Motivic Homotopy

Nov. 18, 2004

Today’s lecturer is Marc Levine.

Our goal today is to prove the following.

Theorem. For Z a fibrant 0-connected space, Z is 𝔸^{1}-local if and only if the homotopy sheaves π_{n}(Z) are strictly 𝔸^{1}-invariant for all m ≥ 1.

(We’re going to cheat slightly; we will actually assume Z is simply connected.)

Along the way, we’ll prove:

Theorem. (Morel) If M is a sheaf of abelian groups on V = Sm/k, and H^{p}(X, M) ≅ H^{p}(X × 𝔸^{1}, M) for p = 0, 1, and all X, then M is strictly 𝔸^{1}- invariant.

It’s interesting to contrast this with:

Theorem. (Voevodsky) If M is a sheaf with transfers, then M is strictly 𝔸^{1}- invariant if and only if M is 𝔸^{1}-invariant.

Our main technical tool will be the Gersten complex and Gersten’s lemma. This is valid (within some range) for π_{n}(Z) where Z is fibrant and 𝔸^{1}-local.

The basic theme of today’s talk is to start in the derived category D(Ab(V)) and move through various steps:

D(Ab(V)) → D_{≥0}(Ab(V)) → HΔ^{op}(Ab(V)) → HShv(V) = H_{s}(k)

Recall that D(Ab(V)) is the homotopy category HC(Ab(V)) of the category of chain complexes of abelian sheaves on V. The model category structure on C(Ab(V)) has termwise monomorphisms as cofibrations and quasi-isomorphisms as weak equivalences. A chain complex C_{∗} is 𝔸^{1}-local if the natural transformation (−, C_{∗})_{D} → (− ⊗ ℤ(𝔸^{1}), C_{∗})_{D} is an isomorphism. An 𝔸^{1}-weak equivalence is a map f : X → Y such that (f, C_{∗})_{D} is an isomorphism for all C_{∗} 𝔸^{1}-local.

We have a full subcategory D_{𝔸^{1}}(Ab(V)) of 𝔸^{1}-local objects and a localization functor L_{𝔸^{1}} : D → D_{𝔸^{1}}. We proved in a previous lecture that we can take

L_{𝔸^{1}}C_{∗} = colim Σ^{N}Ω_{𝔸^{1}}^{N}C_{∗}.

More explicitly, define L_{𝔸^{1}}^{(1)}C_{∗} = Cone(ev_{1} : C_{∗}^{(𝔸^{1})} → C_{∗}), where C_{∗}^{(𝔸^{1})} = Hom(ℤ(𝔸_{•}^{1}), C_{∗}). ℤ(𝔸_{•}^{1}) is the cone on the map ℤ → ℤ[𝔸^{1}].

Iterating gives us a sequence of maps C_{∗} → L_{𝔸^{1}}^{(1)}C_{∗} → L_{𝔸^{1}}^{(2)}C_{∗} → · · ·. The colimit of this sequence is L_{𝔸^{1}}C_{∗}.

We have already proven the connectedness theorem: If C_{∗} is N-connected, then so is L_{𝔸^{1}}C_{∗}.

As an aside, we should always be assuming C_{∗} is fibrant in these constructions. Now let’s move on to new things.

We have an inclusion i : C_{≥0}(Ab(V)) ↪ C(Ab(V)). Here C_{≥0} = {C_{∗} : C_{n} = 0 for n < 0}.

In addition, there is a functor back, left adjoint to i, denoted by τ_{≥0}. Explicitly,

(τ_{≥0}C_{∗})_{n} = \begin{cases}C_{n} & if n > 0 \\ ker(C_{0} → C_{−1}) & if n = 0 \\ 0 & if n < 0 \end{cases} .

τ_{≥0}C_{∗} is often referred to as the (−1)-connected cover of C_{∗}. (We already saw this functor appearing in the t-structure.) The counit of the adjunction is the obvious map τ_{≥0}C_{∗} → C_{∗} which induces an isomophism of H_{n} for n ≥ 0.

We note at this point that we can make C_{≥0} into a model category just as we did with the category C. The main point is that the map i preservescofibrations and weak equivalences, so τ_{≥0} preserves fibrations and weak equivalences.

If we want to find a fibrant replacement for C_{∗} in C_{≥0}, we first find a fibrant replacement C_{∗} → I_{∗} in C. Then τ_{≥0}I_{∗} is a fibrant replacement for C_{∗} in C_{≥0}. As a somewhat unusual special case, if C_{∗} is concentrated in dimension 0, it is already fibrant. (One can check this directly.)

Example. For X ∈ V, Hom_{D_{≥0}}(ℤ(x)[n], C_{∗}[m]) = ℍ^{m−n}(X, C_{∗}) for m, n ≥ 0.

(Recall that our convention is that C_{∗}[m] = Σ^{m}C_{∗}, or equivalently (C_{∗}[m])_{n} = C_{n−m}.)

Next, we have a pair of functors as below.

C_{≥0}(Ab(V)) ⇆_{K}^{N} Δ^{op}Ab(V)

These are sheaf versions of the corresponding maps for abelian groups. They give inverse equivalences between these two categories. The functor N is the normalized chain complex functor and K is the inverse equivalence.

For posterity, we will record what these functors actually are.

Given A_{∗} ∈ Δ^{op}Ab, (NA_{∗}, ∂) is the complex with (NA_{∗})_{n} = ∩_{i>0} ker d_{i}, and ∂ = d_{0}.

For C_{∗} ∈ C_{≥0}Ab, K(C_{∗})_{n} = Hom_{C_{≥0}}(N(ℤΔ^{n}), C_{∗}).

These equivalences have the property that π_{∗}∘K = H_{∗}. Therefore, they preserve weak equivalences.

Note: If f : A → B is a map in C_{≥0}, then we get an exact triangle Cone(f) → A[1] → B[1]. Applying K to this sequence gives us a homotopy fiber sequence K(Cone(f)) → K(A[1]) → K(B[1]).

The Borel construction. Given a (topological) group G with EG a free contractible G-space, we can form the classifying space BG = EG/G. If G is abelian, we can pick things so that BG has a group structure, and iterate this construction. This gives us a construction of K(A, n) for any n:

B^{n}G = K(G, n) = K(G[n])

Given A → A′, we get a diagram:

[figure: K(A[1]); K(A′[1]); fib(BKf); BKA; BKA′; EKA′ ×_{BKA′} BKA; EKA]

In other words, EKA′ ×_{BKA′} BKA ≃ K(Cone(f)).

Theorem 1. Take C_{∗} in C_{≥0}(Ab(V)) which is 0-connected and fibrant. Then K(C_{∗}) is 𝔸^{1}-local if and only if C_{∗} is 𝔸^{1}-local.

Note that by the definitions, C_{∗} is 𝔸^{1}-local in C_{≥0}(Ab(V)) if and only if C_{∗} is 𝔸^{1}-local in C(Ab(V)).

Corollary. Let M be a sheaf of abelian groups on V. Then M is strictly 𝔸^{1}-invariant if and only if M[1] is 𝔸^{1}-invariant.

To prove the corollary, apply the theorem to C_{∗} = M[1]. Since M[1] is 0connected, M[1] is 𝔸^{1}-local in C_{≥0}(Ab(V)) if and only if H^{p}(X, M) ≅ H^{p}(X × 𝔸^{1}, M) for p ≤ 1. On the other hand, M[1] is 𝔸^{1}-local in C(Ab(V)) if and only if M is strictly 𝔸^{1}-invariant.

Proof of theorem 1. Take any C_{∗}. Then C_{∗}^{(𝔸^{1})} is 𝔸^{1}-contractible in C(Ab(V)), as follows. Consider the sequence

C_{∗}^{(𝔸^{1})} ⟶^{i_{1}} C_{∗}^{(𝔸^{1})} ⊗ ℤ(𝔸_{•}^{1}) ⟶^{μ} C_{∗}^{(𝔸^{1})}.

The composite map is the identity, and the center object is 𝔸^{1}-contractible. Applying τ_{≥0} to these maps gives the following diagram:

\xymatrix{τ_{≥0}C_{∗}^{(𝔸^{1})} ⊗ ℤ(𝔸_{•}^{1}) \ar[r] & τ_{≥0}C_{∗}^{(𝔸^{1})} \\ τ_{≥0}C_{∗}^{(𝔸^{1})} \ar[u] \ar@{-->}[ur]}

This shows that C_{≥0}^{(𝔸^{1})} is 𝔸^{1}-contractible. Therefore, the map C → U_{𝔸^{1}}^{(1)}C =

Cone(ev_{1} : C_{≥0}^{(𝔸^{1})} → C) is an 𝔸^{1}-weak equivalence. The map C^{(𝔸^{1})} → C_{≥0}^{(𝔸^{1})} induces a factorization

C → U_{𝔸^{1}}^{(1)}C → L_{𝔸^{1}}^{(1)}C.

Let U_{𝔸^{1}}^{(n)}C be the n’th iterate of this procedure, and U_{𝔸^{1}}^{∞}C the colimit. We get a diagram of 𝔸^{1}-weak equivalences:

\xymatrix{C \ar[r] \ar[dr] & U_{𝔸^{1}}^{(1)} \ar[r] \ar[d] & U_{𝔸^{1}}^{(2)} \ar[r] \ar[d] & ⋯ \ar[r] & U_{𝔸^{1}}^{∞} \ar[d] \\  & L_{𝔸^{1}}^{(1)} \ar[r] & L_{𝔸^{1}}^{(2)} \ar[r] & ⋯ \ar[r] & L_{𝔸^{1}}^{∞}}

Now assume H_{0}C = 0. (Up to weak equivalence, we can then choose C such that C_{0} = 0.) The U_{𝔸^{1}}^{(1)}C is 0-connected, so U_{𝔸^{1}}^{(1)}C = τ_{≥1}L_{𝔸^{1}}^{(1)}C. Iterating this, we find that U_{𝔸^{1}}^{∞}C = τ_{≥1}L_{𝔸^{1}}C, which is quasi-isomorphic to L_{𝔸^{1}}C.

Therefore, under the assumption that C is 0-connected, C → U_{𝔸^{1}}^{∞}C is an 𝔸^{1}- weak equivalence in C_{≥0} and U_{𝔸^{1}}^{∞}C is 𝔸^{1}-local. (This follows from adjointness.) Therefore, it is an 𝔸^{1}-localization of C in C_{≥0}(Ab(V)).

Therefore, if C is 0-connected, we find that L_{𝔸^{1}}C is still the 𝔸^{1}-localization of C in C_{≥0}. If C is 0-connected, we then find that K(C) is 𝔸^{1}-local if and only if C is 𝔸^{1}-local in C_{≥0}(Ab(V)), which is equivalent to C being local in C(Ab(V)).

Our next main step is to discuss Gersten complexes and Gersten’s lemma.

Fix Z in Δ^{op}Shv(V), fibrant.

For W a closed subset of X ∈ Sm/k, i ≤ 0, we define

H_{W}^{i} (X) = [S^{−i} ∧ (X/X \ W), Z]_{H_{s}}.

This is “cohomology with supports in W”. In particular, H^{i}(X) = H_{X}^{i}(X) = ℍ^{i}(X, Z).

This definition is set up so that the cofiber sequence X \ W → X → X/(X \ W) yields a half-long exact sequence of cohomology with supports. Similarly, if W_{1} ⊂ W_{2} ⊂ X are closed subsets, we then have X \ W_{2} ⊂ X \ W_{1} ⊂ X, so there is a cofiber sequence (X \ W_{1})/(X \ W_{2}) → X/(X \ W_{2}) → X/(X \ W_{1}) that yields a long exact sequence of cohomology with supports:

⋯ → H_{w_{1}}^{i}(X) → H_{W_{2}}^{i}(X) → H_{W_{2}\W_{1}}^{i}(W_{2}) → H_{W_{1}}^{i+1}(X) → ⋯

At the end of the sequence, the objects are no longer abelian groups. For x ∈ X,

H_{x}^{i}(X) = colim H_{x̄∩U}^{i}(U) = H_{x}^{i}(X^{x}),

where U ranges over open subsets of X containing x, and X^{x} = Spec(O_{X,x}). We then consider flags of these objects. Definition. The cohomology of X with support in codimension j is defined as

H_{X^{(j)}}^{i}(X) = colim_{W_{j+1}⊂W_{j}⊂X} H_{W_{j}\W_{j+1}}^{i}(X \ W_{j+1}),

where W_{i} ranges over subsets of X of codimension i. (Here X^{(j)} is the set of codimension j points of X.) A lemma says that this is isomorphic to ⊕_{x∈X^{(j)}}H_{x}^{i}(X). Consider chains of closed subsets ⋯ ⊂ W_{2} ⊂ W_{1} ⊂ X, where W_{i} has codimension i. From each pair of spaces we get a long exact sequence of cohomology with supports. Passing to the limit over all W_{i} gives us the following exact couple.

\xymatrix{H_{W_{n+2}}^{p} \ar[rr] &  & H_{W_{n+1}}^{p} \ar[rr] \ar[dl] &  & H_{W_{n}}^{p} ⋯ \ar[dl] \\  & H_{W_{n+1}\W_{n+2}}^{p} \ar[ul] &  & H_{W_{n}\W_{n+1}}^{p} \ar@{-->}[ul]}

The Gersten complex in the limit is

C_{Z}^{(n)}(X) : H^{n}(X) = H_{X^{(0)}}^{n} → H_{X^{(1)}}^{n+1} → ⋯ → H_{X^{(n)}}^{0},

for n < 0. Sheafifying this on X, in either the Zariski or Nisnevich topology, gives us the following complex of sheaves.

π_{−n}Z|_{X} = H^{n} → ⨁_{x∈X^{(0)}}(i_{x})_{∗}H_{x}^{n} → ⨁_{x∈X^{(1)}}(i_{x})_{∗}H_{x}^{n+1} → ⋯ → ⨁_{x∈X^{(}n)}(i_{x})_{∗}H_{x}^{0}

Gersten’s Lemma. For X = Spec(O_{Y,S}), S ⊂ Y finite, Y ∈ Sm/k, this augmented Gersten complex is exact in global sections. Corollary. For Z 𝔸^{1}-local, the sheaf Gersten complex is a partial acyclic resolution of π_{−n}Z on X_{Zar} and X_{Nis}, and H^{i}(C_{Z}^{(n)}(X)) = H^{i}(X, π_{−n}Z). Here is the main lemma to prove this. Let Z be 𝔸^{1}-local and fibrant. Suppose we have a diagram

[figure: W; closed; finite; 𝔸_{V}^{1}; π; \overline{π(W)}; V]

with V ∈ Sm/k, semilocal (having finitely many closed points), k infinite. If we define W′ = π^{−1}(π(W)), then H_{W}^{p} (𝔸_{V}^{1} ) → H_{W′}^{p}(𝔸_{V}^{1} ) is the zero map. Proof. The map π : W → π(W) is finite and V is semilocal. Therefore, we can find a section s : V → 𝔸_{V}^{1} such that s(V) ∩ W = ∅. Since Z is 𝔸^{1}-local, and W′ = 𝔸_{π(W)}^{1}, the top row in the following diagram is an isomorphism.

\xymatrix{H_{W′}^{p}(𝔸_{V}^{1} ) \ar[r]^{s^{∗}}_{∼} & H_{π(W)}^{p}(V) \\ H_{W}^{p} (𝔸_{V}^{1} ) \ar[u] \ar[r]^{s^{∗}} & H_{∅}^{p}(V) = 0 \ar[u]}

Now use Gabber’s presentation lemma to go from W ⊂ 𝔸_{V}^{1} to the situation W ⊂ U ⊂ Y, where S ⊂ U. We can find a digram

[figure: W; 𝔸_{V}^{1}; W; U.; étale]

We can compute H^{∗} with the Gersten complex, and use it to show that the homotopy sheaves π_{m}(Z), for Z 𝔸^{1}-local, are 1-𝔸^{1}-local. Lemma. Let F be a field, X = 𝔸_{F}^{1} , Z 𝔸^{1}-local, and n ≤ 2. Then

0 → H^{n}(F, Z) → C^{(n)}(𝔸_{F}^{1} ) → C^{(n−1)}(⅁_{F}^{1} ) → 0

is exact. (Note H^{n}(F, Z) = π_{−n}(Z)_{F}.) Equivalently, H^{0}(𝔸_{F}^{1} , π_{−n}Z) = π_{−n}(Z)_{F}, and H^{1}(𝔸_{F}^{1} , π_{−n}Z) = H^{1}(F, π_{−n}Z) = 0. Proof. Use the main lemma with W some point of 𝔸_{F}^{1} . A similar argument gives: Lemma’. For x ∈ X = Spec(O_{Y,x}),

0 → H_{x}^{n+p}(Y) → C_{x×𝔸^{1}}^{(n)}(Y × 𝔸^{1})^{p} → C_{x×𝔸^{1}}^{(n)}(Y × 𝔸^{1})^{p+1} → 0

is exact. For W ⊂ Y, C_{W}^{(n)}(Y)^{p} = ⊕_{x∈Y^{(p)}∩W} H_{x}^{n+p}(Y). What you get out of this is the following. For X arbitrary, Z 𝔸^{1}-local, we have that the map H^{p}(X, π_{n}(Z)) → H^{p}(X × 𝔸^{1}, π_{n}(Z)) is an isomorphism for p = 0, 1. The proof consists of computing both sides using the Gersten complex. We can filter the Gersten complex on X × 𝔸^{1} by taking the subcomplexes

⨁_{y→x∈X^{(q)}} H_{y}^{p+n}(X × 𝔸^{1}).

The p’th graded pieces are the complexes as in Lemma’. Therefore, the associated graded is the Gersten complex for X, and the spectral sequence degenerates.