Motivic Homotopy

Sept. 24, 2004

Morel’s results on algebraic varieties.

Algebraic Topology is the study of invariants; for example, the degree of a map, Euler characteristic, and homotopy/homology groups. These are homotopy invariants. As a “universal” homotopy invariant, we can study the homotopy category of spaces.

What we’d like to do is have a corresponding homotopy theory of varieties.

Reference: Milnor: Algebraic K-theory and quadratic forms, Inv. vol 9.

Some invariants that appear in this context:

• K_{0}(R) is the Grothendieck group of finitely generated projective modules over R.

• Suppose F is a field. Then we can define the Milnor K-theory, K_{∗}^{M}(F),

as the algebra

Tens(F^{×})/[a][1 − a] = 0.

In other words, it is the free algebra on the abelian group F^{×} subject to the relation [a][1 − a] = 0 for a ≠ 0, 1. It is a graded algebra with F^{×} in grading 1.

• GW (F) is the Grothendieck-Witt ring of quadratic forms over F. It is the group completion of the group of quadratic forms over F, mod isomorphism, with addition operation ⊕. As a group, it is the free abelian group on “irreducible” quadratic forms.

• The Witt ring W (F) is GW (F)/ℤ(x^{2} − y^{2}).

There is a short exact sequence I → GW (F) → ℤ. Milnor conjectured the following:

• K_{n}^{M}(F)/2 ≅ I^{n}/I^{n+1}. (This is now proven by Orlov-Vishik-Voevodsky and Morel.)

• K_{n}^{M}(F)/2 ≅ H^{n}(G, ℤ/2) (This has been proven by Voevodsky.)

Here G is the Galois group Gal(F̄/F)).

It turns out the second conjecture implies the first.

The proofs of these facts use motivic homotopy theory. Algebraic Topology. We have at our disposal the following principles:

We can accumulate local facts to global facts, using:

• open covers

• triangulation

We also have the values of certain computations. For example, we have the Hurewicz theorem: If X is a pointed n−1-connected space, n > 1, then H̃_{i}(X) = 0 for i < n, and π_{n}(X) ≅ H̃_{n}(X). As a result, π_{n}(S^{n}) = ℤ. (These are basic facts; we could think of them as “physical constants” for the theory.)

Algebraic Geometry.

In algebraic geometry, we have something that could play the role of a sphere: ℙ^{1}. In topology, ℙ^{1} = S^{2} is simply connected. Over ℝ, however, ℙ^{1} is not simply connected; it is equivalent to 𝕊^{1}.

Consider the degree map [ℙ^{1}, ℙ^{1}] → ℤ. This is computed as follows. Given f: ℙ^{1} → ℙ^{1}, choose y ∈ ℙ^{1} a regular value. Then deg(f) = ∑_{x→y} ±1.

Over ℂ, all these signs will be +1; the degree is positive. Over ℝ, however, there could be complex points in the preimage of a real point. There is an inequality −deg(f) ≤ deg^{ℝ}f ≤ deg(f), and deg^{ℝ}(f) ≡ deg^{ℂ}(f) (mod 2).

This gives a map deg^{ℝ} : [ℙ^{1}, ℙ^{1}] → ℤ.

Similarly, given a quadratic form V/ℝ, i.e. a vector space with a quadratic form, we have two numbers: a rank and a sign.

rank(V) := dim(V) = n.

sign = σ, −n ≤ σ ≤ n, and σ ≡ n (mod 2).

(The sign is the number of positive eigenvalues of the form minus the number of negative eigenvalues.)

So if one wants to thing about self-maps of ℙ^{1} over ℝ, degree is replaced with a quadratic form valued functor. So π_{n}(S^{n}) “becomes” a Grothendieck-Witt ring. This is Morel’s theorem; π_{n}(S^{n}) is a Grothendieck-Witt ring in the homotopy theory of algebraic varieties. Spheres in algebraic geometry.

There are two different candidates for spheres in algebraic geometry:

S^{1} = 𝔸^{1}/{0, 1} =: S^{1,0}

S^{1} = 𝔸^{1} \ {0} =: S^{1,1}

The numbers here indicate “Hodge weights.” The problem with this numbering system is that over ℝ, 𝔸^{1}/{0, 1} ≃ S^{1}, and 𝔸^{1} \ {0} ≃ {±1}. The connectivity of the sphere is not properly indicated.

Morel’s notation rectifies this problem. We redefine S^{1,0} = S^{1}(0), S^{1,1} = S^{0}(1).

The smash product.

Just as in topology, the smash product is defined by X ∧ Y = X × Y/(∗ × Y) ∪ (X × ∗). As a result, we define the n-sphere S^{n} as S^{1} ∧ S^{1} ⋯ S^{1}.

Similarly, we define S^{p}(q) = (S^{1} ∧ S^{1} ⋯ S^{1}) ∧ (𝔾_{m} ∧ 𝔾_{m} ⋯ 𝔾_{m}). (p copies of S^{1}, q copies of 𝔾_{m}.) As a special case, 𝔾_{m} = S^{0}(1).

We have the identity S^{p}(q) ∧ S^{p′}(q′) ≅ S^{p+p′}(q + q′).

It turns out that in 𝔸^{1}-homotopy theory, there are also the following identities:

ℙ^{1} ≃ S^{1}(1).

𝔸^{n} \ {0} = S^{2n−1,n} = S^{n−1}(n).

ℙ^{n}/ℙ^{n−1} ≃ S^{n}(n).

In topology, ℙ^{1} has a path of self-maps given by rotation through the angle θ, 0 ≤ θ ≤ 2π. However, this isn’t an “algebraic path”.

Suppose we are working over a field k and θ ∈ k^{×}. Then there is a “rotate by θ” map θ : ℙ^{1} → ℙ^{1}, given by:

[x : y] ↦ [θx : y].

This clearly takes multiplication in k^{×} to composition.

By smashing with the identity, we get induced maps R_{u} : S^{p}(q) → S^{p}(q) for p, q ≥ 1, u ∈ k^{×}. In other words, there is a group homomorphism R : k^{×} → [S^{p}(q), S^{p}(q)] given by u ↦ R_{u}.

Consider the map from k^{×} to the Grothendieck-Witt ring GW (k), given by u ↦ [ux^{2}] =:< u >. This fits into a diagram as below.

\xymatrix{k^{×} \ar[d] \ar[r] & [S^{p}(q), S^{p}(q)] \\ GW (k) \ar@{-->}[ur]}

The upper arrow of this is not a group homomorphism for the additive structure on [S^{p}(q), S^{p}(q)]. It takes multiplication in k^{×} to the composition structure in [S^{p}(q), S^{p}(q)]. (This corresponds to multiplication.) The extension GW (k) → [S^{p}(q), S^{p}(q)] is, however, a group homomorphism.

Theorem. (Morel) For p ≥ 2, q ≥ 1, this is an isomorphism

GW (k) ≅ [S^{n}(i), S^{n}(i)]

if char(k) ≠ 2. (This is our goal!)

• It’s an interesting result that degree is replaced by quadratic forms.

• It’s also interesting that the Grothendieck-Witt ring GW (−) is homotopy invariant.

In fact, we actually know the groups [S^{n}(i), S^{n}(j)] and [S^{n}(i), S^{n+k}(j)]. Homotopy theory of algebraic varieties: A simple-minded guess for the definition of the 𝔸^{1}-homotopy category might be following: Let C be the category of smooth varieties over k. Define [X, Y] = C(X, Y)/homotopy, where two maps f and g are homotopic if there is a diagram of the form

\xymatrix{X × 𝔸^{1} \ar[r]^{h} & Y \\ X × \{0, 1\} \ar@{^{(}->}[u] \ar[ur]_{(f,g)}}

What’s wrong with this definition? The first source of trouble is the following: Suppose f : X → Y, and {U_{α}} → Y is a covering closed under intersections. If the resrictions f^{−1}U_{α} → U_{α} are all homotopy equivalences, it does not follow that f : X → Y is. This is true in Top. The remedy: We will work with sheaves. The second source of trouble: We have an expression ℙ^{1} = 𝔸^{1} ∪_{𝔸^{1}\{0}} 𝔸^{1}. In other words, ℙ^{1} is supposed to look like ΣS^{0}(1). Now: We have a map ΣX → ΣX ∨ ΣX (a homotopy comultiplication, which gives group structure to homotopy groups). However, ℙ^{1} does not have a coproduct. The remedy for this is to “formally introduce Σ and mapping cones.” Rather than just working with sheaves, we will work with sheaves of simplicial sets. The topology we will be using is the Nisnevich topology, defined as follows. Suppose X is a smooth variety over k. Definition: A collection of étale maps {U_{α}} → X is an étale cover if X = ∪f_{α}(U_{α}).

It is a Nisnevich cover if X = ∪f(U_{α}) and for each x ∈ X, there is y ∈ U_{α} for some α such that f_{α}(y) = x and k(y) ← k(x) is an isomorphism of residue fields. It is a Zariski cover if U_{α} ↪ X, and X = ∪U_{α}. From now on, let V be the category of smooth varieties over k. For X ∈ V, Map(−, X) is an étale/Nisnevich/Zariski sheaf on V. This gives us a sequence of inclusions of topologies: V ↪ V_{et} ↪ V_{Nis} ↪ V_{Zar} ↪ Presheaves. Some good things about the Nisnevich topology: Suppose M is a sheaf of abelian groups, and X is variety of dimension n. Then H_{Nis}^{p}(X, M) = 0 for p > n. (This is also true in the Zariski topology. For the étale topology, however, H^{p} doesn’t vanish above p = n; for example, the cohomology of Spec(F) for F a field is Galois cohomology.) Suppose x ∈ X, k(x) = k. The sheaf quotient X/ (X \ {x}) is isomorphic to 𝔸^{n}/(𝔸^{n} \ {0}) as Nisnevich sheaves. So we have the proper behavior locally. We will work in the category of sheaves of simplicial sets in the Nisnevich topology. We will refer to these as “spaces” or “simplicial sheaves”. (As an analogy, we could re-interpret Steenrod’s convenient category of topological spaces as a category of sheaves on compact spaces.) The homotopy theory of simplicial sheaves is gotten by declaring X × 𝔸^{1} → X is an equivalence. In order to do this rigorously, we need:

• Quillen’s theory of model categories.

• Simplicial sets.

We will cover this next week.