VOEVODSKY’S NORDFJORDEID LECTURES ON MOTIVIC HOMOTOPY THEORY

OLIVER RÖNDIGS AND PAUL ARNE ØSTVÆR

Preliminary version

1. Introduction

This note is a report on Voevodsky’s lectures at the summer school on motivic homotopy theory in Nordfjordeid.

2. Motivic stable homotopy theory

In this section we introduce the motivic stable homotopy category of a field k.

2.1. Spaces. Let Sm/k be the category of separated smooth schemes of finite type over a field k. The category of spaces is the category Δ^{op}Shv_{Nis}(Sm/k) of Nisnevich sheaves on Sm/k with values in simplicial sets. Any scheme X in Sm/k determines a representable space via the Yoneda embedding. This follows since the Nisnevich topology is sub-canonical. On the other hand, any simplicial set can be viewed as a constant Nisnevich sheaf on Sm/k.

A pointed space is a space X together with a map x : Spec(k) → X, where Spec(k) is the representable sheaf with constant simplicial structure. We denote the category of pointed spaces by Δ^{op}Shv_{Nis}(Sm/k)_{∗}. If X is a space, let X_{+} denote the pointed space X ⨿ Spec(k). This defines a left adjoint to the forgetful functor from pointed spaces to spaces. The category of pointed spaces has a symmetric monoidal structure: Suppose X and Y are pointed spaces. Their smash product X ∧ Y is the space associated to the presheaf U ↦ X(U) ∧ Y (U). Note that Spec(k)_{+} is the unit for the smash product.

To define the stable motivic homotopy category we employ the notion of bispectra for pointed spaces. In classical stable homotopy theory one ‘inverts’ the simplicial circle to construct spectra of pointed simplicial sets. This works slightly differently in the motivic context. Most notably; for reasons which will become apparent later, we need two ‘circles’ to obtain an adequate stable homotopy theory. Let Δ[n] denote the standard simplicial n-simplex. The simplicial circle is the coequalizer of the diagram Δ[0] ⇉ Δ[1]. By analogy, we denote the corresponding pointed space S_{s}^{1}. Let 𝔸^{1} ∈ Sm/k denote the affine line. The Tate circle S_{t}^{1} is the space 𝔸^{1} −0 pointed by the global section given by the identity. Note that S_{t}^{1} is the underlying scheme of the multiplicative group. We let S_{t}^{n} denote the n-fold smash product of the Tate circle, and likewise for the n-simplicial sphere S_{s}^{n}.

2.2. The motivic s-stable homotopy category SH_{s}^{𝔸^{1}}(k). To invert S_{s}^{1} we form spectra of pointed spaces. This is analogous to classical spectra where the simplicial circle is inverted.

Definition 2.3. 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 2.4. 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. Let (X, x) be a pointed space. If n ≥ 1, then π_{n}(X, x) is 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). This is a sheaf of Abelian groups if n ≥ 2. The classical suspension homomorphism for a pointed simplicial set induces 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 n ∈ ℤ, we obtain a 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] & ⋯}

of sheaves of Abelian groups where m > n. The sheaves of stable homotopy groups of E are the sheaves of Abelian groups

π_{n}(E) := colim_{m} π_{n+m}(E_{m})

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

Definition 2.5. Let SH_{s}(k) be the category obtained from Spt_{s}(k) by inverting the s-stable weak equivalences.

Remark 2.6. One can show that Spt_{s}(k) has the structure of a proper simplicial stable model category, with s-stable weak equivalences as weak equivalences. The homotopy category of this model category is SH_{s}(k).

There is an obvious way to smash an s-spectrum with a pointed space. When smashing with S_{s}^{1}, the induced simplicial suspension functor becomes an equivalence of SH_{s}(k). The word ‘s-stable’ refers to this fact.

Remark 2.7. Let Spt denote the category of ordinary spectra. 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 achieve this is as follows: By evaluating the spaces in an s-spectrum E at a scheme U ∈ Sm/k we obtain an ordinary spectrum E(U). An s-spectrum E is called 𝔸^{1}-local if the map E(U) → E(U × 𝔸^{1}) induced by the projection U × 𝔸^{1} → U is a stable weak equivalence for every scheme U ∈ Sm/k. Next, 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 bijective.

Definition 2.8. 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.

Lemma 2.9. Assume X ∈ Sm/k. Then the canonical map X × 𝔸^{1} → X induces an 𝔸^{1}-stable weak equivalence of s-suspension spectra. In particular, Σ_{s}^{∞} (𝔸^{1}, 0) is zero in the motivic s-stable homotopy category of k.

We note that the simplicial suspension functor is an equivalence of SH_{s}^{𝔸^{1}}(k). The last step in the construction of the motivic stable homotopy category is to invert the Tate circle. This will be made precise in the next section.

2.10. The motivic stable homotopy category SH(k). The definition we will give of SH(k) combines the category of s-spectra and the Tate circle S_{t}^{1}. In order to make this statement precise we use bispectra as a convenient technical device. 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 have to be compatible in the sense that the following diagram is required to commute.

\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 bispectra. An (s, t)-bispectrum can be interpreted as a 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 let Spt_{s,t}(k) denote the category of (s, t)-bispectra. If X is a pointed space, we denote the corresponding suspension (s, t)-bispectrum by Σ_{s,t}^{∞}X.

If E is an (s, t)-bispectrum and p, q are integers, we let π_{p,q}(E) denote 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}).

In this expression we use the observation that 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 2.11. 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 2.12. 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 2.13. We remark that there is an underlying model category structure on Spt_{s,t}(k) where the weak equivalences are the stable weak equivalences defined in 2.11. Moreover, this model structure is stable, proper and simplicial.

By construction, suspension Σ_{t}^{1} with the Tate circle induces an equivalence on the motivic stable homotopy category. The same is true for the simplicial suspension functor Σ_{s}^{1}, and the analogue of 2.9 holds in SH(k).

Lemma 2.14. The canonical map 𝔸^{1} → Spec(k) induces a stable weak equivalence of suspension (s, t)-bispectra.

In 2.15 we show that maps between suspension spectra in SH(k) can be expressed in terms of maps in SH_{s}^{𝔸^{1}}(k). This result depends essentially on the fact that schemes have finite cohomological dimension in the Nisnevich topology. For example, it is not clear whether there is a similar result for the etale topology.

Proposition 2.15. For X ∈ Sm/k and E ∈ Spt_{s,t}(k) there is an isomorphism

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

Suppose X, Y ∈ Sm/k. As a special case of 2.15 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 2.16. We will not discuss the notoriously difficult notion of an adequate smash product for (bi)spectra. One solution is to consider symmetric spectra instead of spectra. An alternative solution is discussed in the lectures of B. I. Dundas. For our purposes it suffices to know that the 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 (s, t)-suspension spectrum 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 X, Y ∈ Sm/k.

Let us briefly summarize the above constructions:

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

The spaces in our context are the 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. From the principle that 𝔸^{1} should be contractible we use spectra in the s-direction to define the motivic s-stable homotopy category SH_{s}^{𝔸^{1}}(k). Finally, the motivic stable homotopy category SH(k) is obtained from (s, t)-bispectra by formally inverting the class of stable weak equivalences. If Σ_{s,t}^{∞} : Sm/k → SH(k) denotes the (s, t)-suspension spectrum functor, then Σ_{s,t}^{∞} = Σ_{t}^{∞} ∘ Σ_{s}^{∞}.

Remark 2.17. Work in progress by Voevodsky suggests a different construction of the motivic stable homotopy category. The idea is to use framed correspondences, an algebraic notion of tubular neighborhoods, to construct a category equivalent to SH(k).

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

The category Spt_{s,t}(k) of (s, t)-bispectra is complete and cocomplete by forming limits and colimits degreewise in pointed spaces. In particular, arbitrary coproducts exist. Furthermore, the coproduct of (s, t)-bispectra induces a coproduct ∨ in the motivic stable homotopy category. It is easy to show that SH(k) is an additive category: The simplicial circle is a cogroup object in the classical unstable homotopy category. Since every spectrum in SH(k) is a twofold simplicial suspension, it follows that all objects in SH(k) are Abelian cogroup objects. The set of maps out of any object is thus an Abelian group.

Let us define the distinguished triangles and the shift functor: If E is an (s, t)- bispectrum, then the shift E[1] of E is Σ_{s}^{1}E. A map f : E → E′ of (s, t)-bispectra has an associated cofibration sequence E → E′ → Cone(f) → Σ_{s}^{1}E. Here, the cone of f is defined by a push-out square in Spt_{s,t}(k) where Δ[1] is pointed by 0:

\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). Suppose E is an (s, t)-bispectrum and let [−, −] denote Hom_{SH(k)}. It follows that any distinguished triangle X → Y → Z → X[1] in SH(k) induces 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] & ⋯ \\ ⋯ \ar[r] & [Z[n], E] \ar[r] & [Y [n], E] \ar[r] & [X[n], E] \ar[r] & [Z[n − 1], E] \ar[r] & ⋯}

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

We include some examples of distinguished triangles.

Lemma 2.18. 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}^{1}Σ_{s,t}^{∞} X}

is a distinguished triangle in SH(k).

Proof. It follows from the model structure in 2.13 that the canonically induced map Cone(Σ_{s,t}^{∞}(X ↪ Y)) → Σ_{s,t}^{∞} Y/X is a stable weak equivalence. □

Remark 2.19. Note that 2.18 applies to open and closed embeddings in Sm/k.

The following objects are of fundamental interest in motivic homotopy theory: An upper distinguished square in Sm/k is a pullback square

(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. For example, a Zariski open cover U ∪ V of X with W = U ∩ V gives an upper distinguished square. The Nisnevich topology is generated by coverings {i : U → X, p : V → X} where i and p give an upper distinguished square with W = p^{−1}(U). Since spaces are sheaves in the Nisnevich topology, we have

Lemma 2.20. The square of representable spaces corresponding to an upper distinguished square is a pushout square.

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

Corollary 2.21. 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}^{1}Σ_{s,t}^{∞} W_{+}.}

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

\xymatrix{Σ_{s,t}^{∞} V/W_{+} \ar[r] & Σ_{s,t}^{∞} X/U_{+}.}

Proof. By 2.19, the pushout square of representable spaces associated to an upper distinguished square is also a homotopy pushout square. This implies (a) since the suspension spectrum functor from spaces to bispectra preserves homotopy pushout squares. To prove (b) we use 2.20 to conclude that there is even an isomorphism between the underlying pointed spaces. □

In the following, we consider the projective line ℙ^{1} pointed by the rational point at infinity. Corollary 2.21 allows us to give different models for the mixed spheres.

Lemma 2.22. 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. Part (a) follows from 2.14 and the following distinguished triangle obtained from 2.18

\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}^{1}Σ_{s,t}^{∞} (𝔸^{1} − 0, 1).}

To prove (b), one covers the projective line by two affine lines. The choice of a point on ℙ^{1} is not important since all points are 𝔸^{1}-homotopic to each other. At any rate, by 2.21(b), we have 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}^{1}Σ_{s,t}^{∞} (𝔸^{1} − 0)_{+}.}

To get a distinguished triangle of spaces without disjoint basepoints, we point all spaces by 1 : Spec(k) → 𝔸^{1} − 0 and form quotients, e.g.

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

In this way we obtain a distinguished triangle, and the claim follows from homotopy invariance 2.14. □

Remark 2.23. The pointed space T := 𝔸^{1}/𝔸^{1} − 0 is called the Tate sphere. There is an ‘unstable motivic homotopy category’ for spaces and 2.22 holds unstably.

We place emphasize on a typical instance of 2.21(b):

Corollary 2.24. Suppose p : V → X is an etale morphism and Z is a closed subscheme of X such 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 2.25. Suppose E → X is a vector bundle over a smooth scheme X. 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 2.26. We list some examples of Thom spaces.

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

(b) If E_{1} → X_{1} and E_{2} → X_{2} are vector bundles over smooth k-schemes, then Th(E_{1} × E_{2}/X_{1} × X_{2}) ≅ Th(E_{1}/X_{1}) ∧ Th(E_{2}/X_{2}) as pointed spaces.

For Thom spaces we have the homotopy purity theorem.

Theorem 2.27. Let i : Z ↪ X be a closed embedding of smooth schemes over the field k. Denote the corresponding normal vector bundle of Z in X by ℕ_{X,Z}. Then there is a canonical isomorphism in SH(k):

Σ_{s,t}^{∞} Th(ℕ_{X,Z}) ≅ Σ_{s,t}^{∞} X/X − i(Z).

Let us prove the homotopy purity theorem in the following special case: Assume k′ is a finite separable extension of k. By the Primitive Element Theorem there exists an element α such that k′ = k(α). We obtain a surjective map ϕ : k[x] → k′, x ↦ α. Let f be the minimal polynomial of α. Then the induced closed embedding Spec(k′) ↪ 𝔸_{k}^{1} sends the closed point of Spec(k′) to the closed point f of 𝔸_{k}^{1}.

Lemma 2.28. Suppose k′/k is a finite separable extension, and let f denote the minimal polynomial of a primitive element α for k′ over k. Then there is a canonical isomorphism in SH(k)

Σ_{s,t}^{∞} Th(ℕ_{𝔸_{k}^{1},k′}) ≅ Σ_{s,t}^{∞} 𝔸_{k}^{1}/𝔸_{k}^{1} − f.

Proof. We can identify the normal bundle of the closed embedding i : Spec(k′) ↪ 𝔸_{k}^{1} with the affine line 𝔸_{k′}^{1}. By definition, the pointed space Th(ℕ_{𝔸_{k}^{1},k′}) is 𝔸_{k′}^{1}/𝔸_{k′}^{1} −{0}. Recall from above that 𝔸_{k}^{1}/𝔸_{k}^{1} − i(Spec(k′)) = 𝔸_{k}^{1}/𝔸_{k}^{1} − {f}. In k′[x], f factors into irreducible polynomials, say f_{1}, …, f_{q}. Since k′/k is a separable extension we may assume f_{1} = x − α, and f_{i}(α) ≠ 0 if i ≠ 1. Define ρ_{α} : k′[x] → k′[x] by x ↦ x − α. There is an induced upper distinguished square:

\xymatrix{𝔸_{k′}^{1} − \{f_{1}\} \ar[r] \ar[d]_{ρ_{α}^{∗}} & 𝔸_{k′}^{1} \ar[d]^{ρ_{α}^{∗}} \\ 𝔸_{k′}^{1} − \{0\} \ar[r] & 𝔸_{k′}^{1}}

By 2.21(b) we conclude that there is a canonical isomorphism

Σ_{s,t}^{∞} Th(ℕ_{𝔸_{k}^{1},k′}) ≅ Σ_{s,t}^{∞} 𝔸_{k′}^{1}/𝔸_{k′}^{1} − {f_{1}}

in SH(k). The closed embedding 𝔸_{k′}^{1} − {f_{1}} ↪ 𝔸_{k′}^{1} fits into the upper distinguished square:

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

Similarly to the above, we obtain a canonical isomorphism

Σ_{s,t}^{∞} Th(ℕ_{𝔸_{k}^{1},k′}) ≅ Σ_{s,t}^{∞} 𝔸_{k′}^{1} − {f_{2}, …, f_{q}}/𝔸_{k′}^{1} − {f_{1}, f_{2}, …, f_{q}}

in SH(k). The right hand side of the isomorphism is related to 𝔸_{k}^{1}/𝔸_{k}^{1} − {f} via the following upper distinguished square.

\xymatrix{𝔸_{k′}^{1} − \{f_{1}, f_{2}, …, f_{q}\} \ar[r] \ar[d] & 𝔸_{k′}^{1} − \{f_{2}, …, f_{q}\} \ar[d] \\ 𝔸_{k}^{1} − \{f\} \ar[r] & 𝔸_{k}^{1}}

There results a canonical isomorphism

Σ_{s,t}^{∞} 𝔸_{k′}^{1} − {f_{2}, …, f_{q}}/𝔸_{k′}^{1} − {f_{1}, f_{2}, …, f_{q}} ≅ Σ_{s,t}^{∞} 𝔸_{k}^{1}/𝔸_{k}^{1} − {f}

in SH(k). This allows us to conclude the proof. □

3. Cohomology theories

The introduction of the motivic stable homotopy category via spectra provides a framework for defining cohomology theories of algebraic varieties. We will consider the following examples: Motivic cohomology, algebraic K-theory and algebraic cobordism.

In 2.22(b) we noted that the (s, t)-suspension spectra of (ℙ^{1}, ∞) and S_{s}^{1} ∧ S_{t}^{1} are canonically isomorphic in SH(k). In fact something better is a foot; one can replace the two suspension coordinates S_{s}^{1} and S_{t}^{1} by the single suspension coordinate ℙ^{1} without changing the stable motivic homotopy category. This approach is often convenient in order to construct spectra which represent cohomology theories on Sm/k.

3.1. 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

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

To describe a specific model, 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 basepoint. As K(ℤ, n) one can choose the underlying simplicial set of the simplicial Abelian group ℤ(Δ^{n}/∂Δ^{n}). The K(ℤ, n)’s define a spectrum Hℤ which represents singular cohomology. If X is a simplicial set, then the integral singular cohomology groups H^{n}(X, ℤ) coincide with Hom_{SH}(Σ^{∞}X_{+}, Hℤ).

Motivic cohomology is supposed to be an algebro-geometric analogue of singular cohomology. In motivic homotopy theory it is clear that (2) fails; however, one can construct a spectrum Hℤ that represents motivic cohomology. This uses the theory of algebraic cycles. Next, we will indicate how one defines the integral motivic Eilenberg-MacLane spectrum.

For every scheme X ∈ Sm/k we will describe a pointed space L(X). One should think of L as a version of the functor ℤ(−). The value of L(X) at U ∈ Sm/k is the free Abelian group generated by cycles, that is, closed irreducible subsets of U × X which are finite over U and surjective on a connected component. The graph Γ(f) of a morphism f : U → X is an example of a cycle in U × X. One can verify that L(X) is a Nisnevich sheaf, and that the graph yields a map of Nisnevich sheaves X → L(X). One can extend^{2} L to take values on the mixed spheres S_{s}^{p} ∧ S_{t}^{q}. For example, L(ℙ^{1}, ∞) is the quotient L(ℙ^{1})/L(Spec(k)) of Nisnevich sheaves of Abelian groups considered as a pointed space. To construct the structure maps of the motivic Eilenberg-MacLane spectrum, one uses the exterior product map of cycles to define a natural map:

\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, the compositions

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

yield the structure maps of the motivic Eilenberg-MacLane spectrum.

Definition 3.2. The Eilenberg-MacLane spectrum Hℤ is the (s, t)-bispectrum with pointed spaces Hℤ_{m,n} := L(S_{s}^{m} ∧ S_{t}^{n}) and structure maps as above.

Remark 3.3. To define the Eilenberg-MacLane spectrum with finite coefficients ℤ/ℓ 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 3.4. 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 Bloch’s higher Chow group CH^{q}(X, 2q − p). There is 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. In section 4 we will outline an alternative approach to construct such a spectral sequence depending on some conjectures about Hℤ.

2To be more precise, L is required to preserve colimits as a functor from pointed spaces to Nisnevich sheaves with values in simplicial Abelian groups. Since every pointed space is a colimit of representable functors, it suffices to describe the values of L on Sm/k.

3.5. The algebraic K-theory spectrum KGL. If m is a non-negative integer, we denote the Grassmannian of n-planes in the n+m-dimensional affine space over k by Gr_{n}(𝔸^{n+m}). By letting m tend to infinity one obtains a directed system of spaces; its colimit is denoted by BGL_{n}. There are canonical monomorphisms BGL_{n} ↪ BGL_{n+1} for all n. We write BGL for the colimit. Instead of letting ℤ × BGL be the n-th term of the algebraic K-theory spectrum, one defines KGL as a fibrant replacement of ℤ × BGL in the unstable motivic homotopy theory of k. The reason is that the classifying map

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

for the canonical Bott element of K_{0}(ℙ^{1} ∧ ℤ × BGL) is the lift of an isomorphism ℙ^{1} ∧ ℤ × BGL →^{≅} ℤ × BGL in the unstable motivic homotopy category. We will not describe this isomorphism in detail.

Definition 3.6. Let KGL be the ℙ^{1}-spectrum (KGL, KGL, ⋯) with structure maps given by (3).

As for 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 ℙ^{1} ∧ KGL = KGL in SH(k).

3.7. The algebraic cobordism spectrum MGL. Recall the notation in 3.5, 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}. For example, 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 is a bundle map 𝔸^{1} × γ_{n} → γ_{n+1}. By taking Thom spaces, we obtain the map

(4)  Th(𝔸^{1}) ∧ Th(γ_{n}) = (𝔸^{1}/𝔸^{1} − 0) ∧ Th(γ_{n}) → Th(γ_{n+1}).

Note that we may apply part (b) of 2.26, since γ_{n} → BGL_{n} is a colimit of vector bundles of smooth schemes. By the identification of ℙ^{1} with 𝔸^{1}/𝔸^{1} − 0 in 2.22 and by (4), there results maps for n ≥ 0

(5)  c_{n} : ℙ^{1} ∧ Th(γ_{n}) → Th(γ_{n+1}).

Definition 3.8. The algebraic cobordism spectrum MGL consists of the sequence of pointed spaces (Th(γ_{0}), Th(γ_{1}), …, Th(γ_{n}), …), together with structure maps given by (5).

The analogy between MGL and the ordinary complex cobordism spectrum should be obvious. We remark that results about orientations, maps of rings and formal group laws for classical spectra carry over to ℙ^{1}-spectra. Properties of the algebraic cobordism spectrum are used in the proof of the Milnor conjecture. (((Bloch-Kato-conjecture)))

4. 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. Thus one has an internal description of the classical Eilenberg-MacLane spectrum in the stable homotopy category. But the situation in motivic homotopy theory is quite different. For example, the homotopy groups π_{p,q}Hℤ are often non-zero. One can hope that the so-called slice filtration provides an internal description of the spectrum Hℤ. We will discuss some conjectures for the slice filtrations of the sphere spectrum 1 = Σ_{s,t}^{∞}Spec(k)_{+} and of the Eilenberg-MacLane and algebraic K-theory spectra of k.

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_{+} where X ∈ Sm/k. We point out that this category does not contain the desuspension spectra Σ_{t}^{−n}Σ_{s,t}^{∞}X_{+} for any n ≥ 1. The ‘effective’ s-stable homotopy category SH_{s}^{eff}(k) is constructed similarly by replacing Σ_{s,t}^{∞} with Σ_{s}^{∞} in the above definition.

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

(6)  \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] & ⋯}

Each Σ_{s,t}^{n}SH^{eff}(k) is a triangulated category with arbitrary direct sums and a set of compact generators. The sequence (6) is called the slice filtration of k. It is a filtration of SH(k) in the sense that the smallest triangulated category which contains every Σ_{s,t}^{n}SH^{eff}(k) and is closed under arbitrary direct sums is the motivic stable homotopy category of k. By the theory of triangulated categories, the full inclusion 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) which satisfies r_{n} ∘ i_{n} ≅ Id. We denote the other composition by

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

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

(7)  \xymatrix{f_{n}E \ar[r] & f_{n−1}E \ar[r] & s_{n}E \ar[r] & Σ_{s}f_{n}E}

in SH(k) for all n ∈ ℤ.

Definition 4.1. The n-th slice of E is s_{n}E.

The slice tower (7) is analogous to the Postnikov tower and the slices are the cofibers or quotients. Recall that the Postnikov tower of an ordinary spectrum E is a diagram of cofibrations with colimit E:

\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 fundamental conjecture:

Conjecture 4.2. s_{0}1 = Hℤ.

The collection of the functors s_{n} is compatible with the smash product: 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 Conjecture 4.2 holds, this shows that each slice of an (s, t)-bispectrum has a natural and unique structure of a module over the motivic cohomology spectrum Hℤ.

At present, the following is known about Conjecture 4.2.

Theorem 4.3. Conjecture 4.2 holds for fields of characteristic zero.

To sketch a proof of Theorem 4.3, we will first describe the zeroth slice of Hℤ.

Theorem 4.4. If k is a field of characteristic zero, then s_{0}Hℤ = Hℤ.

The proof of 4.4 uses that r_{0}Hℤ = Hℤ, and r_{1}Hℤ = 0. By the first statement, Hℤ can be build out of (s, t)-suspension spectra. The result for r_{1} is a consequence of the cancellation theorem. To finish the sketch of the proof of 4.3, it remains by 4.4 to show that the fiber of the unit map 1 → Hℤ belongs to Σ_{s,t}^{1}SH^{eff}(k). This is known if the base field k has characteristic zero: The proof uses properties of the infinite symmetric products of 𝔸^{n} and 𝔸^{n} − 0.

There is a slice filtration on SH_{s}^{𝔸^{1}}(k) constructed analogous to (6). One can show that the Tate suspension functor Σ_{t}^{∞} preserves the slice filtrations in the sense that Σ_{t}^{∞}(Σ_{t}^{n}SH_{s}^{𝔸^{1}}(k)) ⊆ Σ_{s,t}^{n}SH(k)). To get back into SH_{s}^{𝔸^{1}} we use that the Tate suspension functor has a right adjoint Ω_{t}^{∞} : SH(k) → SH_{s}^{𝔸^{1}}(k). For Ω_{t}^{∞} there is the following conjecture.

Conjecture 4.5. There is an inclusion Ω_{t}^{∞}(Σ_{s,t}^{n}SH^{eff}(k)) ⊆ Σ_{s}^{n}SH_{s}(k).

Recall from topology that taking the suspension and then the loop space preserves connectivity. One can use an inductive argument to show that the following conjecture would imply 4.5.

Conjecture 4.6. 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 possible approach to prove 4.6 is to develop a theory of operads, in order to have a detection principle for the Tate loop spaces Ω_{t}X_{+}. Framed correspondences might be a useful tool in working out such a theory.

Let us finally relate the above machinery to a possibly new approach to some recent results in algebraic K-theory. For twenty years ago, Beilinson conjectured the existence of an Atiyah-Hirzebruch type spectral sequence for the algebraic K-groups of nice schemes. For fields, there is a construction of such a spectral sequence due to Bloch and Lichtenbaum. More general versions were developed later by Friedlander and Suslin, and also by Levine.

Assume X ∈ Sm/k. From the slice tower (7) of an (s, t)-spectrum E one can construct a spectral sequence with input terms the cohomology with coefficients in the stable homotopy groups of E and target groups the cohomology groups represented by E. In the example of algebraic K-theory we get the spectral sequence

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

The problem one encounters is to identify the input terms with motivic cohomology. This is essentially the content of the following conjecture.

Conjecture 4.7. s_{0}KGL = Hℤ.

Because of the Bott periodicity, this conjecture describes indeed all slices of KGL. If Conjecture 4.7 holds, we would immediately obtain an Atiyah-Hirzebruch type spectral sequence for algebraic K-theory as conjectured by Beilinson:

(9)  H^{p−q,q}(X, ℤ) ⇒ K_{−p−q}(X).

VOEVODSKY’S NORDFJORDEID LECTURES ON MOTIVIC HOMOTOPY THEORY 15

To give some further evidence that this approach leads to the spectral sequence in (9), we remark that Conjectures 4.2 and 4.5 imply Conjecture 4.7.

Department of Mathematics, University of Western Ontario, Canada.

E-mail address: oroendig@math.uwo.ca

Isaac Newton Institute for Mathematical Sciences, Cambridge, U.K.

E-mail address: P.Ostvaer@newton.cam.ac.uk