A possible new approach to the motivic spectral sequence for algebraic K-theory

Vladimir Voevodsky^{1}^{,}^{2}

January 2001

Abstract

We describe a simple construction of the spectral sequence relating algebraic K-theory and motivic cohomology modulo two general conjectures on the structure of the motivic homotopy category. The first conjecture is the motivic analog of the fact that the zero stage of the Postnikoff tower for the (topological) sphere spectrum is the Eilenberg-Maclane spectrum H_{Z}. The second is the motivic analog of the fact that the functor Ω^{1}Σ^{1} takes n-connected spaces to n-connected spaces.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 The s-stable motivic homotopy category . . . . . . . . . . . . 2 3 A connectivity result . . . . . . . . . . . . . . . . . . . . . . . 5 4 Computation of s_{0}(Ω_{t}^{∞}(KGL)) . . . . . . . . . . . . . . . . . 7 5 The slice spectral sequence for algebraic K-theory . . . . . . . 10

1 Introduction

Despite the considerable progress in motivic cohomology and motivic homotopy theory achieved in recent years we still do not have a simple construction of the spectral sequence relating motivic cohomology and algebraic K-theory. The construction invented by Dan Grayson (see [3]) is simple and elegant but we are still unable to identify the E_{2}-term of the resulting spectral sequence with the motivic cohomology groups. The approach pioneered by Spencer Bloch and Steven Lichtenbaum in [1] and further developed by Eric Friedlander and Andrei Suslin in [2] gives a spectral sequence of the required form but is technically and conceptually very involved.

1Supported by the NSF grants DMS-97-29992 and DMS-9901219 and The Ambrose Monell Foundation

2School of Mathematics, Institute for Advanced Study, Princeton NJ, USA. e-mail:

vladimir@ias.edu

In [9] we suggested a different approach to this problem. Its first ingredient is a construction of a canonical Postnikov tower for any motivic spectrum E. The quotients of this tower s_{i}(E) are called the slices of E and, by construction, there is a spectral sequence whose E_{2}-term is given by the cohomology theories represented by the slices and which attempts to converge to the theory represented by E. For KGL, the spectrum representing algebraic K-theory, the main problem is to identify the slices with the motivic cohomology spectra H_{Z} i.e. to prove the following conjecture:

Conjecture 1 s_{n}(KGL) ≅ Σ_{T}^{n}H_{Z}

Since algebraic K-theory is periodic i.e. Σ_{T}^{1} KGL = KGL, it is sufficient to prove this conjecture for n = 0.

At the end of [9] we outlined a possible approach to such an identification. It depends on two conjectures:

Conjecture 2 Let 1 be the sphere spectrum. Then s_{0}(1) = H_{Z}.

Conjecture 3 Ω_{t}^{∞}(Σ_{T}^{n}SH^{eff}) ⊂ Σ_{t}^{n}SH_{s}

The notations used in the second conjecture are explained below. Note that these conjectures concern only general properties of the motivic stable homotopy categories and do not refer to any specifics of the spectrum representing algebraic K-theory. The main goal of this short paper is to give a complete proof that Conjecture 2 and Conjecture 3 imply Conjecture 1. We will use freely the formalism of slices, and refer the reader to [9] for the corresponding discussion.

This paper is an expanded version of the talk given at the Great Lakes Algebraic K-theory conference in Toronto (2000). It was written while I was a member of the Institute for Advanced Study in Princeton. I am very grateful to the institute for its support.

2 The s-stable motivic homotopy category

We denote by HS_{s}(S) the stable A^{1}-homotopy category of S_{s}^{1}-spectra over a base scheme S and by SH(S) the stable A^{1}-homotopy category of T -spectra over S. These categories are considered in detail in [4]. Given an S_{s}^{1}-spectrum (E_{i}, S_{s}^{1} ∧ E_{i} → E_{i+1}), the sequence of pointed sheaves S_{t}^{i} ∧ E_{i} together with the morphisms

S_{s}^{1} ∧ S_{t}^{1} ∧ (S_{t}^{i} ∧ E_{i}) → S_{t}^{i+1} ∧ S_{s}^{1} ∧ E_{i} → S_{t}^{i+1} ∧ E_{i+1}

form an S_{t}^{1} ∧ S_{s}^{1}-spectrum. Since the homotopy category of S_{s}^{1} ∧ S_{t}^{1}-spectra is equivalent to the homotopy category of T -spectra we get a functor

Σ_{t}^{∞} : SH_{s}(S) → SH(S)

This functor has the usual properties of a suspension spectrum functor. In particular, it has a right adjoint

Ω_{t}^{∞} : SH(S) → SH_{s}(S)

which takes a fibrant T -spectrum (E_{i}, T ∧E_{i} → E_{i+1}) to the s-spectrum with terms Ω_{t}^{i}(E_{i}) and the structure maps adjoint to the composition

S_{t}^{1} ∧ S_{s}^{1} ∧ Ω_{t}^{i}(E_{i}) → T ∧ Ω_{t}^{i}(E_{i}) → Ω_{t}^{i}(T ∧ E_{i}) → Ω_{t}^{i}(E_{i+1})

Let

Σ_{s}^{∞} : H_{•}(S) → SH_{s}(S)

be the usual suspension spectrum functor from the pointed unstable homotopy category to SH_{s}. Then Σ_{t}^{∞}Σ_{s}^{∞} = Σ_{T}^{∞} where Σ_{T}^{∞} is the suspension spectrum functor from H_{•} to SH(S).

Recall that SH^{eff} is the smallest triangulated subcategory in SH which contains suspension spectra and is closed with respect to all direct sums.

Lemma 2.1 If f : E → F is a morphism in SH^{eff} and Ω_{t}^{∞}(f) is an isomorphism then f is an isomorphism.

Proof: The definition of SH^{eff} implies that a morphism f : E → F in this category such that for any pointed simplicial sheaf X_{•} and any n ∈ Z the map

Hom_{SH}(Σ_{s}^{n}Σ_{T}^{∞}(X_{•}), E) → Hom_{SH}(Σ_{s}^{n}Σ_{T}^{∞}(X_{•}), F)

defined by f is bijective, is an isomorphism. Since Σ_{T}^{∞} = Σ_{t}^{∞}Σ_{s}^{∞}, this map is isomorphic to

Hom_{SH_{s}}(Σ_{s}^{n}Σ_{s}^{∞}(X_{•}), Ω_{t}^{∞}(E)) → Hom_{SH_{s}}(Σ_{s}^{n}Σ_{s}^{∞}(X_{•}), Ω_{t}^{∞}(F)).

For n ≥ 0, let Σ_{t}^{n}SH_{s} be the localizing subcategory of SH_{s}(S) generated by objects of the form Σ_{t}^{n}Σ_{∞}^{s}(X_{•}) for all pointed simplicial sheaves X_{•}. We get a filtration:

… ⊂ Σ_{t}^{n}SH_{s} ⊂ Σ_{t}^{n−1}SH_{s} … ⊂ … Σ_{t}^{0}SH_{s} = SH_{s}

All the categories appearing in this filtration are compactly generated triangulated categories with all direct sums and the functors respect direct sums and distinguished triangles. Therefore, by [6], the inclusion functors have right adjoints and we denote the composition

SH_{s} → Σ_{t}^{n}SH_{s} → SH_{s}

by f_{n}. For any E in SH_{s} we get natural distinguished triangles

f_{q+1}E → f_{q}(E) → s_{q}(E) → Σ_{s}^{1}f_{q+1}(E)

where s_{q}(E) belongs to Σ_{t}^{q}SH_{s} and is right orthogonal to Σ_{t}^{q+1}SH_{s}. These triangles form the SH_{s}-analog of the slice tower (see [9, Theorem 2.2]) in SH.

It is clear that the functor Σ_{t}^{∞} maps the slice filtration in SH_{s} to the slice filtration in SH:

Σ_{t}^{∞}(Σ_{t}^{n}SH_{s}) ⊂ Σ_{T}^{n}SH^{eff}

In particular, the whole SH_{s} is mapped to SH^{eff}. It is also clear that the functor Ω_{t}^{∞} respects the “adjoint” filtration i.e. that it maps objects in SH which are right orthogonal to Σ_{T}^{n}SH^{eff} to objects which are right orthogonal to Σ_{t}^{n}SH_{s}. We can now restate Conjecture 3 from the introduction:

Conjecture 4 The functor Ω_{t}^{∞} respects the slice filtration i.e.

Ω_{t}^{∞}(Σ_{T}^{n}SH^{eff}) ⊂ Σ_{t}^{n}SH_{s}

Since all the functors involved in Conjecture 4 are triangulated and commute with direct sums it is sufficient to check that for a smooth scheme X over S one has

Ω_{t}^{∞}(Σ_{s}^{∞}(T^{n} ∧ X_{+})) ∈ Σ_{t}^{n}SH_{s}

The object on the left hand side can be represented by a homotopy colimit of objects of the form Ω_{t}^{i}Σ_{t}^{i+n}(Σ_{s}^{∞}(X_{+})) where both Σ_{t} and Ω_{t} are taken in SH_{s}. A simple inductive argument shows now that Conjecture 4 follows from the following:

Conjecture 5 For any smooth scheme X over S and any n ≥ 0 one has

Ω_{t}^{1}Σ_{t}^{1}Σ_{s}^{∞}(S_{t}^{n} ∧ X_{+}) ∈ Σ_{t}^{n}SH_{s}

The topological analog of this statement (where SH is replaced by the stable homotopy category and SH_{s} by the unstable one) asserts that ΩΣ takes nconnected objects to n-connected objects. One way to see it is to use the fact that ΩΣ(X) has a model (“James construction”) possessing a filtration whose quotients are X^{∧i}. This is the starting point of the operadic theory of loop spaces and it appears that any such theory for t-loop spaces in SH_{s} would provide a proof of Conjectures 4 and 5.

Lemma 2.2 Assume that Conjecture 4 holds. Then for any E in SH one has:

Ω_{t}^{∞}(f_{n}(E)) = f_{n}(Ω_{t}^{∞}(E))
Ω_{t}^{∞}(s_{n}(E)) = s_{n}(Ω_{t}^{∞}(E))

Proof: To prove the first equality it is sufficient to show that Ω_{t}^{∞}(f_{n}(E)) is in Σ_{t}^{n}SH_{s} and that the cones of the morphism Ω_{t}^{∞}(f_{n}(E) → E) is right orthogonal to Σ_{t}^{n}SH_{s}. The former is Conjecture 4. The later is clear because Ω_{t}^{∞} is adjoint to Σ_{t}^{∞}. The second equality follows from the first one since s_{n} is the cone of the morphism f_{n+1} → f_{n} and Ω_{t}^{∞} commutes with cones.

3 A connectivity result

The goal of this section is to prove Theorem 3.2. Its corollary 3.4 will be used below to prove the convergence of the slice spectral sequence for algebraic K-theory. Everywhere in this section we assume that S is the spectrum of a field. We need this assumption in order to have the following lemma.

Lemma 3.1 Let E be an s-spectrum such that π_{<0}(E) = 0. Then one has,

π_{<0}(Σ_{t}^{1}E) = 0
π_{<0}(Ω_{t}^{1}E) = 0

Proof: See [4].

Theorem 3.2 Let E be an object of Σ_{t}^{q}SH_{s} such that π_{<0}(Ω_{t}^{q}E) = 0. Then π_{<0}(E) = 0.

Proof: Consider the adjunction morphism Σ_{t}^{q}Ω_{t}^{q}E → E and let E^{(1)} be its cone. Applying this construction inductively we get a sequence of distinguished triangles

Σ_{t}^{q}Ω_{t}^{q}E^{(n)} → E^{(n)} → E^{(n+1)} → …  (1)

and therefore a sequence of morphisms

E → E^{(1)} → … → E^{(n)} → …

Let E^{(∞)} be the homotopy colimit of this sequence. This object is in Σ_{t}^{q}SH_{s}. Applying the functor Ω_{t}^{q} to the triangles (1) we get split triangles. Therefore, since Ω_{t}^{q} commutes with the homotopy colimits of sequences, Ω_{t}^{q}E^{(∞)} is zero as the homotopy colimit of a sequence of zero morphisms. The following straightforward lemma implies that E^{(∞)} = 0.

Lemma 3.3 Let E be an object in Σ_{t}^{q}SH_{s} such that Ω_{t}^{q}E = 0. Then E = 0.

Proof: Under the assumptions of the lemma the class E of objects F in Σ_{t}^{q}SH_{s} such that Hom(Σ_{s}^{n}F, E) = 0 for all n, contains objects of the form Σ_{t}^{q}Σ_{s}^{∞}(X_{+}) and is closed under triangles and direct sums. Therefore, E coincides with Σ_{t}^{q}SH_{s}.

Since objects of the form Σ_{s}^{i}Σ_{s}^{∞}(X_{+}) are compact, the fact that E^{(∞)} is zero implies that for any i one has

colim_{n} π_{i}(E^{(n)}) = 0

It remains to show that for i < 0 the maps

π_{i}(E^{(n)}) → π_{i}(E^{(n+1)})

are monomorphisms. The long exact sequence defined by (1) implies that it is sufficient to check that π_{<0}(Σ_{t}^{q}Ω_{t}^{q}E^{(n)}) = 0. Proceed by induction on n. By Lemma 3.1 it is sufficient to check that π_{<0}(Ω_{t}^{q}E^{(n)}) = 0. For n = 0 this follows from our assumption on E. Assume that π_{<0}(Ω_{t}^{q}E^{(n−1)}) = 0. Applying π_{i}(Ω_{t}^{q}(−)) to the triangle (1) we get a short exact sequence

0 → π_{i}(Ω_{t}^{q}(E^{(n)})) → π_{i−1}(Ω_{t}^{q}Σ_{t}^{q}Ω_{t}^{q}(E^{(n−1)})) → π_{i−1}(Ω_{t}^{q}(E^{(n−1)})) → 0

Using again Lemma 3.1 and the inductive assumption we conclude that π_{i}(Ω_{t}^{q}(E^{(n)})) = 0. This finishes the proof of Theorem 3.2.

Recall that for E in SH(S), π_{p,q}(E) denote the sheaf associated to the presheaf

X ↦ Hom_{SH}(Σ_{t}^{q}Σ_{s}^{p−q}Σ_{T}^{∞}(X), E)

Corollary 3.4 Assume that Conjecture 4 holds and that S = Spec(k) where k is a field. Let E be an object of SH(S) and q ≥ 0 an integer such that π_{p,q}(E) = 0 for p < q. Then π_{<0,0}(f_{q}(E)) = 0.

Proof: By adjunction we have π_{n,0}(f_{q}(E)) = π_{n}(Ω_{t}^{∞}(f_{q}(E))). Since f_{q}(E) is in Σ_{T}^{q} SH^{eff}, Conjecture 4 implies that Ω_{t}^{∞}(f_{q}(E)) is in Σ_{t}^{q}SH_{s}. By Theorem 3.2 it is sufficient to show that π_{n}(Ω_{t}^{q}Ω_{t}^{∞}(f_{q}(E))) = 0 for n < 0. By adjunction this group equals

π_{n,0}(Ω_{t}^{q}Ω_{t}^{∞}(f_{q}(E))) = π_{n+q,q}(f_{q}(E))

4 Computation of s_{0}(Ω_{t}^{∞}(KGL))

The results of this section do not depend on any conjectures. We assume here that the base scheme S is regular.

Denote the spectrum Ω_{t}^{∞}(KGL) by KGL_{s}. Since S is regular

KGL^{p,q}(X) = K_{2q−p}^{Q}(X)

where K_{∗}^{Q} is the usual (Quillen’s) K-theory (see [7] where we use BGL instead of KGL), and therefore,

Hom_{SH_{s}}(Σ_{s}^{∞}(X_{+}), Σ_{s}^{n}KGL_{s}) = KGL^{n,0}(X) = K_{−n}^{Q}(X)

For a sheaf of abelian groups \underline{A} denote by H_{A,s} the Eilenberg-MacLane sspectrum defined by \underline{A}, i.e. the sequence of the simplicial sheaves K(\underline{A}, n) = K(\underline{A}[n]) together with the obvious structure morphisms. The goal of this section is to prove the following theorem.

Theorem 4.1 There exists an isomorphism s_{0}(KGL_{s}) = H_{Z,s} which takes the identity map 1 → s_{0}(KGL_{s}) to the identity map of H_{Z,s}.

For any s-spectrum E, denote by π_{i}(E), i ∈ Z the Nisnevich sheaf associated with the presheaf

X ↦ Hom_{SH_{s}}(Σ_{s}^{∞}(X_{+}), Σ_{s}^{−i}(E)).

If π_{i}(E) = 0 for i < 0 then we have a canonical morphism E → H_{π_{0}(E),s}. In particular, since K^{−n}(X) = 0 for X regular and n > 0 and the sheaf associated with K_{0} is Z we get a canonical morphism ϕ : KGL_{s} → H_{Z,s}. To prove the theorem it is sufficient to show that s_{0}(H_{Z,s}) = H_{Z,s} and that the fiber of ϕ is in Σ_{t}^{1}SH_{s}. The former is shown in Lemma 4.2, the later in Lemma 4.6.

Lemma 4.2 One has s_{0}(H_{Z,s}) = H_{Z,s}.

Proof: We need to show that H_{Z,s} is orthogonal to Σ_{t}^{1}SH_{s}, i.e. that for a smooth scheme X over S and any n ∈ Z we have

Hom_{SH_{s}}(Σ_{t}^{1}Σ_{s}^{∞}(X_{+}), Σ_{s}^{n}H_{Z,s}) = 0

Since Nisnevich cohomology with coefficients in Z are homotopy invariant, H_{Z,s} is A^{1}-local and this group equals ker(H_{Nis}^{n}(X ×G_{m}, Z) → H_{Nis}^{n}(X, Z)). Since S is regular, X and X × G_{m} are smooth and H_{Nis}^{n}(−, Z) can only be non zero for n = 0. For n = 0 the kernel is zero because for a henselian local X the scheme X × G_{m} is connected.

Remark 4.3 The assumption that S is regular is not necessary for the proof of Lemma 4.2. In the proof, for a general S, the reduction to H_{Nis}^{0} should be replaced by the fact that for a local henselian S one has H_{Nis}^{i}(S ×G_{m}, Z) = 0 for i > 0.

Proposition 4.4 Let E = (E_{n}) be a fibrant s-spectrum such that π_{i}(E) = 0 for i < 0. Then E belongs to the smallest subcategory closed under distinguished triangles, direct sums and smash products containing Σ_{s}^{∞}(E_{0}).

Proof: Denote the smallest subcategory satisfying the conditions listed above by C. Note that E = hocolim_{n} Σ_{s}^{−n}Σ_{s}^{∞}(E_{n}) and, since homotopy colimits of sequences can be expressed in terms of cones and direct sums, it is sufficient to show that Σ_{s}^{∞}(E_{n}) is in C. By induction we may assume that Σ_{s}^{∞}(E_{n−1}) is in C. Since E is fibrant we have E_{n−1} = Ω_{s}^{1}(E_{n}). Using an appropriate model for loop spaces we may assume that E_{n−1} has a monoid structure. Let B(E_{n−1}) be the classifying space of E_{n−1} defined as the diagonal of the bisimplicial sheaf B_{••}(E_{n−1}) whose rows are E_{n−1}^{m}. Our connectivity assumption on E implies that E_{n} = B(E_{n−1}). The formula

Σ_{s}^{∞}(X_{•} × Y_{•}) = (Σ_{s}^{∞}(X_{•}) ∧ Σ_{s}^{∞}(Y_{•})) ∨ Σ_{s}^{∞}(X_{•}) ∨ Σ_{s}^{∞}(Y_{•})

implies that the suspension spectra of the rows of B_{••}(E_{n−1}) are in C. By Lemma 4.5 we conclude that the suspension spectrum of E_{n} is in C.

Lemma 4.5 Let B_{••} be a pointed bisimplicial sheaf with rows B_{i}. Then Σ_{s}^{∞}(Δ(B_{••})) belongs to the localizing subcategory C generated by Σ_{s}^{∞}(B_{i}).

Proof: Consider B = B_{••} as a simplicial object over Δ^{op}Shv_{•} with terms B_{i}. Let W r(B) be the degeneracy free simplicial object obtained by first forgetting the degeneracies of B and then adding new ones freely (see [8, p.45]) such that

W r(B)_{i} = ⋁_{[i]→[j]} B_{j}

where [i] → [j] runs through all monomorphisms in Δ. We have a canonical map W r(B) → B which is a weak equivalence column-wise because for a pointed simplicial set X the map W r(X) → X is a weak equivalence. Therefore, it remains to check that the suspension spectrum of Δ(W r(B)) is in C. Since Δ(W r(B)) = colim_{n} Δ(sk_{n}(W r(B))) and C is closed under triangles and direct sums it is sufficient to prove that the suspension spectrum of Δ(sk_{n}(W r(B))) is in C for each n. The simplicial object W r(B) is a degeneracy free object based on the sequence B_{i}. Therefore, for each n we have a push-out square

B_{n} ⊗ ∂Δ^{n} ⟶ sk_{n−1}(W r(B))
↓  ↓
B_{n} ⊗ Δ^{n} ⟶ sk_{n}(W r(B))

Applying the diagonal functor we get a push-out square in pointed simplicial objects of the form

B_{n} ∧ ∂Δ_{+}^{n} ⟶ Δ(sk_{n−1}(W r(B)))
↓  ↓
B_{n} ∧ Δ_{+}^{n} ⟶ Δ(sk_{n}(W r(B)))

which implies that the cone of the morphism

Σ_{s}^{∞}Δ(sk_{n−1}(W r(B))) → Σ_{s}^{∞}Δ(sk_{n}(W r(B)))

is isomorphic to the cone of the morphism

Σ_{s}^{∞}(B_{n} ∧ ∂Δ_{+}^{n}) → Σ_{s}^{∞}(B_{n} ∧ Δ_{+}^{n})

i.e. to Σ_{s}^{n}Σ_{s}^{∞}(B_{n}). This finishes the proof of the lemma.

Lemma 4.6 The fiber of ϕ is in Σ_{t}^{1}SH_{s}.

Proof: The fiber of ϕ satisfies the connectivity assumption of Proposition 4.4. It remains to see that Σ_{s}^{∞}(fiber(KGL_{s,0} → Z)) is in Σ_{t}^{1}SH_{s}. Since KGL_{s,0} represents K_{0} it is A^{1}-weakly equivalent, by [5, Theorem 4.3.13], to BGL×Z where BGL is the geometric infinite Grassmannian. Our result follows now from Lemma 4.7 and the fact that Σ_{t}^{1}SH_{s} is closed under homotopy colimits of sequences.

Lemma 4.7 Let BGL(n, m) be the Grassmannian of rank n submodules in O^{m} which we consider as a pointed scheme by means of any point. Then Σ_{s}^{∞}(BGL(n, m)) is in Σ_{t}^{1}SH_{s}.

Proof: Let X be a smooth scheme over S and U be a dense open subscheme in X such that X − U is a divisor with normal crossings. A simple inductive argument together with the homotopy purity theorem ([5, Theorem 3.2.23]) shows that then the cone of the map Σ_{s}^{∞}(U_{+}) → Σ_{s}^{∞}(X_{+}) is in Σ_{t}^{1}SH_{s}. Let U be an open subscheme in BGL(n, m) such that:

1. U contains the distinguished point

2. U ≅ A^{N}

3. BGL(n, m) − U is a divisor with normal crossings.

Applying the previous remark we conclude that Σ_{s}^{∞}(BGL(n, m)) is in Σ_{t}^{1}SH_{s}.

5 The slice spectral sequence for algebraic K-theory

Consider the slice tower

f_{q+1}KGL → f_{q}(KGL) → s_{q}(KGL) → Σ_{s}^{1}f_{q+1}(KGL)  (2)

By construction, the algebraic K-theory spectrum KGL is periodic i.e. we have an isomorphism Σ_{T}^{1} ∧ KGL = KGL. This isomorphism defines isomorphisms

s_{q}(KGL) = Σ_{T}^{q} s_{0}(KGL)
f_{q}(KGL) = Σ_{T}^{q} f_{0}(KGL)

For any smooth scheme X over S consider the spectral sequence for

KGL^{p,q}(X) = Hom_{SH}(Σ_{T}^{∞}(X_{+}), Σ_{t}^{q}Σ_{s}^{p−q}KGL)

defined by the tower (2). If we index this spectral sequence such that it starts with the E_{2}-term then we have

E_{2}^{p,q} = s_{0}(KGL)^{p−q,−q}(X)  (3)

where s_{0}(KGL)^{∗,∗} denotes the cohomology theory defined by the spectrum s_{0}(KGL). Theorem 5.1 below identifies s_{0}(KGL) with the integral motivic Eilenberg-MacLane spectrum and (3) becomes

E_{2}^{p,q} = H^{p−q,−q}(X, Z)

Proposition 5.5 implies that our spectral sequence strongly converges to KGL^{p+q,0}(X) = K_{−p−q}(X).

Theorem 5.1 Assume that S = Spec(k) where k is a field and that Conjectures 2 and 3 hold over S. Then there is an isomorphism s_{0}(KGL) = H_{Z} in SH(S) which takes the unit of s_{0}(KGL) to the unit of H_{Z}.

Proof: By Conjecture 2, we have s_{0}(1) = H_{Z}. Therefore, the unit map 1 → KGL defines a morphism H_{Z} → s_{0}(KGL) which takes the unit morphism 1 → H_{Z} to the unit morphism 1 → KGL. We are going to prove that any such morphism is an isomorphism.

Conjecture 2 implies in particular that H_{Z} ∈ SH^{eff}. Therefore, by Lemma 2.1 it is sufficient to prove that

Ω_{t}^{∞}(H_{Z} → s_{0}(KGL))  (4)

is an isomorphism.

Lemma 5.2 Let S = Spec(k) where k is a field and let H_{Z} be the motivic Eilenberg-MacLane spectrum. Then Ω_{t}^{∞}(H_{Z}) = H_{Z,s}.

Proof: The spectrum H_{Z,s} can be characterized by the property that

π_{i}(H_{Z,s}) = \{ \begin{matrix}Z & for i = 0 \\ 0 & for i ≠ 0 \end{matrix}

We have:

π_{i}(Ω_{t}^{∞}(H_{Z})) = π_{i,0}(H_{Z}) = ( \begin{matrix}The sheaf associated \\ to the presheaf \\ X ↦ H^{i,0}(X, Z) \end{matrix} )

and the lemma follows from the fact that for a smooth connected scheme X over a field one has:

H^{i,0}(X, Z) = \{ \begin{matrix}Z & for i = 0 \\ 0 & for i ≠ 0 \end{matrix}  (5)

By Lemma 2.2 (this is where Conjecture 3 is used) we have Ω_{t}^{∞}(s_{0}(KGL)) = s_{0}(Ω_{t}^{∞}(KGL)) and by Theorem 4.1 we have s_{0}(Ω_{t}^{∞}(KGL)) = H_{Z,s}. Therefore, (4) is a morphism from H_{Z,s} to H_{Z,s}. The fact that it is an isomorphism follows from the lemma below.

Lemma 5.3 Let f : H_{Z,s} → H_{Z,s} be an endomorphism which takes the unit map 1 → H_{Z,s} to itself. Then f is the identity.

Proof: Since H_{Z,s} is A^{1}-local the endomorphisms of this object in SH_{s} can be computed in the stable homotopy category of simplicial presheaves (without A^{1}-localization). Our result follows from the fact that in this category the endomorphisms of H_{Z,s} are given by H^{0}(S, Z).

Remark 5.4 The only place in the proof of Theorem 5.1 where we used the assumption that S is the spectrum of a field is in the proof of Lemma 5.2. If we knew that the motivic cohomology of weight zero of all regular connected schemes is given by (5) then we could prove Theorem 5.1 for any regular S.

Proposition 5.5 Let S be the spectrum of a field and assume that Conjecture 4 holds. Then for a smooth scheme X over S of absolute dimension d and n > d − q one has:

Hom_{SH}(Σ_{T}^{∞}(X_{+}), Σ_{s}^{n}f_{q}(KGL)) = 0.

Proof: The sheaf π_{p,q}(KGL) is the sheaf associated to the presheaf K_{p−2q}(−). In particular it is zero for p < 2q. Applying Corollary 3.4 we conclude that π_{<q}(f_{q}(KGL)) = 0 for all q ≥ 0. The statement of the proposition follows from the Connectivity Theorem (see [4]).

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