VOEVODSKY’S NORDFJORDEID LECTURES ON MOTIVIC STABLE HOMOTOPY THEORY

OLIVER RÖNDIGS AND PAUL ARNE ØSTVÆR

This note is an extract from Voevodsky’s lectures at the summer school on motivic homotopy theory, held in Nordfjordeid. The reader will find a leisurely introduction to motivic stable homotopy theory, examples of cohomology theories for algebraic varieties and a sample of current research problems. As general background material, we recommend the Nordfjordeid lectures of Dundas [Dun02] and Levine [Lev02]. A reference for motivic homotopy theory is Voevodsky’s ICM address [Voe98].

In section 1, we recall the construction of the motivic stable homotopy category. To illustrate the general theory we consider motivic cohomology, algebraic K-theory and algebraic cobordism; see section 2. In March 2000, Voevodsky [Voe00a] posted a list of open problems concerning motivic homotopy theory. There has been so much work done in the interim that an update of the status of these conjectures may be useful to the practitioners of motivic homotopy theory. It is our understanding that section 3 gives a concise summary of the recent developments.

Finally, we would like to thank Voevodsky for helpful discussions on the content of this note and his kind permission to include a sketch proof of Theorem 3.4. The actual wording here, and the responsibility for any misinterpretations, are our own.

1. Motivic stable homotopy theory

In this section, we introduce the motivic stable homotopy category. Although the construction may be carried out more generally, the only base schemes we consider are fields. A final word about precursors: We assume the reader is familiar with the basics of model categories and triangulated categories. Introductory textbooks on these subjects include [Hov99] and [Nee01].

1.1. Spaces. Let k be a field and consider the category Sm/k of smooth separated schemes of finite type over Spec(k). From a homotopical point of view, the category Sm/k is intractable because we are not allowed to form arbitrary colimits. The spaces we shall consider, are the objects in the category Spc(k) := Δ^{op}Shv_{Nis}(Sm/k) of Nisnevich sheaves on Sm/k with values in simplicial sets. Let us mention two types of examples of such sheaves. First, any scheme in Sm/k determines a representable space via the Yoneda embedding. This follows since the Nisnevich topology is subcanonical. Second, any simplicial set can be viewed as a constant Nisnevich sheaf on Sm/k (and also as a constant sheaf in any other Grothendieck topology).

A pointed space consists of a space X together with a map x : Spec(k) → X. Here, we consider Spec(k) as a representable sheaf with constant simplicial structure. Let Spc_{•}(k) denote the category of pointed spaces. If X is a space, we let X_{+} denote the canonically pointed space X ⨿ Spec(k). By adding disjoint base-points in the prescribed way, we get a left adjoint to the forgetful functor from Spc_{•}(k) to Spc(k). It is important to note that the category of pointed spaces has a symmetric monoidal structure: Suppose X and Y are pointed spaces. Then their smash product X ∧Y is the space associated to the presheaf U ↦ X(U)∧Y (U). As the sheaf represented by Spec(k) is U ↦ ∗; the discrete category with one object, the pointed space Spec(k)_{+} is the unit for the smash product.

In classical stable homotopy theory, one ‘inverts’ the simplicial circle to construct spectra of pointed simplicial sets. This part works slightly differently in the motivic context. To define the stable motivic homotopy category, we shall employ the notion of bispectra for pointed spaces. In particular, we will need two different ‘circles’ in order to obtain an adequate stable homotopy theory; this will become apparent later. One of the circles is well-known to topologists: Let Δ[n] denote the standard simplicial n-simplex. Recall that the simplicial circle S^{1} is the coequalizer of the diagram Δ[0] ⇉ Δ[1]. We let S_{s}^{1} denote the corresponding pointed space. The second circle is well-known to algebraic geometers: Let 𝔸^{1} ∈ Sm/k denote the affine line Spec(k[X]). The Tate circle S_{t}^{1} is the space 𝔸^{1} ∖0 pointed by the global section given by the identity; this is the underlying scheme of the multiplicative group. Since Spc_{•}(k) has a smash product, we may form the n-fold smash products S_{s}^{n} and S_{t}^{n} of the simplicial circle and the Tate circle, respectively. A mixed sphere refers to a smash product of S_{s}^{m} and S_{t}^{n}.

1.2. The motivic s-stable homotopy category SH_{s}^{𝔸^{1}}(k). To invert S_{s}^{1}, we shall consider spectra of pointed spaces. This is analogous to the situation with ordinary spectra and the simplicial circle.

Definition 1.3. An s-spectrum E is a sequence of pointed spaces {E_{n}}_{n≥0} together with structure maps S_{s}^{1}∧E_{n} → E_{n+1}. A map E → E′ of s-spectra consists of degreewise maps E_{n} → E′_{n} of spaces which are compatible with the structure maps. Let Spt_{s}(k) denote the category of s-spectra.

Example 1.4. The s-suspension spectrum of a pointed space X is the s-spectrum Σ_{s}^{∞}X with n-th term S_{s}^{n} ∧ X and identity structure maps.

The next step is to define weak equivalences of s-spectra. If n ≥ 1 and (X, x) is a pointed space, let π_{n}(X, x) be the sheaf of homotopy groups associated to the presheaf U ↦ π_{n}(X(U), x_{U}), where x_{U} is the image of x in X(U). If n ≥ 2, this is a sheaf of Abelian groups. The suspension homomorphism for pointed simplicial sets yields a suspension homomorphism π_{n}(X) → π_{n+1}(S_{s}^{1} ∧ X) of sheaves of homotopy groups.

If E is an s-spectrum and m > n are integers, we may consider the sequence

\xymatrix{π_{n+m}(E_{m}) \ar[r] & π_{n+m+1}(S_{s}^{1} ∧ E_{m}) \ar[r] & π_{n+m+1}(E_{m+1}) \ar[r] & · · · .}

The sheaves of stable homotopy groups of E are the sheaves of Abelian groups

π_{n}(E) := colim_{m>n} π_{n+m}(E_{m}).

A map E → E′ of s-spectra is called an s-stable weak equivalence if for all n ∈ ℤ, the induced map π_{n}(E) → π_{n}(E′) is a sheaf isomorphism.

Definition 1.5. Let SH_{s}(k) be the category obtained from Spt_{s}(k) by inverting the s-stable weak equivalences.

Remark 1.6. One can show that Spt_{s}(k) has the structure of a proper simplicial stable model category. The associated homotopy category is SH_{s}(k). In this model structure, the weak equivalences are the s-stable weak equivalences.

There is an obvious way to smash an s-spectrum with any pointed space. If we smash with the simplicial circle, the induced simplicial suspension functor Σ_{s}^{1} becomes an equivalence of SH_{s}(k): Indeed, an s-spectrum is a Nisnevich sheaf on Sm/k with values in the category Spt of ordinary spectra. Since the simplicial suspension functor is an equivalence of the stable homotopy category SH, it is also an equivalence of SH_{s}(k). The word ‘s-stable’ refers to this fact.

Remark 1.7. There is a canonical functor Spt → Spt_{s}(k) obtained by considering pointed simplicial sets as pointed spaces. A stable weak equivalence of spectra induces an s-stable weak equivalence of s-spectra, and there is an induced functor SH → SH_{s}(k) between the corresponding homotopy categories.

The basic organizing principle in motivic homotopy theory is to make the affine line contractible. One way to obtain this is as follows: An s-spectrum F is 𝔸^{1}-local if for all U ∈ Sm/k and n ∈ ℤ, the map

\xymatrix{Hom_{SH_{s}(k)}(Σ_{s}^{∞}U_{+}, Σ_{s}^{n}F) \ar[r] & Hom_{SH_{s}(k)}(Σ_{s}^{∞}(U × 𝔸^{1})_{+}, Σ_{s}^{n}F)}

defined by the projection U × 𝔸^{1} → U is bijective. We say that a map E → E′ of s-spectra is an 𝔸^{1}-stable weak equivalence if for any 𝔸^{1}-local s-spectrum F, the canonical map Hom_{SH_{s}(k)}(E′, F) → Hom_{SH_{s}(k)}(E, F) is a bijection.

Definition 1.8. Let the motivic s-stable homotopy category SH_{s}^{𝔸^{1}}(k) be the category obtained from Spt_{s}(k) by inverting the 𝔸^{1}-stable weak equivalences.

The following Lemma is now evident.

Lemma 1.9. Assume X ∈ Sm/k. Then the canonical map X × 𝔸^{1} → X induces an 𝔸^{1}-stable weak equivalence of s-spectra. In particular, Σ_{s}^{∞} (𝔸^{1}, 0) is contractible.

We note that the simplicial suspension functor is an equivalence of SH_{s}^{𝔸^{1}}(k). The last step in the construction of Voevodsky’s motivic stable homotopy category is to invert the Tate circle in the sense that smashing with S_{t}^{1} becomes an equivalence. To indicate as to why further constructions are required, we shall briefly discuss an unsatisfactory facet of the motivic s-stable homotopy category. In topology, recall that a finite unramified covering map Y → Z of topological spaces induces a transfer map Σ^{∞}Z_{+} → Σ^{∞}Y_{+} in the ordinary stable homotopy category. The algebraic analogue is a finite Galois extension k′/k of fields. However, in SH_{s}^{𝔸^{1}}(k) there is in general no non-trivial transfer map Σ_{s}^{∞}Spec(k)_{+} → Σ_{s}^{∞}Spec(k′)_{+}. If E is an s-spectrum and π^{s} denotes ordinary stable homotopy groups, then

Hom_{SH_{s}(k)}(Σ_{s}^{∞}S_{s}^{n}, E) = Hom_{SH}(Σ^{∞}S^{n}, E(Spec(k)))
= π_{n}^{s}(E(Spec(k))).

Now, let X ∈ Sm/k and suppose that Hom_{Sm/k}(𝔸^{1}, X) = Hom_{Sm/k}(Spec(k), X). Then the above implies an isomorphism

Hom_{SH_{s}^{𝔸^{1}}(k)}(Σ_{s}^{∞}S_{s}^{n}, Σ_{s}^{∞}X_{+}) = Hom_{SH}(Σ^{∞}S^{n}, Σ^{∞}X(Spec(k))_{+}).

When X = Spec(k′), note that Hom_{Sm/k}(Spec(k), Spec(k′)) = ∅. By specializing the above remarks to the case n = 0, we find

Hom_{SH_{s}^{𝔸^{1}}(k)}(Σ_{s}^{∞}Spec(k)_{+}, Σ_{s}^{∞}Spec(k′)_{+}) = Hom_{SH}(Σ^{∞}S^{0}, Σ^{∞}Spec(k′)(Spec(k))_{+})
= 0.

Remark 1.10. It turns out that the existence of transfer maps for finite Galois extensions and the Tate circle being invertible in the homotopy category are closely related issues. The existence of transfer maps is incorporated in Voevodsky’s derived category DM_{−}^{eff}(k) of effective motivic complexes over k [Voe00b]. If k is a perfect field, the cancellation theorem [Voe02a] shows that the map

\xymatrix{Hom_{DM_{−}^{eff}(k)}(C, D) \ar[r] & Hom_{DM_{−}^{eff}(k)}(C ⊗ ℤ(1), D ⊗ ℤ(1))}

induced by tensoring with the Tate object ℤ(1) in DM_{−}^{eff}(k) is an isomorphism.

1.11. The motivic stable homotopy category SH(k). The definition of SH(k) combines the category of s-spectra and the Tate circle. In order to make this precise we use bispectra. An (s, t)-bispectrum E consists of pointed spaces E_{m,n} together with structure maps σ_{s} : S_{s}^{1} ∧ E_{m,n} → E_{m+1,n} and σ_{t} : S_{t}^{1} ∧ E_{m,n} → E_{m,n+1}. In addition, the structure maps are required to be compatible in the sense that the following diagram commutes.

\xymatrix{S_{s}^{1} ∧ S_{t}^{1} ∧ E_{m,n} \ar[rr]^{τ∧E_{m,n}} \ar[d]_{S_{s}^{1}∧σ_{t}} &  & S_{t}^{1} ∧ S_{s}^{1} ∧ E_{m,n} \ar[d]^{S_{t}^{1}∧σ_{s}} \\ S_{s}^{1} ∧ E_{m,n+1} \ar[r]^{σ_{s}} & E_{m+1,n+1} & S_{t}^{1} ∧ E_{m+1,n} \ar[l]_{σ_{t}}}

Here, τ flips the copies of S_{s}^{1} and S_{t}^{1}. There is an obvious notion of maps between such bispectra. An (s, t)-bispectrum can be interpreted as a t-spectrum object in the category of s-spectra, that is, a collection E_{n} := E_{∗,n} of s-spectra together with maps of s-spectra S_{t}^{1} ∧ E_{n} → E_{n+1} induced by the structure maps in the t-direction. We write Spt_{s,t}(k) for the category of (s, t)-bispectra. If X is a pointed space, let Σ_{s,t}^{∞}X denote the corresponding suspension (s, t)-bispectrum.

If E is an (s, t)-bispectrum and p, q are integers, denote by π_{p,q}(E) the sheaf of bigraded stable homotopy groups associated to the presheaf

U [figure]colim_{m} Hom_{SH_{s}^{𝔸^{1}}(k)}(S_{s}^{p−q} ∧ S_{t}^{q+m} ∧ Σ_{s}^{∞}U_{+}, E_{m}).

This expression makes sense for p < q since smashing with S_{s}^{1} yields an equivalence of categories SH_{s}^{𝔸^{1}}(k) → SH_{s}^{𝔸^{1}}(k), E ↦ S_{s}^{1} ∧ E. Moreover, we assume q + m ≥ 0.

Definition 1.12. A map E → E′ of (s, t)-bispectra is a stable weak equivalence if for all p, q ∈ ℤ, the induced map π_{p,q}(E) → π_{p,q}(E′) is an isomorphism of sheaves.

We are ready to define our main object of study:

Definition 1.13. The motivic stable homotopy category SH(k) of k is the category obtained from Spt_{s,t}(k) by inverting the stable weak equivalences.

Remark 1.14. There is an underlying model category structure on Spt_{s,t}(k); the weak equivalences are the stable weak equivalences defined in 1.12. Moreover, this model structure is stable, proper and simplicial.

By construction, the suspension functor Σ_{t}^{1} with respect to the Tate circle induces an equivalence of the motivic stable homotopy category. The same is true for the s-suspension functor Σ_{s}^{1} and the analogous result of 1.9 holds in SH(k).

Lemma 1.15. The canonical map 𝔸^{1} → Spec(k) induces a stable weak equivalence of suspension (s, t)-bispectra.

In 1.16, we note that maps between suspension spectra in SH(k) can be expressed in terms of maps in SH_{s}^{𝔸^{1}}(k). As the proof in the Nisnevich topology hinges on finite cohomological dimension, it is not clear whether there is a similarly general result in the etale topology.

Proposition 1.16. If X ∈ Sm/k and E ∈ Spt_{s,t}(k), there is an isomorphism

Hom_{SH(k)}(Σ_{s,t}^{∞} X_{+}, E) = colim_{n} Hom_{SH_{s}^{𝔸^{1}}(k)}(S_{t}^{n} ∧ Σ_{s}^{∞}X_{+}, E_{n}).

Proof. If E = (E_{m,n}) is some (s, t)-bispectrum, recall that E_{n} is the s-spectrum defined by the sequence (E_{0,n}, E_{1,n}, …). By model category theory, there exists a fibrant replacement E^{f} of E in Spt_{s,t}(k) and isomorphisms

Hom_{SH(k)}(Σ_{s,t}^{∞} X_{+}, E) = Hom_{Spt_{s,t}(k)}(Σ_{s,t}^{∞} X_{+}, E^{f})/_{≃} = Hom_{Spt_{s}(k)}(Σ_{s}^{∞} X_{+}, E_{0}^{f})/_{≃}.

The relation ≃ is the homotopy relation on maps; in our setting, homotopies are parametrized by the affine line 𝔸^{1}. By using properties of the Nisnevich topology (every scheme in Sm/k has finite cohomological dimension, coverings are generated by upper distinguished squares (1)), one can choose the fibrant replacement E^{f} so that

E_{0}^{f} = colim_{n} ((E_{0})^{f}  [figure]Ω_{t}((E_{1})^{f})  [figure]Ω_{t}(Ω_{t}((E_{2})^{f}))  [figure]⋯),

where (E_{n})^{f} is a fibrant replacement of E_{n} in Spt_{s}(k) and Ω_{t} is the right adjoint of the functor \xymatrix{S_{t}^{1} ∧ −: Spt_{s}(k) \ar[r] & Spt_{s}(k)} . Since the s-suspension spectra Σ_{s}^{∞} X_{+} and Σ_{s}^{∞} X_{+} ∧ 𝔸_{+}^{1} are finitely presentable objects in Spt_{s}(k), we have

Hom_{Spt_{s}(k)}(Σ_{s}^{∞} X_{+}, E_{0}^{f})/_{≃} = colim_{n} Hom_{Spt_{s}(k)}(Σ_{s}^{∞} X_{+}, Ω_{t}^{n}((E_{n})^{f}))/_{≃}.

The latter and the isomorphism

Hom_{SH_{s}(k)}(Σ_{s}^{∞} X_{+}, Ω_{t}^{n}(E_{n})) = Hom_{SH_{s}(k)}(S_{t}^{n} ∧ Σ_{s}^{∞} X_{+}, E_{n}),

obtained from the adjunction, imply the claimed group isomorphism. □

Suppose X, Y ∈ Sm/k. As a special case of 1.16 we obtain the isomorphism

Hom_{SH(k)}(Σ_{s,t}^{∞} X_{+}, Σ_{s,t}^{∞} Y_{+}) = colim_{m} Hom_{SH_{s}^{𝔸^{1}}(k)}(S_{t}^{m} ∧ Σ_{s}^{∞}X_{+}, S_{t}^{m} ∧ Σ_{s}^{∞}Y_{+}).

Remark 1.17. We will not discuss the notoriously difficult notion of an adequate smash product for (bi)spectra. One solution is to consider symmetric spectra rather than spectra [Jar00]. An alternate solution is discussed in the lectures of Dundas [Dun02]. For our purposes it suffices to know that a handicrafted smash product of bispectra induces a symmetric monoidal structure on SH(k). The proof of this fact is tedious but straight-forward. The unit for the monoidal structure is the ‘sphere spectrum’ or the suspension (s, t)-bispectrum Σ_{s,t}^{∞} Spec(k)_{+} of k and there is an isomorphism between the smash product Σ_{s,t}^{∞}X ∧ Σ_{s,t}^{∞}Y and Σ_{s,t}^{∞}(X × Y) for all X, Y ∈ Sm/k.

Let us briefly summarize the above constructions

\xymatrix{Spc_{•}(k) \ar[r]^{Σ_{s}^{∞}} & SH_{s}^{𝔸^{1}}(k) \ar[r]^{Σ_{t}^{∞}} & SH(k).}

The spaces in the motivic setting are the pointed simplicial sheaves on the Nisnevich site of Sm/k. By using the circles S_{s}^{1} and S_{t}^{1}, one defines spectra of pointed spaces as for spectra of simplicial pointed sets. The notion of 𝔸^{1}-stable weak equivalences of s-spectra forces the s-suspension spectrum of (𝔸^{1}, 0), the affine line pointed by zero, to be contractible. We use the same class of maps to define the motivic s-stable homotopy category SH_{s}^{𝔸^{1}}(k). Finally, the motivic stable homotopy category SH(k) is obtained by considering (s, t)-bispectra and formally inverting the class of stable weak equivalences; such maps are defined in terms of sheaves of bigraded homotopy groups. If Σ_{s,t}^{∞} : Sm/k → SH(k) denotes the suspension (s, t)-bispectrum functor, there is a natural equivalence of functors Σ_{s,t}^{∞} = Σ_{t}^{∞} ∘ Σ_{s}^{∞}.

Remark 1.18. Work in progress by Voevodsky suggests another construction of SH(k). This approach uses the theory of framed correspondences; a distant algebraic relative of framed cobordisms, which could have computational advantages.

Next, we shall specify a triangulated structure on SH(k). As for a wide range of other examples, this additional structure provides a convenient tool to construct long exact sequences.^{1}

The category Spt_{s,t}(k) is obviously complete and cocomplete: Limits and colimits are formed degree-wise in Spc_{•}(k). In particular, there is an induced coproduct ∨ in SH(k). We also note that the latter is an additive category: Since the simplicial circle is a cogroup object in the ordinary unstable homotopy category, we conclude that every spectrum in SH(k) is a two-fold simplicial suspension. Thus all objects in SH(k) are Abelian cogroup objects, and the set of maps out of any object is an Abelian group.

To define a triangulated category structure on SH(k), we need to specify a class of distinguished triangles and also a shift functor. Let E be an (s, t)-bispectrum. Its shift E[1] is the s-suspension of E. Any map f : E → E′ of (s, t)-bispectra has an associated cofibration sequence E → E′ → Cone(f) → Σ_{s}^{1}E. The cone of f is defined in terms of a push-out square in Spt_{s,t}(k), where Δ[1] is pointed by zero:

\xymatrix{E \ar[r]^{E∧0} \ar[d]_{f} & E ∧ Δ[1] \ar[d] \\ E′ \ar[r] & Cone(f)}

The map Cone(f) → Σ_{s}^{1}E collapses E′ to a point. A distinguished triangle in SH(k) is a sequence that is isomorphic to the image of a cofibration sequence in Spt_{s,t}(k). It follows that to any distinguished triangle X → Y → Z → X[1] in SH(k), there are long exact sequences of Abelian groups

\xymatrix{⋯ \ar[r] & [E, X[n]] \ar[r] & [E, Y [n]] \ar[r] & [E, Z[n]] \ar[r] & [E, X[n + 1]] \ar[r] & ⋯, and \\ ⋯ \ar[r] & [Z[n], E] \ar[r] & [Y [n], E] \ar[r] & [X[n], E] \ar[r] & [Z[n − 1], E] \ar[r] & · · · .}

Here, [−, −] denotes Hom_{SH(k)}(−, −).

The next lemma points out an important class of distinguished triangles.

Lemma 1.19. If X ↪ Y is a monomorphism of pointed spaces, then

\xymatrix{Σ_{s,t}^{∞} X \ar[r] & Σ_{s,t}^{∞} Y \ar[r] & Σ_{s,t}^{∞} Y/X \ar[r] & Σ_{s,t}^{∞} X[1]}

is a distinguished triangle in SH(k).

1The homotopy categories SH_{s}(k) and SH_{s}^{𝔸^{1}}(k) are also triangulated.

Proof. It follows from the model structure in 1.14 that the canonically induced map Cone(Σ_{s,t}^{∞}(X ↪ Y)) → Σ_{s,t}^{∞} Y/X is a stable weak equivalence. □

Remark 1.20. Note that 1.19 applies to open and closed embeddings in Sm/k.

An upper distinguished square is a pullback square in Sm/k

(1)  \xymatrix{W \ar[r] \ar[d] & V \ar[d]^{p} \\ U \ar[r]^{i} & X}

where i is an open embedding, p is an etale map, and p^{−1}(X ∖ U) → (X ∖ U) induces an isomorphism of reduced schemes. Any Zariski open covering gives an example of an upper distinguished square: If X = U ∪ V, let W = U ∩ V. In the Nisnevich topology, the generating coverings are {i : U → X, p : V → X} where i, p and W = p^{−1}(U) form an upper distinguished square (1). Since spaces are sheaves in the Nisnevich topology, we have

Lemma 1.21. A square of representable spaces obtained from an upper distinguished square is a pushout square.

In the following, we will show that upper distinguished squares give examples of distinguished triangles in SH(k). Part (a) of the next result can be thought of as a generalized Mayer-Vietoris property:

Corollary 1.22. Consider the upper distinguished square (1).

(a) There is a distinguished triangle in SH(k):

\xymatrix{Σ_{s,t}^{∞} W_{+} \ar[r] & Σ_{s,t}^{∞} U_{+} ∨ Σ_{s,t}^{∞} V_{+} \ar[r] & Σ_{s,t}^{∞} X_{+} \ar[r] & Σ_{s,t}^{∞} W_{+}[1].}

(b) There is a naturally induced stable weak equivalence:

\xymatrix{Σ_{s,t}^{∞} V/W \ar[r] & Σ_{s,t}^{∞} X/U.}

Proof. 1.20 shows that the pushout square of representable spaces associated to an upper distinguished square is a homotopy pushout square. This implies (a) because the suspension functor Σ_{s,t}^{∞} preserves homotopy pushout squares. To prove (b), we use 1.21 to conclude that there is even an isomorphism between the underlying pointed spaces. □

In the next result we consider the projective line ℙ^{1} to be pointed by the rational point at infinity. Another piece of fundamental information about the mixed spheres is:

Lemma 1.23. In SH(k), there are canonical isomorphisms:

(a) Σ_{s,t}^{∞} (𝔸^{1}/𝔸^{1} ∖ 0) = Σ_{s,t}^{∞} (S_{s}^{1} ∧ S_{t}^{1}).

(b) Σ_{s,t}^{∞} (ℙ^{1}, ∞) = Σ_{s,t}^{∞} (S_{s}^{1} ∧ S_{t}^{1}).

Proof. To prove (a), we use 1.15 and the following distinguished triangle obtained from 1.19,

\xymatrix{Σ_{s,t}^{∞} (𝔸^{1} ∖ 0, 1) \ar[r] & Σ_{s,t}^{∞} (𝔸^{1}, 1) \ar[r] & Σ_{s,t}^{∞} (𝔸^{1}/𝔸^{1} ∖ 0) \ar[r] & Σ_{s,t}^{∞} (𝔸^{1} ∖ 0, 1)[1].}

To prove (b), let us cover the projective line by two affine lines. We note that the choice of a point on ℙ^{1} is not important here, since all such points are 𝔸^{1}-homotopic. Moreover, by 1.22(a), there is the distinguished triangle

\xymatrix{Σ_{s,t}^{∞} (𝔸^{1} ∖ 0)_{+} \ar[r] & Σ_{s,t}^{∞} 𝔸_{+}^{1} ∨ Σ_{s,t}^{∞} 𝔸_{+}^{1} \ar[r] & Σ_{s,t}^{∞} ℙ_{+}^{1} \ar[r] & Σ_{s,t}^{∞} (𝔸^{1} ∖ 0)_{+}[1].}

To remove the disjoint base-points, we point all spaces by 1 : Spec(k) → 𝔸^{1} ∖ 0 and consider the quotients. For instance,

Σ_{s,t}^{∞}(𝔸^{1} ∖ 0)_{+}/Σ_{s,t}^{∞}Spec(k)_{+} = Σ_{s,t}^{∞}(𝔸^{1} ∖ 0, 1).

By considering the resulting distinguished triangle, the claim follows immediately from homotopy invariance 1.15. □

Remark 1.24. The pointed space T := 𝔸^{1}/𝔸^{1} ∖ 0 is called the Tate sphere. There is also an ‘unstable motivic homotopy category’ for spaces and 1.23 holds unstably.

We place emphasize on a typical instance of 1.22(b):

Corollary 1.25. Suppose p : V → X is an etale morphism and Z is a closed subscheme of X, so that p^{−1}(Z) → Z is an isomorphism. Then there is a canonical isomorphism in SH(k):

Σ_{s,t}^{∞} (V/V ∖ p^{−1}(Z)) = Σ_{s,t}^{∞} (X/X ∖ Z).

As in classical homotopy theory, there is a notion of Thom spaces in motivic homotopy theory.

Definition 1.26. Suppose E → X is a vector bundle and let i : X → E denote the zero-section. Then the Thom space Th(E/X) is the space E/E ∖ i(X) pointed by the image of E ∖ i(X).

Example 1.27. We list some examples of Thom spaces.

(a) Suppose E_{1} → X_{1} and E_{2} → X_{2} are vector bundles. As pointed spaces, we have that Th(E_{1} × E_{2}/X_{1} × X_{2}) = Th(E_{1}/X_{1}) ∧ Th(E_{2}/X_{2}).

(b) The Thom space of the trivial 1-bundle 𝔸^{1} × X → X is the smash product T ∧ X_{+}.

Theorem 1.28 (Homotopy Purity). Let i : Z ↪ X be a closed embedding in Sm/k. Denote the corresponding normal vector bundle of Z in X by ℕ_{X,Z}. Then there is a canonical isomorphism in SH(k):

(2)  Σ_{s,t}^{∞} Th(ℕ_{X,Z}) = Σ_{s,t}^{∞} (X/X ∖ i(Z)).

To prove the general case of the homotopy purity theorem, one employs the wellknown deformation to the normal cone construction based on the blow-up of X ×𝔸^{1}. In what follows, by ‘isomorphism’ we mean a canonical isomorphism in SH(k). Let us show by elementary means how to construct the isomorphism (2) in the case of a finite separable field extension k′/k. By the Primitive Element Theorem, there exists an element α such that k′ = k(α). Let ϕ : k[X] → k′ be the surjective map defined by X ↦ α. If f is the minimal polynomial of α, then the induced closed embedding i : Spec(k′) ↪ 𝔸_{k}^{1} sends the closed point of Spec(k′) to the closed point x of 𝔸_{k}^{1}, corresponding to the prime ideal (f). We claim to have an isomorphism

Σ_{s,t}^{∞} Th(ℕ_{𝔸_{k}^{1},k′}) = Σ_{s,t}^{∞} (𝔸_{k}^{1}/𝔸_{k}^{1} ∖ {x}).

With respect to the identification of the normal bundle of i with 𝔸_{k′}^{1}, the Thom space Th(ℕ_{𝔸_{k}^{1},k′}) is isomorphic to 𝔸_{k′}^{1}/𝔸_{k′}^{1} ∖ {0}. In k′[X], f factors into irreducible polynomials, say f_{1}, …, f_{q}. Let x_{1}, …, x_{q} denote the corresponding closed points. Since the extension k′/k is separable, we may assume f_{1} = X − α and f_{i}(α) ≠ 0 if i ≠ 1. Note that k′[X] → k′[X], defined by X ↦ X − α, is an automorphism. By the above, it results an isomorphism

Σ_{s,t}^{∞} Th(ℕ_{𝔸_{k}^{1},k′}) = Σ_{s,t}^{∞} (𝔸_{k′}^{1}/𝔸_{k′}^{1} ∖ {x_{1}}).

We note that 𝔸_{k′}^{1} ∖ {x_{1}} ↪ 𝔸_{k′}^{1} fits into the upper distinguished square:

\xymatrix{𝔸_{k′}^{1} ∖ \{x_{1}, x_{2}, …, x_{q}\} \ar[r] \ar[d] & 𝔸_{k′}^{1} ∖ \{x_{2}, …, x_{q}\} \ar[d] \\ 𝔸_{k′}^{1} ∖ \{x_{1}\} \ar[r] & 𝔸_{k′}^{1}}

By applying 1.22(b), we find

(3)  Σ_{s,t}^{∞} Th(ℕ_{𝔸_{k}^{1},k′}) = Σ_{s,t}^{∞} (𝔸_{k′}^{1} ∖ {x_{2}, …, x_{q}}/𝔸_{k′}^{1} ∖ {x_{1}, x_{2}, …, x_{q}}).

The right hand side of the isomorphism (3) is related to 𝔸_{k}^{1}/𝔸_{k}^{1} ∖ {x} via the upper distinguished square

\xymatrix{𝔸_{k′}^{1} ∖ \{x_{1}, x_{2}, …, x_{q}\} \ar[r] \ar[d] & 𝔸_{k′}^{1} ∖ \{x_{2}, …, x_{q}\} \ar[d] \\ 𝔸_{k}^{1} ∖ \{x\} \ar[r] & 𝔸_{k}^{1}}

which defines an isomorphism

Σ_{s,t}^{∞} (𝔸_{k′}^{1} ∖ {x_{2}, …, x_{q}}/𝔸_{k′}^{1} ∖ {x_{1}, x_{2}, …, x_{q}}) = Σ_{s,t}^{∞} (𝔸_{k}^{1}/𝔸_{k}^{1} ∖ {x}).

By using the fact that the isomorphism of 1.28 is compatible in the obvious sense with etale morphisms, one can show that the isomorphism which we have constructed above indeed coincides with (2).

Remark 1.29. If Y → X is a finite etale map, the proof of 1.28 implies the existence of a transfer map Σ_{s,t}^{∞} X_{+} → Σ_{s,t}^{∞} Y_{+} in SH(k).

2. Cohomology theories

The introduction of the motivic stable homotopy category via spectra provides a convenient framework for defining cohomology theories of algebraic varieties. Such theories encode important data about the input, often in a form which allows to make algebraic manipulations. In what follows, we consider the examples of motivic cohomology, algebraic K-theory and algebraic cobordism.

2.1. The motivic Eilenberg-MacLane spectrum Hℤ. Let us first motivate the construction of the motivic Eilenberg-MacLane spectrum Hℤ. In stable homotopy theory, the key to understand cohomology theories is to study their representing spectra. An important example is singular cohomology. We let K(ℤ, n) denote the Eilenberg-MacLane simplicial set with homotopy groups

(4)  π_{i}K(ℤ, n) = \begin{cases}ℤ & i = n, \\ 0 & i ≠ n. \end{cases}

The simplicial set K(ℤ, n) is uniquely determined up to weak equivalence by (4). A specific model may be constructed as follows: Let ℤ(−) be the functor which associates to a pointed simplicial set (X, x) the simplicial Abelian group ℤ[X]/ℤ[x], that is, the simplicial free Abelian group generated by X modulo the copy of the integers generated by the base-point. Then K(ℤ, n) is the underlying simplicial set of the simplicial Abelian group ℤ(Δ^{n}/∂Δ^{n}). Moreover, the K(ℤ, n)’s assemble to define a spectrum Hℤ which represents singular cohomology: If X is a simplicial set, then its integral singular cohomology groups H^{n}(X, ℤ) coincide with Hom_{SH}(Σ^{∞}X_{+}, Hℤ).

In motivic homotopy theory, there is supposed to be a closely related algebrogeometric analog of the spectrum Hℤ. As it turns out, it is impossible to construct an (s, t)-bispectrum whose constituent terms satisfy a direct motivic analog of (4); however, by using the theory of algebraic cycles one can define a spectrum Hℤ that represents motivic cohomology. Next, we shall indicate the construction.

If X ∈ Sm/k, let L(X) be the following functor: Its value on U ∈ Sm/k is the free Abelian group generated by closed irreducible subsets of U × X which are finite over U and surjective over a connected component of U. An element in this group is called a cycle. The graph Γ(f) of a morphism f : U → X is an example of a cycle in U × X. One can show that L(X) is a Nisnevich sheaf, hence a pointed space by forgetting the Abelian group structure. Moreover, there is a map Γ(X) : X → L(X). One can extend L(−) to a functor from pointed spaces to Nisnevich sheaves with values in simplicial Abelian groups.^{2} For example, L(S_{s}^{0} ∧ S_{t}^{1}) = L((𝔸^{1} ∖ 0, 1)) is the quotient Nisnevich sheaf of Abelian groups L(𝔸^{1} ∖ 0)/L(Spec(k)), considered as a pointed space. The pointed space L(S_{s}^{1} ∧ S_{t}^{1}) turns out to be equivalent to L(ℙ^{1}, ∞) = L(ℙ^{1})/L(Spec(k)). By using the exterior product of cycles, there exists a pairing

\xymatrix{L(S_{s}^{p} ∧ S_{t}^{q}) ∧ L(S_{s}^{m} ∧ S_{t}^{n}) \ar[r] & L(S_{s}^{p+m} ∧ S_{t}^{q+n}).}

In particular, we obtain the composite maps

(5) [figure: σ_{s}: S_{s}^{1} ∧ L(S_{s}^{m} ∧ S_{t}^{n}); Γ(S_{s}^{1})∧id; L(S_{s}^{1}) ∧ L(S_{s}^{m} ∧ S_{t}^{n}); L(S_{s}^{m+1}, S_{t}^{n}), and; σ_{t}: S_{t}^{1} ∧ L(S_{s}^{m} ∧ S_{t}^{n}); Γ(S_{t}^{1})∧id; L(S_{t}^{1}) ∧ L(S_{s}^{m} ∧ S_{t}^{n}); L(S_{s}^{m}, S_{t}^{n+1}).]

Remark 2.2. The topologically inclined reader might find it amusing to compare the above with Bökstedt’s notion of an FSP.

Definition 2.3. The Eilenberg-MacLane spectrum Hℤ is the (s, t)-bispectrum with constituent pointed spaces Hℤ_{m,n} := L(S_{s}^{m} ∧ S_{t}^{n}) and structure maps given by (5).

Remark 2.4. Let ℓ be a prime number. To define the Eilenberg-MacLane spectrum with mod-ℓ coefficients, one considers the reduction of L(X) modulo ℓ in the category of Abelian sheaves and proceeds as above.

We can now define the motivic cohomology groups of (s, t)-bispectra.

Definition 2.5. Let E be an (s, t)-bispectrum and let p, q be integers. The integral motivic cohomology groups of E are defined by

Hℤ^{p,q}(E) := Hom_{SH(k)}(E, S_{s}^{p−q} ∧ S_{t}^{q} ∧ Hℤ).

Suppose X ∈ Sm/k. The integral motivic cohomology group H^{p,q}(X, ℤ) of X in degree p and weight q is by definition Hℤ^{p,q}(Σ_{s,t}^{∞}X_{+}). One can show that H^{p,q}(X, ℤ) is isomorphic to the higher Chow group CH^{q}(X, 2q − p). In section 3, we outline an approach to construct a spectral sequence whose E_{2}-terms are the integral motivic cohomology groups of X and whose target groups are the algebraic K-groups of X. Since there is a spectral sequence for topological K-theory with input terms singular cohomology, this would allow to make more precise the analogy between motivic and singular cohomology.

2.6. The algebraic K-theory spectrum KGL. In 1.23(b), we noted that the suspension (s, t)-bispectra of (ℙ^{1}, ∞) and S_{s}^{1} ∧ S_{t}^{1} are canonically isomorphic in SH(k). In fact, we may replace the suspension coordinates S_{s}^{1} and S_{t}^{1} by the single suspension coordinate ℙ^{1} without changing the motivic stable homotopy category. In order to represent cohomology theories on Sm/k, it turns out to be convenient to consider ℙ^{1}-spectra. To define the category of such spectra, one replaces in 1.3 every occurance of the simplicial circle by the projective line. If E is a ℙ^{1}-spectrum, we may associate a bigraded cohomology theory by setting

2The extension is dictated by the requirement that L preserves colimits. Since every pointed space is a colimit of representable functors, it suffices to describe the values of L on Sm/k.

(6)  E^{p,q}(X) := Hom_{SH(k)}(Σ_{ℙ^{1}}^{∞}X_{+}, E ∧ S_{s}^{p−2q} ∧ (ℙ^{1})^{∧q}).

An important example is the spectrum representing algebraic K-theory, which we describe next.

If m is a non-negative integer, we denote the Grassmannian of vector spaces of dimension n in the n+m-dimensional vector space over k by Gr_{n}(𝔸^{n+m}). By letting m tend to infinity, it results a directed system of spaces; we denote its colimit by BGL_{n}. There are canonical monomorphisms ⋯ ↪ BGL_{n} ↪ BGL_{n+1} ↪ ⋯, and we write BGL for the sequential colimit of this diagram. Define KGL to be a fibrant replacement of ℤ × BGL in the unstable motivic homotopy theory of k. Then there exists a map

(7)  \xymatrix{β : ℙ^{1} ∧ KGL \ar[r] & KGL,}

which represents the canonical Bott element of K_{0}(ℙ^{1} ∧ ℤ × BGL). More precisely, the map (7) is a lift of an isomorphism ℙ^{1} ∧ ℤ × BGL → ℤ × BGL in the unstable motivic homotopy category. Details are recorded in section 6.2 of [Voe98].

Definition 2.7. The algebraic K-theory spectrum KGL is the ℙ^{1}-spectrum

(KGL, KGL, ⋯, KGL, ⋯),

together with the structure maps in (7).

As in topological K-theory, there is also Bott periodicity in algebraic K-theory: Since the structure maps of KGL are defined by lifting an isomorphism in the unstable motivic homotopy category, there is clearly an isomorphism (8) in SH(k)

(8)  ℙ^{1} ∧ KGL = KGL.

If k admits a complex embedding, taking ℂ-points defines a realization functor t_{ℂ} : SH(k) → SH which maps Σ_{s,t}^{∞} S^{p,q} to the suspension spectrum of the p-sphere. This functor takes KGL to the ordinary complex topological K-theory spectrum.

2.8. The algebraic cobordism spectrum MGL. Recall the notation in 2.6, and let γ_{n,m} → Gr_{n}(𝔸^{n+m}) be the tautological vector bundle. The canonical morphism Gr_{n}(𝔸^{n+m}) → Gr_{n}(𝔸^{n+m+1}) is covered by a bundle map γ_{n,m} → γ_{n,m+1}. By taking the colimit over m we get the universal n-dimensional vector bundle γ_{n} → BGL_{n}. The product 𝔸^{1}×γ_{n} → BGL_{n} with the trivial one-dimensional bundle 𝔸^{1} → Spec(k) is classified by the canonical map BGL_{n} → BGL_{n+1}. In particular, there exists a bundle map 𝔸^{1} × γ_{n} → γ_{n+1}. In terms of Thom spaces, we obtain the map

(9)  \xymatrix{Th(𝔸^{1}) ∧ Th(γ_{n}) = (𝔸^{1}/𝔸^{1} ∖ 0) ∧ Th(γ_{n}) \ar[r] & Th(γ_{n+1}).}

Here, note that we may apply 1.27(a) since γ_{n} → BGL_{n} is a colimit of vector bundles of smooth schemes. By using (9) and 1.24, there result maps

(10)  \xymatrix{c_{n} : ℙ^{1} ∧ Th(γ_{n}) \ar[r] & Th(γ_{n+1}).}

Definition 2.9. The algebraic cobordism spectrum MGL is the ℙ^{1}-spectrum

(Th(γ_{0}), Th(γ_{1}), …, Th(γ_{n}), …),

together with the structure maps in (10).

There is a strong analogy between MGL and the ordinary complex cobordism spectrum MU. For example, we have t_{ℂ}(MGL) = MU. The notions of orientation, map of rings and formal group laws in ordinary stable homotopy theory have direct analogs in the world of ℙ^{1}-spectra.

3. The slice filtration

In classical homotopy theory, the Eilenberg-MacLane space K(ℤ, n) has a unique non-trivial homotopy group and up to homotopy equivalence there is a unique such space for each n (4). The situation in motivic homotopy theory is quite different. For example, the homotopy groups π_{p,q}(Hℤ) are often non-zero. To give an internal description of Hℤ within the stable motivic homotopy category, we employ the socalled slice filtration. In what follows, we recall and discuss the status of Voevodsky’s conjectures about the slices of the sphere spectrum 1 = Σ_{s,t}^{∞}Spec(k)_{+}, Hℤ and KGL. For more details we refer to the original papers [Voe00a] and [Voe02b].

Let SH^{eff}(k) be the smallest triangulated sub-category of SH(k) which is closed under direct sums and contains all the (s, t)-bispectra Σ_{s,t}^{∞}X_{+}. In particular if n ≥ 1, the desuspension spectra Σ_{t}^{−n}Σ_{s,t}^{∞}X_{+} are not contained in SH^{eff}(k). The ‘effective’ s-stable homotopy category SH_{s}^{eff}(k) is constructed similarly, by replacing Σ_{s,t}^{∞} with Σ_{s}^{∞} in the definition above.

In this section, we shall study the sequence of full embeddings of categories:

(11)  \xymatrix{⋯ \ar@{^{(}->}[r] & Σ_{s,t}^{1}SH^{eff}(k) \ar@{^{(}->}[r] & SH^{eff}(k) \ar@{^{(}->}[r] & Σ_{s,t}^{−1}SH^{eff}(k) \ar@{^{(}->}[r] & ⋯}

The sequence (11) is called the slice filtration. The above is a filtration in the sense that SH(k) is the smallest triangulated category which contains Σ_{s,t}^{n}SH^{eff}(k) for all n and is closed under arbitrary direct sums in SH(k). For each n, Σ_{s,t}^{n}SH^{eff}(k) is a triangulated category having arbitrary direct sums and a set of compact generators. By a general result for triangulated categories [Nee96], the full inclusion functor i_{n} : Σ_{s,t}^{n}SH^{eff}(k) ↪ SH(k) allows a right adjoint r_{n} : SH(k) → Σ_{s,t}^{n}SH^{eff}(k) and the unit of the adjunction Id → r_{n} ∘ i_{n} is an isomorphism. Consider now the reverse composition

\xymatrix{f_{n} : SH(k) \ar[r]^{r_{n}} & Σ_{s,t}^{n}SH^{eff}(k) \ar[r]^{i_{n}} & SH(k).}

The counit f_{n+1} → Id applied to the functor f_{n} determines a natural transformation f_{n+1} = f_{n+1} ∘ f_{n} → f_{n}. If E is an (s, t)-bispectrum and n ∈ ℤ, the slice tower of E consists of the distinguished triangles

(12)  \xymatrix{f_{n+1}E \ar[r] & f_{n}E \ar[r] & s_{n}E \ar[r] & f_{n+1}E[1]}

in SH(k).

Definition 3.1. The n-th slice of E is s_{n}E.

Remark 3.2. If E ∈ SH(k), then s_{n}E is contained in Σ_{s,t}^{n}SH^{eff}(k). This uses that f_{n+1}E, f_{n}E ∈ Σ_{s,t}^{n}SH^{eff}(k). The adjunction above implies that s_{n}E receives only trivial maps from Σ_{s,t}^{n+1}SH^{eff}(k). A standard argument shows that the mentioned properties characterize the triangulated functors s_{n} : SH(k) → SH(k) up to canonical isomorphism. Suppose we have E ∈ Σ_{s,t}^{n}SH^{eff}(k) and let k ≤ n. Then f_{k}E = E and moreover, the k-th slice s_{k}E is trivial for all k < n.

The slice tower (12) is analogous to a Postnikov tower; the slices corresponding to the cofibers or quotients. If E is an ordinary spectrum, recall that its Postnikov tower expresses E as the sequential colimit of a diagram of cofibrations

\xymatrix{⋯ \ar[r] & P_{−1}E \ar[r]^{p_{0}} & P_{0}E \ar[r]^{p_{1}} & P_{1}E \ar[r] & ⋯ \ar[r]^{p_{n}} & P_{n}E \ar[r]^{p_{n+1}} & · · · .}

Here, the canonical map P_{n}E → E induces isomorphisms on all stable homotopy groups of index less or equal to n, and π_{i}(P_{n}E) = 0 for i > n. Thus the cofiber of the map p_{n} is an Eilenberg-MacLane spectrum Σ^{n}Hπ_{n}(E). In particular, it is a module over Hℤ. The zeroth stage of the Postnikov tower of the ordinary sphere spectrum is the Eilenberg-MacLane spectrum Hℤ. This motivates the following conjecture:

Conjecture 3.3. In SH(k) we have s_{0}1 = Hℤ.

The collection of the functors s_{n} is compatible with the smash product, meaning that if E and F are objects of SH(k), there is a natural map

\xymatrix{s_{n}(E) ∧ s_{m}(F) \ar[r] & s_{n+m}(E ∧ F).}

In particular, we get a map s_{0}(1) ∧ s_{n}(E) → s_{n}(E). If 3.3 holds, the above shows that each slice of an (s, t)-bispectrum has a natural and unique structure of a module over the motivic cohomology spectrum.

Theorem 3.4. Conjecture 3.3 holds for all fields of characteristic zero.

In our sketch proof of 3.4, we start with the zeroth slice of Hℤ.

Lemma 3.5. Let k be a field of characteristic zero. Then s_{0}Hℤ = Hℤ.

To prove 3.5 we use the following facts: First, the motivic Eilenberg-MacLane spectrum can be shown to be an effective spectrum. For example, infinite symmetric products of the mixed spheres constitute a model for Hℤ, provided that k is a field of characteristic zero. More details are given below. This implies immediately that f_{0}Hℤ = Hℤ. Second, if X is a scheme in Sm/k, then every map in SH(k) from Σ_{s,t}^{1}Σ_{s,t}^{∞} X_{+} to Hℤ is trivial by Voevodsky’s cancellation theorem [Voe02a].^{3} Hence for every E ∈ Σ_{s,t}^{1}SH^{eff}(k), we get isomorphisms

Hom_{Σ_{s,t}^{1}SH^{eff}(k)}(E, r_{1}Hℤ) = Hom_{SH(k)}(i_{1}E, Hℤ) = 0,

which proves r_{1}Hℤ = 0. In particular, f_{1}Hℤ = 0 and hence s_{0}Hℤ = f_{0}Hℤ = Hℤ.

Denote by cone the cone of the unit map 1 → Hℤ. To finish the proof of 3.4, by 3.5 it remains to show that s_{0}cone = 0. As one can see from 3.2, it suffices to prove that cone is contained in Σ_{s,t}^{1}SH^{eff}(k). If k is a field of characteristic zero, Hℤ has an explicit description in terms of infinite symmetric products of 𝔸^{n} and 𝔸^{n} ∖ 0. This allows to conclude the statement about cone: We shall sketch a proof. Recall that Hℤ_{n,n} = L(S_{s}^{n} ∧ S_{t}^{n}), which is weakly equivalent to the quotient sheaf L(𝔸^{n})/L(𝔸^{n} ∖ 0). Let L^{eff}(𝔸^{n}) denote the sheaf which maps U ∈ Sm/k to the free Abelian monoid generated by closed irreducible subsets of U × 𝔸^{n} which are finite over U and surjective over a connected component of U. The sheaf L^{eff}(𝔸^{n} ∖ 0) is defined similarly. In particular, L^{eff} consists of cycles with nonnegative coefficients. Denote the quotient L^{eff}(𝔸^{n})/L^{eff}(𝔸^{n} ∖ 0) by Hℤ_{n,n}^{eff}. It is straightforward to define Hℤ_{m,n}^{eff} and moreover, along the same lines as above, one can show that these pointed spaces form an (s, t)-bispectrum Hℤ^{eff}. If n ≥ 1, the canonical map Hℤ_{n,n}^{eff} → Hℤ_{n,n} turns out to be a weak equivalence of pointed spaces. Furthermore, the canonical inclusion 1 ↪ Hℤ factors through 1 ↪ Hℤ^{eff}. Hence the quotient of the latter map is equivalent to cone.

Let us consider T^{n} = 𝔸^{n}/𝔸^{n} ∖ 0 and let Hℤ_{n,n}^{≤d} be the subsheaf of Hℤ_{n,n}^{eff} mapping U ∈ Sm/k to the cycles of degree ≤ d over U. Then the natural inclusion T^{n} ↪ Hℤ_{n,n}^{eff} has a filtration

\xymatrix{T^{n} = Hℤ_{n,n}^{≤1} \ar@{^{(}->}[r] & Hℤ_{n,n}^{≤2} \ar@{^{(}->}[r] & ⋯ \ar@{^{(}->}[r] & Hℤ_{n,n}^{≤d} \ar@{^{(}->}[r] & ⋯ \ar@{^{(}->}[r] & Hℤ_{n,n}^{eff}.}

By the above, it results a filtration on the quotient sheaf Hℤ_{n,n}^{eff}/T^{n} given in terms of the quotients Hℤ_{n,n}^{≤d}/Hℤ_{n,n}^{≤d−1}.

¿From the characteristic zero assumption and results in [SV96], Hℤ_{n,n}^{≤d}/Hℤ_{n,n}^{≤d−1} is isomorphic to the sheaf (T^{n})^{∧d}/S_{d}; the d-th symmetric power of T^{n}. As one can prove by analyzing (T^{n})^{∧d}/S^{d}, this sheaf is contained in the smallest class of pointed spaces which is closed under homotopy colimits and contains all sheaves X/(X −Z), where Z is a closed subscheme of X ∈ Sm/S of codimension ≥ n + 1. However, 1.28 implies that the (s, t)-suspension spectrum of any space in this class is contained in Σ_{s,t}^{n+1}SH^{eff}(k). This completes the sketch proof of 3.4.

3In other words, motivic cohomology of X in weight −1 is zero.

There is also a slice filtration of SH_{s}^{𝔸^{1}}(k) which is constructed analogously to (11): If n ≥ 0, we let Σ_{t}^{n}SH_{s}^{𝔸^{1}}(k) be the smallest compactly generated triangulated subcategory of SH_{s}^{𝔸^{1}}(k) which is closed under arbitrary direct sums and generated by the objects Σ_{t}^{n}Σ_{s}^{∞}X_{+}. It is clear that we obtain a filtration

(13) \xymatrix{⋯ \ar@{^{(}->}[r] & Σ_{t}^{n}SH_{s}^{𝔸^{1}}(k) \ar@{^{(}->}[r] & Σ_{t}^{n−1}SH_{s}^{𝔸^{1}}(k) ⋯ \ar@{^{(}->}[r] & Σ_{t}^{0}SH_{s}^{𝔸^{1}}(k) = SH_{s}^{𝔸^{1}}(k).}

Given the above, the resulting slice filtration of SH_{s}^{𝔸^{1}}(k) is obtained similarly to (12). One can show that the t-suspension functor Σ_{t}^{∞} preserves the slice filtrations, that is, Σ_{t}^{∞}(Σ_{t}^{n}SH_{s}^{𝔸^{1}}(k)) ⊆ Σ_{s,t}^{n}SH^{eff}(k). Denote by Ω_{t}^{∞} : SH(k) → SH_{s}^{𝔸^{1}}(k) its right adjoint.

Conjecture 3.6. There is an inclusion Ω_{t}^{∞}(Σ_{s,t}^{n}SH^{eff}(k)) ⊆ Σ_{t}^{n}SH_{s}^{𝔸^{1}}(k).

In topology, recall that first applying the suspension functor and second the loop space functor preserve connectivity. By using an inductive argument, one can show the following conjecture implies 3.6.

Conjecture 3.7. If X ∈ Sm/k and n ≥ 0, then Ω_{t}^{1}Σ_{t}^{1}Σ_{s}^{∞}(S_{t}^{n} ∧ X_{+}) ∈ Σ_{t}^{n}SH_{s}^{𝔸^{1}}(k).

This conjecture is not known at present, even if k is a field of characteristic zero. A proof seems to aquire a fair amount of work. A possible approach to prove 3.7 is to develop an analog of the theory of operads, at least for A_{∞}-operads, in order to have an explicit model for Ω_{ℙ^{1}}Σ_{ℙ^{1}}(X). Framed correspondences might be a useful tool in working out such a theory, cp. 1.18.

Let us finally relate the above machinery to a possibly new approach to some recent results in algebraic K-theory. In [Bl87], Beilinson conjectured the existence of an Atiyah-Hirzebruch type spectral sequence for the algebraic K-groups of nice schemes. In [BL95], Bloch and Lichtenbaum constructed such a spectral sequence. Their work has been expanded by Friedlander and Suslin [FS00], by Levine [Lev01] and by Suslin [Sus02]. By construction, there is a spectral sequence associated to the slice tower (12). In the example of KGL, one obtains the spectral sequence

(14)  Hom_{SH(k)}(Σ_{s,t}^{∞}X_{+}, S_{s}^{p−q} ∧ S_{t}^{q} ∧ s_{0}KGL) ⇒ K_{−p−q}(X).

The main problem one encounters here, is to identify the input terms in (14) with motivic cohomology. This is essentially the content of the next conjecture.

Conjecture 3.8. s_{0}KGL = Hℤ.

Because of Bott periodicity (8), the above describes indeed all the slices of KGL. More precisely, one expects to have s_{n}KGL = Σ_{s,t}^{n}Hℤ. As a consequence, we have that (14) and 3.8 imply the Atiyah-Hirzebruch type spectral sequence

(15)  H^{p−q,q}(X, ℤ) ⇒ K_{−p−q}(X).

To give some further evidence as to why this approach should lead to the spectral sequence (15), we remark that 3.3 and 3.6 would imply 3.8. A proof of this claim can be found in [Voe02b].

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Department of Mathematics, University of Western Ontario, London, Canada.

E-mail address: oroendig@uwo.ca

Department of Mathematics, University of Oslo, Oslo, Norway.

E-mail address: paularne@math.uio.no