Convergence of Voevodsky’s Slice Tower

Marc Levine^{1}

Received: May 7, 2012 Revised: March 3, 2013

Communicated by Alexander Merkurjev

Abstract. We consider Voevodsky’s slice tower for a finite spectrum E in the motivic stable homotopy category over a perfect field k. In case k has finite cohomological dimension, we show that the slice tower converges, in that the induced filtration on the bi-graded homotopy sheaves Π_{a,b}f_{n}E is finite, exhaustive and separated at each stalk (after inverting the exponential characteristic of k). This partially verifies a conjecture of Voevodsky.

2010 Mathematics Subject Classification: Primary 14F42; Secondary 55P42 Keywords and Phrases: Morel-Voevodsky stable homotopy category, slice filtration, motivic homotopy theory

Contents

Introduction  908
1. Background and notation  911
2. Voevodsky’s slice tower  914
3. The homotopy coniveau tower  917
4. The simplicial filtration  919
5. The bottom of the filtration  923
6. Finite spectra and cohomologically finite spectra  925
7. The proof of the convergence theorem  930
Appendix A. Norm maps  933
Appendix B. Inverting integers in a triangulated category  936
References  939

1Research supported by the Alexander von Humboldt Foundation

Introduction

We continue our investigation, begun in [12], of the slice filtration on the bigraded homotopy sheaves Π_{∗,∗}(E) for objects E in the motivic stable homotopy category SH(k). We refer the reader to §1 for the notation to be used in this introduction. Let k be a perfect field, let SH(k) denote Voevodsky’s motivic stable homotopy category of T -spectra over k, SH the classical stable homotopy category of spectra. For a spectrum E ∈ SH, the Postnikov tower of E,

… → E^{(n+1)} → E^{(n)} → … → E

consists of the n − 1-connected covers E^{(n)} → E of E, that is, π_{m}E^{(n)} → π_{m}E is an isomorphism for m ≥ n and π_{m}E^{(n)} = 0 for m < n. Sending E to E^{(n)} defines a functor from SH to the full subcategory Σ^{n}SH^{eff} of n − 1-connected spectra that is right adjoint to the inclusion Σ^{n}SH^{eff} → SH. Replacing Σ^{n}SH^{eff} with a certain triangulated subcategory Σ_{T}^{n} SH^{eff}(k) of SH(k) that measures a kind of “ℙ^{1}-connectedness” (in a suitable sense, see [30, 31], [23, 24]) or §2 of this paper), Voevodsky has defined a motivic analog of the Postnikov tower; for an object E of SH(k) this yields the Tate-Postnikov tower (or slice tower)

… → f_{n+1}E → f_{n}E → … → E

for E. For integers a, b, we have the stable homotopy sheaf Π_{a,b}(E), defined as the Nisnevich sheaf associated to the presheaf

U ∈ Sm/k ↦ [Σ_{S^{1}}^{a}Σ_{𝔾_{m}}^{b}Σ_{T}^{∞}U_{+}, E]_{SH(k)}

and the Tate-Postnikov tower for E gives rise to the filtration

Fil_{Tate}^{n} Π_{a,b}(E) := im(Π_{a,b}f_{n}E → Π_{a,b}E).

Let SH_{fin}(k) ⊂ SH(k) be the thick subcategory of SH(k) generated by the objects Σ_{T}^{n} Σ_{T}^{∞}X_{+}, with X smooth and projective over k, n ∈ ℤ. For example, the motivic sphere spectrum 𝕊_{k} := Σ_{T}^{∞} Spec k_{+} is in SH_{fin}(k). Voevodsky has stated the following conjecture:

Conjecture 1 ([30, conjecture 13]). Let k be a perfect field. Then for E ∈ SH_{fin}(k), the Tate-Postnikov tower of E is convergent in the following sense: for all a, b, m ∈ ℤ, one has

∩_{n}F_{Tate}^{n} Π_{a,b}f_{m}E = 0.

The cases E = Σ_{𝔾_{m}}^{q}𝕊_{k}, a = m = 0 gives some evidence for this conjecture, as we shall now explain. For k a perfect field, Morel has given a natural isomorphism of Π_{0,−p}(𝕊_{k}) with the Milnor-Witt sheaf K_{p}^{MW}; this is a certain sheaf on Sm/k with value on each field F over k given by the Milnor-Witt group K_{p}^{MW} (F).^{2} For F a field, K_{0}^{MW} (F) is canonically isomorphic to the Grothendieck-Witt group GW(F) of non-degenerate symmetric bilinear forms over F [14, lemma 3.10]. More generally, Morel has constructed a natural isomorphism^{3} for p, q ∈ ℤ

2A presentation of the graded ring K_{∗}^{MW} (F) may be found in [14, definition 3.1].

Π_{0,p}(Σ_{𝔾_{m}}^{q}𝕊_{k}) ≅ K_{q−p}^{MW} .

The isomorphism K_{0}^{MW} (F) ≅ GW(F) makes K_{∗}^{MW} (F) a GW(F)-module; let I(F) ⊂ GW(F) denote the augmentation ideal. Our main result of loc. cit. is

Theorem 2 ([12, theorem 1]). Let F be a perfect field extension of k of characteristic ≠ 2. Then

Fil_{Tate}^{n} Π_{0,p}(Σ_{𝔾_{m}}^{q}𝕊_{k})(F) = I(F)^{M}K_{q−p}^{MW} (F) ⊂ K_{q−p}^{MW} (F) = Π_{0,p}(Σ_{𝔾_{m}}^{q}𝕊_{k})(F)

where M = 0 if n ≤ p or n ≤ q, and M = min(n − p, n − q) if n ≥ p and n ≥ q.

The following consequence of theorem 2 gives some evidence for Voevodsky’s convergence conjecture:

Proposition 3. Let k be a perfect field with char. k ≠ 2. For all p, q ≥ 0, and all perfect field extensions F of k, we have

∩_{n}F_{Tate}^{n} Π_{0,p}Σ_{𝔾_{m}}^{q}𝕊_{k}(F) = 0.

Proof. In light of theorem 2, this is asserting that the I(F)-adic filtration on K_{q−p}^{MW} (F) is separated. By [16, théorème 5.3], for m ≥ 0, K_{m}^{MW} (F) fits into a cartesian square of GW(F)-modules

\xymatrix{K_{m}^{MW} (F) \ar[r] \ar[d] & K_{m}^{M}(F) \ar[d]^{P f} \\ I(F)^{m} \ar[r]_{q} & I(F)^{m}/I(F)^{m+1},}

where K_{m}^{M}(F) is the Milnor K-group, q is the quotient map and P f is the map sending a symbol {u_{1}, …, u_{m}} to the class of the Pfister form <<u_{1}, …, u_{m}>> mod I(F)^{m+1}. For m < 0, K_{m}^{MW} (F) is isomorphic to the Witt group W (F) of F, that is, the quotient of GW(F) by the ideal generated by the hyperbolic form x^{2} − y^{2}. Also, the map GW(F) → W (F) gives an isomorphism of I(F)^{r} with its image in W (F) for all r ≥ 1. Thus, for n ≥ 1,

I(F)^{n}K_{m}^{MW} (F) = \begin{cases}I(F)^{n} ⊂ W (F) & for m < 0 \\ I(F)^{n+m} ⊂ W (F) & for m ≥ 0. \end{cases}

The fact that ∩_{n}I(F)^{n} = 0 in W (F) is a theorem of Arason and Pfister [1, Korollar 1]. □

Remarks. 1. The proof in [16] that K_{m}^{MW} (F) fits into a cartesian square as above relies on the Milnor conjecture.

2. As pointed out to me by Igor Kriz, Voevodsky’s convergence conjecture in the generality as stated above is false. In fact, take E to be the Moore spectrum 𝕊_{k}/ℓ for some prime ℓ ≠ 2. Since Π_{a,q}𝕊_{k} = 0 for a < 0, proposition 6.9 below shows that Π_{a,q}f_{n}𝕊_{k} = 0 for a < 0, and thus we have the right exact sequence for all n ≥ 0

3This follows from [14, theorem 6.13, theorem 6.40], using the argument of [14, theorem 6.43].

Π_{0,0}f_{n}𝕊_{k} ⟶^{×ℓ} Π_{0,0}f_{n}𝕊_{k} → Π_{0,0}f_{n}E → 0.

In particular, we have

F_{Tate}^{n}Π_{0,0}E(k) = im (F_{Tate}^{n}Π_{0,0}𝕊_{k}(k) → Π_{0,0}𝕊_{k}(k)/ℓ) = = im (I(k)^{n} → GW(k)/ℓ) .

Take k = ℝ. Then I(ℝ) ⊂ GW(ℝ) is isomorphic to ℤ via the virtual negative index, and I(ℝ)^{n} = (2^{n−1}) ⊂ ℤ = I(ℝ). Thus Π_{0,0}E = ℤ/ℓ ⊕ ℤ/ℓ and the filtration F_{Tate}^{n}Π_{0,0}E is constant, equal to ℤ/ℓ = I(ℝ)/ℓ, and is therefore not separated. The convergence property is thus not a “triangulated" one in general, and therefore seems to be a subtle one. However, if the I-adic filtration on GW(F) is finite for all finitely generated F over k (possibly of varying length depending on F), then the augmentation ideal in GW(F) is two-primary torsion. Our computations (at least in characteristic ≠ 2) show that the filtration F_{Tate}^{∗}Π_{0,p}Σ_{T}^{∞}𝔾_{m}^{∧q} is in this case at least locally finite, and thus has better triangulated properties. In particular, for ℓ ≠ 2,

Π_{0,0}(𝕊_{k}/ℓ) = ℤ/ℓ, F_{Tate}^{n}Π_{0,0}(𝕊_{k}/ℓ) = 0 for n > 0.

One can therefore ask if Voevodsky’s convergence conjecture is true if one assumes the finiteness of the I(F)-adic filtration on GW(F) for all finitely generated fields F over k. The main theorem of this paper is a partial answer to the convergence question along these lines.

Theorem 4. Let k be a perfect field of finite cohomological dimension and let p denote the exponential characteristic.^{4} Take E in SH_{fin}(k) and take x ∈ X ∈ Sm/k with X irreducible. Let d = dim_{k} X. Then for every r, q, m ∈ ℤ, there is an integer N = N(E, r, d, q) such that

(Fil_{Tate}^{n} Π_{r,q}f_{m}E)_{x}[1/p] = 0

for all n ≥ N. In particular, if F is a field extension of k of finite transcendence dimension d over k, then Fil_{Tate}^{n} Π_{r,q}f_{m}E(F)[1/p] = 0 for all n ≥ N.

For a more detailed and perhaps more general statement, we refer the reader to theorem 7.3.

Remarks. 1. The proof of theorem 4 relies on the Bloch-Kato conjecture.

2. As we have seen, Voevodsky’s convergence conjecture is not true for all base fields k. An interesting class of fields strictly larger than the class of fields of finite cohomological dimension is those of finite virtual cohomological dimension (e.g., ℝ). We suggest the following formulation:

4That is, p = char. k if char. k > 0, p = 1 if char. k = 0.

Conjecture 5. Let k be a field of finite virtual 2-cohomological dimension. Then the I(k)-completed slice tower is weakly convergent: after I(k)-completion, the filtration Fil_{Tate}^{∗} Π_{r,q}f_{m}E is stalkwise separated for each E in SH_{fin}(k) and each r, q, m.

This modified conjecture is equivalent to Voevodsky’s convergence conjecture in case k has finite 2-cohomological dimension, as in this case I(k) is nilpotent. One could also ask for a weaker version, in which one assumes that k has finite p-cohomological dimension for all odd primes p.

3. It would be interesting to be able to say something about the p-torsion in (Fil_{Tate}^{∗} Π_{r,q}E)_{x}.

The paper is organized as follows: We set the notation in §1. In §2 we recall some basic facts about the slice tower, the truncation functors f_{n} in SH(k) and SH_{S^{1}}(k), and the associated filtration Fil_{Tate}^{∗} Π_{a,b}. We recall the construction and basic properties of the homotopy coniveau tower, a simplicial model for the slice tower in SH_{S^{1}}(k), in §3. In §4 we use the simplicial nature of the homotopy coniveau tower to analyze the terms in the slice tower. This leads to the main inductive step in our argument (lemma 4.5), and isolates the particular piece that we need to study. This is analyzed further in §5, where we more precisely identify this piece in terms of a K_{∗}^{MW} -module structure on the bigraded homotopy sheaves (see theorem 5.3). In §6 we use a decomposition theorem of Morel and results of Cisinski-Déglise to prove some boundedness properties of the homotopy sheaves Π_{p,q}E and their ℚ-localizations Π_{p,q}E_{ℚ} for E in SH_{fin}(k), under the assumption that the base-field k has finite 2cohomological dimension. In the final section 7, we assemble all the pieces and prove our main result. We conclude with two appendices; the first collects some results on norm maps for finite field extensions that are used throughout the paper and the second assembles some basic facts on the localization of compactly generated triangulated categories with respect to a collection of non-zero integers. I am grateful to the referee for a number of comments and suggestions for improving an earlier version of this paper and to S. Kelly for discussions on aspects of localization.

1. Background and notation

Unless we specify otherwise, k will be a fixed perfect base field, without restriction on the characteristic. For details on the following constructions, we refer the reader to [5, 7, 8, 14, 15, 17, 18]. We write [n] for the set {0, …, n} with the standard order (including [−1] = ∅) and let Δ be the category with objects [n], n = 0, 1, …, and morphisms [n] → [m] the order-preserving maps of sets. Given a category C, the category of simplicial objects in C is as usual the category of functors Δ^{op} → C, and the category of cosimplicial objects the functor category C^{Δ}. Spc will denote the category of simplicial sets, Spc_{•} the category of pointed simplicial sets, H := Spc[W E^{−1}] the classical unstable homotopy category and H_{•} := Spc_{•}[W E^{−1}] the pointed version; here W E is the usual class of weak equivalences, that is, morphisms A → B that induce an isomorphism on all π_{n} for all choice of base-point. Spt is the category of spectra, that is, spectrum objects in Spc_{•} with respect to the left suspension functor Σ_{S^{1}}^{ℓ} := S^{1} ∧ (−). With sW E denoting the class of stable weak equivalences, that is, morphisms f : E → F in Spt that induce an isomorphism on all stable homotopy groups, SH := Spt[sW E^{−1}] is the classical stable homotopy category. For a simplicial object in Spc, resp. Spc_{•}, resp. Spt, S : Δ^{op} → Spc, Spc_{•}, Spt, we let |S| ∈ Spc, Spc_{•}, Spt denote respective homotopy colimit hocolim_{Δ^{op}} S. The motivic versions are as follows: Sm/k is the category of smooth finite type k-schemes. Spc(k) is the category of Spc-valued presheaves on Sm/k, Spc_{•}(k) the Spc_{•}-valued presheaves, and Spt_{S^{1}}(k) the Spt-valued presheaves. These all come with “motivic” model structures as simplicial model categories (see for example [8]); we denote the corresponding homotopy categories by H(k), H_{•}(k) and SH_{S^{1}}(k), respectively. Sending X ∈ Sm/k to the sheaf of sets on Sm/k represented by X (which we also denote by X) gives an embedding of Sm/k to Spc(k); we have the similarly defined embedding of the category of smooth pointed schemes over k into Spc_{•}(k). Sending a (pointed) simplicial set A to the constant presheaf with value A (also denoted by A) defines an embedding of Spc in Spc(k) and of Spc_{•} in Spc_{•}(k). Let 𝔾_{m} be the pointed k-scheme (𝔸^{1} \ 0, 1). We let T := 𝔸^{1}/(𝔸^{1} \ {0}) and let Spt_{T} (k) denote the category of T -spectra, i.e., spectra in Spc_{•}(k) with respect to the left T -suspension functor Σ_{T}^{ℓ} := T ∧ (−). Spt_{T} (k) has a motivic model structure (see [8]) and SH(k) is the homotopy category. We can also form the category of spectra in Spt_{S^{1}}(k) with respect to Σ_{T}^{ℓ}; with an appropriate model structure the resulting homotopy category is equivalent to SH(k). We will identify these two homotopy categories without further mention. For each A ∈ Spc_{•}(k), the suspension functor Σ_{A} : Spc_{•}(k) → Spc_{•}(k), Σ_{A}(B) := B ∧A, extends to the suspension functor Σ_{A} : Spt_{S^{1}}(k) → Spt_{S^{1}}(k) or Σ_{A} : Spt_{T} (k) → Spt_{T} (k). For A cofibrant, this gives the suspension functors Σ_{A} : H_{•}(k) → H_{•}(k), Σ_{A} : SH_{S^{1}}(k) → SH_{S^{1}}(k) and Σ_{A} : SH(k) → SH(k) by applying Σ_{A} to a cofibrant replacement. Both SH_{S^{1}}(k) and SH(k) are triangulated categories with translation functor Σ_{S^{1}}. On H_{•}(k), SH_{S^{1}}(k) and SH(k), we have Σ_{T} ≅ Σ_{S^{1}} ∘Σ_{𝔾_{m}}; the suspension functors Σ_{T} and Σ_{𝔾_{m}} on SH(k) are invertible. For A ∈ Spc_{•}(k), we have an enriched Hom on Spt_{S^{1}}(k) and Spt_{T} (k) with values in spectra; we denote the enriched Hom functor by Hom(A, −). This passes to the homotopy categories H_{•}(k), SH_{S^{1}}(k) and SH(k) to give for A ∈ H_{•}(k) an enriched Hom Hom(A, −) with values in SH. For X ∈ Sm/k, E ∈ Spt_{S^{1}}(k), Hom(X_{+}, E) = E(X).

We have the triangle of infinite suspension functors Σ^{∞} and their right adjoints Ω^{∞}

\xymatrix{H_{•}(k) \ar[r]^{Σ_{S^{1}}^{∞}} \ar[dr]_{Σ_{T}^{∞}} & SH_{S^{1}}(k) \ar[d]^{Σ_{T}^{∞}} & H_{•}(k) & SH_{S^{1}}(k) \ar[l]_{Ω_{S^{1}}^{∞}} \\  & SH(k) &  & SH(k) \ar[u]_{Ω_{T}^{∞}} \ar@{-->}[ul]^{Ω_{T}^{∞}}}

both commutative up to natural isomorphism. These are all left, resp. right derived versions of Quillen adjoint pairs of functors on the underlying model categories. For X ∈ H_{•}(k), we have the bi-graded homotopy sheaf Π_{a,b}X, defined for a, b ≥ 0, as the Nisnevich sheaf associated to the presheaf on Sm/k

U ↦ Hom_{H_{•}(k)}(Σ_{S^{1}}^{a}Σ_{𝔾_{m}}^{b}U_{+}, X);

note the perhaps non-standard indexing. We have the bi-graded homotopy sheaves Π_{a,b}E for E ∈ SH_{S^{1}}(k), b ≥ 0, a ∈ ℤ, and Π_{a,b}E for E ∈ SH(k), a, b ∈ ℤ, by taking the Nisnevich sheaf associated to

U ↦ Hom_{SH_{S^{1}} (k)}(Σ_{S^{1}}^{a}Σ_{𝔾_{m}}^{b}Σ_{S^{1}}^{∞}U_{+}, E) or U ↦ Hom_{SH(k)}(Σ_{S^{1}}^{a}Σ_{𝔾_{m}}^{b}Σ_{T}^{∞}U_{+}, E),

as the case may be. We write π_{n} for Π_{n,0}; for E ∈ Spt_{S^{1}}(k) fibrant, π_{n}E is the Nisnevich sheaf associated to the presheaf U ↦ π_{n}(E(U)). SH(k) has the set of compact generators

{Σ_{S^{1}}^{n}Σ_{T}^{m}Σ_{S^{1}}^{∞}X_{+}, n, m ∈ ℤ, X ∈ Sm/k}

and SH_{S^{1}}(k) has the set of compact generators

{Σ_{S^{1}}^{n}Σ_{T}^{m}Σ_{S^{1}}^{∞}X_{+}, n ∈ ℤ, m ≥ 0, X ∈ Sm/k}.

For SH(k), this is [4, theorem 9.2]; the proof of this result goes through without change to yield the statement for SH_{S^{1}}(k). As these triangulated categories are both homotopy categories of stable model categories, both admit arbitrary small coproducts. For F a finitely generated field extension of k, we may view Spec F as the generic point of some X ∈ Sm/k (since k is perfect). Thus, for a Nisnevich sheaf S on Sm/k, we may define S(F) as the stalk of S at Spec F ∈ X. For an arbitrary field extension F of k (not necessarily finitely generated over k), we define S(F) as the colimit over S(F_{α}), as F_{α} runs over subfields of F containing k and finitely generated over k. For a finitely generated field F over k, we consider objects such as Spec F, or 𝔸_{F}^{n} as pro-objects in Spc(k) by the usual system of finite-type models; the same holds for related objects such as Spec F_{+} in H_{•}(k) or Σ_{S^{1}}^{∞} Spec F_{+} in SH_{S^{1}}(k), etc. We extend this to arbitrary field extensions of k by taking the system of finitely generated subfields. We will usually not explicitly insert the “pro-” in the text, but all such objects, as well as morphisms and isomorphisms between them, should be so understood.

2. Voevodsky’s slice tower

We begin by recalling definition and basic properties of the Tate-Postnikov tower in SH_{S^{1}}(k) and in SH(k). We then define the main object of our study: the filtration on the bi-graded homotopy sheaves of a T -spectrum or an S^{1}- spectrum induced by the respective Tate-Postnikov towers. For n ≥ 0, we let Σ_{T}^{n} SH_{S^{1}}(k) be the localizing subcategory of SH_{S^{1}}(k) generated by the (compact) objects Σ_{T}^{m}Σ_{S^{1}}^{∞}X_{+}, with X ∈ Sm/k and m ≥ n. We note that Σ_{T}^{0} SH_{S^{1}}(k) = SH_{S^{1}}(k). The inclusion functor i_{n} : Σ_{T}^{n} SH_{S^{1}}(k) → SH_{S^{1}}(k) admits, by results of Neeman [21, theorem 4.1], a right adjoint r_{n}; define the functor f_{n} : SH_{S^{1}}(k) → SH_{S^{1}}(k) by f_{n} := i_{n} ∘ r_{n}. The co-unit for the adjunction gives us the natural morphism

ρ_{n} : f_{n}E → E

for E ∈ SH_{S^{1}}(k); similarly, the inclusion Σ_{T}^{m}SH_{S^{1}}(k) ⊂ Σ_{T}^{n} SH_{S^{1}}(k) for n < m gives the natural transformation f_{m}E → f_{n}E, forming the Tate-Postnikov tower

… → f_{n+1}E → f_{n}E → … → f_{0}E = E;

we define f_{n} := id for n < 0. We complete f_{n+1}E → f_{n}E to a distinguished triangle

f_{n+1}E → f_{n}E → s_{n}E → f_{n+1}E[1];

this distinguished triangle actually characterizes s_{n}E up to unique isomorphism, hence this defines a distinguished triangle that is functorial in E. The object s_{n}E is the nth slice of E. There is an analogous construction in SH(k): For n ∈ ℤ, let

Σ_{T}^{n} SH^{eff}(k) ⊂ SH(k)

be the localizing category generated by the T -suspension spectra Σ_{T}^{m}Σ_{T}^{∞}X_{+}, for X ∈ Sm/k and m ≥ n; write SH^{eff}(k) for Σ_{T}^{0} SH^{eff}(k). As above, the inclusion i_{n} : Σ_{T}^{n} SH^{eff}(k) → SH(k) admits a right adjoint r_{n}, giving us the truncation functor f_{n}, n ∈ ℤ, and the Tate-Postnikov tower

… → f_{n+1}E → f_{n}E → … → E.

We define the layer s_{n}E by a distinguished triangle as above. For integers N ≥ n, we let ρ_{n,N} : f_{N} → f_{n} and ρ_{n} : f_{n} → id denote the canonical natural transformations. We mention the following elementary but useful result.

Lemma 2.1. For integers N, n, the diagram of natural endomorphisms of SH(k)

\xymatrix{f_{n} ∘ f_{N} \ar[d]_{f_{n}(ρ_{N})} \ar[r]^{ρ_{n}(f_{N})} & f_{N} \ar[d]^{ρ_{N}} \\ f_{n} \ar[r]_{ρ_{n}} & id}

commutes. Moreover, for N ≥ n, the map ρ_{n}(f_{N}) is a natural isomorphism, and for N ≤ n, the map f_{n}(ρ_{N}) is a natural isomorphism. The same holds with SH_{S^{1}}(k) replacing SH(k).

Proof. The first assertion is just the naturality of ρ_{n} with respect to the morphism ρ_{N} : f_{N} → id. Suppose N ≥ n. Then Σ_{T}^{N}SH^{eff}(k) ⊂ Σ_{T}^{n} SH^{eff}(k) and thus for all E ∈ SH(k), id : f_{N}E → f_{N}E satisfies the universal property of ρ_{n}(f_{N}E) : f_{n}(f_{N}E) → f_{N}E, namely, f_{N}E is in Σ_{T}^{n} SH^{eff}(k) and id : f_{N}E → f_{N}E is universal for maps T → f_{N}E with T ∈ Σ_{T}^{n} SH^{eff}(k). Thus, ρ_{n}(f_{N}E) is an isomorphism. If N ≤ n, then for E ∈ SH(k), f_{n}(f_{N}E) is in Σ_{T}^{n} SH^{eff}(k) and ρ_{n}(f_{N}E) : f_{n}(f_{N}E) → f_{N}E is universal for maps T → f_{N}E with T ∈ Σ_{T}^{n} SH^{eff}(k). Since Σ_{T}^{n} SH^{eff}(k) ⊂ Σ_{T}^{N}SH^{eff}(k), the universal property of ρ_{N}(E) : f_{N}E → E shows that ρ_{N}(E) ∘ ρ_{n}(f_{N}E) : f_{n}(f_{N}E) → E is universal for maps T → E with T ∈ Σ_{T}^{n} SH^{eff}(k), and thus f_{n}(ρ_{N}(E)) is an isomorphism. The proof for SH_{S^{1}}(k) is the same. □

Lemma 2.2. For n ∈ ℤ, there is a natural isomorphism

(2.1)  f_{n}Ω_{T}^{∞}E ≅ Ω_{T}^{∞}f_{n}E.

Proof. First suppose that n ≥ 0. It follows from [10, theorem 7.4.1] that Ω_{T}^{∞}f_{n}E is in Σ_{T}^{n} SH_{S^{1}}(k) and thus we need only show that Ω^{∞}ρ_{n} : Ω_{T}^{∞}f_{n}E → Ω_{T}^{∞}E satisfies the universal property of f_{n}Ω_{T}^{∞}E → Ω_{T}^{∞}E. Σ_{T}^{n} SH_{S^{1}}(k) is generated as a localizing subcategory of SH_{S^{1}}(k) by objects Σ_{T}^{n} G, G ∈ SH_{S^{1}}(k), so it suffices to check for objects of this form. We have

Hom_{SH_{S^{1}} (k)}(Σ_{T}^{n} G, Ω_{T}^{∞}f_{n}E) ≅ Hom_{SH(k)}(Σ_{T}^{∞}Σ_{T}^{n} G, f_{n}E)

≅ Hom_{SH(k)}(Σ_{T}^{n} Σ_{T}^{∞}G, f_{n}E) [figure: ∼; ρ_{n∗}]Hom_{SH(k)}(Σ_{T}^{n} Σ_{T}^{∞}G, E)

≅ Hom_{SH(k)}(Σ_{T}^{∞}Σ_{T}^{n} G, E) ≅ Hom_{SH_{S^{1}} (k)}(Σ_{T}^{n} G, Ω_{T}^{∞}E).

It is easy to check that this sequence of isomorphisms is induced by (Ω_{T}^{∞}ρ_{n})_{∗}. Now suppose that n < 0. Then f_{n}Ω_{T}^{∞}E ≅ f_{0}Ω_{T}^{∞}E ≅ Ω_{T}^{∞}f_{0}E, so it suffices to show that the map f_{0}E → f_{n}E induces an isomorphism Ω_{T}^{∞}f_{0}E → Ω_{T}^{∞}f_{n}E. But for F ∈ SH_{S^{1}}(k), Σ_{T}^{∞}F is in SH^{eff}(k) and

Hom_{SH_{S^{1}} (k)}(F, Ω_{T}^{∞}f_{0}E) ≅ Hom_{SH(k)}(Σ_{T}^{∞}F, f_{0}E)

[figure: ρ_{n,0}; ∼]Hom_{SH(k)}(Σ_{T}^{∞}F, f_{n}E) ≅ Hom_{SH_{S^{1}} (k)}(F, Ω_{T}^{∞}f_{n}E).

□

For E ∈ SH_{S^{1}}(k), we have (by [10, theorem 7.4.2]) the canonical isomorphism

(2.2)  Ω_{𝔾_{m}}^{r}f_{n}E ≅ f_{n−r}Ω_{𝔾_{m}}^{r}E

for r ≥ 0. As Ω_{𝔾_{m}} : SH(k) → SH(k) is an auto-equivalence, and restricts to an equivalence

Ω_{𝔾_{m}} : Σ_{T}^{n} SH^{eff}(k) → Σ_{T}^{n−1}SH^{eff}(k),

the analogous identity in SH(k) holds as well, for all r ∈ ℤ.

Definition 2.3. For a ∈ ℤ, b ≥ 0, E ∈ SH_{S^{1}}(k), define the filtration F_{Tate}^{∗}Π_{a,b}E of Π_{a,b}E by

F_{Tate}^{n}Π_{a,b}E := im(Π_{a,b}f_{n}E → Π_{a,b}E); n ∈ ℤ.

Similarly, for E ∈ SH(k), a, b, n ∈ ℤ, define

F_{Tate}^{n}Π_{a,b}E := im(Π_{a,b}f_{n}E → Π_{a,b}E).

The main object of this paper is to understand F_{Tate}^{n}Π_{a,b}E for suitable E. For later use, we note the following:

Lemma 2.4. 1. For E ∈ SH_{S^{1}}(k), n, p, a, b ∈ ℤ with p, b, b − p ≥ 0, the adjunction isomorphism Π_{a,b}E ≅ Π_{a,b−p}Ω_{𝔾_{m}}^{p}E induces an isomorphism

F_{Tate}^{n} Π_{a,b}E ≅ F_{Tate}^{n−p}Π_{a,b−p}Ω_{𝔾_{m}}^{p}E.

Similarly, for E ∈ SH(k), n, p, a, b ∈ ℤ, the adjunction isomorphism Π_{a,b}E ≅ Π_{a,b−p}Ω_{𝔾_{m}}^{p}E induces an isomorphism

F_{Tate}^{n} Π_{a,b}E ≅ F_{Tate}^{n−p}Π_{a,b−p}Ω_{𝔾_{m}}^{p}E.

2. For E ∈ SH(k), a, b, n ∈ ℤ, with b ≥ 0, we have a canonical isomorphism

φ_{E,a,b,n} : Π_{a,b}f_{n}E → Π_{a,b}f_{n}Ω_{T}^{∞}E,

inducing an isomorphism F_{Tate}^{n} Π_{a,b}E ≅ F_{Tate}^{n} Π_{a,b}Ω_{T}^{∞}E.

Proof. (1) By (2.2), adjunction induces isomorphisms

F_{Tate}^{n}Π_{a,b}E :=im(Π_{a,b}f_{n}E → Π_{a,b}E)
≅im(Π_{a,b−p}Ω_{𝔾_{m}}^{p}f_{n}E → Π_{a,b−p}Ω_{𝔾_{m}}^{p}E)
=im(Π_{a,b−p}f_{n−p}Ω_{𝔾_{m}}^{p}E → Π_{a,b−p}Ω_{𝔾_{m}}^{p}E) = F_{Tate}^{n−p}Π_{a,b−p}Ω_{𝔾_{m}}^{p}E.

The proof for E ∈ SH(k) is the same. For (2), the isomorphism φ_{E,a,b,n} arises from (2.1) and the adjunction isomorphism

Hom_{SH_{S^{1}} (k)}(Σ_{S^{1}}^{a}Σ_{𝔾_{m}}^{b}Σ_{S^{1}}^{∞}U_{+},f_{n}Ω_{T}^{∞}E)
≅ Hom_{SH_{S^{1}} (k)}(Σ_{S^{1}}^{a}Σ_{𝔾_{m}}^{b}Σ_{S^{1}}^{∞}U_{+}, Ω_{T}^{∞}f_{n}E)
≅ Hom_{SH(k)}(Σ_{S^{1}}^{a}Σ_{𝔾_{m}}^{b}Σ_{T}^{∞}U_{+}, f_{n}E).
□

3. The homotopy coniveau tower

Our computations rely heavily on our model for the Tate-Postnikov tower in SH_{S^{1}}(k), which we briefly recall (for details, we refer the reader to [10]). We start with the cosimplicial scheme n ↦ Δ^{n}, with Δ^{n} the algebraic n-simplex Spec k[t_{0}, …, t_{n}]/ ∑_{i} t_{i} − 1. The cosimplicial structure is given by sending a map g : [n] → [m] to the map Δ(g) : Δ^{n} → Δ^{m} determined by

Δ(g)^{∗}(t_{i}) = \begin{cases}∑_{j,g(j)=i} t_{j} & if g^{−1}(i) ≠ ∅ \\ 0 & else. \end{cases}

A face of Δ^{m} is a closed subscheme F defined by equations t_{i_{1}} = … = t_{i_{r}} = 0; we let ∂Δ^{n} ⊂ Δ^{n} be the closed subscheme defined by ∏_{i=0}^{n} t_{i} = 0, i.e., ∂Δ^{n} is the union of all the proper faces. Take X ∈ Sm/k. We let S_{X}^{(q)}(m) denote the set of closed subsets W ⊂ X ×Δ^{m} such that

codim_{X×F} W ∩ X × F ≥ q

for all faces F ⊂ Δ^{m} (including F = Δ^{m}). We make S_{X}^{(q)}(m) into a partially ordered set via inclusions of closed subsets. Sending m to S_{X}^{(q)}(m) and g : [n] → [m] to Δ(g)^{−1} : S_{X}^{(q)}(m) → S_{X}^{(q)}(n) gives us the simplicial poset S_{X}^{(q)}. Now take E ∈ Spt_{S^{1}}(k). For X ∈ Sm/k and closed subset W ⊂ X, we have the spectrum with supports E^{W} (X) defined as the homotopy fiber of the restriction map E(X) → E(X \ W). This construction is functorial in the pair (X, W), where we define a map f : (Y, T) → (X, W) as a morphism f : Y → X in Sm/k with f^{−1}(W) ⊂ T. We usually denote the map induced by f : (Y, T) → (X, W) by f^{∗} : E^{W} (X) → E^{T} (Y), but for f = id_{X} : (X, T) → (X, W), i : W → T the resulting inclusion, we write i_{∗} : E^{W} (X) → E^{T} (X) for id_{X}^{∗}. Define

E^{(q)}(X, m) := hocolim_{W ∈S_{X}^{(q)}(m)} E^{W} (X × Δ^{m}).

The fact that m ↦ S_{X}^{(q)}(m) is a simplicial poset, and (Y, T) ↦ E^{T} (Y) is a functor from the category of pairs to spectra shows that m ↦ E^{(q)}(X, m) defines a simplicial spectrum. We define the spectrum E^{(q)}(X) by

E^{(q)}(X) := |m ↦ E^{(q)}(X, m)| := hocolim_{Δ^{op}} E^{(q)}(X, −).

For q ≥ q′, the inclusions S_{X}^{(q)}(m) ⊂ S_{X}^{(q′)}(m) induce a map of simplicial posets S_{X}^{(q)} ⊂ S_{X}^{(q′)} and thus a morphism of spectra i_{q′,q} : E^{(q)}(X) → E^{(q′)}(X). Since E^{(0)}(X, 0) = E(X), we have the canonical map

ϵ_{X} : E(X) → E^{(0)}(X),

which is a weak equivalence if E is homotopy invariant. Together, this forms the augmented homotopy coniveau tower

E^{(∗)}(X) := … → E^{(q+1)}(X) ⟶^{i_{q}} E^{(q)}(X) ⟶^{i_{q−1}}

… E^{(1)}(X) ⟶^{i_{0}} E^{(0)}(X) ⟵^{ϵ_{X}} E(X)

with i_{q} := i_{q,q+1}. For homotopy invariant E, this gives us the homotopy coniveau tower in SH

E^{(∗)}(X) := … → E^{(q+1)}(X) ⟶^{i_{q}} E^{(q)}(X)

⟶^{i_{q−1}} … E^{(1)}(X) ⟶^{i_{0}} E^{(0)}(X) ≅ E(X).

Letting Sm//k denote the subcategory of Sm/k with the same objects and with morphisms the smooth morphisms, it is not hard to see that sending X to E^{(∗)}(X) defines a functor from Sm//k^{op} to augmented towers of spectra. On the other hand, for E ∈ Spt_{S^{1}}(k), we have the (augmented) Tate-Postnikov tower

f_{∗}E := … → f_{q+1}E → f_{q}E → … → f_{0}E ≅ E

in SH_{S^{1}}(k), which we may evaluate at X ∈ Sm/k, giving the tower f_{∗}E(X) in SH, augmented over E(X). Call E ∈ Spt_{S^{1}}(k) quasi-fibrant if, for E → E^{fib} a fibrant replacement in the motivic model structure, the map E(X) → E^{fib}(X) is a stable weak equivalence in Spt for all X ∈ Sm/k. As a general rule, we will represent an E ∈ SH_{S^{1}}(k) by a fibrant object in Spt_{S^{1}}(k), also denoted E, without making explicit mention of this choice. As a direct consequence of [10, theorem 7.1.1] we have

Theorem 3.1. Let E be a quasi-fibrant object in Spt_{S^{1}}(k), and take X ∈ Sm/k. Then there is an isomorphism of augmented towers in SH

(f_{∗}E)(X) ≅ E^{(∗)}(X)

over the identity on E(X), which is natural with respect to smooth morphisms in Sm/k.

In particular, we may use the model E^{(q)}(X) to understand (f_{q}E)(X).

Remark 3.2. For X, Y ∈ Sm/k with given k-points x ∈ X(k), y ∈ Y (k), we have a natural isomorphism in SH_{S^{1}}(k)

Σ_{S^{1}}^{∞}(X ∧ Y) ⊕ Σ_{S^{1}}^{∞}(X ∨ Y) ≅ Σ_{S^{1}}^{∞}(X × Y),

using the additivity of the category SH_{S^{1}}(k). Thus, Σ_{S^{1}}^{∞}(X∧Y) is a canonically defined summand of Σ_{S^{1}}^{∞}(X × Y). In particular for E a quasi-fibrant object of Spt_{S^{1}}(k), we have a natural isomorphism in SH

Hom(X ∧ Y, E) ≅ hofib (E(X × Y) → hofib(E(X) ⊕ E(Y) → E(k)))

where the maps are induced by the evident restriction maps. In particular, we may define E(X ∧ Y) via the above isomorphism, and our comparison results for Tate-Postnikov tower and homotopy coniveau tower extend to values at smash products of smooth pointed schemes over k.

4. The simplicial filtration

In this section, we study the filtration on π_{r}f_{n}E(X) induced by the simplicial structure of the model E^{(n)}(X).

Lemma 4.1. Let S be a smooth k-scheme, W ⊂ S × 𝔸^{1} a closed subset such that p : W → S is finite. Let E ∈ Spt_{S^{1}}(k) be quasi-fibrant. Then the map induced by the inclusion i : W → p^{−1}(p(W)) induces the zero map

i_{∗} : π_{∗}(E^{W} (S × 𝔸^{1})) → π_{∗}(E^{p^{−1}(p(W))}(S × 𝔸^{1})).

Proof. We steal a proof of Morel’s: Let Z = p(W), and let j_{0} : S ×𝔸^{1} → S ×ℙ^{1} be the standard open neighborhood of S × 0 in S × ℙ^{1}. Since W is finite over S, W is closed in S × ℙ^{1}, so we have the following commutative diagram

(4.1)  \xymatrix{π_{r}(E^{W} (S × ℙ^{1})) \ar[r]^{ī_{∗}} \ar[d]_{j_{0}^{∗}} & π_{r}(E^{Z×ℙ^{1}}(S × ℙ^{1})) \ar[d]^{j_{0}^{∗}} \\ π_{r}(E^{W} (S × 𝔸^{1})) \ar[r]_{i_{∗}} & π_{r}(E^{Z×𝔸^{1}}(S × 𝔸^{1})),}

where ī : W → Z × ℙ^{1} is the inclusion. Let i_{∞} : S → S × ℙ^{1} be the infinity section. Since W ∩ S × ∞ = ∅, the composition

π_{r}(E^{W} (S × ℙ^{1})) ⟶^{ī_{∗}} π_{r}(E^{Z×ℙ^{1}}(S × ℙ^{1})) ⟶^{i_{∞}^{∗}} π_{r}(E^{Z×∞}(S × ∞))

is the zero map. Letting j_{∞} : S × 𝔸^{1} → S × ℙ^{1} be the standard open neighborhood of S × ∞ in S × ℙ^{1}, the restriction map

i_{∞}^{∗} : π_{r}(E^{Z×𝔸^{1}}(S × 𝔸^{1})) → π_{r}(E^{Z×∞}(S × ∞))

is an isomorphism, hence

j_{∞}^{∗} ∘ ī_{∗} : π_{r}(E^{W} (S × ℙ^{1})) → π_{r}(E^{Z×𝔸^{1}}(S × 𝔸^{1}))

is the zero map. Write j̃_{∞} for the inclusions of S × ℙ^{1} \ {0, ∞} into j_{0}(S × 𝔸^{1}) and j̃_{0} for the inclusions of S × ℙ^{1} \ {0, ∞} into j_{∞}(S × 𝔸^{1}). Combining (4.1) with the commutativity of the diagram

\xymatrix{π_{r}(E^{W} (S × ℙ^{1})) \ar[r]^{j_{∞}^{∗}∘ī_{∗}} \ar[d]_{j_{0}^{∗}} & π_{r}(E^{Z×𝔸^{1}}(S × 𝔸^{1})) \ar[d]^{j̃_{0}^{∗}} \\ π_{r}(E^{W} (S × 𝔸^{1})) \ar[r]_{j̃_{∞}^{∗}∘i_{∗}} & π_{r}(E^{Z×ℙ^{1}\\{0,∞\}}(S × ℙ^{1} \ \{0, ∞\}))}

we see that j̃_{∞}^{∗} ∘ i_{∗} = 0. From the long exact localization sequence

… → π_{r}(E^{Z×0}(S × 𝔸^{1})) ⟶^{i_{0∗}} π_{r}(E^{Z×𝔸^{1}}(S × 𝔸^{1}))

⟶^{j̃_{∞}^{∗}} π_{r}(E^{Z×𝔸^{1}\{0}}(S × 𝔸^{1} \ {0})) → …

we see that

i_{∗}(π_{r}(E^{W} (S × 𝔸^{1}))) ⊂ i_{0∗}(π_{r}(E^{Z×0}(S × 𝔸^{1}))) ⊂ π_{r}(E^{Z×𝔸^{1}}(S × 𝔸^{1})).

But i_{0∗} : π_{r}(E^{Z×0}(S × 𝔸^{1})) → π_{r}(E^{Z×𝔸^{1}}(S × 𝔸^{1})) is the zero map, since the map i_{1}^{∗} : π_{r}(E^{Z×𝔸^{1}}(S × 𝔸^{1})) → π_{r}(E^{Z×1}(S × 1)) is an isomorphism and i_{1}^{∗} ∘ i_{0∗} = 0. □

Lemma 4.2. Suppose F is infinite. Take W ∈ S_{F}^{(n)}(p) and suppose codim_{Δ_{F}^{p}} (W) > n. Then the canonical map E^{W} (Δ_{F}^{p}) → E^{(n)}(Spec F, p) induces the zero map on π_{∗}.

Proof. We identify Δ^{p} with 𝔸^{p} via the barycentric coordinates t_{1}, …, t_{p}. Suppose W has dimension d < p − n. Then d ≤ p − 1 and, as F is infinite, a general linear projection L : 𝔸^{p} → 𝔸^{p−1} restricts to W to a finite morphism W → 𝔸^{p−1}. In addition, W′ := L^{−1}(L(W)) is in S_{F}^{(n)}(p) for L suitably general. Letting i : W → W′ be the inclusion, it suffices to show that the map

i_{∗} : π_{∗}E^{W} (Δ_{F}^{p}) → π_{∗}E^{W′}(Δ_{F}^{p})

is the zero map. Via an affine linear change of coordinates on Δ^{p}, we may identify Δ^{p} with 𝔸^{p−1}×𝔸^{1} and L : 𝔸^{p} → 𝔸^{p−1} with the projection 𝔸^{p−1}×𝔸^{1} → 𝔸^{p−1}. The result thus follows from lemma 4.1. □

Let (Δ_{F}^{p} , ∂Δ^{p})^{(n)} be the set of codimension n points w of Δ_{F}^{p} such that \overline{\{w\}} is in S_{F}^{(n)}(p).

Lemma 4.3. Let F be an infinite field. Then the restriction maps

E^{W} (Δ_{F}^{p}) → ⊕_{w∈(Δ_{F}^{p} ,∂Δ^{p})^{(n)}∩W} E^{w}(Spec O_{Δ_{F}^{p} ,w})

for W ∈ S_{F}^{(n)}(p) defines an injection

π_{r}(E^{(n)}(F, p)) → ⊕_{w∈(Δ_{F}^{p} ,∂Δ^{p})^{(n)}}π_{r}E^{w}(Spec O_{Δ_{F}^{p} ,w})

for each r ∈ ℤ.

Proof. Take W ∈ S_{F}^{(n)}(p). Since Δ_{F}^{p} is affine, we can find a W′ ∈ S_{F}^{(n)}(p) of pure codimension n with W′ ⊃ W: just take a sufficiently general collection of n functions f_{1}, …, f_{n} vanishing on W and let W′ be the common zero locus of the f_{i}. Thus the set of pure codimension n subsets W′ of Δ_{F}^{p} with W′ ∈ S_{F}^{(n)}(p) is cofinal in S_{F}^{(n)}(p).

Let W ∈ S_{F}^{(n)}(p) have pure codimension n on Δ_{F}^{p} and let W_{0} ⊂ W be any closed subset. Then W_{0} is also in S_{F}^{(n)}(p) and we have the long exact localization sequence

… → π_{r}E^{W_{0}}(Δ_{F}^{p}) ⟶^{i_{W_{0}∗}} π_{r}E^{W} (Δ_{F}^{p}) → π_{r}E^{W \W_{0}}(Δ_{F}^{p} \ W_{0}) → …

Let S_{F}^{(n)}(p)_{0} ⊂ S_{F}^{(n)}(p) be the set of all W_{0} ∈ S_{F}^{(n)}(p) with codim_{Δ_{F}^{p}} W_{0} > n. Let

E^{(n)}(F, p)_{0} = hocolim_{W_{0}∈S_{F}^{(n)}(p)_{0}} E^{W_{0}}(Δ_{F}^{p}).

Passing to the limit over the above localization sequences gives us the long exact sequence

… → π_{r}E^{(n)}(F, p)_{0} ⟶^{i_{0∗}} π_{r}E^{(n)}(F, p)

→ ⊕_{w∈(Δ_{F}^{p} ,∂Δ^{p})^{(n)}}π_{r}E^{w}((Spec O_{Δ_{F}^{p} ,w})) → …

By lemma 4.2, the map i_{0∗} is the zero map, which proves the lemma. □

Let S : Δ^{op} → Spt be a simplicial spectrum, |S| = hocolim_{Δ^{op}} S ∈ Spt the associated spectrum, giving us the spectral sequence

E_{p,q}^{1} = π_{q}S(p) ⟹ π_{p+q}|S|.

This spectral sequence induces an increasing filtration Fil_{∗}^{simp} π_{r}|S| on π_{r}|S|. We have the q-truncated simplicial spectrum S_{≤q} and Fil_{q}^{simp} π_{r}|S| is just the image of π_{r}|S_{≤q}| in π_{r}|S|. In particular Fil_{−1}^{simp} π_{r}|S| = 0 and ∪_{q=0}^{∞} Fil_{q}^{simp} π_{r}|S| = π_{r}S, so the spectral sequence is weakly convergent, and is strongly convergent if for instance there is an integer q_{0} such that S(p) is q_{0}-connected for all p. The isomorphism of theorem 3.1 thus gives us the weakly convergent spectral sequence

(4.2)  E_{p,q}^{1}(X, E, n) = π_{q}E^{(n)}(X, p) ⟹ π_{p+q}f_{n}E(X)

which is strongly convergent if π_{q}E^{(n)}(X, p) = 0 for q ≤ q_{0}, independent of p. This defines the increasing filtration Fil_{∗}^{simp}(E)π_{r}f_{n}E(X) of π_{r}f_{n}E(X) with associated graded gr_{p}^{simp}(E)π_{r}f_{n}E(X) = E_{p,r−p}^{∞}.

Lemma 4.4. Suppose that k is infinite and that Π_{a,∗}E(K) = 0 for a < 0 and all fields K over k. Let F ⊃ k be a field extension of k. Then

(1) E_{p,r−p}^{1}(F, E, n) = 0 for p > r + n and Fil_{r+n}^{simp}(E)π_{r}f_{n}E(F) = π_{r}f_{n}E(F).

(2) E_{p,q}^{1}(F, E, n) is isomorphic to a subgroup of ⊕_{w∈(Δ_{F}^{p} ,∂Δ^{p})^{(n)}}Π_{q+n,n}E(F (w)).

(3) The spectral sequence (4.2) is strongly convergent.

Proof. Since the spectral sequence is weakly convergent, to prove (3) it suffices to show that E_{p,q}^{1}(F, E, n) = 0 for q < −n and for (1) it suffices to show that E_{p,r−p}^{1} = 0 for p > r +n. These both follows from (2) as our hypothesis implies that Π_{a,∗}E(F (w)) = 0 for a < 0, w ∈ Δ_{F}^{p}. For (2), lemma 4.3 gives us an inclusion

E_{p,q}^{1} = π_{q}E^{(n)}(F, p) ⊂ ⊕_{w∈(Δ_{F}^{p} ,∂Δ^{p})^{(n)}}π_{q}E^{w}(Spec O_{Δ_{F}^{p} ,w}).

Take w ∈ (Δ_{F}^{p} , ∂Δ^{p})^{(n)}. By the Morel-Voevodsky purity isomorphism [18, loc.cit.], we have E^{w}(Spec O_{Δ_{F}^{p} ,w}) ≅ Hom(Σ_{S^{1}}^{n}Σ_{𝔾_{m}}^{n}w_{+}, E), hence

π_{q}E^{w}(Spec O_{Δ_{F}^{p} ,w}) ≅ Π_{q+n,n}E(F (w)),

which proves (2). □

For a field extension K of k, we write tr. dim_{k}K for the transcendence dimension of K over k.

Lemma 4.5. Let E be in SH_{S^{1}}(k) and suppose Π_{a,∗}E(K) = 0 for a < 0 and all fields K over k. Let p be the exponential characteristic of k. Let r be an integer. Suppose we have functions

(d, q) ↦ N_{j}(d, q; E) ≥ 0; d, q ≥ 0, j = 0, …, r − 1,

such that, for each field extension K of k with tr. dim_{k}K ≤ d, each j = 0, …, r − 1, and all integers q, M ≥ 0, m ≥ N_{j}(d, q; E), we have

(4.3)  F_{Tate}^{m} Π_{j,q}f_{M}E(K)[1/p] = 0.

Let F be a field extension of k with tr. dim_{k}F ≤ d and fix an integer n ≥ 0. For r > 0, let N = max_{j=0}^{r−1}N_{j}(r − j + d, n; E); for r ≤ 0, set N = 0. Then for all integers m ≥ N, n ≥ 0, , we have

π_{r}(f_{n}(f_{m}E))(F))[1/p] = Fil_{n}^{simp}(f_{m}E)π_{r}(f_{n}(f_{m}E))(F)[1/p].

Proof. We delete the “[1/p]" from the notation in the proof, using the convention that we have inverted the exponential characteristic p throughout. If k is a finite field, fix a prime ℓ and let k_{ℓ} be the union of all ℓ-power extensions of k. If we know the result for k_{ℓ}, then using proposition A.1(2) for each k ⊂ k′ ⊂ k_{ℓ} with k′ finite over k proves the result for k, after inverting ℓ. Doing the same for some ℓ′ ≠ ℓ, we reduce to the case of an infinite field k. By [12, proposition 3.2], the hypothesis Π_{a,∗}E(K) = 0 for a < 0 and all K implies Π_{a,∗}f_{m}E(K) = 0 for a < 0, all K, and all m ≥ 0. In particular, π_{r}(f_{n}(f_{m}E))(F) = 0 for r < 0 and all n, m ≥ 0, so for r < 0, the lemma is trivially true. We therefore assume r ≥ 0. In addition, it follows from lemma 4.4(1) that the spectral sequence (4.2), E_{∗,∗}^{∗}(F, f_{m}E, n), is strongly convergent for all m, n ≥ 0 and E_{p,r−p}^{1}(F, f_{m}E, n) = 0 for p > r + n. Since Fil_{∗}^{simp}(f_{m}E)π_{r}(f_{n}(f_{m}E))(F) is by definition the filtration on π_{r}(f_{n}(f_{m}E))(F) induced by this spectral sequence, we need only show that

E_{p,r−p}^{1}(F, f_{m}E, n) = 0 for n < p ≤ r + n and m ≥ N.

In particular, the result is proved for r = 0; we now assume r > 0.

Let p be an integer, n < p ≤ r + n. By lemma 4.4(2),

E_{p,r−p}^{1}(F, f_{m}E, n) ⊂ ⊕_{w∈(Δ_{F}^{p} ,∂Δ^{p})^{(n)}}Π_{r−p+n,n}f_{m}E(F (w)).

For w ∈ (Δ_{F}^{p} , ∂Δ^{p})^{(n)}, w has codimension n on Δ_{F}^{p}, hence tr. dim_{F} F (w) = p−n and thus tr. dim_{k}F (w) ≤ p−n+d. We have m ≥ N_{r−p+n}(p−n+d, n; E) since 0 ≤ r − p + n < r, and so our hypothesis (4.3) implies

F_{Tate}^{m}Π_{r−p+n,n}f_{m}E(F (w)) = 0.
But F_{Tate}^{m}Π_{a,b}f_{m}E(F (w))  =  im(Π_{a,b}f_{m}f_{m}E(F (w))  ⟶^{ρ_{m}(f_{m}E)_{∗}}

Π_{a,b}f_{m}E(F (w))) by definition, and ρ_{m}(f_{m}E) is an isomorphism (lemma 2.1), hence

F_{Tate}^{m}Π_{a,b}f_{m}E(F (w)) = Π_{a,b}f_{m}E(F (w)).

Thus Π_{r−p+n,n}f_{m}E(F (w)) = 0 and hence E_{p,r−p}^{1} = 0 for n < p ≤ r + n, as desired. □

5. The bottom of the filtration

In this section, k will be a fixed perfect base field. We study the subgroup Fil_{n}^{simp}(E)π_{r}f_{n}E(F) isolated in lemma 4.5.

Lemma 5.1. Let E be in SH_{S^{1}}(k). Then Fil_{n−1}^{simp}(E)π_{r}f_{n}E(F) = 0 for all fields F over k.

Proof. For any X ∈ Sm/k, Fil_{q}^{simp} π_{r}E^{(n)}(X) is by definition the image of

π_{r}|E^{(n)}(X, − ≤ q)| → π_{r}E^{(n)}(X),

where E^{(n)}(X, − ≤ q) is the q-truncation (or q-skeleton) of E^{(n)}(X, −). For X = Spec F, we clearly have S_{F}^{(n)}(p) = ∅ for p < n, as Δ_{F}^{p} has no closed subsets of codimension > p. Thus |E^{(n)}(X, − ≤ q)| is the 0-spectrum for q < n and hence Fil_{n−1}^{simp} π_{r}E^{(n)}(F) = 0. □

To study the first non-zero layer Fil_{n}^{simp}(E)π_{r}f_{n}E(F) in Fil_{∗}^{simp}(E)π_{r}f_{n}E(F), we apply the results of [12]. For this, we recall some of these results and constructions. We let V_{n} = (Δ_{F}^{1} \∂Δ^{1})^{n}. The function −t_{1}/t_{0} on Δ^{1} gives an open immersion ρ_{n} : V_{n} → 𝔸_{F}^{n}, identifying V_{n} with (𝔸_{F}^{1} \ {0, 1})^{n}. Suppose that E is an n-fold T -loop spectrum, that is, there is an object ω_{T}^{−n}E ∈ Spt(k) and an isomorphism E ≅ Ω_{T}^{n} ω_{T}^{−n}E in SH_{S^{1}}(k). Given an n-fold delooping ω_{T}^{−n}E of E , we have explained in [12, §5] how to construct a “transfer map”

Tr_{F (w)/F}^{∗} : π_{∗}E(w) → π_{∗}E(F),

for each closed point w ∈ 𝔸_{F}^{n}, separable over F. If now E = Ω_{T}^{∞}E for some T -spectrum E ∈ SH(k), then the bi-graded homotopy sheaves Π_{∗,∗}E admit a canonical right action by the bi-graded homotopy sheaves of the sphere spectrum 𝕊_{k} ∈ SH(k):

Π_{a,b}E ⊗ Π_{p,q}(𝕊_{k}) → Π_{a+p,b+q}E

Hopkins and Morel [17, theorem 6.3.3] have defined a graded ring homomorphism

(5.1)  θ_{∗}(F) : ⊕_{n∈ℤ}K_{n}^{MW} (F) → ⊕_{n∈ℤ}Π_{0,−n}𝕊_{k}(F),

natural in the field extension F of k.^{5} Via θ_{∗}(F), the right Π_{p,q}(𝕊_{k})-action gives a right action, natural in F,

Π_{a,b}E(F) ⊗ K_{∗}^{MW} (F) → Π_{a,b−∗}E(F).

This gives us the filtration F_{MW}^{n} Π_{a,b}E(F) of Π_{a,b}E, defined by

F_{MW}^{n} Π_{a,b}E(F) := im[Π_{a,b+n}E(F) ⊗ K_{n}^{MW} (F) → Π_{a,b}E(F)]; n ≥ 0.

Completing with respect to the transfer maps yields the filtration F_{MW^{T r}}^{n}Π_{a,b}E(F):

Definition 5.2 (see [12, definition 7.9]). Let E = Ω_{T}^{∞}E for some E ∈ SH(k), F a field extension of k. Take integers a, b, n with n, b ≥ 0. Let F_{MW^{T r}}^{n}Π_{a,b}E(F) denote the subgroup of Π_{a,b}E(F) generated by elements of the form

T r_{F} (w)^{∗}(x); x ∈ F_{MW}^{n} Π_{a,b}E(F (w))

as w runs over closed points of 𝔸_{F}^{n}, separable over F. We write F_{MW^{T r}}^{∗}π_{r}E(F) for the filtration F_{MW^{T r}}^{∗}Π_{r,0}E(F) on π_{r}E(F) = Π_{r,0}E(F)

Theorem 7.11 of [12] expresses the “Tate-Postnikov” filtration F_{Tate}^{∗}π_{0}E(F) on π_{0}E(F) in terms of the “Milnor-Witt” filtration F_{MW}^{∗} π_{0}E(F), under the connectivity assumption Π_{a,∗}E = 0 for a < 0. With some minor changes, the proof of this result goes through to show:

Theorem 5.3. Let E be in SH(k) and let E = Ω_{T}^{∞}E. Suppose that Π_{a,∗}E = 0 for a < 0. Let F be a perfect field extension of k. Then for all r ≥ 0, n ≥ 0, we have an equality of subgroups of Π_{r,q}E(F):

ρ_{n}(E)(Fil_{n}^{simp}(E)π_{r}f_{n}E(F)) = F_{MW^{T r}}^{n}π_{r}E(F)

Sketch of proof. We briefly indicate the changes that need to be made in the arguments for theorem 7.11 loc. cit.: In [12, prop. 4.3 and thm. 7.6], replace π_{0} with π_{r} and F_{Tate}^{n}π_{0}(E)(F) with ρ_{n}(E)(Fil_{n}^{simp}(E)π_{r}f_{n}E(F)). Also, instead of using the fact that θ_{∗}(F) is an isomorphism, we need only use that θ_{∗}(F) is a ring homomorphism, and that for u ∈ F^{×}, θ_{1}([u]) is the element of Π_{0,−1}𝕊_{k}(F) coming from the map u : Spec F → 𝔾_{m} corresponding to u (see [17, theorem 6.3.3]). □

5A result of Morel [14, corollary 6.41] implies that θ_{∗}(F) is an isomorphism for all fields F, but we will not need this.

6. Finite spectra and cohomologically finite spectra

In this section we will have occasion to use some localizations of SH(k) with respect to multiplicatively closed subsets of ℤ \ {0}. To make our discussion precise, we have collected notations and some elementary facts concerning such localizations in Appendix B; we will use these without further mention in this section. For a field F, let F^{sep} denote the separable closure of F and G_{F} the absolute Galois group Gal(F^{sep}/F). For p a prime, let cd_{p}(G_{F}) denote the pcohomological dimension of the profinite group G_{F}, as defined in [26, I, §3.1]. We write cd_{p}(F) for cd_{p}(G_{F}), and call cd_{p}(F) the p-cohomological dimension of F. The cohomological dimension of F, cd(F), is the supremum of the cd_{p}(F) over all primes p. We turn to our main topic in this section: the study of finite and cohomologically finite objects in SH(k).

Definition 6.1. 1. Let SH(k)_{fin}(k) be the thick subcategory of SH(k) generated by objects Σ_{T}^{n} Σ_{T}^{∞}X_{+} for X a smooth projective k-scheme and n ∈ ℤ.

2. Let SH(k)_{coh.fin}(k) be the full subcategory of SH(k) with objects those E such that

(i) there is an integer d such that, for n > d, Π_{r,n}(E)_{ℚ} = 0 for all r, (ii) there is an integer c such that Π_{r,q}E = 0 for r ≤ c, q ∈ ℤ.

3. For E ∈ SH(k)_{coh.fin}(k), define d(E) to be the infimum among integers d such that Π_{r,n}(E)_{ℚ} = 0 for all r and for all n > d.

4. We say that E ∈ SH(k) is topologically c-connected if Π_{r,∗}E = 0 for r ≤ c. We let c(E) be the supremum among integers c such that E is topologically c-connected.

5. For E ∈ SH_{S^{1}}(k), we say that E is topologically c-connected if Π_{r,n}E = 0 for r ≤ c and n ≥ 0; we say that E is c-connected if π_{r}E = 0 for r ≤ c.

Remarks 6.2. 1. SH(k)_{coh.fin}(k) is a thick subcategory of SH(k).

2. As Σ_{T}^{n} Σ_{T}^{∞}X_{+} is compact for all n ∈ ℤ, X ∈ Sm/k, SH(k)_{fin}(k) is contained in the category SH(k)^{c} of compact objects in SH(k). If k admits resolution of singularities, it is not hard to show that Σ_{T}^{n} Σ_{T}^{∞}X_{+} is in SH(k)_{fin}(k) for all X ∈ Sm/k, n ∈ ℤ; as these objects (and their translates) form a set of compact generators for SH(k), it follows that SH(k)_{fin}(k) = SH(k)^{c} if k admits resolution of singularities, that is, if k has characteristic zero.

Remark 6.3. For a presheaf A of abelian groups on Sm/k, we have the presheaf A_{−1} defined by

A_{−1}(U) := ker(i_{1}^{∗} : A(U × 𝔸^{1} \ {0}) → A(U)).

Let E be in SH_{S^{1}}(k). By [17, lemma 4.3.11], there is a natural isomorphism π_{n}(Ω_{𝔾_{m}}E) ≅ π_{n}(E)_{−1}. In particular, if E is c-connected, then so is Ω_{𝔾_{m}}E. As Π_{a,b}E = Π_{a,b−1}Ω_{𝔾_{m}}E and Π_{r,0} = π_{r}, we see that E is c-connected if and only if E is topologically c-connected.

Lemma 6.4. Take E ∈ SH_{fin}(k). Then there is an integer n(E) such that ρ_{n}(E) : f_{n}(E) → E is an isomorphism for all n ≤ n(E).

Proof. As f_{n} is exact, it suffices to prove the result for E = Σ_{T}^{m}Σ_{T}^{∞}X_{+} with X smooth and projective over k and m ∈ ℤ. As f_{n} ∘ Σ_{T}^{m} = Σ_{T}^{m} ∘ f_{n−m}, we need only prove the result for E = Σ_{T}^{∞}X_{+}. But Σ_{T}^{∞}X_{+} is in SH^{eff}(k), so ρ_{n}(Σ_{T}^{∞}X_{+}) is an isomorphism for all n ≤ 0. □

Lemma 6.5. Let E be in SH(k)_{coh.fin}(k).

1. For U ∈ Sm/k, we have Hom_{SH(k)_{ℚ}}((Σ_{S^{1}}^{p}Σ_{𝔾_{m}}^{q}Σ_{T}^{∞}U_{+})_{ℚ}, E_{ℚ}) = 0 for all p ∈ ℤ, q > d(E).

2. For all n > d(E), (f_{n}E)_{ℚ} ≅ 0 in SH(k)_{ℚ}.

Proof. Let i_{n} : Σ_{T}^{n} SH^{eff}(k) → SH(k) be the inclusion, r_{n} the right adjoint to i_{n}; recall that f_{n} := i_{n}∘r_{n}. By example B.5, the ℚ-localization Σ_{T}^{n} SH^{eff}(k)_{ℚ} of Σ_{T}^{n} SH^{eff}(k) is the localizing subcategory of SH(k)_{ℚ} generated by the objects (Σ_{𝔾_{m}}^{q}Σ_{T}^{∞}U_{+})_{ℚ}, q ≥ n, U ∈ Sm/k, the inclusion Σ^{n}SH^{eff}(k)_{ℚ} → SH(k)_{ℚ} is given by i_{nℚ}, the right adjoint to i_{nℚ} is r_{nℚ} and f_{nℚ}(E_{ℚ}) = (f_{n}E)_{ℚ} for all E ∈ SH(k). Using this, we see that (1) implies (2), as (1) implies that for F ∈ Σ_{T}^{n} SH^{eff}(k)_{ℚ}, n > d(E),

Hom_{SH(k)_{ℚ}}(F, E_{ℚ}) = 0.

Since (f_{n}E)_{ℚ} → E_{ℚ} is universal for maps F → E_{ℚ} with F ∈ Σ_{T}^{n} SH^{eff}(k)_{ℚ}, it follows that (f_{n}E)_{ℚ} = 0 for n > d(E). For (1), by lemma B.2(2), we have

Hom_{SH(k)_{ℚ}}((Σ_{S^{1}}^{p}Σ_{𝔾_{m}}^{q}Σ_{T}^{∞}U_{+})_{ℚ}, E_{ℚ}) = Hom_{SH(k)}(Σ_{S^{1}}^{p}Σ_{𝔾_{m}}^{q}Σ_{T}^{∞}U_{+}, E)_{ℚ}.

Thus, tensoring the usual Gersten-Quillen spectral sequence with ℚ gives the strongly convergent spectral sequence

E_{1}^{a,b} = ⊕_{u∈U^{(a)}}Π_{−b,q+a}E(k(u))_{ℚ} ⟹ Hom_{SH(k)_{ℚ}}((Σ_{S^{1}}^{−a−b}Σ_{𝔾_{m}}^{q}Σ_{T}^{∞}U_{+})_{ℚ}, E_{ℚ}),

concentrated in the range 0 ≤ a ≤ dim U. The assumption q > d(E) implies that E_{1}^{a,b} = 0 for all a, b, proving (1). □

Lemma 6.6. Let E be in SH(k)_{coh.fin}(k). Then f_{n}E is in SH(k)_{coh.fin}(k), d(f_{n}E) ≤ d(E) and c(f_{n}E) ≥ c(E), for all n ∈ ℤ.

Proof. We have shown in [12, proposition 3.2] that, for E ∈ SH_{S^{1}}(k), if E is topologically −1-connected, then f_{p}E is also topologically −1-connected for all p. For E := Ω_{T}^{∞}Σ_{S^{1}}^{−c−1}Σ_{𝔾_{m}}^{q}E, we have

Π_{r−c−1,m+q}f_{n+q}E ≅ Π_{r,m+q}f_{n+q}Σ_{𝔾_{m}}^{q}E ≅ Π_{r,m+q}Σ_{𝔾_{m}}^{q}f_{n}E ≅ Π_{r,m}f_{n}E

for m ≥ −q. Similarly, Π_{r−c−1,m+q}E ≅ Π_{r,m}E for m ≥ −q. Thus, if Π_{r,m}E = 0 for r ≤ c and m ≥ −q, the same holds for f_{n}E. As q was arbitrary, we see that if E is topologically c-connected, so is f_{n}E for all n, that is, c(f_{n}E) ≥ c(E). We have already seen in lemma 6.5 that f_{n}E_{ℚ} = 0 for n > d(E), hence f_{n}E is in SH(k)_{coh.fin}(k) and d(f_{n}E) = −∞ for n > d(E). For n ≤ d(E), take m > d(E) ≥ n. Then Π_{r,m}(f_{n}E) ≅ Π_{r,m}(E), hence Π_{r,m}(f_{n}E)_{ℚ} = 0. Thus f_{n}E is in SH(k)_{coh.fin}(k) and d(f_{n}E) ≤ d(E). □

In order to proceed further with our study of SH_{fin}(k) and SH(k)_{coh.fin}(k), we need to recall Morel’s decomposition of SH(k)[\frac{1}{2}]. We recall the ring homomorphism (5.1)

θ_{k} := θ_{0}(k) : K_{0}^{MW} (k) → End_{SH(k)}(𝕊_{k}).

In addition, Morel [14, lemma 3.10] has defined an isomorphism of rings K_{0}^{MW} (k) ≅ GW(k), with 1 + η · [u] corresponding to the one-dimensional form ux^{2} if char. k ≠ 2. Morel [17, section 6.1] has considered the action of ℤ/2 on the sphere spectrum 𝕊_{k} arising from the exchange of factors

τ : T ∧ T → T ∧ T.

We also write τ for the induced automorphism of the sphere spectrum 𝕊_{k}. Morel [17, remark 6.3.5] identifies the corresponding element of End(𝕊_{k}) (denoted −ϵ in loc. cit.) as

(6.1)  τ := θ_{k}(1 + η · [−1]).

After inverting 2, the action of τ decomposes 𝕊_{k}[\frac{1}{2}] into its +1 and −1 eigenspaces

𝕊_{k}[\frac{1}{2}] = 𝕊_{k}^{+}[\frac{1}{2}] ⊕ 𝕊_{k}^{−}[\frac{1}{2}];

as 𝕊_{k}[\frac{1}{2}] is the unit in the tensor category SH(k)[\frac{1}{2}], this induces a decomposition of SH(k)[\frac{1}{2}] as

SH(k)[\frac{1}{2}] = SH(k)^{+} × SH(k)^{−}.

This extends to a decomposition of SH(k)_{ℚ} as SH(k)_{ℚ}^{+} × SH(k)_{ℚ}^{−}. For an object E of SH(k), we write the corresponding factors of E_{ℚ} as E_{ℚ}^{+}, E_{ℚ}^{−}.

Lemma 6.7. Suppose that either (1) char. k = 0 and cd_{2}(k) < ∞, or

(2) char. k > 0.

Then 2^{N+1} · η = 0, where in case (1), N = cd_{2}(k), and in case (2), N = 0 if char. k = 2 or if char. k = p is odd and p ≡ 1 mod 4, N = 1 if p ≡ 3 mod 4. Moreover, letting I ⊂ GW(k) be the augmentation ideal, we have I^{n} = 0 for n > cd_{2}(k), assuming k has characteristic ≠ 2.

Proof. Suppose that char. k ≠ 2. Sending (a_{1}, …, a_{n}) ∈ (k^{×})^{n} to the n-fold Pfister form <<a_{1}, …, a_{n}>> descends to a well-defined surjective homomorphism p_{n} : K_{n}^{M}(k)/2 → I^{n}/I^{n+1} [13, §4.1]; in particular 2I^{n} ⊂ I^{n+1} for all n ≥ 0. By the Milnor conjecture [29, theorem 6.6 and theorem 7.4] and [19, theorem 4.1], p_{n} is an isomorphism, and induces an isomorphism

I^{n}/I^{n+1} ≅ H_{ét}^{n}(k, μ_{2}^{⊗n}).

Thus I^{n} = I^{n+1} for n ≥ N + 1, where N := cd_{2} k. By the theorem of Arason-Pfister [1, Korollar 1], ∩_{n}I^{n} = {0}, hence I^{N+1} = 0, and thus 2^{N+1} kills the Witt group W (k). As a K_{0}^{MW} (k)-module, K_{−1}^{MW} (k) is cyclic with generator η. However,

K_{−1}^{MW} (k) ≅ W (k)

by [14, lemma 3.10], and thus 2^{N+1}η = 0. This handles the case (1). In case k has characteristic p > 0, then as η comes from base extension from the prime field 𝔽_{p}, it suffices to show that 2^{N+1}η = 0 in SH(𝔽_{p}), with N as in the statement of the lemma. For p odd, we have GW(𝔽_{p}) ≅ K_{0}^{MW} (𝔽_{p}), with 1 + η[−1] corresponding to the form −x^{2}; the hyperbolic form x^{2}−y^{2} corresponds to 2+η[−1]. If p ≡ 1 mod 4, then −1 is a square, hence −x^{2} and x^{2} are isometric forms and 1 + η[−1] = 1 in K_{0}^{MW} (𝔽_{p}). The relation η(2 + η[−1]) = 0 in K_{−1}^{MW} (𝔽_{p}) thus simplifies to 2η = 0. If p ≡ 3 mod 4, then −1 is a sum of two squares, and hence the quadratic form x^{2} + y^{2} + z^{2} is isotropic. Thus the Pfister form x^{2} + y^{2} + z^{2} + w^{2} is also isotropic and hence hyperbolic, and hence the form −x^{2} − y^{2} − z^{2} − w^{2} is hyperbolic as well. Translating this back to K_{0}^{MW} (𝔽_{p}) gives the relation 4(1 + η[−1]) = 2(2 + η[−1]) or 2η[−1] = 0. Combining this with relation η(2 + η[−1]) = 0 yields 4η = 0. In case k has characteristic 2, −1 = +1. The relation η(2+η[−1]) = 0 simplifies to 2η = 0 (as [1] = 0 in K_{1}^{MW} (F) for all fields F [14, lemma 3.5]). □

Lemma 6.7 and (6.1) imply:

Lemma 6.8. Suppose that k has finite 2-cohomological dimension or that char. k > 0. Then SH(k)[\frac{1}{2}] = SH(k)^{+}.

Proposition 6.9. Take X ∈ Sm/k. Then

1. Π_{r,n}(Σ_{T}^{∞}X_{+}) = 0 for r < 0 and n ∈ ℤ

2. Suppose X is smooth and projective over k of dimension d and that k has finite 2-cohomological dimension or char. k > 0. Then Π_{r,n}(Σ_{T}^{∞}X_{+})_{ℚ} = 0 for n > d, r ∈ ℤ.

3. Suppose k has finite 2-cohomological dimension or char. k > 0. Then each E in SH(k)_{fin}(k) is also in SH(k)_{coh.fin}(k).

Proof. Noting that SH(k)_{coh.fin}(k) is a thick subcategory of SH(k), and that

Π_{r,n}(Σ_{T}^{m}Σ_{T}^{∞}X_{+}) = Π_{r−m,n−m}(Σ_{T}^{∞}X_{+}),

we see that (3) follows from (1) and (2). We first prove (1). We have the S^{1}-spectrum Σ_{S^{1}}^{∞}X_{+} ∈ SH_{S^{1}}(k). By [8, theorem 10] we have

Π_{r,n}(Σ_{T}^{∞}X_{+}) = \underrightarrow{lim}_{b} Π_{r,n+b}Σ_{S^{1}}^{∞}Σ_{𝔾_{m}}^{b}X_{+}

so it suffices to see that Σ_{S^{1}}^{∞}Σ_{𝔾_{m}}^{b}X_{+} is topologically -1 connected for all b ≥ 0. By remark 6.3, we need only see that Σ_{S^{1}}^{∞}Σ_{𝔾_{m}}^{b}X_{+} is -1 connected. This follows from Morel’s S^{1}-stable 𝔸^{1}-connectedness theorem [17, theorem 4.2.10]. For (2), lemma 6.8 and our assumption on k imply that SH(k)_{ℚ} = SH(k)_{ℚ}^{+}. By a result of Morel (proved in detail by Cisinski-Déglise [3, theorem 15.2.13]), there is an equivalence of triangulated categories SH(k)_{ℚ}^{+} ≅ DM(k)_{ℚ}, and hence

Hom_{SH(k)}(Σ_{S^{1}}^{a}Σ_{𝔾_{m}}^{b}Σ_{T}^{∞}U_{+},Σ_{T}^{∞}X_{+})_{ℚ}
≅ Hom_{SH(k)_{ℚ}^{+}}((Σ_{S^{1}}^{a}Σ_{𝔾_{m}}^{b}Σ_{T}^{∞}U_{+})_{ℚ}^{+}, (Σ_{T}^{∞}X_{+})_{ℚ}^{+})
≅ Hom_{DM(k)_{ℚ}}(M(U)(b)_{ℚ}[a + b], M(X)_{ℚ}),

where M : Sm/k → DM(k) is the canonical functor. Since M(U)(b) is compact in DM(k), we have

Hom_{DM(k)_{ℚ}}(M(U)(b)_{ℚ}[a + b], M(X)_{ℚ}) ≅ Hom_{DM(k)}(M(U)(b)[a + b], M(X))_{ℚ}.

As X is smooth and projective, we may use duality (via, e.g., duality in Chow motives over k and [33, V, prop. 2.1.4])

Hom_{DM(k)}(M(U)(b)[c], M(X)) ≅ Hom_{DM(k)}(M(U × X), ℤ(d − b)[2d − c])
= H^{2d−c}(U × X, ℤ(d − b)).

But for all Y ∈ Sm/k, H^{p}(Y, ℤ(q)) = 0 for q < 0, p ∈ ℤ, by [28, corollary 2]. Thus the presheaf

U ↦ Hom_{SH(k)}(Σ_{S^{1}}^{a}Σ_{𝔾_{m}}^{b}Σ_{T}^{∞}U_{+}, Σ_{T}^{∞}X_{+})_{ℚ}

is zero for b > d, and hence the associated sheaf Π_{a,b}(Σ_{T}^{∞}X_{+})_{ℚ} is zero as well. □

Remark 6.10. The statement of the result of Cisinski-Déglise [3, theorem 16.2.13] cited above is not the same as given here; Cisinski-Déglise show that the 𝔸^{1}-derived category over a base-scheme S, with ℚ-coefficients, D_{𝔸^{1},ℚ}(S), has plus part D_{𝔸^{1},ℚ}(S)^{+} equivalent to the category of “Beilinson motives” DM_{�} (S). For S geometrically unibranch, they show [3, theorem 16.1.4] that DM_{�} (S) ≅ DM(S)_{ℚ}. Furthermore, the well-known adjunction between simplicial sets and chain complexes extends to give an equivalence of SH(S)_{ℚ} with D_{𝔸^{1},ℚ}(S) (see e.g. [3, §5.3.35]). Putting these equivalences all together gives us the equivalence SH(k)_{ℚ}^{+} ≅ DM(k)_{ℚ} we use in the proof of proposition 6.9. Alternatively, one also can repeat the argument used by Cisinski-Déglise, replacing D_{𝔸^{1},ℚ}(S) with SH(k)_{ℚ}, and DM_{�} (S) with Ho Mℤ-Mod_{ℚ}, where Mℤ is the commutative monoid in symmetric motivic spectra over k defined in [25], and representing motivic cohomology in SH(k). This shows that the forgetful functor Ho Mℤ-Mod_{ℚ} → SH(k)_{ℚ} induces an equivalence Ho Mℤ-Mod_{ℚ} → SH(k)_{ℚ}^{+}. One can then use the theorem of Röndigs-Østvær [25, theorem 1.1, discussion preceding theorem 1.2], which gives an equivalence of Ho Mℤ-Mod_{ℚ} with DM(k)_{ℚ} for k a perfect field.

7. The proof of the convergence theorem

The following result combines with lemma 4.5 to form the heart of the proof of our main theorem.

Lemma 7.1. Let k be a perfect field. Suppose that k has finite cohomological dimension D_{k}. Let F be a perfect field extension of k with tr. dim_{k}F ≤ d. Let E = Ω_{T}^{∞}E for some E ∈ SH_{coh.fin}(k) and suppose that Π_{a,∗}E = 0 for a < 0. Consider the map

ρ_{n}(f_{M}E) : π_{r}f_{n}(f_{M}E)(F) → π_{r}f_{M}E(F).

Then, for all r, M ≥ 0, n > max(D_{k} + d, d(E)), we have

ρ_{n}(f_{M}E)(Fil_{n}^{simp}(f_{M}E)π_{r}f_{n}(f_{M}E)(F)) = 0.

Proof. Let A = max(D_{k} + d, d(E)) ≥ 0. By theorem 5.3, and the definition of the filtration F_{MW^{T r}}^{n}π_{r}f_{M}E(F), it suffices to show that, for every finite extension F′ of F, the product

∪ : Π_{r,n}f_{M}E(F′) ⊗ K_{n}^{MW} (F′) → Π_{r,0}f_{M}E(F′)

is zero if n > A. We note that f_{M}E ≅ Ω_{T}^{∞}f_{M}E for M ≥ 0 (lemma 2.2). Thus, by lemma 6.6, (Π_{r,n}f_{M}E)_{ℚ} = 0 for n > max(d(E), −1) and the image of ∪ is the same as the subgroup generated by the images of the maps

∪_{N} : Π_{r,n}f_{M}E(F′)_{N−tor} ⊗ K_{n}^{MW} (F′)/N → π_{r}f_{M}E(F′); N ∈ ℕ.

First suppose char. k ≠ 2. By Morel’s theorem [16, theorem 5.3], we have a cartesian diagram

\xymatrix{K_{n}^{MW} (F′) \ar[r] \ar[d] & K_{n}^{M}(F′) \ar[d] \\ I(F′)^{n} \ar[r] & I(F′)^{n}/I(F′)^{n+1}}

But as tr. dim_{k}F′ ≤ d, we have cd_{2}(F) ≤ cd(F) ≤ D_{k} + d [26, II, §4.2, proposition 11]. Thus, for n > D_{k} + d, lemma 6.7 tells us that K_{n}^{MW} (F′) = K_{n}^{M}(F′). Furthermore, for N prime to the characteristic, the Bloch-Kato conjecture^{6} gives the isomorphism

K_{n}^{M}(F′)/N ≅ H_{ét}^{n}(F′, μ_{N}^{⊗n}),

and hence K_{n}^{M}(F′)/N = 0 for n > D_{k} +d. For N = p = char. k, it follows from [2, theorem 2.1] that K_{n}^{M}(F′)/p = 0, since F′ is perfect. Thus, for n > D_{k} + d, K_{n}^{MW} (F′)/N = 0, and hence the image of ∪ is zero if n > A. If char. k = 2, then as F′ is perfect, each u ∈ F′ is a square. Thus, by [14, proposition 3.13], the quotient map K_{n}^{MW} (F′) → K_{n}^{M}(F′) is an isomorphism for n ≥ 0. The remainder of the discussion is the same. □

6Proved by Rost and Voevodsky [32]. For a presentation of some of Rost’s results that go into the proof of Bloch-Kato, see, e.g., [6, 27].

Relying on lemma 7.1, here is the main step in the proof of theorem 4:

Proposition 7.2. Let k be a perfect field of finite cohomological dimension and let p be the exponential characteristic of k. Then there is a function N_{k} : ℤ^{3} → ℤ, with N_{k}(e, r, d) ≥ 0 for all e, d, r, such that, given an integer e, and an E in SH_{coh.fin}(k) with c(E) ≥ −1 and d(E) ≤ e, we have

Fil_{Tate}^{N} Π_{r,q}f_{M}Ω_{T}^{∞}E(F)[1/p] = 0

for all r, M ∈ ℤ, all N ≥ N_{k}(e − q, r, d) + q, all q ≥ 0, and all field extension F of k with tr. dim_{k}F ≤ d.

Proof. For a field F, let F^{per} be the perfect closure of F. By proposition A.1, Fil_{Tate}^{N} Π_{a,b}G(F)[1/p] ≅ Fil_{Tate}^{N} Π_{a,b}G(F^{per})[1/p] for all G ∈ SH(k), so we may replace F with F^{per} whenever necessary. From our hypothesis c(E) ≥ −1 and lemma 6.6 follows c(f_{M}E) ≥ −1, so Π_{r,q}Ω_{T}^{∞}f_{M}E = Π_{r,q}f_{M}Ω_{T}^{∞}E = 0 for r < 0, q ≥ 0. Thus, taking N_{k}(e, r, d) = 0 for r < 0, and all d and e settles the cases r < 0, so we may proceed by induction on r, defining N_{k}(e, r, d) recursively. We omit the [1/p] in the notation, using the convention that we invert p throughout the proof of this proposition. Assume r ≥ 0 and that we have defined N_{k}(e, j, d) for j ≤ r − 1 (and all e, d) so that the proposition holds for Fil_{Tate}^{N} Π_{j,q}f_{M}Ω_{T}^{∞}E(F), j < r; we assume in addition that N_{k}(e, j, d) = 0 for j < 0. Define N_{j}(d, q; E) := N_{k}(e − q, j, d) + q for j < r. Then (4.3) is satisfied for q, M ≥ 0, m ≥ N_{j}(d, q; E), j = 0, …, r−1. By lemma 4.5, it follows that for all n ≥ 0, all fields F with tr. dim_{k}F ≤ d and all m ≥ max_{j=0}^{r−1}N_{j}(r − j + d, n; E) (or m ≥ 0 in case r = 0), we have

π_{r}(f_{n}f_{m}E)(F) = Fil_{n}^{simp}(f_{m}E)π_{r}(f_{n}f_{m}E)(F).

As the filtration Fil_{∗}^{simp}(G)π_{r}(f_{n}G)(F) is functorial in G, this implies that

f_{n}(ρ_{m}(E))(π_{r}(f_{n}f_{m}E)(F)) ⊂ Fil_{n}^{simp}(E)π_{r}(f_{n}E)(F)

for n, m, F as above. If we take n ≥ max(D_{k} + d, d(E)) + 1, then by lemma 7.1

im(Fil_{n}^{simp}(E)π_{r}(f_{n}E)(F^{per}) ⟶^{ρ_{n}(E)} π_{r}(E)(F^{per})) = 0.

For m ≥ n, we have f_{m}f_{n} = f_{m} and ρ_{n}(E) ∘ f_{n}(ρ_{m}(E)) = ρ_{m}(E) (lemma 2.1). Thus, for

n = max(D_{k} + d, e) + 1, m ≥ max_{j=−1}^{r−1}N_{j}(r − j + d, n; E),

we have

0 = ρ_{n}(E) ∘ f_{n}(ρ_{m}(E))(π_{r}(f_{m}f_{n}E)(F^{per})) = ρ_{m}(π_{r}(f_{m}E)(F^{per})) ⊂ π_{r}E(F^{per}).

Let n = max(D_{k} + d, e) + 1 and define

N_{k}(e, r, d) := n + max_{j=−1}^{r−1}N_{k}(e − n, j, r − j + d).

Using the isomorphism F_{Tate}^{N}π_{r}E(F) ≅ F_{Tate}^{N}π_{r}E(F^{per}), we thus have

F_{Tate}^{N}π_{r}E(F) = 0

for N ≥ N_{k}(e, r, d) and F a field extension of k with tr. dim_{k}F ≤ d. Fix q ≥ 0 and let E′ = Ω_{T}^{q} E, E′ = Ω_{T}^{∞}E′. Then Π_{a,b}E′ ≅ Π_{a,b+q}E for all b ∈ ℤ, hence c(E′) ≥ −1. Also, d(E′) = d(E) − q ≤ e − q, and we may apply our result with E′ replacing E, and e − q replacing e. Since π_{r}E′ = Π_{r,q}E and F_{Tate}^{N−q}π_{r}E′ = F_{Tate}^{N}Π_{r,q}E (lemma 2.4), we thus have

F_{Tate}^{N}Π_{r,q}E(F) = 0

for N ≥ N_{k}(e − q, r, d) + q, q ≥ 0, and F as above. Now take an integer M. Then c(f_{M}E) ≥ −1 and d(f_{M}E) ≤ d(E) (by lemma 6.6), so the same result holds for E_{M} := Ω_{T}^{∞}f_{M}E. By lemma 2.2, E_{M} ≅ f_{M}E , so

F_{Tate}^{N}Π_{r,q}f_{M}E(F)[1/p] = 0

for q ≥ 0, N ≥ N_{k}(e−q, r, d)+q, and all field extensions F of k with tr. dim_{k}F ≤ d. As M was arbitrary, the induction thus goes through, completing the proof.

□

Now for the proof of the main theorem. For x ∈ X ∈ Sm/k, and F a sheaf on Sm/k_{Nis}, we let F_{x} denote the Nisnevich stalk of F at x.

Theorem 7.3. Let k be a perfect field of finite cohomological dimension. Let p be the exponential characteristic of k, let x ∈ X ∈ Sm/k and let d = dim_{k} X. Let E be in SH_{coh.fin}(k). Then for all integers r, q, and M, we have

(Fil_{Tate}^{n} Π_{r,q}f_{M}E)_{x}[1/p] = 0

for n ≥ N_{k}(d(E)−q, r −c(E)−1, d)+q, where N_{k} is the integer-valued function given by proposition 7.2.

As SH(k)_{fin} is a subcategory of SH(k)_{coh.fin} in case k has finite cohomological dimension (proposition 6.9(3)), our main theorem 4 follows immediately from theorem 7.3, with N(E, r, d, q) := N_{k}(d(E) − q, r − c(E) − 1, d) + q.

Proof of theorem 7.3. As before, we invert p throughout the proof and omit the “[1/p]” from the notation. The proof of [15, lemma 6.1.4] shows that, for x ∈ X ∈ Sm/k, X_{x} := Spec O_{X,x} and U ⊂ X_{x} open, the map X_{x} → X_{x}/U in H(k) is equal to the map sending X_{x} to the base-point of X_{x}/U. This implies that for any F ∈ SH(k), a, b ∈ ℤ, the restriction map

[Σ_{S^{1}}^{a}Σ_{𝔾_{m}}^{b}Σ^{∞}X_{x+}, F]_{SH(k)} → [Σ_{S^{1}}^{a}Σ_{𝔾_{m}}^{b}Σ^{∞}U_{+}, F]_{SH(k)}

is injective. Passing to the limit over U, this shows that the restriction map

Π_{a,b}(F)_{x} → Π_{a,b}(F)(k(X))

is injective. From this it easily follows that the restriction map

(Fil_{Tate}^{n} Π_{r,q}f_{M}E)_{x} → (Fil_{Tate}^{n} Π_{r,q}f_{M}E)(k(X)) is injective. Thus, it suffices to show that Fil_{Tate}^{n} Π_{r,q}f_{M}E(F) = 0 for M ∈ ℤ, for n ≥ N(d(E) − q, r − c(E) − 1, d) + q, and for all finitely generated fields F over k with tr. dim_{k}F ≤ d. As

Fil_{Tate}^{n} Π_{r+p,q}Σ_{S^{1}}^{p}E = Fil_{Tate}^{n} Π_{r,q}E

we may replace E with Σ_{S^{1}}^{−c(E)−1}E and assume that c(E) ≥ −1. Similarly, if q < 0 we may replace E with Σ_{𝔾_{m}}^{−q}E, since

Fil_{Tate}^{n−q} Π_{r,0}f_{M−q}Σ_{𝔾_{m}}^{−q}E = Fil_{Tate}^{n} Π_{r,q}f_{M}E

and d(Σ_{𝔾_{m}}^{−q}E) = d(E) − q. This reduces us to the case q ≥ 0 and c(E) ≥ −1. Letting E = Ω_{T}^{∞}E, we have

Fil_{Tate}^{n} Π_{r,q}f_{M}E = Fil_{Tate}^{n} Π_{r,q}Ω_{T}^{∞}f_{M}E = Fil_{Tate}^{n} Π_{r,q}f_{M}E

for all M ∈ ℤ, q ≥ 0, n ≥ 0, so the result follows from proposition 7.2. □

Corollary 7.4. Take k, x, d and E as in theorem 7.3. Then for all q ∈ ℤ, (Π_{r,q}f_{M}E)_{x}[1/p] = 0 for all M ≥ N_{k}(d(E) − q, r − c(E) − 1, d) + q.

Proof. Indeed, for M ≥ n, f_{n}f_{M} ≅ f_{M}, hence Fil_{Tate}^{n} Π_{r,q}f_{M}E = Π_{r,q}f_{M}E. The result thus follows from theorem 7.3. □

Remark 7.5. Althoug N_{k}(e, r, d) is defined recursively, it has a simple expression: For D_{k} + d ≥ 0,

N_{k}(e, r, d) = \begin{cases}(r + 1)(D_{k} + d) + \frac{1}{2}(r + 1)(r + 2) & if e ≤ D_{k} + d, r ≥ 0, \\ e + r(D_{k} + d) + \frac{1}{2}(r + 1)(r + 2) & if e > D_{k} + d, r ≥ 0, \\ 0 & if r < 0. \end{cases}

In particular, N_{k}(e, r, d) is an increasing function in each variable (assuming D_{k} + d ≥ 0). Thus, one can apply theorem 7.3 or corollary 7.4 even if one only has an upper bound for d(E) and a lower bound for c(E). For instance, for Y ∈ Sm/k, we have d(Σ_{T}^{∞}Y_{+}) ≤ dim_{k} Y and c(Σ_{T}^{∞}Y_{+}) ≥ −1. Thus, for x ∈ X ∈ Sm/k, r ≥ 0, M ∈ ℤ, we have (F_{Tate}^{N}Π_{r,q}f_{M}Σ_{T}^{∞}Y_{+})_{x} = 0 for

N ≥ max(dim_{k} Y − q, D_{k} + dim_{k} X) + r(D_{k} + dim_{k} X) + \frac{1}{2}(r + 1)(r + 2) + q.

We do not know if any of these bounds are sharp.

Appendix A. Norm maps

Suppose our perfect base-field k has characteristic p > 0. For an abelian group A, we write A′ for A ⊗_{ℤ} ℤ[1/p]. Our task in this section is to prove

Proposition A.1. 1. Suppose that k is a perfect field of characteristic p > 0. Take E ∈ SH(k) and let α : F → L be a purely inseparable extension of finitely generated fields over k. Then the map

α^{∗} : Π_{a,b}E(F)′ → Π_{a,b}E(L)′

is injective.

2. Suppose that k is a finite field. Take E ∈ SH(k) and let α : k → k′ be a finite extension of degree n, let F be a field extension of k, and let F′ = F ⊗_{k} k′. Then the kernel of

α^{∗} : Π_{a,b}E(F) → Π_{a,b}E(F′)

is n-torsion.

As one would expect, we construct a quasi-inverse to α^{∗} by constructing transfers. We first discuss (1). Take a ∈ F^{×} \ (F^{×})^{p} and consider the extension F_{a} := F (a^{1/p}). We have the corresponding closed point p(a) of 𝔸_{F}^{1} defined by the homogeneous ideal (X^{p} − a) ⊂ F [X]. This gives us the closed point p(a) of ℙ_{F}^{1} via the standard open immersion x ↦ (1 : x) of 𝔸^{1} into ℙ^{1}. Suppose F = k(U) for some finite type k-scheme U. Since k is perfect, U has a dense open subscheme smooth over k; shrinking U and changing notation, we may assume that U is affine and smooth over k, and also that a is a global unit on U. Let V ⊂ 𝔸^{1} × U be the closed subscheme defined by X^{p} − a. Then V is reduced and irreducible (since a ∉ (F^{×})^{p}) and is finite over U. Shrinking U again, we may assume that V is also smooth and affine over k. Let I_{V} ⊂ k[U][t] be the ideal defining V in 𝔸^{1} × U. Using X^{p} − a as generator for I_{V} /I_{V}^{2} , we have the Morel-Voevodsky purity isomorphism [18, theorem 3.2.23]

𝔸^{1} × U/(𝔸^{1} × U \ V) ≅ ℙ_{V}^{1} .

Combining with the excision isomorphism 𝔸^{1} ×U/(𝔸^{1} ×U \V) ≅ ℙ^{1} ×U/(ℙ^{1} × U \V) and passing to the limit over open subschemes of U gives us the sequence of maps of pro-objects in H_{•}(k):

(ℙ_{F}^{1} , ∞) → ℙ_{F}^{1} /ℙ_{F}^{1} \ {p(a)} ≅ (ℙ_{F_{a}}^{1}, ∞);

we denote the composition by Tr_{p(a)/F}. Passing to SH(k) gives us the morphism

Tr_{p(a)/F} : 𝕊_{k} ∧ Spec F_{+} → 𝕊_{k} ∧ p(a)_{+} = 𝕊_{k} ∧ Spec F_{a+},

where we consider these objects as pro-objects in SH(k). Composing with the map induced by the structure morphism π_{a} : p(a) → Spec F gives us the endomorphism π_{a} ∘ Tr_{p(a)/F} of 𝕊_{k} ∧ Spec F_{+}. Proposition A.1(1) is an immediate consequence of

Lemma A.2. After inverting p, the endomorphism

π_{a} ∘ Tr_{p(a)/F} : 𝕊_{k} ∧ Spec F_{+} → 𝕊_{k} ∧ Spec F_{+}

is an isomorphism.

Proof. We consider a deformation of Tr_{F_{a}/F}. Let P (a) ⊂ Spec F [t][X] be the closed subscheme defined by the ideal (X^{p} + t(X + 1) + (t − 1)a). We have

d(X^{p} + t(X + 1) + (t − 1)a) = (X + a + 1)dt + tdX ∈ Ω_{F [t][X]/F} .

Thus the singular locus of P (a) (over F) is given by

(X + a + 1 = t = 0) ∩ (X^{p} + t(X + 1) + (t − 1)a = 0);

since a ∉ (F^{×})^{p}, the singular locus is empty, i.e., P (a) is smooth over F. Using X^{p} + t(X + 1) + (t − 1)a as the generator for I_{P (a)}/I_{P (a)}^{2}, we have as above the map

Tr_{P (a)/F [t]} : ℙ_{k}^{1} ∧ Spec F [t]_{+} → ℙ_{k}^{1} ∧ P (a)_{+}

of pro-objects in H_{•}(k), and the endomorphism

π_{P (a)/F [t]} ∘ Tr_{P (a)/F [t]} : 𝕊_{k} ∧ Spec F [t]_{+} → 𝕊_{k} ∧ Spec F [t]_{+}

of pro-objects in SH(k). Setting t = 1 gives us the closed subscheme p_{0} of Spec F [X] defined by the ideal (X^{p} + X + 1). Clearly π_{0} : p_{0} → Spec F is finite and étale of degree p. Using X^{p} + X + 1 as generator of I_{p_{0}}/I_{p_{0}}^{2} gives us the map

Tr_{p_{0}/F} : (ℙ_{k}^{1}, ∞) ∧ Spec F_{+} → (ℙ_{k}^{1}, ∞) ∧ p_{0+}

of pro-objects in H_{•}(k), and the endomorphism

π_{0} ∘ Tr_{p_{0}/F} : 𝕊_{k} ∧ Spec F_{+} → 𝕊_{k} ∧ Spec F_{+}

of pro-objects in SH(k). The map π_{P (a)/F [t]} ∘ Tr_{P (a)/F [t]} thus gives us an 𝔸^{1}-homotopy between the maps π_{a} ∘ Tr_{p(a)/F} and π_{0} ∘ Tr_{p_{0}/F}. Thus, these maps are equal in SH(k), and it suffices to show that π_{0} ∘ Tr_{p_{0}/F} is an isomorphism after inverting p. As X^{p} + X + 1 has coefficients in 𝔽_{p}, π_{0} ∘ Tr_{p_{0}/F} arises as base-extension from the similarly defined map

π_{0} ∘ Tr_{p_{0}/𝔽_{p}} : 𝕊_{𝔽_{p}} → 𝕊_{𝔽_{p}}

in SH(𝔽_{p}), that is, from the corresponding element [π_{0} ∘ Tr_{p_{0}/𝔽_{p}}] ∈ [𝕊_{𝔽_{p}}, 𝕊_{𝔽_{p}}]_{SH(𝔽_{p})}. By Morel’s theorem [𝕊_{𝔽_{p}}, 𝕊_{𝔽_{p}}]_{SH(𝔽_{p})} ≅ GW(𝔽_{p}) (see [14, lemma 3.10, corollary 6.41]). Since [𝔽_{p}(p_{0}) : 𝔽_{p}] = p, it follows that the image of π_{0} ∘ Tr_{p_{0}/𝔽_{p}} ∈ [𝕊_{𝔽_{p}}, 𝕊_{𝔽_{p}}]_{SH(𝔽_{p})} under the rank homomorphism GW(𝔽_{p}) → ℤ is p. Since the augmentation ideal I ⊂ GW(𝔽_{p}) is nilpotent (in fact I^{2} = 0), it follows that π_{0} ∘ Tr_{p_{0}/𝔽_{p}} is a unit in GW(𝔽_{p})[1/p], completing the proof of (1). The proof of proposition A.1(2) is similar. Write k′ = k(a), let f(x) ∈ k[x] be the minimal polynomial of a and let p ∈ 𝔸_{k}^{1} be the closed point defined by the ideal (f(x)). Using f(x) as the generator for I_{p}/I_{p}^{2}, the Morel-Voevodsky purity isomorphism [18, loc. cit.] defines the map

Tr_{p/k} : (ℙ_{k}^{1}, ∞) → (ℙ_{k}^{1}, ∞) ∧ p_{+}

in H_{•}(k), and the endomorphism

π_{p} ∘ Tr_{p/k} : 𝕊_{k} → 𝕊_{k}

in SH(k). We can use Morel’s theorem again and identify End(𝕊_{k}) with GW(k). Under this identification, π_{p} ∘ Tr_{p/k} corresponds to an element γ ∈ GW(k), mapping to the degree n under the rank homomorphism GW(k) → ℤ. Since the augmentation ideal I ⊂ GW(k) is nilpotent, this implies that there is an element β ∈ GW(k) with β · γ = n. Viewing β as an endomorphism of 𝕊_{k}, this gives π_{p} ∘ (Tr_{p/k} ∘ β) = n × id. As α^{∗} = π_{p}^{∗} ∧ id_{Spec F} on Π_{a,b}E(F), this implies n · ker α^{∗} = 0, completing the proof. □

Appendix B. Inverting integers in a triangulated category

We collect some notations and elementary facts concerning the localization of a triangulated category with respect to a multiplicatively closed subset of ℤ\{0}. If every object of the triangulated category is compact, there is no issue about what such a localization is and how to define it, however for “large" triangulated categories, there are a number of definitions possible, as well as some questions involving extending adjoints. Although this material is quite elementary, these facts on localization do not appear to be in the literature in the setting of compactly generated triangulated categories admitting arbitrary coproducts. Some of this material appears in an appendix to S. Kelly’s doctoral thesis [9]; I am grateful to him for pointing out these results to me. In what follows, T will be a triangulated category admitting arbitrary small coproducts.

Remark B.1. Suppose R is a localizing subcategory of T with a set of compact generators. Then (see e.g. [22, construction 1.6]), the Verdier localization q : T → T /R exists and admits a right adjoint r : T /R → T . r induces an equivalence of T /R with R^{⊥}, this being the full subcategory of objects X ∈ T such that Hom_{T} (A, X) = 0 for all A ∈ R. If R is a set of compact generators for R, then R^{⊥} = R^{⊥}. Let R be a set of compact objects in T and R the localizing subcategory of T generated by R, that is, the smallest localizing subcategory of T containing R. We recall from [21, theorem 2.1] that R is a set of generators for R, that is, if X is an object of R such that Hom_{R}(A, X) = 0 for all A ∈ R, then X ≅ 0. Furthermore, the subcategory R^{c} of compact objects in R is the thick subcategory of R generated by R.

Let S be a multiplicatively closed subset of ℤ\{0} containing 1. Call an object X in T S-torsion if n · id_{X} = 0 for some n ∈ S. We let T_{S-tor} be the localizing subcategory of T generated by the compact S-torsion objects of T and let T_{S^{−1}ℤ} denote the Verdier localization T /T_{S-tor}. If S is the set of powers of some integer n, we write ℤ[\frac{1}{n}] for S^{−1}ℤ, T_{n^{∞}-tor} for T_{S-tor} and T [\frac{1}{n}] for T_{S^{−1}ℤ}. For an object A of T write A_{S^{−1}ℤ} or A[\frac{1}{n}] for the image of A in T_{S^{−1}ℤ} or T [\frac{1}{n}]; for an abelian group or (pre)sheaf of abelian groups M, we write M_{S^{−1}ℤ} for M ⊗_{ℤ} S^{−1}ℤ. For S = ℤ \ {0}, we will of course write ℚ for S^{−1}ℤ. If T is a compactly generated tensor triangulated category such that the tensor product of compact objects is compact, then T_{S-tor} is a tensor ideal (since n · id_{A⊗B} = (n · id_{A}) ⊗ id_{B}) so T_{S^{−1}ℤ} inherits a tensor structure from T , and the localization functor T → T_{S^{−1}ℤ} is an exact tensor functor of tensor triangulated categories.

For A an object in T , let A⊗^{L} ℤ/n denote an object fitting into a distinguished triangle

A ⟶^{n·id} A → A ⊗^{L} ℤ/n → A[1];

this defines A ⊗^{L} ℤ/n up to non-unique isomorphism. In addition, A ⊗^{L} ℤ/n is an n^{2}-torsion object.

Lemma B.2. Let S be a subset of ℤ \ {0} containing 1, and suppose T has a set C of compact generators.

1. The set C_{S} := {A ⊗^{L} ℤ/n | A ∈ C, n ∈ S} is a set of compact generators for T_{S-tor}.

2. For A, X objects in T with A compact, there is a canonical isomorphism

Hom_{T} (A, X)_{S^{−1}ℤ} ≅ Hom_{T_{S^{−1}ℤ}}(A_{S^{−1}ℤ,}X_{S^{−1}ℤ}).

In addition, for A compact in T , A_{S^{−1}ℤ} is compact in T_{S^{−1}ℤ}, and T_{S^{−1}ℤ} is compactly generated, with C^{S} := {A_{S^{−1}ℤ}, A ∈ C} a set of compact generators.

3. T_{S-tor} is equal to the full subcategory T (S) of T with objects X such that Hom_{T} (A, X)_{S^{−1}ℤ} = 0 for all compact A in T .

4. SH(k)_{S-tor} is the full subcategory of SH(k) with objects those K such that Π_{a,b}(K)_{S^{−1}ℤ} = 0 for all a, b ∈ ℤ.

5. A map f : E → F in SH(k) becomes an isomorphism in SH(k)_{S^{−1}ℤ} if and only if the induced map on the localized homotopy sheaves

f_{∗} ⊗ id : Π_{a,b}(E)_{S^{−1}ℤ} → Π_{a,b}(F)_{S^{−1}ℤ}

is an isomorphism for all a, b ∈ ℤ.

6. SH(k)_{S-tor} is a tensor ideal, hence SH(k)_{S^{−1}ℤ} is a tensor triangulated category and the localization functor SH(k) → SH(k)_{S^{−1}ℤ} is an exact tensor functor.

Proof. For (1), clearly C_{S} consists of compact objects of T_{S-tor}. If X is n-torsion in T , then X is a summand of X ⊗^{L} ℤ/n, hence the localizing subcategory of T generated by C_{S} contains all the compact objects in T_{S-tor}, hence is equal to T_{S-tor}, proving (1). To prove (2), let Y ⟶^{v} A ⟶^{u} Z → Y [1] be a distinguished triangle in T with A compact and Z in T_{S-tor}. Then n · u = 0 for some n ∈ S, so there is a map w : A → Y with v ∘ w = n · id_{A}. Thus the category of maps {n · id : A → A} is cofinal in the category of maps Y → A with cone in T_{S-tor}, from which the isomorphism in (2) follows directly. As the localization functor q : T → T_{S^{−1}Z} admits a right adjoint, q preserves all small coproducts; the isomorphism we have just proved shows that A_{S^{−1}ℤ} is compact if A is compact. If Hom_{T} (A, X)_{S^{−1}ℤ} = 0 for all A ∈ C, then clearly X is in T_{S-tor} hence X_{S^{−1}ℤ} = 0, completing the proof of (2). (3) follows immediately from (2). Clearly (5) follows from (4). To prove (4), let SH(k)_{S-tor}^{∗} ⊂ SH(k) be the full subcategory of objects K as in (4). We recall that SH(k) has the set of compact generators {Σ_{S^{1}}^{a}Σ_{T}^{b} Σ_{T}^{∞}X_{+} | a, b ∈ ℤ, X ∈ Sm/k}.

Take K ∈ SH(k)_{S-tor}^{∗}, X ∈ Sm/k, and let f : Σ_{S^{1}}^{a}Σ_{T}^{b} Σ_{T}^{∞}X_{+} → K be a morphism. Using the Gersten spectral sequence on X and the assumption that K is in SH(k)_{S-tor}^{∗}, we see there is an n ∈ S such that n · f = 0. Thus SH(k)_{S-tor}^{∗} ⊂ SH(k)(S); the reverse inclusion follows from the definition of the homotopy sheaf Π_{a,b}(K) as the Nisnevich sheaf associated to the presheaf

X ↦ [Σ_{S^{1}}^{a−b}Σ_{T}^{b} Σ_{T}^{∞}X_{+}, K]_{SH(k)}.

By (3), this shows that SH(k)_{S-tor}^{∗} = SH_{S-tor}. For (6), we note that Σ_{T}^{∞}X_{+} ∧ Σ_{T}^{∞}Y_{+} ≅ Σ_{T}^{∞}X × Y_{+}. As the subcategory of compact objects of SH(k) is the thick subcategory generated by C (see remark B.1), it follows that the ∧-product of compact objects is compact, which suffices to prove (6). □

Remark B.3. Lemma B.2(4)-(6) hold with SH(k) replaced by SH_{S^{1}}(k), with the obvious modification in the statements.

Lemma B.4. Let L : T_{1} → T_{2} be an exact functor of compactly generated triangulated categories, with right adjoint R : T_{2} → T_{1}; we assume that T_{1} and T_{2} admit arbitrary small coproducts. Suppose that L(A) is compact for A in T_{1} compact. Then

1. R is compatible with small coproducts.

2. Let S ⊂ ℤ \ {0} be a multiplicatively closed subset. Then (L, R) descends to an adjoint pair L_{S^{−1}ℤ} : T_{1S^{−1}ℤ} [figure]T_{2S^{−1}ℤ} : R_{S^{−1}ℤ}.

3. Suppose that L is the inclusion functor for a full triangulated subcategory T_{1} of T_{2}. Then L_{S^{−1}ℤ} : T_{1S^{−1}ℤ} → T_{2S^{−1}ℤ} is an isomorphism of T_{1S^{−1}ℤ} with its image in T_{2S^{−1}ℤ}.

Proof. For (1), it is easy to show that L sends compact objects of T_{1} to compact objects of T_{2} if and only if R preserves small coproducts; we leave the details of the proof to the reader. For (2), if C_{2} is a set of compact generators for T_{2}, and A is in C_{2}, then for n ∈ S, R(A) ⊗^{L} ℤ/n ≅ R(A ⊗^{L} ℤ/n), hence R maps C_{2S} to T_{1S-tor}. As R is compatible with small coproducts, it follows that R(T_{2S-tor}) ⊂ T_{1S-tor}. As L is a left adjoint, L is compatible with small coproducts, so the same argument shows that L(T_{1S-tor}) ⊂ T_{2S-tor}, giving the induced functors on the localizations

L_{S^{−1}ℤ} : T_{1S^{−1}ℤ} [figure]T_{2S^{−1}ℤ} : R_{S^{−1}ℤ}.

To show that (L_{S^{−1}ℤ}, R_{S^{−1}ℤ}) is an adjoint pair, we may replace T_{iS^{−1}ℤ} with the equivalent full subcategory C_{iS}^{⊥} of T_{i}, i = 1, 2. R maps C_{2S}^{⊥} to C_{1S}^{⊥}, and R_{S^{−1}ℤ} is identified with the functor R_{S} : C_{2S}^{⊥} → C_{1S}^{⊥} induced from R. Letting f : T_{2} → C_{2S}^{⊥} be the left adjoint to the inclusion i_{2} : C_{2S}^{⊥} → T_{2}, L_{S^{−1}ℤ} is identified with the restriction L_{S} of f ∘ L to C_{1S}^{⊥}. Letting i_{1} : C_{1S}^{⊥} → T_{1} be the inclusion, we have

Hom_{C_{2S}^{⊥}}(L_{S}(X), Y) ≅ Hom_{T_{2}}(L(i_{1}X), i_{2}Y)

≅ Hom_{T_{1}}(i_{1}X, R(i_{2}Y)) = Hom_{C_{1S}^{⊥}}(X, R_{S}(Y)) for X ∈ C_{1S}^{⊥}, Y ∈ C_{2S}^{⊥}. For (3), as L and L_{S^{−1}ℤ} are the same on objects, we need only show that L_{S^{−1}ℤ} is fully faithful. Let X, Y be objects of T_{1}. If X is compact, then L(X) is compact by assumption, and thus

Hom_{T_{2S^{−1}ℤ}}(L_{S^{−1}ℤ}(X_{S^{−1}ℤ}),L_{S^{−1}ℤ}(Y_{S^{−1}ℤ}))
= Hom_{T_{2S^{−1}ℤ}}(L(X)_{S^{−1}ℤ}, L(Y)_{S^{−1}ℤ})

≅ Hom_{T_{2}}(L(X), L(Y))_{S^{−1}ℤ} ≅ Hom_{T_{1}}(X, Y)_{S^{−1}ℤ}

≅ Hom_{T_{1S^{−1}ℤ}}(X_{S^{−1}ℤ}, Y_{S^{−1}ℤ}).

Furthermore, as L_{S^{−1}ℤ} is a left adjoint, it preserves small coproducts, hence for fixed Y, the full subcategory T_{1S^{−1}ℤ}^{Y} of X_{S^{−1}ℤ} in T_{1S^{−1}ℤ} such that

Hom_{T_{2S^{−1}ℤ}}(L_{S^{−1}ℤ}(X_{S^{−1}ℤ}), L_{S^{−1}ℤ}(Y_{S^{−1}ℤ})) ≅ Hom_{T_{1S^{−1}ℤ}}(X_{S^{−1}ℤ}, Y_{S^{−1}ℤ})

is a localizing subcategory of T_{1S^{−1}ℤ} containing the objects A_{S^{−1}ℤ} for A a compact object of T_{1}. As these form a set of compact generators for T_{1S^{−1}ℤ}, we see that T_{1S^{−1}ℤ}^{Y} = T_{1S^{−1}ℤ}, hence L_{S^{1} ℤ} is fully faithful. □

Example B.5. We consider the example of i_{n} : Σ_{T}^{n} SH^{eff}(k) [figure]SH(k) : r_{n}. Lemma B.4 gives us the full subcategory Σ_{T}^{n} SH^{eff}(k)_{S^{−1}ℤ} of SH(k)_{S^{−1}ℤ} with inclusion functor i_{nS^{−1}ℤ}, the adjoint pair of functors

i_{nS^{−1}ℤ} : Σ_{T}^{n} SH^{eff}(k)_{S^{−1}ℤ} [figure]SH(k)_{S^{−1}ℤ} : r_{nS^{−1}ℤ},

the truncation functor f_{nS^{−1}ℤ} := i_{nS^{−1}ℤ} ∘ r_{nS^{−1}ℤ}, and for E ∈ SH(k), the canonical isomorphism

(f_{n}E)_{S^{−1}ℤ} ≅ f_{nS^{−1}ℤ}(E_{S^{−1}ℤ}).

In addition, by lemma B.2, Σ_{T}^{n} SH^{eff}(k)_{S^{−1}ℤ} is equal to the localizing subcategory of SH^{eff}(k)_{S^{−1}ℤ} generated by the set of compact objects (Σ_{T}^{q} Σ_{T}^{∞}X_{+})_{S^{−1}ℤ}, for q ≥ n and X ∈ Sm/k. The analogous results hold for SH_{S^{1}}(k).

References

[1] Arason, J. K., and Pfister, A., Beweis des Krullschen Durchschnittsatzes für den Wittring. Invent. Math. 12 (1971), 173–176.

[2] Bloch, S., and Kato, K., p-adic étale cohomology. Publ. Math. IHES 63 (1986), 147-164.

[3] Cisinski, D.C., and Déglise, F., Triangulated categories of mixed motives,

   preprint 2012 (version 3), arXiv:0912.2110v3 [math.AG].

[4] Dugger, D., Isaksen, D.C., Motivic cell structures. Algebr. Geom. Topol. 5 (2005), 615–652.

[5] Goerss, P. G., Jardine, J. F., Localization theories for simplicial presheaves. Canad. J. Math. 50 (1998), no. 5, 1048–1089.

[6] Haesemeyer, C., Weibel, C., Norm varieties and the chain lemma (after Markus Rost). Algebraic topology, 95Ð130, Abel Symp., 4, Springer, Berlin, 2009.

[7] Jardine, J.F., Simplicial presheaves, J. Pure Appl. Algebra 47 (1987), no. 1, 35–87.

[8] Jardine, J.F., Motivic symmetric spectra, Doc. Math. 5 (2000), 445–553.

[9] Kelly, S., Triangulated categories of motives in positive characteristic. Doctoral thesis, Univ. Paris 13, Oct. 2012.

http://www.uni-due.de/~hm0159/these_kelly.pdf

[10] Levine, M., The homotopy coniveau tower. J Topology 1 (2008) 217–267.

[11] Levine, M., Slices and transfers. Documenta Mathematica, Extra Volume: Andrei A. Suslin’s Sixtieth Birthday (2010), 393–443.

[12] Levine, M., The slice filtration and Grothendieck-Witt groups, PAMQ 7 no. 4.

[13] Milnor, John Algebraic K-theory and quadratic forms. Invent. Math. 9 1969/1970 318Ð344.

[14] Morel, F., A^{1}-Algebraic topology over a field, Lecture Notes in Mathematics, 2052. Springer, Heidelberg, 2012. x+259 pp.

[15] Morel, F., The stable 𝔸^{1}-connectivity theorems, K-theory, 2005, 35, pp 1-68.

[16] Morel, F., Sur les puissances de l’idéal fondamental de l’anneau de Witt. Comment. Math. Helv. 79 (2004), no. 4, 689–703.

[17] Morel, F., An introduction to 𝔸^{1}-homotopy theory. Contemporary developments in algebraic K-theory 357–441, ICTP Lect. Notes, XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004.

[18] Morel, F., and Voevodsky, V., 𝔸^{1}-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143.

[19] Orlov, D., Vishik, A., Voevodsky, V., An exact sequence for K_{?}^{M}/2 with applications to quadratic forms. Ann. of Math. (2) 165 (2007), no. 1, 1–13.

[20] Naumann, N., Spitzweck, M., Brown representability in 𝔸^{1}-homotopy theory. J. K-Theory 7 (2011), no. 3, 527D539.

[21] Neeman, A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Amer. Math. Soc. 9 (1996), no. 1, 205–236.

[22] Neeman, A., The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 547–566.

[23] Pelaez-Menaldo, J.P. , Multiplicative structure on the motivic Postnikov tower. Astérisque No. 335 (2011), xvi+289 pp.

[24] Pelaez-Menaldo, J.P. , Mixed motives and the slice filtration. C. R. Math. Acad. Sci. Paris 347 (2009), no. 9-10, 541–544.

[25] Röndigs, O., Østvær, P.A., Modules over motivic cohomology. Adv. Math. 219 (2008), no. 2, 689–727.

[26] Serre, J.-P., Galois cohomology. Springer-Verlag, Berlin, 1997. x+210 pp.

[27] Suslin, A., Joukhovitski, S., Norm varieties. J. Pure Appl. Algebra 206 (2006), 245Ð276.

[28] Voevodsky, V., Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic. Int. Math. Res. Not. 2002, no. 7, 351–355.

[29] Voevodsky, V., Motivic cohomology with ℤ/2-coefficients. Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59–104.

[30] Voevodsky, V., Open problems in the motivic stable homotopy theory. I. Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), 3–34, Int. Press Lect. Ser., 3, I, Int. Press, Somerville, MA, 2002.

[31] Voevodsky, V., A possible new approach to the motivic spectral sequence for algebraic K-theory, Recent progress in homotopy theory (Baltimore, MD, 2000) 371–379, Contemp. Math., 293 (Amer. Math. Soc., Providence, RI, 2002).

[32] Voevodsky, V., On Motivic Cohomology with ℤ/l coefficients, Ann. of Math. (2) 174 (2011), no. 1, 401–438.

[33] Voevodsky, V., Suslin, A., Friedlander, E.M., Cycles, transfers, and motivic homology theories. Annals of Mathematics Studies, 143. Princeton University Press, Princeton, NJ, 2000.

Marc Levine

Universität Duisburg-Essen Fakultät Mathematik Campus Essen 45117 Essen Germany marc.levine@uni-due.de 942

Documenta Mathematica 18 (2013)