Motivic Homotopy

Nov. 4, 2004

So far, we have “proved” the 𝔸^{1}-Hurewicz theorem: for X an (n − 1)-connected simplicial sheaf in the Nisnevich topology, n ≥ 2, π_{n}L^{𝔸^{1}}X is the free strictly 𝔸^{1}-invariant sheaf generated by π_{n}X, and π_{i}L_{𝔸^{1}}X = 0 for i < n. There are some gaps in what we’ve done so far, but we’re going to forge ahead.

Our goal is to calculate ℤ^{𝔸^{1}}(𝔾_{m} ∧ … ∧ 𝔾_{m}), the free strictly 𝔸^{1}-invariant sheaf on a wedge of copies of 𝔾_{m}.

(Recall F ∧G = F ×G/F ∨G for F, G pointed sheaves, and F ∨G = F ∪_{Speck} G.)

There is a cofiber sequence F ∨G → F ×G → F ∧G. More, there are projection maps on F ×G; if addition exists, we could use the map π_{1} +π_{2} to get a splitting of this sequence.

In particular, there is a splitting of the exact triangle ℤ(F ∨ G) → ℤ(F × G) → ℤ(F ∧ G), which gives an isomorphism ℤ(F × G) = ℤ(F) ⊕ ℤ(G) ⊕ ℤ(F ∧ G). (Here ℤ is “reduced” ℤ for pointed spaces.)

By the 𝔸^{1}-Hurewicz theorem, ℤ^{𝔸^{1}}(𝔾_{m} ∧ … ∧ 𝔾_{m}) = π_{n−1}^{𝔸^{1}}(𝔸^{n} \ {0}).

Strictly 𝔸^{1}-invariant sheaves of abelian groups.

Definition. A is strictly 𝔸^{1}-invariant if for all smooth Y, all i, H^{i}(Y; A) → H^{i}(Y × 𝔸^{1}; A) is an isomorphism.

Theorem. (Morel) If H^{0}(−; A) and H^{1}(−; A) are 𝔸^{1}-invariant, then A is strictly 𝔸^{1}-invariant. (This is a special case of the theorem that if C_{∗} is 0connected, then C_{∗} is 𝔸^{1}-local if and only if the associated simplicial sheaf K(C_{∗}) is 𝔸^{1}-local.)

Example. A = 𝔾_{m} is 𝔸^{1}-invariant. (Strictly speaking, we need a fibrant replacement of 𝔾_{m}.)

Example. Let K_{n}(Y) be the Quillen algebraic K-theory. If K_{n} is the sheaf associated to the presheaf K_{n}, then K_{n} is strictly 𝔸^{1}-invariant. (In fact, we have that K_{n} is the homotopy sheaf of some object.)

Theorem. (Voevodsky) If A has transfers for branched covers and is 𝔸^{1}- invariant, then A is strictly 𝔸^{1}-invariant.

A general question: Given some kind of association K ↦ A(K) which takes fields to abelian groups (not necessarily a functor), what kind of data do we need in order to find a strictly 𝔸^{1}-invariant sheaf \underline{A} with \underline{A}(SpecK) = A(K)?

(We only need this for K the field of rational functions on something smooth over the base field k.)

Here is some topological motivation.

Suppose we have such a strictly 𝔸^{1}-invariant sheaf, and R is a discrete valuation ring with total field K and residue field F. Let Z = SpecF, U = SpecK, M = SpecR. Then U = M \ Z. There is a uniformizer in R, i.e. a map out to 𝔸^{1}, such that Z is the zero set.

Then M/U should be something like Z^{ν}, the Thom complex of the normal bundle of Z in M.

However, Z is the zero set of a function on M, so the normal bundle is trivial. Therefore, we might expect

M/U ≃ Σ^{𝔸^{1}}Z = Z × 𝔸^{1}/Z × 𝔾_{m}.

This would give us a long exact sequence in cohomology.

⋯ → H^{k}(M/U, A) → H^{k}(M, A) → H^{k}(U, A) → H^{k+1}(M/U, A) → ⋯

The map H^{k}(M/U, A) → H^{k}(M, A) will always be zero. (Marc proved this in his talk.) Using the above isomorphism for M/U, we get short exact sequences

0 → H^{k}(M, A) → H^{k}(U, A) → H^{k}(Z, A^{𝔾_{m}}) → 0.

Then as motivation, in order to define the sheaf A we also need a series of functors representing A^{𝔾_{m}}, and residue maps A(K) → A^{𝔾_{m}}(F) for discrete valuation rings.

Example. A = 𝔾_{m}. Then 𝔾_{m}(U) = K^{×}, and

A^{𝔾_{m}}(F) = ker (F [u, u^{−1}]^{×} → F^{×}) .

In other words, A^{𝔾_{m}}(F) is the group ℤ of possible valuations of units in F [u, u^{−1}]. The map A(K) → A^{𝔾_{m}}(F) is the valuation map K^{×} → ℤ, whose kernel is R^{×}. The kernel H^{k}(M) is the “unramified part” of H^{k}(U).

This gives us some instruction as to how to define things.

Example. A = K_{∗}^{M}, the Milnor K-theory functor. What is A^{𝔾_{m}}?

In fact, Milnor defined residue maps res : K_{n}^{M}(K) → K_{n−1}^{M}(F). These are determined by the formula

res ([π][u_{1}] … [u_{n−1}]) = [ū_{1}] … [ū_{n−1}]

when π is a uniformizer of the discrete valuation ring and u_{i} ∈ R^{×}.

If Y is a smooth variety, we can define

K_{n}^{M}(Y) = ker (K_{n}^{M}(K(Y)) → ⊕_{codimp=1}K_{n−1}^{M}(K(p))) .

This is “unramified Milnor K-theory.”

Theorem. Unramified Milnor K-theory is a strictly 𝔸^{1}-invariant sheaf.

(First, it’s apparently not too hard to show that it is a sheaf.)

Why is this theorem true? It is true for at least two reasons:

• Unramified Milnor K-theory shows up as the motivic cohomology of Y with coefficients in Voevodsky’s motivic complexes.

• Unramified Milnor K-theory has transfers.

Markus Rost, in his paper “Chow groups with coefficients,” has another approach. He axiomatizes this setup, and has a general theory that shows you how to construct strictly 𝔸^{1}-invariant sheaves.

However, it is not immediately applicable to some sheaves we might want. Here is why:

Suppose Z ⊂ Y is the zero set of a “uniformizer” map f : Y → 𝔸^{1}. Then Y/(Y \ Z) ≃ Σ^{𝔸^{1}}Z, but this isomorphism is not canonical: it depends on the choice of uniformizer.

The story in Rost’s paper assumes that the residue map is independent of the choice of uniformizer. This is what we need to get away from.

In a sense, Milnor K-theory is the universal object for Rost’s construction; it’s universal for trivialized Thom spaces. We want the corresponding universal gadget that doesn’t depend on trivialization.

Philosophically, in Rost’s theory, the objects are some analogues of Mackey functors with Milnor K-theory playing the role of the Burnside ring. We’re about to describe an “unoriented” version of Milnor K-theory which drops the identification of Y/Y \ Z with Σ^{𝔸^{1}}Z.

Milnor-Witt K-theory.

Let F be a field. We define a ℤ-graded ring K_{∗}^{MW} (F) as follows.

K_{∗}^{MW} (F) is generated as a ring by an element η in dimension −1 and elements [u] in dimension 1 for u ∈ F^{×}, subject to the following relations:

• [a][1 − a] = 0 for a ∈ F^{×}, a ≠ 1.

• [ab] = [a] + [b] + η[a][b].

• [u]η = η[u].

• ηh = 0, where h = η[−1] + 2 ∈ K_{0}^{MW}.

Where on earth do these come from?

We’re going to show K_{0}^{MW} (F) = GW (F), the Grothendieck-Witt ring of quadratic forms on F. Recall that GW (F) is the ring generated by ⟨u⟩, for u ∈ F^{×}, subject to these relations:

• ⟨u⟩ = ⟨uv^{2}⟩.

• ⟨u⟩ + ⟨v⟩ = ⟨u + v⟩ + ⟨(u + v)uv⟩.

(The element ⟨u⟩ corresponds to the quadratic form ux^{2}. The second relation comes from examining the form ux^{2} + vy^{2} in the basis [1, 1], [v, −u].)

We can show using these relations that ⟨u⟩ + ⟨−u⟩ = ⟨1⟩ + ⟨−1⟩ = h. (This corresponds to the “hyperbolic plane” quadratic form.)

The relationship between K_{0}^{MW} and GW is that they are isomorphic; the isomorphism is given by ⟨a⟩ ↦ η[a] + 1.

One can show that the relations hold, and that this map is an isomorphism.

Now, we can also see from the relations defining K_{∗}^{MW} that K_{∗}^{MW} (F)/(η) = K_{∗}^{M}(F). The groups K_{∗}^{MW} are going to be motivic homotopy groups of spheres; η is going to be the Hopf map.

What is K_{∗}^{MW} in negative degrees? The elements in K_{−n}^{MW} are all gotten by multiplying elements in K_{0}^{MW} by η^{n}; one can check that the kernel of this multiplication is precisely η. Therefore, the negative Milnor-Witt groups are all isomorphic to W (F), the Witt ring of F, which is GW (F)/h. The map η : GW (F) → W (F) from degree 0 to degree −1 is the quotient map, and further multiplications by η are all isomorphisms.

We forgot to mention something important about K_{∗}^{MW}: it is almost commutative. If we define ϵ = − ⟨−1⟩, then [u][v] = ϵ[v][u] for v, u ∈ F^{×}.

Morel has the following description of higher Milnor-Witt groups. Consider the augmentation dim : GW (F) → ℤ, ⟨u⟩ ↦ 1. This map does not factor through W (F), since h ↦ 2. However, there is a diagram as follows.

\xymatrix{J \ar[d] \ar[r] & GW (F) \ar[r] \ar[d] & ℤ/2 \\ I \ar[r] & W (F) \ar[ur]}

Here J and I are the kernels of the maps out to ℤ/2. Milnor produced maps K_{n}^{M} ⊗ ℤ/2 → I^{n}/I^{n+1}, and these have been shown to be isomorphisms. (This was the Milnor conjecture, and it was hard to prove.)

Morel defines a pullback square:

\xymatrix{J_{n} \ar[r] \ar[d] & K_{n}^{M} \ar[d] \\ I^{n} \ar[r] & I^{n}/I^{n+1}}

It is easy to show that J_{0} = K_{0}^{MW}: in this case, the pullback square is

\xymatrix{GW \ar[r] \ar[d] & ℤ \ar[d] \\ W \ar[r] & ℤ/2}

Theorem. (Morel) J_{n}(F) = K_{n}^{MW} (F).

There’s a map K_{n}^{MW} (F) → J_{n}, which is induced from the apparent map on F^{×} → J_{1}. (Note that the pullback diagrams are compatible with the product structure.)

This theorem of Morel’s is hard to prove; we won’t do it today.

Our aim is to extend K_{∗}^{MW} to a strictly 𝔸^{1}-invariant sheaf.

Fact: Unramified K_{∗}^{MW} is a strictly 𝔸^{1}-invariant sheaf.

There are 2 proofs of this: one good, one bad.

• The good proof: One could use the formulae defining K_{∗}^{MW} to define residue maps and go through Rost’s construction. This is relatively lowtech. Nobody has ever actually gone through this, however.

• The bad proof (and the one we are going to use): Use some high-powered machinery.

Proof.

• Step 1: 𝕎(−) - the unramified Witt ring sheaf - is strictly 𝔸^{1}-invariant. (Balmer, Hornbostel, Morel)

• Step 2: By induction, using 0 → I^{n+1} → I^{n} → K_{n}^{M} ⊗ ℤ/2 → 0, we conclude that the I^{n+1} are strictly 𝔸^{1}-invariant. (This uses the Milnor conjecture, and so might feel slightly amoral.)

• Step 3: Use Mayer-Vietoris on the pullback square. In order for this to work, we need to know that the map I^{n} → K_{n}^{M} ⊗ ℤ/2 is surjective.

Our debt load for things we’ve used without proving them is now huge.

As an aside, it seems unlikely that there are transfers for Milnor-Witt K-theory. This would seem to force η = 0.

Theorem. ℤ^{𝔸^{1}}(𝔾_{m} ∧ … ∧ 𝔾_{m}) ≅ K_{n}^{MW} for n ≥ 0.

Proof. The proof consists of producing maps in both directions.

Define ℤ^{𝔸^{1}}(n) = ℤ^{𝔸^{1}}(∧^{n}𝔾_{m}) for n ≥ 0. (This is not terribly good notation because it conflicts with the notation for Voevodsky’s motivic complexes ℤ(n), but we will only be writing it this way temporarily.) Simiarly, define ℤ^{𝔸^{1}}(−n) = Hom(ℤ^{𝔸^{1}}(n+1), ℤ^{𝔸^{1}}(1)). It can be shown that this is the same as Hom(ℤ^{𝔸^{1}}(n+ k), ℤ^{𝔸^{1}}(k)) for all k > 0.

The group ⨁ ℤ^{𝔸^{1}}(n) forms a graded ring, using the smash product and composition operations. We will find a map K_{∗}^{MW} → ⊕ℤ^{𝔸^{1}}(n) on fields, and use the (as yet undefined) residue maps to extend to a map on unramified Milnor-Witt K-theory.

For u ∈ F^{×}, we get a map SpecF → 𝔾_{m}, which is an element of 𝔾_{m}(F). This hence gives us an element of ℤ^{𝔸^{1}}(𝔾_{m})(F).

On the other hand, to define η we consider the “Hopf bundle” 𝔸^{2} \ {0} → ℙ^{1}, given by (x, y) ↦ [x : y]. In terms of 𝔸^{1}-homotopy theory, this gives a map Σ𝔾_{m} ∧ 𝔾_{m} → Σ𝔾_{m}. Applying ℤ^{𝔸^{1}} to this map gives a map Σℤ^{𝔸^{1}}𝔾_{m} ∧ 𝔾_{m} → Σℤ^{𝔸^{1}}𝔾_{m}, which represents an element of Hom(ℤ^{𝔸^{1}}(2), ℤ^{𝔸^{1}}(1)).

As an alternative, we could have defined η as the map

μ − π_{1} − π_{2} : ℤ^{𝔸^{1}}(𝔾_{m} × 𝔾_{m}) → ℤ^{𝔸^{1}}(𝔾_{m}),

which factors through the summand ℤ^{𝔸^{1}}(𝔾_{m} ∧ 𝔾_{m}). Here μ, π_{1}, and π_{2} are induced by the maps 𝔾_{m} × 𝔾_{m} → 𝔾_{m}.

Aside. Why are these two definitions the same?

Consider the map 𝔸^{2}\{0} → ℙ^{1}. This map is a pushout of the following diagram of affine maps:

\xymatrix{𝔸^{1} × 𝔾_{m} \ar[d]^{μ} & 𝔾_{m} × 𝔾_{m} \ar[r]^{(v,v)} \ar[d]^{μ} \ar[l]_{(1,1)} & 𝔾_{m} × 𝔸^{1} \ar[d]^{μ} \\ 𝔸^{1} & 𝔾_{m} \ar[l]_{1} \ar[r]^{v} & 𝔸^{1}}

Here v is the inversion map on 𝔾_{m}. Applying ℤ^{𝔸^{1}} to this diagram gives the following:

\xymatrix{ℤ^{𝔸^{1}}(𝔾_{m}) \ar[d] & ℤ^{𝔸^{1}}(𝔾_{m} × 𝔾_{m}) \ar[r]^{vπ_{1}} \ar[d]^{μ} \ar[l]_{π_{2}} & ℤ^{𝔸^{1}}(𝔾_{m}) \ar[d] \\ 0 & ℤ^{𝔸^{1}}(𝔾_{m}) \ar[l] \ar[r] & 0}

Taking pushouts of chain complexes gives a map of exact triangles.

\xymatrix{ℤ^{𝔸^{1}}(𝔾_{m} × 𝔾_{m}) \ar[r]^{vπ_{1}−π_{2}} \ar[d]^{μ} & ℤ^{𝔸^{1}}(𝔾_{m} ∨ 𝔾_{m}) \ar[r] \ar[d] & Σℤ^{𝔸^{1}}(𝔾_{m} ∧ 𝔾_{m}) \ar[d]^{Ση} \\ ℤ^{𝔸^{1}}(𝔾_{m}) \ar[r] & 0 \ar[r] & Σℤ^{𝔸^{1}}(𝔾_{m})}

and desuspending gives an equivalent map of exact triangles

\xymatrix{ℤ^{𝔸^{1}}(𝔾_{m} ∧ 𝔾_{m}) \ar[d]^{η} \ar[r] & ℤ^{𝔸^{1}}(𝔾_{m} × 𝔾_{m}) \ar[r]^{vπ_{1}−π_{2}} \ar[d]^{μ} & ℤ^{𝔸^{1}}(𝔾_{m} ∨ 𝔾_{m}) \ar[d] \\ ℤ^{𝔸^{1}}(𝔾_{m}) \ar[r] & ℤ^{𝔸^{1}}(𝔾_{m}) \ar[r] & 0}

The map vπ_{1}−π_{2} is a splitting of the summand ℤ^{𝔸^{1}}(𝔾_{m}∨𝔾_{m}) of ℤ^{𝔸^{1}}(𝔾_{m}×𝔾_{m}). The map η, therefore, is the unique map which agrees with μ on 𝔾_{m} ∧ 𝔾_{m} and sends 𝔾_{m} ∨ 𝔾_{m} to zero; however, η − π_{1} − π_{2} has this property.

In order to show that the elements u and η define a map K_{∗}^{MW} → ⊕ℤ^{𝔸^{1}}(n), we now need to show that the relations hold.

That the relation [u][1−u] = 0 holds is an observation of Hu and Kriz. Consider the map 𝔸^{1} \ {0, 1} → 𝔾_{m} × 𝔾_{m}, given by x ↦ (x, 1 − x). Suspending gives a map Σ𝔸^{1} \{0, 1} → Σ𝔾_{m} ×𝔾_{m}, which can then be composed with the canonical map Σ𝔾_{m} × 𝔾_{m} → Σ𝔾_{m} ∧ 𝔾_{m}.

The composite map Σ𝔸^{1} \ {0, 1} → Σ𝔾_{m} ∧ 𝔾_{m} is null.

In fact, purity implies the following:

Σ𝔸^{1} \ {0, 1} ≃ 𝔸^{1}/𝔸^{1} \ {0, 1}
≃ (𝔸^{1}/𝔸^{1} \ {0}) ∨ (𝔸^{1}/𝔸^{1} \ {1})
≃ (Σ𝔾_{m}) ∨ (Σ𝔾_{m})

One can check that the morphism out to Σ𝔾_{m} ∨𝔾_{m} → Σ𝔾_{m} ×𝔾_{m} is equivalent to the inclusion. In other words, the sequence of maps

𝔸^{1} \ {0, 1} → 𝔾_{m} × 𝔾_{m} → 𝔾_{m} ∧ 𝔾_{m}

is a cofiber sequence after suspending. This implies the desired relation.

For the relation [ab] = [a] + [b] + η[a][b], use the formula η = μ − π_{1} − π_{2} for the Hopf map.

The relations η[a] = [a]η and η^{2}[−1] + 2η = 0 are left to the reader.

We can also define a map 𝔾_{m} → J_{1} using this pullback diagram:

\xymatrix{𝔾_{m} \ar@{-->}[dr] \ar[drr]^{1} \ar[ddr]_{u↦⟨u⟩−1} \\  & J_{1} \ar[r] \ar[d] & K_{1}^{M} = 𝔾_{m} \ar[d] \\  & I \ar[r] & K_{2}^{M} = I/I^{2}}

Since J_{1} is strictly 𝔸^{1}-invariant, we get a map ℤ^{𝔸^{1}}(1) → J_{1}, which we can extend by multiplication to a map ℤ^{𝔸^{1}}(n) → J_{n}.

Now, by defining the residue map for K_{n}^{MW} we can show that the map K_{n}^{MW} (F) → ℤ^{𝔸^{1}}(n)(F) comes from a map of sheaves, K_{n}^{MW} → ℤ^{𝔸^{1}}(n).

To check one composite is the identity, consider these maps.

\xymatrix{∏^{n} 𝔾_{m} \ar[dr] \ar[d] \\ ∧^{n}𝔾_{m} \ar[r] \ar[dr] \ar[d] & K_{n}^{MW} \ar[d] \\ ℤ^{𝔸^{1}}(∧^{n}𝔾_{m}) \ar[r] & ℤ^{𝔸^{1}}(n)}

We want to check that the lower map is the identity, which we can do by checking the composite map from ∏ 𝔾_{m}.

The other half is similar; Morel shows that the composite map K_{n}^{MW} → ℤ^{𝔸^{1}}(n) → J_{n} is an isomorphism. We are out of time and will not be able to give any more details of this proof.