for all K ∈ Σ_{T}^{n} SH^{eff}

Hom_{SH}(K, E) = 0

Let SH^{⊥}(n) denote the full subcategory of SH consisting of the n-orthogonal objects.

The slice filtration admits an alternative definition in terms of (left and right) Bousfield localization of Spt(M) [11, 12]. The Bousfield localizations are constructed following Hirschhorn’s approach [2]. In order to be able to apply Hirschhorn’s techniques, 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 to Hirschhorn’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 category which is cellular and proper. Let L be a set of maps in A and let K be a set of objects 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-colocal equivalences exists.

Now, we can describe the slice filtration in terms of suitable Bousfield localizations of Spt(M).

Theorem 1.4 (see [12]). 1. Let R_{C_{eff}^{q}} Spt(M) be the right Bousfield localization of Spt(M) with respect to the set of objects C_{eff}^{q} (see Eqn. (1.1)). Then its homotopy category R_{C_{eff}^{q}} SH is triangulated and naturally equiv-

alent to Σ_{T}^{q} SH^{eff} . Moreover, the functor f_{q} is canonically isomorphic to the following composition of triangulated functors:

\xymatrix{SH \ar[r]^{R} & R_{C_{eff}^{q}} SH \ar[r]^{C_{q}} & SH}

where R is a fibrant replacement functor in Spt(M), and C_{q} a cofibrant replacement functor in R_{C_{eff}^{q}} Spt(M).

2. Let L_{<q}Spt(M) be the left Bousfield localization of Spt(M) with respect to the set of maps

{F_{n}(S^{r} ∧ 𝔾_{m}^{s} ∧ U_{+}) → ∗| F_{n}(S^{r} ∧ 𝔾_{m}^{s} ∧ U_{+}) ∈ C_{eff}^{q} }

Then its homotopy category L_{<q}SH is triangulated and naturally equivalent to SH^{⊥}(q). Moreover, the functor s_{<q} is canonically isomorphic to the following composition of triangulated functors:

\xymatrix{SH \ar[r]^{Q} & L_{<q}SH \ar[r]^{W_{q}} & SH}

where Q is a cofibrant replacement functor in Spt(M), and W_{q} a fibrant replacement functor in L_{<q}Spt(M).

3. Let S^{q}Spt(M) be the right Bousfield localization of L_{<q+1}Spt(M) with respect to the set of objects

{F_{n}(S^{r} ∧ 𝔾_{m}^{s} ∧ U_{+}) | n, r, s ≥ 0; s − n = q; U ∈ Sm_{X}}

Then its homotopy category S^{q}SH is triangulated and the identity functor

id : R_{C_{eff}^{q}} Spt(M) → S^{q}Spt(M)

is a left Quillen functor. Moreover, the functor s_{q} is canonically isomorphic to the following composition of triangulated functors:

\xymatrix{SH \ar[r]^{R} & R_{C_{eff}^{q}} SH \ar[r]^{C_{q}} & S^{q}SH \ar[r]^{W_{q+1}} & R_{C_{eff}^{q}} SH \ar[r]^{C_{q}} & SH}

Proof. (1) and (3) follow directly from [12, Thms. 3.3.9, 3.3.25, 3.3.50, 3.3.68]. On the 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]

2 Birational and Weakly Birational Cohomology Theories

In this section, we construct the birational and weakly birational motivic stable homotopy categories. These are defined as left Bousfield localizations of Spt(M) with respect to maps which are induced by open immersions with a numerical condition in the codimension of the closed complement (which is assumed to be smooth in the weakly birational case). The existence of the left Bousfield localizations considered in this section follows immediately from theorems 1.2 and 1.3.

Lemma 2.1. Let a, a′, b, b′, p, p′ ≥ 0 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 weak equivalence in Spt(M), which is natural with respect to Y

g_{p,p′}^{a,b}(Y) : F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ Y_{+}) → F_{p′}(S^{a′} ∧ 𝔾_{m}^{b′} ∧ 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_{p,p′}^{a,b}(Y) as adjoint to the identity map:

S^{a} ∧ 𝔾_{m}^{b} ∧ Y_{+} ⟶^{id} ev_{p}(F_{p′}(S^{a′} ∧ 𝔾_{m}^{b′} ∧ Y_{+})) ≅ S^{p−p′} ∧ 𝔾_{m}^{p−p′} ∧ S^{a′} ∧ 𝔾_{m}^{b′} ∧ Y_{+}
≅ S^{a} ∧ 𝔾_{m}^{b} ∧ Y_{+}

Thus, it is clear that g_{p,p′}^{a,b}(Y) is natural in Y, and it follows from [12, Prop. 2.4.26] that it is a weak equivalence in Spt(M).

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 ≥ 0, and consider the following set of 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.

Now we consider the left Bousfield localization of Spt(M) with respect to a suitable set of maps induced by the families of open immersions B_{n} described above.

Definition 2.4. Let n ∈ ℤ be an arbitrary integer.

1. Let B_{n}Spt(M) denote the left Bousfield localization of Spt(M) with respect to the set of maps

sB_{n} = {F_{p}(𝔾_{m}^{b} ∧ ι_{U,Y}) : b, p, r ≥ 0, b − p ≥ n − r; ι_{U,Y} ∈ B_{r}}.

2. Let b^{(n)} denote its fibrant replacement functor and SH(B_{n}) its associated homotopy category.

For n ≠ 0 we will call SH(B_{n}) the codimension n + 1-birational motivic stable homotopy category, and for n = 0 we will call it the birational motivic stable homotopy category.

Lemma 2.5. Let n ∈ ℤ be an arbitrary integer. Then for every a ≥ 0, the maps

S^{a} ∧ sB_{n} = {F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ ι_{U,Y}) : b, p, r ≥ 0, b − p ≥ n − r; ι_{U,Y} ∈ B_{r}}

are weak equivalences in B_{n}Spt(M).

Proof. Let F_{p}(𝔾_{m}^{b} ∧ ι_{U,Y}) ∈ sB_{n} with ι_{U,Y} ∈ B_{r}. Both F_{p}(𝔾_{m}^{b} ∧ U_{+}) and F_{p}(𝔾_{m}^{b} ∧ Y_{+}) are cofibrant in Spt(M) (see [12, Props. 2.4.17, 2.6.18 and Thm. 2.6.30]) and hence also in B_{n}Spt(M). By construction, F_{p}(𝔾_{m}^{b}∧ι_{U,Y}) is a weak equivalence in B_{n}Spt(M); and [2, Thm. 4.1.1.(4)] implies that B_{n}Spt(M) is a simplicial model category. Thus, it follows from Ken Brown’s lemma (see [4, lemma 1.1.12]) that F_{p}(S^{a}∧𝔾_{m}^{b}∧ι_{U,Y}) is also a weak equivalence in B_{n}Spt(M) for every a ≥ 0.

Proposition 2.6. Let E be an arbitrary symmetric T -spectrum. Then E is fibrant in B_{n}Spt(M) if and only if the following conditions hold:

1. E is fibrant in Spt(M).

2. For every a, b, p, r ≥ 0 such that b − p ≥ n − r; and every ι_{U,Y} ∈ B_{r}, the induced map

\xymatrix{Hom_{SH}(F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ Y_{+}), E) \ar[r]_{≅}^{ι_{U,Y}^{∗}} & Hom_{SH}(F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ U_{+}), E)}

is an isomorphism.

Proof. (⇒): Since the identity functor

id : Spt(M) → B_{n}Spt(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_{0}, η_{0} denote the base points of the pointed simplicial sets Map_{∗}(F_{p}(𝔾_{m}^{b} ∧ Y_{+}), E) and Map_{∗}(F_{p}(𝔾_{m}^{b} ∧ U_{+}), E) respectively. Since F_{p}(𝔾_{m}^{b} ∧ Y_{+}) and F_{p}(𝔾_{m}^{b} ∧ 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} induces a weak equivalence of simplicial sets:

\xymatrix{Map_{∗}(F_{p}(𝔾_{m}^{b} ∧ Y_{+}), E) \ar[r]^{ι_{U,Y}^{∗}} & Map_{∗}(F_{p}(𝔾_{m}^{b} ∧ U_{+}), E)}

Since Spt(M) is a pointed simplicial model category, we observe that lemma 6.1.2 in [4] and remark 2.4.3(2) in [12] imply that the following diagram is commutative for a ≥ 0 and all the vertical arrows are isomorphisms

\xymatrix{π_{a,ω_{0}}Map_{∗}(F_{p}(𝔾_{m}^{b} ∧ Y_{+}), E) \ar@{-->}[dr]_{ι_{U,Y}^{∗}} \ar[dd]_{≅} \\  & π_{a,η_{0}}Map_{∗}(F_{p}(𝔾_{m}^{b} ∧ U_{+}), E) \ar[dd]^{≅} \\ Hom_{SH}(F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ Y_{+}), E) \ar@{-->}[dr]^{ι_{U,Y}^{∗}} \\  & Hom_{SH}(F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ U_{+}), E)}

by hypothesis, the bottom row is an isomorphism, hence the top row is also an isomorphism. This implies that for every map in sB_{n}, the induced map

\xymatrix{Map_{∗}(F_{p}(𝔾_{m}^{b} ∧ Y_{+}), E) \ar[r]^{ι_{U,Y}^{∗}} & Map_{∗}(F_{p}(𝔾_{m}^{b} ∧ U_{+}), E)}

is a weak equivalence when it is restricted to the path component of Map_{∗}(F_{p}(𝔾_{m}^{b} ∧ Y_{+}), E) containing ω_{0}. This holds in particular for

\xymatrix{Map_{∗}(F_{p+1}(𝔾_{m}^{b+1} ∧ Y_{+}), E) \ar[r]^{ι_{U,Y}^{∗}} & Map_{∗}(F_{p+1}(𝔾_{m}^{b+1} ∧ U_{+}), E)}

Therefore, the following map is a weak equivalence of pointed simplicial sets, since taking S^{1}-loops kills the path components that do not contain the base point

Map_{∗}(S^{1}, Map_{∗}(F_{p+1}(𝔾_{m}^{b+1} ∧ Y_{+}), E))

Map_{∗}(S^{1}, Map_{∗}(F_{p+1}(𝔾_{m}^{b+1} ∧ U_{+}), E))

Now, since Spt(M) is a simplicial model category we deduce that the rows in the following commutative diagram are isomorphisms

\xymatrix{Map_{∗}(S^{1}, Map_{∗}(F_{p+1}(𝔾_{m}^{b+1} ∧ Y_{+}), E)) \ar@{-->}[dr]_{≅} \ar[dd]_{ι_{U,Y}^{∗}} \\  & Map_{∗}(F_{p+1}(S^{1} ∧ 𝔾_{m}^{b+1} ∧ Y_{+}), E) \ar[dd]^{ι_{U,Y}^{∗}} \\ Map_{∗}(S^{1}, Map_{∗}(F_{p+1}(𝔾_{m}^{b+1} ∧ U_{+}), E)) \ar@{-->}[dr]_{≅} \\  & Map_{∗}(F_{p+1}(S^{1} ∧ 𝔾_{m}^{b+1} ∧ U_{+}), E)}

Thus, by the three out of two property for weak equivalences, we conclude that

\xymatrix{Map_{∗}(F_{p+1}(S^{1} ∧ 𝔾_{m}^{b+1} ∧ Y_{+}), E) \ar[r]^{ι_{U,Y}^{∗}} & Map_{∗}(F_{p+1}(S^{1} ∧ 𝔾_{m}^{b+1} ∧ U_{+}), E)}

is also a weak equivalence of pointed simplicial sets. Finally, lemma 2.1 implies that the following diagram is commutative and the vertical arrows are weak equivalences in Spt(M)

\xymatrix{Map_{∗}(F_{p+1}(S^{1} ∧ 𝔾_{m}^{b+1} ∧ Y_{+}), E) \ar[r]^{ι_{U,Y}^{∗}} & Map_{∗}(F_{p+1}(S^{1} ∧ 𝔾_{m}^{b+1} ∧ U_{+}), E) \\ Map_{∗}(F_{p}(𝔾_{m}^{b} ∧ Y_{+}), E) \ar[r]^{ι_{U,Y}^{∗}} \ar[u]^{g_{p+1,p}^{1,b+1}(Y)^{∗}} & Map_{∗}(F_{p}(𝔾_{m}^{b} ∧ U_{+}), E) \ar[u]_{g_{p+1,p}^{1,b+1}(U)^{∗}}}

Thus, we conclude by the two out of three property for weak equivalences that the bottom arrow is also a weak equivalence in Spt(M).

Proposition 2.7. The homotopy category SH(B_{n}) is a compactly generated triangulated 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 is enough to show that the smash product with the simplicial circle induces a Quillen equivalence (see [14, sections I.2, I.3])

(− ∧ S^{1}, Ω_{S^{1}}−, φ) : B_{n}Spt(M) → B_{n}Spt(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] with proposition 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) → B_{n}Spt(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 generators as SH.

Definition 2.8. We fix an arbitrary integer n ≥ 0, and consider the following set of 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 is smooth, 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 normal bundle for the closed immersion Y \U → Y.

Definition 2.9. Let n ∈ ℤ be an arbitrary integer.

1. Let W B_{n}Spt(M) denote the left Bousfield localization of Spt(M) with respect to the set of maps

sW B_{n} = {F_{p}(𝔾_{m}^{b} ∧ ι_{U,Y}) : b, p, r ≥ 0, 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 associated homotopy category.

For n ≠ 0 we will call SH(W B_{n}) the codimension n + 1-weakly birational motivic stable homotopy category, and for n = 0 we will call it the weakly birational motivic stable homotopy category.

Proposition 2.10. Let E be an arbitrary symmetric T -spectrum. Then E is fibrant in W B_{n}Spt(M) if and only if the following conditions hold:

1. E is fibrant in Spt(M).

2. For every a, b, p, r ≥ 0 such that b − p ≥ n − r; and every ι_{U,Y} ∈ W B_{r}, the induced map

\xymatrix{Hom_{SH}(F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ Y_{+}), E) \ar[r]_{≅}^{ι_{U,Y}^{∗}} & Hom_{SH}(F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ U_{+}), E)}

is an isomorphism.

Proof. The proof is exactly the same as in proposition 2.6.

Proposition 2.11. The homotopy category SH(W B_{n}) is a compactly generated triangulated category in the sense of Neeman.

Proof. The proof is exactly the same as in proposition 2.7.

Proposition 2.12. Assume that the base scheme X = Spec k, with k a perfect field, then the Quillen adjunction:

(id, id, φ) : W B_{n}Spt(M) → B_{n}Spt(M)

is a Quillen equivalence.

Proof. Consider the following commutative diagram

\xymatrix{ & Spt(M) \ar[dl]_{id} \ar[dr]^{id} \\ W B_{n}Spt(M) \ar@{-->}[rr]_{id} &  & B_{n}Spt(M)}

where the solid arrows are left Quillen functors. Clearly, W B_{r} ⊆ B_{r} for every r ≥ 0, so 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 localizations implies that the horizontal arrow is also a left Quillen functor. The universal property for left Bousfield localizations also implies that it is enough to show that all the maps in

sB_{n} = {F_{p}(𝔾_{m}^{b} ∧ ι_{U,Y}) : b, p, r ≥ 0, b − p ≥ n − r; ι_{U,Y} ∈ B_{r}}

become weak equivalences in W B_{n}Spt(M). Given F_{p}(𝔾_{m}^{b} ∧ 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 reduced scheme structure), hence F_{p}(𝔾_{m}^{b} ∧ ι_{U,Y}) ∈ sW B_{n} and then a weak equivalence in W B_{n}Spt(M). If dim Z > 0, then we consider the singular locus Z_{s} of Z over X. We have that dim Z_{s} < dim Z since k is a perfect field. Therefore, by induction on the dimension F_{p}(𝔾_{m}^{b} ∧ ι_{V,Y}) is a weak equivalence in W B_{n}Spt(M), where V = Y \Z_{s}. On the other hand, F_{p}(𝔾_{m}^{b} ∧ ι_{U,V}) is also a weak equivalence in W B_{n}Spt(M) since ι_{U,V} is also in B_{r} and its closed complement V \U = Z\Z_{s} is smooth over X, by construction of Z_{s}. But F_{p}(𝔾_{m}^{b} ∧ ι_{U,Y}) = F_{p}(𝔾_{m}^{b} ∧ ι_{V,Y}) ∘ F_{p}(𝔾_{m}^{b} ∧ ι_{U,V}), so by the two out of three property for weak equivalences we conclude that F_{p}(𝔾_{m}^{b} ∧ ι_{U,Y}) is a weak equivalence in W B_{n}Spt(M).

3 A Characterization of the Slices

This section contains our main results. We give a characterization of the slices in terms of effectivity and birational conditions (in the sense of definition 3.1), and we also show that there is an equivalence between the notion of orthogonality (see definition 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 B_{n}Spt(M) (respectively W B_{n}Spt(M)). If n = 0, we will simply 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′) for some symmetric T -spectrum E′.

Definition 3.2. 1. Let ι_{U,Y} be an open immersion in Sm_{X}. Let Y/U denote the pushout of the following diagram in M (i.e. the homotopy cofibre of ι_{U,Y} in M)

\xymatrix{U_{+} \ar[r]^{ι_{U,Y}} \ar[d] & Y_{+} \ar[d] \\ X \ar[r] & Y/U}

2. Given a vector bundle π : V → Y with Y ∈ Sm_{X}, let T h(V) denote the Thom space of V, i.e. V/(V \σ_{0}(Y)), where σ_{0} : Y → V denotes the zero section of V.

Lemma 3.3. Let ι_{U,Y} ∈ W B_{r} for some r ≥ 0, and let a, b, p ≥ 0 be arbitrary integers such that b − p ≥ n − r. Then

F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ Y/U) ∈ Σ_{T}^{n+1}SH^{eff}

Proof. Since Σ_{T}^{n+1}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_{0}(Y/U) ∈ Σ_{T}^{r+1}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_{0}(Y/U) → F_{0}(T h(N))

where Th(N) is the Thom space of the normal bundle N of the (smooth) complement 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 bundle we conclude from [8, Prop. 2.17(2)] that

F_{0}(T h(N)) ≅ F_{0}(S^{c} ∧ 𝔾_{m}^{c} ∧ Z_{+}) ∈ Σ_{T}^{c} SH^{eff} ⊆ Σ_{T}^{r+1}SH^{eff}

Finally, we conclude in the general case by choosing a Zariski cover of Z which trivializes N and using the Mayer-Vietoris property for Zariski covers.

Lemma 3.4. Let U ∈ Sm_{X}. Consider the open immersion in Sm_{X}

m_{U} : 𝔸_{U}^{1}\U → 𝔸_{U}^{1}

given by the complement of the zero section. Then m_{U} ∈ W B_{0}, and there exists a weak equivalence in Spt(M) between its homotopy cofibre in M, 𝔸_{U}^{1}/(𝔸_{U}^{1}\U) and S^{1} ∧ 𝔾_{m} ∧ U_{+}

t_{U} : 𝔸_{U}^{1}/(𝔸_{U}^{1}\U) → S^{1} ∧ 𝔾_{m} ∧ U_{+}

Proof. Since the zero section i_{0} : U → 𝔸_{U}^{1} is a closed embedding of codimension 1 between smooth schemes over X, it follows from the definition of W B_{0} that m_{U} ∈ W B_{0}. Finally, [8, Prop. 2.17(2)] implies the existence of the weak equivalence t_{U}.

Proposition 3.5. Let E ∈ Spt(M) be a symmetric T -spectrum and n ∈ ℤ. Consider the following conditions:

1. E is fibrant in L_{<n+1}Spt(M).

2. E is weakly n + 1-birational (see definition 3.1(1)).

3. E is n + 1-birational (see definition 3.1(1)).

Then (1) and (2) are equivalent. In addition, if the base scheme X = Spec k, with k a perfect field, then (1), (2) and (3) are equivalent.

Proof. (1)⇒(2): Assume that E is fibrant in L_{<n+1}Spt(M). By proposition 2.10 it suffices to show that for every a, b, p, r ≥ 0 with b − p ≥ n − r, and every ι_{U,Y} ∈ W B_{r}; the induced map

\xymatrix{Hom_{SH}(F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ Y_{+}), E) \ar[r]_{≅}^{ι_{U,Y}^{∗}} & Hom_{SH}(F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ U_{+}), E)}

is an isomorphism. We observe that

F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ −) : M → Spt(M)

is a left Quillen functor, therefore the following

\xymatrix{F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ U_{+}) \ar[r]^{F_{p}(S^{a}∧𝔾_{m}^{b}∧ι_{U,Y} )} & F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ Y_{+}) \ar[r] & F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ Y/U)}

is a cofibre sequence in Spt(M). However, SH is a triangulated category and lemma 2.1 implies that

F_{p+1}(S^{a} ∧ 𝔾_{m}^{b+1} ∧ Y/U) ≅ Ω_{S^{1}} ∘ R ∘ F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ Y/U)

are isomorphic in SH, where R denotes a fibrant replacement functor in Spt(M). Hence it suffices to show that

Hom_{SH}(F_{p+1}(S^{a} ∧ 𝔾_{m}^{b+1} ∧ Y/U), E) = Hom_{SH}(F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ Y/U), E) = 0 But this follows from lemma 3.3 together with [12, Prop. 3.3.30], since we are assuming that E is fibrant in L_{<n+1}Spt(M). (2)⇒(1) Assume that E is n + 1-weakly birational. Then, proposition 3.3.30 in [12] implies that it suffices to show that

Hom_{SH}(F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ U_{+}), E) = 0

for every F_{p}(S^{a} ∧ 𝔾_{m}^{b} ∧ U_{+}) ∈ C_{eff}^{n+1}. The same argument as in lemma 2.5 implies that it is enough to consider the case when F_{p}(𝔾_{m}^{b} ∧ U_{+}) ∈ C_{eff}^{n+1}. Moreover, we can further reduce to the case where b, p ≥ 1 and F_{p}(S^{1}∧𝔾_{m}^{b}∧U_{+}) ∈ C_{eff}^{n+1}. In effect, if F_{p}(𝔾_{m}^{b}∧U_{+}) ∈ C_{eff}^{n+1}, then lemma 2.1 implies that the natural map

g_{p+1,p}^{1,b+1}(U) : F_{p+1}(S^{1} ∧ 𝔾_{m}^{b+1} ∧ U_{+}) → F_{p}(𝔾_{m}^{b} ∧ U_{+})

is a weak equivalence in Spt(M). Now, it follows from lemma 3.4, that if b ≥ 1, and 0 − p + (b − 1) ≥ n (i.e. b − p ≥ n + 1); then F_{p}(𝔾_{m}^{b−1} ∧ m_{U}) ∈ sW B_{n}, i.e. a weak equivalence in W B_{n}Spt(M). Since SH(W B_{n}) is a triangulated category, id : Spt(M) → W B_{n}Spt(M) is a left Quillen functor, and F_{p}(𝔾_{m}^{b−1} ∧ (𝔸_{U}^{1}/(𝔸_{U}^{1}\U_{+}))) is the homotopy cofibre of F_{p}(𝔾_{m}^{b−1} ∧ m_{U}); we deduce that E being n + 1-weakly birational implies that

Hom_{SH}(F_{p}(𝔾_{m}^{b−1} ∧ (𝔸_{U}^{1}/(𝔸_{U}^{1}\U_{+}))), E) = 0

Finally, it follows from lemma 3.4 that the following groups are isomorphic

0 = Hom_{SH}(F_{p}(𝔾_{m}^{b−1} ∧ (𝔸_{U}^{1}/(𝔸_{U}^{1}\U_{+}))), E)
≅ Hom_{SH}(F_{p}(S^{1} ∧ 𝔾_{m}^{b} ∧ U_{+}), E)

(2)⇔(3): This follows directly from proposition 2.12.

Theorem 3.6. The Quillen adjunction

(id, id, φ) : W B_{n}Spt(M) → L_{<n+1}Spt(M)

is a Quillen equivalence. In addition, if the base scheme X = Spec k, with k a perfect field, then the Quillen adjunction

(id, id, φ) : B_{n}Spt(M) → L_{<n+1}Spt(M)

is also a Quillen equivalence.

Proof. We show first that W B_{n}Spt(M) and L_{<n+1}Spt(M) are Quillen equivalent. Since W B_{n}Spt(M), L_{<n+1}Spt(M) are both left Bousfield localizations of Spt(M), we deduce that they are simplicial model categories with the same cofibrant replacement functor Q. Thus, it suffices to show that they have the same class of weak equivalences.

However, proposition 3.5 implies that W B_{n}Spt(M), and L_{<n+1}Spt(M) also have the same class of fibrant objects. Therefore, it follows from [2, Thm. 9.7.4] that they have exactly the same class of weak equivalences. Finally, if the base scheme is a perfect field, by proposition 2.12 we conclude that W B_{n}Spt(M) and B_{n}Spt(M) are Quillen equivalent.

Theorem 3.7. Let E be fibrant in Spt(M). Then E is an n-slice (see definition 3.1(2)) if and only if the following conditions hold:

S1 E is n-effective, i.e. E ∈ Σ_{T}^{n} 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_{T}^{n} 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_{<n+1}Spt(M). Hence, proposition 3.5 implies that E is also n + 1-weakly birational. Now we assume that E satisfies the conditions S1 and S2 above. Then, proposition 3.5 implies that E is fibrant in L_{<n+1}Spt(M). Therefore, theorem 1.4(3) implies that E is isomorphic in SH to its own n-slice s_{n}(E). Finally, if the base scheme is a perfect field, then by proposition 3.5 the conditions S2 and GSS2 are equivalent; hence we can conclude applying the same argument as above.

Remark 3.8. Notice that theorem 3.6 implies that it is possible to construct the slice filtration directly from the Quillen model categories W B_{n}Spt(M) described in definition 2.9 without making any reference to the effective categories Σ_{T}^{q} SH^{eff} . One of the interesting consequences of this fact is that it is possible to obtain analogues of the slice filtration in the unstable setting, since the suspension with respect to T or S^{1} does not play an essential role in the construction of W B_{n}Spt(M), i.e. we could consider the left Bousfield localization of the motivic unstable homotopy category M with respect to the maps in definition 2.8. We will study the details of this construction in a future work.

4 Some Computations

In this section we use the characterization of the slices obtained in theorem 3.7 to 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_{<j}(K), s_{<j}(f)) denote s_{j}(F_{0}(K)), s_{j}(F_{0}(f)) (respectively s_{<j}(F_{0}(K)), s_{<j}(F_{0}(f))).

Lemma 4.1. Let g : E → F be a map in SH such that s_{<n}(g) and s_{<n+1}(g) are both isomorphisms in SH. Then the n-slice of g, s_{n}(g) is also an isomorphism in SH.

Proof. It follows from [12, Prop. 3.1.19] that the rows in the following commutative diagram are distinguished triangles in SH

\xymatrix{s_{n}(E) \ar[d]_{s_{n}(g)} \ar[r] & s_{<n+1}(E) \ar[d]^{s_{<n+1}(g)} \ar[r] & s_{<n}(E) \ar[d]^{s_{<n}(g)} \ar[r] & S^{1} ∧ s_{n}(E) \ar[d]^{S^{1}∧s_{n}(g)} \\ s_{n}(F) \ar[r] & s_{<n+1}(F) \ar[r] & s_{<n}(F) \ar[r] & S^{1} ∧ s_{n}(F)}

Thus, we conclude that s_{n}(g) is also an isomorphism in SH.

Consider Y ∈ Sm_{X}. Let ℙ^{n}(Y) denote the trivial projective bundle of rank n over Y, and let ℙ^{∞}(Y) denote the colimit in M of the following filtered diagram

ℙ^{0}(Y) → ℙ^{1}(Y) → ⋯ → ℙ^{n}(Y) → ⋯  (4.1)

given by the inclusions of the respective hyperplanes at infinity.

Theorem 4.2. Let Y ∈ Sm_{X}. Then for any integer j ≤ n, the diagram 4.1 induces the following isomorphisms in SH

\xymatrix{s_{j}(ℙ^{n}(Y)_{+}) \ar[r]^{≅} & s_{j}(ℙ^{n+1}(Y)_{+}) \ar[r]^{≅} & ⋯ \ar[r]^{≅} & s_{j}(ℙ^{∞}(Y)_{+})}

Proof. Let k > n, and consider the closed embedding induced by the diagram (4.1) λ_{n}^{k} : ℙ^{n}(Y) → ℙ^{k}(Y). It is possible to choose a linear embedding ℙ^{k−n−1}(Y) → ℙ^{k}(Y) such that its open complement U_{k,n} contains ℙ^{n}(Y) and has the structure of a vector bundle over ℙ^{n}(Y), with zero section σ_{n}^{k}:

\xymatrix{U_{k,n} \ar[r]^{v_{n}^{k}} \ar[d] & ℙ^{k}(Y) & ℙ^{k−n−1}(Y) \ar[l] \\ ℙ^{n}(Y) \ar[u]_{σ_{n}^{k}} \ar[ur]_{λ_{n}^{k}}}

By homotopy invariance s_{<j}(σ_{n}^{k}) is an isomorphism in SH for every integer j. On the other hand, if j ≤ n, then F_{0}(v_{n}^{k}) is a weak equivalence in W B_{j}Spt(M) since the codimension of its closed complement is n + 1. Thus, theorems 1.4(2) and 3.6 imply that if j ≤ n + 1, then s_{<j}(v_{n}^{k}) is also an isomorphism in SH. Therefore, s_{<j}(λ_{n}^{k}) = s_{<j}(v_{n}^{k}) ∘ s_{<j}(σ_{n}^{k}) is an isomorphism in SH for j ≤ n + 1; and using lemma 4.1 we conclude that the induced map on the slices s_{j}(λ_{n}^{k}) is also an isomorphism for j ≤ n. Finally, the result for ℙ^{∞}(Y) follows directly from the fact that the slices commute with filtered homotopy colimits.

Let Hℤ denote Voevodsky’s Eilenberg-MacLane spectrum (see [15, section 6.1]) representing motivic cohomology in SH.

Corollary 4.3. 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 Hℤ:

\xymatrix{Hℤ \ar[r]^{≅} & s_{0}(ℙ^{0}(k)_{+}) \ar[r]^{≅} & s_{0}(ℙ^{1}(k)_{+}) \ar[r]^{≅} & ⋯ \\ ⋯ \ar[r]^{≅} & s_{0}(ℙ^{n}(k)_{+}) \ar[r]^{≅} & ⋯ \ar[r]^{≅} & s_{0}(ℙ^{∞}(k)_{+})}

Proof. This follows immediately from theorem 4.2 together with the computation of Levine [7, Thm. 10.5.1] and Voevodsky [17] for the zero slice of the sphere spectrum.

Theorem 4.4. Let ι_{U,Y} ∈ W B_{n}, π : V → Y a vector bundle of rank r together with a trivialization t : π^{−1}(U) → 𝔸_{U}^{r} 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_{m}^{r} ∧ s_{j−r}(Y_{+})

Proof. Let Z ∈ Sm_{X} be the closed complement of ι_{U,Y}. Consider the following diagram in Sm_{X}, where all the squares are cartesian

\xymatrix{π^{−1}(Z) ∩ (V \σ_{0}(Y)) \ar[r] \ar[d] & V \σ_{0}(Y) \ar[d] & π^{−1}(U) ∩ (V \σ_{0}(Y)) \ar[l]_{β} \ar[d] \\ π^{−1}(Z) \ar[r] \ar[d] & V \ar[d]^{π} & π^{−1}(U) \ar[l]_{α} \ar[d] \\ Z \ar[r] & Y & U \ar[l]^{ι_{U,Y}}}

and let γ : T h(π^{−1}(U)) → T h(V) be the induced map between the corresponding Thom spaces. We observe that α, β also belong to W B_{n}; thus, if j ≤ n we conclude that F_{0}(ι_{U,Y}), F_{0}(α), F_{0}(β) are all weak equivalences in W B_{j}Spt(M). Therefore, theorems 1.4(2) and 3.6 imply that if j ≤ n + 1, then s_{<j}(ι_{U,Y}), s_{<j}(α), s_{<j}(β) are isomorphisms in SH. We claim that if j ≤ n + 1, then

s_{<j}(γ) : s_{<j}(T h(π^{−1}(U))) → s_{<j}(T h(V))

is also an isomorphism in SH. In effect, by construction of the Thom spaces, we deduce that for any integer j ∈ ℤ, the rows in the following commutative diagram in SH are in fact distinguished triangles

\xymatrix{s_{<j}((π^{−1}(U) ∩ (V \σ_{0}(Y)))_{+}) \ar[r] \ar[d]_{s_{<j}(β)} & s_{<j}(π^{−1}(U)_{+}) \ar[r] \ar[d]_{s_{<j}(α)} & s_{<j}(T h(π^{−1}(U))) \ar[d]_{s_{<j}(γ)} \\ s_{<j}((V \σ_{0}(Y))_{+}) \ar[r] & s_{<j}(V_{+}) \ar[r] & s_{<j}(T h(V))}

Documenta Mathematica 18 (2013) 51–70

Since s_{<j}(α), s_{<j}(β) are isomorphisms in SH for j ≤ n + 1, we conclude that for j ≤ n + 1, s_{<j}(γ) is also an isomorphism in SH. Thus, lemma 4.1 implies that for j ≤ n, s_{j}(ι_{U,Y}), s_{j}(γ) are isomorphisms in SH. Now, we use the trivialization t to obtain the following commutative diagram in Sm_{X} where the rows are isomorphisms

\xymatrix{𝔸_{U}^{r}\U \ar[d] &  & π^{−1}(U) ∩ (V \σ_{0}(Y)) \ar[ll]_{t̃}^{≅} \ar[d] \\ 𝔸_{U}^{r} \ar[dr] &  & π^{−1}(U) \ar[dl]^{π_{U}} \ar[ll]_{t}^{≅} \\  & U}

The same argument as above, shows that for every integer j ∈ ℤ, there is an isomorphism in SH

s_{j}(t̄) : s_{j}(T h(π^{−1}(U))) → s_{j}(T h(𝔸_{U}^{r}))

On the other hand, [8, Prop. 2.17(2)] implies that there is a weak equivalence w : F_{0}(T h(𝔸_{U}^{r})) → S^{r} ∧ 𝔾_{m}^{r} ∧ F_{0}(U_{+}) in Spt(M). Thus, for j ≤ n there exist isomorphisms in SH

\xymatrix{s_{j}(T h(π^{−1}(U))) \ar[r]^{s_{j}(t̄)} \ar[d]_{s_{j}(γ)} & s_{j}(T h(𝔸_{U}^{r})) \ar[d]^{s_{j}(w)} \\ s_{j}(T h(V)) & s_{j}(S^{r} ∧ 𝔾_{m}^{r} ∧ U_{+})}

However, there exists a canonical isomorphism in SH

s_{j}(S^{r} ∧ 𝔾_{m}^{r} ∧ U_{+}) → S^{r} ∧ 𝔾_{m}^{r} ∧ s_{j−r}(U_{+})

Finally, we conclude by using the isomorphism s_{j−r}(ι_{U,Y}) (notice that if j ≤ n then certainly j − r ≤ n, since r ≥ 0).

Corollary 4.5. 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^{−1}(U) → 𝔸_{U}^{r} of its restriction to U. Then for every integer j ≤ n, there exists an isomorphism in SH

s_{j}(T h(V)) ≅ S^{r} ∧ 𝔾_{m}^{r} ∧ s_{j−r}(Y_{+})

Proof. Proposition 2.12 implies that F_{0}(ι_{U,Y}) is a weak equivalence in W B_{j}Spt(M) for j ≤ n. Hence, the result follows using exactly the same argument as in theorem 4.4.

Given a closed embedding Z → Y of smooth schemes over X, let Bℓ_{Z}Y denote the blowup of Y with center in Z.

Theorem 4.6. Let ι_{U,Y} ∈ W B_{n} with closed complement Z, and j ∈ ℤ an arbitrary integer. Consider the following cartesian square in Sm_{X}

[figure: D; q; d; Bℓ_{Z}Y; p; U; u; Z; i; Y; U; ι_{U,Y}]  (4.2)

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 cartesian square (4.2) induces the following distinguished triangle in SH

[figure: s_{j}(D_{+}); ^{\binom{−d_{j}}{q_{j}}}; s_{j}(Bℓ_{Z}Y_{+}) ⊕ s_{j}(Z_{+}); (p_{j},i_{j}); s_{j}(Y_{+})]  (4.3)

If j ≤ n, then s_{j}(ι_{U,Y}) is an isomorphism in SH, and the following distinguished triangles in SH split

[figure: s_{j}(D_{+}); ^{\binom{−d_{j}}{q_{j}}}; s_{j}(Bℓ_{Z}Y_{+}) ⊕ s_{j}(Z_{+}); (p_{j},i_{j}); s_{j}(Y_{+}); _{\binom{r_{j}}{0}}]  (4.4)

\xymatrix{s_{j}(Y_{+}) \ar[r]^{r_{j}} & s_{j}(Bℓ_{Z}Y_{+}) \ar[l]^{p_{j}} \ar[r] & s_{j}(T h(O_{D}(1))) \ar[l]}  (4.5)

where r_{j} = s_{j}(u) ∘ (s_{j}(ι_{U,Y}))^{−1}, and O_{D}(1) denotes the canonical line bundle of the projective bundle q : D → Z.

Proof. It follows from [8, Prop. 2.29 and Rmk. 2.30] that the following square is homotopy cocartesian in M

\xymatrix{S^{1} ∧ D_{+} \ar[d]_{id∧q} \ar[r]^{id∧d} & S^{1} ∧ Bℓ_{Z}Y_{+} \ar[d]^{id∧p} \\ S^{1} ∧ Z_{+} \ar[r]_{id∧i} & S^{1} ∧ Y_{+}}

Thus, we deduce that the following diagram is a distinguished triangle in SH

[figure: F_{0}(D_{+}); ^{\binom{−F_{0}(d)}{F_{0}(q)}}; F_{0}(Bℓ_{Z}Y_{+}) ⊕ F_{0}(Z_{+}); (F_{0}(p),F_{0}(i)); F_{0}(Y_{+})]

Since the slices s_{j} are triangulated functors, it follows that diagram (4.3) is a distinguished triangle in SH.

Now, we prove that s_{j}(ι_{U,Y}) is an isomorphism for j ≤ n. By lemma 4.1, it suffices to show that s_{<j}(ι_{U,Y}) is an isomorphism in SH for j ≤ n+1. But this follows directly from theorems 3.6 and 1.4(2) since F_{0}(ι_{U,Y}) is clearly a weak equivalence in W B_{j}Spt(M) for j ≤ n. Thus, r_{j} is well defined for j ≤ n, and the following diagram shows that it gives a splitting for the distinguished triangle (4.4)

[figure: s_{j}(U_{+}); s_{j}(u); s_{j}(Bℓ_{Z}Y_{+}); p_{j}; s_{j}(U_{+}); s_{j}(ι_{U,Y} ); s_{j}(Y_{+})]  (4.6)

Finally, since the normal bundle of the closed embedding d : D → Bℓ_{Z}Y is given by 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

\xymatrix{s_{j}(U_{+}) \ar[r]^{s_{j}(u)} & s_{j}(Bℓ_{Z}Y_{+}) \ar[r] & s_{j}(T h(O_{D}(1)))}

Combining this distinguished triangle with diagram (4.6) above, we conclude that diagram (4.5) is a split distinguished triangle in SH for j ≤ n.

Acknowledgements

The author would like to thank Marc Levine for bringing to his attention the connection between slices and birational cohomology theories, as well as for all his stimulating comments and questions, and also thank Chuck Weibel for his interest in this work and several suggestions which helped to improve the exposition.

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Pablo Pelaez

Department of Mathematics Rutgers University, U.S.A. pablo.pelaez@rutgers.edu