Motivic Homotopy

Oct. 28, 2004

Today’s lecturer is Marc Levine.

Today we’re going to discuss the connectedness theorem; first we will recall what this is. Recall V is the site of smooth varieties over a field k in the Nisnevich topology, and C(Ab(V)) is the category of complexes of sheaves of abelian groups on V.

There is a subcategory of 𝔸^{1}-local objects and a localizing functor L^{𝔸^{1}}. Denote by H_{p}(C) the homology sheaves of the complex C.

Theorem. If H_{p}(C) = 0 for all p < n, then H_{p}(L^{𝔸^{1}}C) = 0 for all p < n.

First, let’s discuss Nisnevich cohomological dimension. (We will be keeping the base field k fixed throughout our discussion.)

Recall a set U = {f_{α} : U_{α} → X} is a Nisnevich cover if:

• the maps f_{α} are all étale and of finite type

• for k → L any field extension, the map of L-valued points ∐ f_{α} : ∐ U_{α}(L) → X(L) is surjective.

The Nisnevich topology has a good theory of stalks. If x ∈ X ∈ V, and F is a Nisnevich sheaf on V, define the stalk of 𝔽 at x to be

F_{x} = lim F(U)

where the limit ranges over all x ∈ U → X, U → X étale.

x is a “point” of Ab(V). The family {x ∈ X ∈ V} is a conservative family, meaning that a map of sheaves F → G is a monomorphism/epimorphism/isomorphism if and only if it is that way on stalks.

The same is true for Ab(X_{Nis}), the category of abelian sheaves on X in the Nisnevich topology. In this case, we only have to check the stalks at points of x.

A similar story also holds for Ab(Sch_{k}), schemes of finite type over k.

Proposition. Suppose X ∈ Sch_{k} has dimension d over k, and F ∈ Ab(X_{Nis}). Then H^{p}(X, F) = 0 for p > d.

Proof. We will actually prove by induction that if F is supported on stalks in dimensions less than or equal to d, H^{p}(X, F) = 0 for p > d. The proof proceeds using several reductions.

1. Suppose X = Spec(F), F a field finitely generated over k. X is the only point of X, and in the Nisnevich topology, the open cover of X by itself is initial among neighborhoods X ∈ U → X. Therefore, the stalk of F at X is F(X). In other words, the global section functor F ↦ F(X) is equivalent to the (exact) stalk functor. As a result, H^{p}(X, F) = 0 for p > 0.

2. Suppose w is a point of X, and F is a sheaf on w_{Nis}. We want to show that H^{p}(X, (i_{w})_{∗}(F)) = H^{p}(w, F) = 0 for p > 0.

We want to compute the stalks of the derived functor sheaves R^{q}(i_{w})_{∗}F at y ∈ X. By a standard argument, these are the same as

lim_{y∈U→X} H^{q}(U ×_{X} w, F),

so these are 0 for q > 0. (U ×_{X} w is étale over w, hence is 0-dimensional.)

Since (i_{w})_{∗} has an exact left adjoint (i_{w})^{∗}, it takes injectives to injectives, and there is a Grothendieck spectral sequence H^{p}(X, R^{q}(i_{w})_{∗}F) ⇒ H^{p+q}(w, F). Since these groups are 0 for q > 0, the spectral sequence collapses to show H^{p}(X, (i_{w})_{∗}F) = H^{p}(w, F) = 0 for p > 0.

3. Supoose X and F are arbitrary, and let Y be the support of F. Then Y is a union ∪Y_{i} of irreducible components, and each Y_{i} has a generic point y_{i}.

Consider the map f : F → ⊕_{i}(i_{y_{i}})_{∗}(i_{y_{i}})^{∗}F, adjoint to the identity maps (i_{y_{i}})^{∗}F → (i_{y_{i}})^{∗}F. The map f is an isomorphism on the stalk at each point y_{i}, and on stalks outside Y both sheaves are 0. If K and C are the kernel and cokernel of f, then K and C are supported on stalks in dimensions less than dim(Y). By induction, H^{p}(X, K) and H^{p}(X, C) vanish in dimensions greater than dim(Y) − 1, and H^{p}(X, ⊕(i_{y_{i}})_{∗}(i_{y_{i}})^{∗}F) = 0 for p > 0. Therefore, using long exact sequences associated to the short exact sequences of sheaves

0 → K → F → Z → 0
0 → Z → ⊕_{i}(i_{y_{i}})_{∗}(i_{y_{i}})^{∗}F → C → 0

gives that H^{p}(X, F) = 0 for p > dim(Y).

The derived category of Ab(V).

(This derived category involves unbounded complexes; we simply won’t be able to get by with bounded complexes.)

The category C_{∗}(Ab(V)) is a model category, as follows. (These definitions will work for C_{∗}(Ab(X_{Nis})) as well.) The weak equivalences are quasi-isomorphisms of chain complexes, and the cofibrations are termwise monomorphisms.

The fibrations are forced. As an aside, any fibrant object is a complex of injectives, and a complex of injectives that is bounded above is fibrant, but a complex of injectives is not necessarily fibrant. Roughly, in order to get a fibrant model for a chain complex C_{∗}, we have to take a limit of bounded above complexes of injectives.

The homotopy category of C_{∗}(Ab(V)) is called the derived category D(Ab(V)). It also includes some extra information in the form of a triangulated structure.

There are some basic facts and operations on the derived category:

For X ∈ V we have an inclusion i_{X} : X_{Nis} → V, which gives a pullback functor i_{X}^{∗} : Ab(V) → Ab(X_{Nis}). i_{X}^{∗} is exact and has an exact left adjoint (i_{X})_{∗} : Ab(X_{Nis}) → Ab(V). It is defined as follows:

(i_{X})_{∗}F(U) = lim_{Y →U→X} F(U)

Here the limit is taken over diagrams Y → U → X with U → X étale.

1. In particular, (i_{X})_{∗}(ℤ) = ℤ(X), where ℤ is the constant sheaf on X and ℤ(X) is the abelianization of X. This follows from the following identities.

Hom_{Ab(V)}(ℤ(X), F) = Hom_{Sh(V)}(X, F)
= F(X)
= Hom_{Ab(X)}(ℤ, i_{X}^{∗}F)
= Hom_{Ab(V)}((i_{X})_{∗}ℤ, F)

2. If C_{∗} is fibrant in C_{∗}(Ab(V)), then i_{X}^{∗}C_{∗} is fibrant in C_{∗}(Ab(X)). (This is formal: The left adjoint (i_{X})_{∗} is exact and so it preserves cofibrations.)

3. If C_{∗} is fibrant in C_{∗}(Ab(X)) and H^{p}(C_{∗}) = 0, then H^{p}(Γ(X, C_{∗})) = 0 for all p. Therefore, the functor F ↦ Γ(X, F) : Ab(X) → Ab extends to a functor RΓ : D(Ab(X)) → D(Ab). It is computed as follows: For C ∈ Ab(X), choose a fibrant replacement C̃. Then RΓ(X, C) = Γ(X, C̃). We can then define hypercohomology groups ℍ^{n}(X, C) = H^{n}(RΓ(X, C)) as usual.

4. ℍ^{n}(X, i_{X}^{∗}C) = Hom_{D(Ab(V))}(ℤ(X), C[n]).

Notice that we’ve already accidentally switched to cohomological grading already.

We define the category C^{b}(ℤ(V)) to be the subcategory of C_{∗}(Ab(V)) of bounded complexes with each term a finite direct sum of the form ⊕ℤ(X_{i}), X_{i} ∈ V. We denote objects coming from this subcategory by K for compact; the functor Hom_{D}(K, −) commutes with limits. (This follows because it is true for ℤ(X_{i}); Hom_{D}(ℤ(X_{i}), −) is a hypercohomology functor which commutes with direct limits.)

For an object K we have a functor − ⊗ K on C_{∗}(Ab(V)). These objects K are flat (they are sums of free objects) and so there are induced functors on the derived category D(Ab(V)).

We also have internal Hom-objects, defined by Hom(ℤ(X), F)(Y) = F(X ×Y). We note that ℤ(X) ⊗ ℤ(Y) = ℤ(X × Y), and

Hom_{C_{∗}(Ab(V))} (C′_{∗}, Hom(ℤ(X), C_{∗})) = Hom_{C_{∗}(Ab(V))}(C′_{∗} ⊗ ℤ(X), C).

As a consequence, if C is fibrant, then Hom(K, C) is still fibrant.

The 𝔸^{1} derived category.

Definition. C ∈ C_{∗}(Ab(V)) is 𝔸^{1}-local if

Hom_{D}(A, C) →^{p^{∗}} Hom_{D}(A ⊗ ℤ(𝔸^{1}), C)

is an isomorphism for all objects A.

Here p^{∗} is the map induced from the projection 𝔸^{1} → Spec(k), which gives a map ℤ(𝔸^{1}) → ℤ(Spec(k)) = ℤ.

A map A → A′ is an 𝔸^{1}-weak equivalence if for all 𝔸^{1}-local C, Hom_{D}(A′, C) → Hom_{D}(A, C) is an isomorphism.

Write D_{𝔸^{1}}(Ab(V)) ⊂ D(Ab(V)) for the full subcategory of 𝔸^{1}-local objects. General localization theory tells us that this is equivalent to the category gotten by inverting the 𝔸^{1}-weak equivalences in D(Ab(V)). There is a localization functor L^{𝔸^{1}}, which we would like to now construct explicitly.

Define

ℤ(𝔸_{•}^{1}) = Cone(ℤ →^{ℤ(i_{0})} ℤ(𝔸^{1})).

Here i_{0} : Spec(k) → 𝔸^{1} is the inclusion of 0.

Lemma 1. Let C be a fibrant object in C_{∗}(Ab(V)). Then the following are equivalent:

1. C is 𝔸^{1}-local.

2. p^{∗} : C → Hom(ℤ(𝔸^{1}), C) is a weak equivalence.

3. i_{0}^{∗} : Hom(ℤ(𝔸^{1}), C) → C is a weak equivalence.

4. Hom(ℤ(𝔸_{•}^{1}), C) ≃ 0.

5. For each X, p, the map

ℍ^{p}(X, Hom(ℤ(𝔸_{•}^{1}), C)) →^{i_{1}^{∗}} ℍ^{p}(X, C)

is the zero map.

6. For each X, p, the map

ℍ^{p}(X, Hom(ℤ(𝔸^{1}), C)) →^{i_{1}^{∗}} ℍ^{p}(X, C)

is an isomorphism.

Notation. Write P_{𝔸^{1}}(C) = Hom(ℤ(𝔸_{•}^{1}), C). Intuitively, it is the “based 𝔸^{1} path space” of C, and the map i_{1}^{∗} is evaluation at the endpoint.

Proof of lemma 1. The equivalence of the first 4 items in the lemma are proved using two facts already proven: the adjunction between Hom(ℤ(𝔸^{1}), −) and − ⊗ ℤ(𝔸^{1}), and the split cofiber sequence ℤ → ℤ(𝔸^{1}) → ℤ(𝔸_{•}^{1}). Similarly,

(5) and (6) are equivalent by using the same cofiber sequence.

That (4) implies (5) is clear, since hypercohomology with coefficients in an object weakly equivalent to 0 vanishes.

The main part of the theorem is to prove that (6) implies (4). To show P_{𝔸^{1}}C = 0, it suffices to show ℍ^{p}(Y, P_{𝔸^{1}}C) = 0 for any Y, p. ((insert a reason why))

However, this hypercohomology group is

ker i_{0}^{∗} : ℍ^{p}(Y, Hom(ℤ(𝔸^{1}), C)) → ℍ^{p}(Y, C),

or equivalently

ker i_{0}^{∗} : ℍ^{p}(Y × 𝔸^{1}, C) → ℍ^{p}(Y, C).

Consider the following diagram.

\xymatrix{(Y × 𝔸^{1}) × 𝔸^{1} \ar[r]^{1×μ} & Y × 𝔸^{1} \\ Y × 𝔸^{1} \ar[u]^{i_{0}} \ar[u]_{i_{1}} \ar[ur]^{i_{0}p} \ar[ur]_{id}}

Here μ is the multiplication map on 𝔸^{1}, and p is projection onto the first factor. (This represents a “homotopy” from the identity map on Y × 𝔸^{1} to i_{0}p.)

Taking X = Y × 𝔸^{1}, by assumption the map i_{1}^{∗} : ℍ^{p}((Y × 𝔸^{1}) × 𝔸^{1}, C) → ℍ^{p}(Y × 𝔸^{1}, C) is an isomorphism. The map p^{∗} is a right inverse to both i_{1}^{∗} and i_{0}^{∗}, so i_{1}^{∗} = i_{0}^{∗}.

Suppose α ∈ ker i_{0}^{∗} : ℍ^{p}(Y × 𝔸^{1}, C) → ℍ^{p}(Y, C). Then

α = i_{1}^{∗}(1 × μ)^{∗}(α) = i_{0}^{∗}(1 × μ)^{∗}(α) = p^{∗}i_{0}^{∗}α = 0.

Now we are in a position to construct explicit 𝔸^{1}-localizations.

Definition. ℤ(S_{𝔸^{1}}^{1}) = Cone [i_{0}^{∗} ⊕ i_{1}^{∗} : ℤ ⊕ ℤ → ℤ(𝔸^{1})]. (Intuitively, we are taking 𝔸^{1} and identifying the points 0 and 1 together.)

Definition. Ω_{𝔸^{1}}(C) = Hom(ℤ(S_{𝔸^{1}}^{1}), C). (Note that we must replace C by a fibrant object if necessary.)

Definition. L_{𝔸^{1}}^{(1)}(C) = ΣΩ_{𝔸^{1}}(C) = Ω_{𝔸^{1}}C[1]. (The bracket notation indicates a shift in homological degree.) Then inductively define L_{𝔸^{1}}^{(n)} = L_{𝔸^{1}}^{(1)} ∘ L_{𝔸^{1}}^{(n−1)}.

Intuitively, we are trying to force suspension and “𝔸^{1}-suspension” to be the same thing.

Explicitly, L_{𝔸^{1}}^{(1)}(C) can be expressed as the total complex of the double complex below.

Tot [ \xymatrix{Hom(ℤ(𝔸^{1}), C) \ar[r]^{i_{0}^{∗}⊕i_{1}^{∗}} & C ⊕ C} ]

There is an inclusion of C as the lefthand summand of C ⊕ C in this double complex, giving a map C → ΣΩ_{𝔸^{1}}C. Iterating this, we get a sequence of maps

C → L_{𝔸^{1}}^{(1)}(C) → L_{𝔸^{1}}^{(2)}(C) → L_{𝔸^{1}}^{(3)}(C) …

Define L_{𝔸^{1}}^{∞}(C) to be the colimit of these maps. L_{𝔸^{1}}^{∞}(C) is still a sheaf, since it is a filtered colimit of monomorphisms.

(As an alternative, we could have taken the cone on the map i_{1}^{∗} : P_{𝔸^{1}}C → C, which has a more obvious inclusion of C as a subcomplex.)

Lemma 2. If C is fibrant, then C → ΣΩ_{𝔸^{1}}C is an 𝔸^{1}-weak equivalence.

Proof. The idea of the proof is to use the fact that ⊗ preserves certain 𝔸^{1}- equivalences to show the corresponding fact for the internal Hom functor.

We have a map μ : 𝔸^{1} ×𝔸^{1} → 𝔸^{1} that sends 𝔸^{1} ∨𝔸^{1} to 0. Taking abelianization and making use of a cofibration sequence gives us a map μ : ℤ(𝔸_{•}^{1}) ⊗ ℤ(𝔸_{•}^{1}) → ℤ(𝔸_{•}^{1}). There is then the following diagram.

\xymatrix{P_{𝔸^{1}}C \ar[rr]^{1⊗i_{1}} \ar[dr]_{id} &  & P_{𝔸^{1}}C ⊗ ℤ(𝔸_{•}^{1}) \ar[dl]^{μ̃} \\  & P_{𝔸^{1}}C}

Here μ̃ is adjoint to the map μ^{∗} : Hom(ℤ(𝔸_{•}^{1}), C) → Hom(ℤ(𝔸_{•}^{1}) ⊗ ℤ(𝔸_{•}^{1}), C).

However, ℤ(𝔸_{•}^{1}) is the cone on the map ℤ → ℤ(𝔸^{1}), hence is 𝔸^{1}-weakly equivalent to 0. Tensoring the cofiber sequence ℤ → ℤ(𝔸^{1}) → ℤ(𝔸_{•}^{1}) with P_{𝔸^{1}}C gives a cofibration sequence P_{𝔸^{1}}C → P_{𝔸^{1}}C ⊗ ℤ(𝔸^{1}) → P_{𝔸^{1}}C ⊗ ℤ(𝔸_{•}^{1}). The first map is an 𝔸^{1}-equivalence, so P_{𝔸^{1}}C ⊗ ℤ(𝔸_{•}^{1}) is 𝔸^{1}-equivalent to 0.

Therefore, in the 𝔸^{1}-homotopy category the previous diagram is a factorization of the identity on P_{𝔸^{1}}C = Hom(ℤ(𝔸_{•}^{1}), C) through 0. Therefore, P_{𝔸^{1}}C = 0 after 𝔸^{1}-localization.

Therefore, the map C → Cone [i_{1}^{∗} : P_{𝔸^{1}}C → C] is an 𝔸^{1}-equivalence. (There is a cofiber sequence P_{𝔸^{1}}C → C → L_{𝔸^{1}}^{(1)}(C), and the left-hand term is 𝔸^{1}-equivalent to 0. Therefore, since localization preserves cofiber sequences, C → L_{𝔸^{1}}^{(1)}(C) is an 𝔸^{1}-equivalence.)

Corollary. C → L_{𝔸^{1}}^{∞}(C) is an 𝔸^{1}-weak equivalence.

Proof. All the maps L_{𝔸^{1}}^{(n)}(C) → L_{𝔸^{1}}^{(n+1)}(C) in the colimit are acyclic cofibrations

in the 𝔸^{1}-model structure. Therefore, [colim L_{𝔸^{1}}^{(n)}(C), Z] = lim [L_{𝔸^{1}}^{(n)}(C), Z] =

[C, Z] for Z any 𝔸^{1}-local object.

Lemma 3. L_{𝔸^{1}}^{∞} is 𝔸^{1}-local.

Proof. The proof will use criterion (5) from lemma 1.

We want to show: For any X, p, the evaluation map i_{1}^{∗} : ℍ^{p}(X, P_{𝔸^{1}}(L_{𝔸^{1}}^{∞}(C))) → ℍ^{p}(X, L_{𝔸^{1}}^{∞}(C)) is the zero map.

First note that P_{𝔸^{1}}(L_{𝔸^{1}}^{∞}(C)) = colim P_{𝔸^{1}}(L_{𝔸^{1}}^{(n)}(C)). By definition, this means

that Hom (ℤ(𝔸_{•}^{1}), colim L_{𝔸^{1}}^{(n)}(C)) = colim Hom (ℤ(𝔸_{•}^{1}), L_{𝔸^{1}}^{(n)}(C)), which fol-

lows because ℤ(𝔸_{•}^{1}) is compact.

The composite map ℍ^{p}(X, P_{𝔸^{1}}(C)) → ℍ^{p}(X, C) → ℍ^{p}(X, L_{𝔸^{1}}^{(1)}(C)) is the zero map, since the maps P_{𝔸^{1}}(C) → C → L_{𝔸^{1}}^{(1)}(C) form an exact triangle. Consider the following diagram:

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

For any n, the map P_{𝔸^{1}}(L_{𝔸^{1}}^{(n)}(C)) → L_{𝔸^{1}}^{∞}(C) is the zero map, since it factors through the composite already shown to be zero. Therefore, the map from the colimit is zero. Therefore, the map P_{𝔸^{1}}(L_{𝔸^{1}}^{∞}(C)) → L_{𝔸^{1}}^{∞}(C) is the zero map; it must therefore induce an isomorphism on hypercohomology.

Weak Gersten’s Lemma. Suppose C is 𝔸^{1}-local, fibrant, X ∈ V, S ⊂ X a finite set of points, X^{S} = Spec(O_{X,S}) ⊂ X.

Take a closed W ⊂ X. Then ℍ^{p}(X^{S}, C) ↪ ℍ^{p}(X^{S} \ W, C).

(Equivalently, ℍ_{W ∩X^{S}}^{p}(X^{S}, C) → ℍ^{p}(X^{S}, C) is the zero map.)

Note. This follows from the following Nisnevich Mayer-Vietoris property:

If f : X′ → X is an étale map, W ⊂ X such that f^{−1}W = W, then if C is any fibrant complex of sheaves,

f^{∗} : ℍ_{W}^{p} (X, C) → ℍ_{W}^{p} (X′, C)

is an isomorphism.

Recall ℍ_{W}^{p} (X, C) = ℍ^{p}(X, Cone(C → (Rj_{∗})j^{∗}C)[−1]), where j is the inclusion X \ W ↪ X.

To prove this, one uses the fact that X′ and X \ W form a cover of X.

Corollary. (of Weak Gersten’s Lemma) If C is 𝔸^{1}-local, fibrant, then H_{p}(C)(F) = 0 for all F = k(x) if and only if H_{p}(C) = 0.

Proof of corollary. We can check H_{p}(C) = 0 stalkwise.

H_{p}(C)_{x} = colim_{x∈U→X} ℍ^{−p}(Spec(O_{U,x}), C)

However, ℍ^{−p}(Spec(O_{U,x})) ↪ ℍ^{−p}(k(U), C) = 0.

To prove Gersten’s lemma, we first need:

Gabber’s presentation lemma. Suppose W ⊂ X ∈ V, dimX = n+1, S ⊂ X a finite set of points. Then there is an open set U with S ⊂ U ⊂ X, V ⊂ 𝔸^{n} open, ϕ : U → V × 𝔸^{1} such that:

• ϕ is étale

• ϕ|_{W′} : W′ → ϕ(W′) is an isomorphism and ϕ^{−1}ϕ(W′) = W′, where W′ = W ∩ U

• p_{1} ∘ ϕ|_{W′} : W′ → V is finite.

Proof. We’ll assume for simplicity that the characteristic of the underlying field is zero, and also that X ⊂ 𝔸^{N}; we can shrink X if necessary.

We can compose with a generic linear projection to get a map π : X → 𝔸^{N} → 𝔸^{n+1} such that:

• π is finite étale on S

• π^{−1}π(W) = W ∪ Y for some Y such that W ∩ Y ∩ S = ∅

• B ⊂ 𝔸^{n+1}, the image of the ramification locus, does not intersect π(S); i.e., π is étale at π^{−1}π(S).

Then take a general π′ : 𝔸^{n+1} → 𝔸^{n} such that π′ restricted to B ∪ π(W ∩ Y) is finite.

Let V = 𝔸^{n} \ (π′(B ∪ π(W ∩ Y))), and U = π^{−1}(V × 𝔸^{1}) \ Y.

Proof of Gersten’s lemma.

Suppose α ∈ ℍ_{W}^{p} (X, C). Take U, V, ϕ : U → V ×𝔸^{1} as in Gabber’s presentation lemma.

[figure: U; ϕ; V × 𝔸^{1}; p; W′; finite; V]

α restricts to ℍ_{W′}^{p}(U, C) ≅ ℍ_{W′}^{p}(V × 𝔸^{1}, C).

We can shrink V in necessary to get a section s of p that misses W′; i.e., s(V) ∩ W′ = ∅.

Then consider the following diagram.

\xymatrix{ℍ_{W′}^{p}(V × 𝔸^{1}, C) \ar[r] \ar[d]^{∼} & ℍ^{p}(V × 𝔸^{1}, C) \ar[d]^{ϕ^{∗}} \ar[r]^{s^{∗}}_{∼} & ℍ^{p}(V, C) \\ ℍ_{W′}^{p}(U, C) \ar[r]_{ϕ^{∗}} & ℍ^{p}(U, C)}

The composite of the maps in the top row is 0, since the section s misses W′. Therefore, the map ℍ_{W′}^{p}(U, C) → ℍ^{p}(U, C) is zero.

Now, finally, we will prove the connectivity theorem.

Theorem. Suppose C ∈ C_{∗}(Ab(V)) is fibrant. If H_{p}(C) = 0 for p < 0 then H_{p}(L_{𝔸^{1}}(C)) = 0 for p < 0.

Proof. It suffices to show, by the weak Gersten’s lemma, that H_{p}(L_{𝔸}^{1}(C))(F) = 0 for all F = k(X) and all p < 0.

Also, if we can show that H_{p} (L_{𝔸^{1}}^{(n)}C) (F) = 0 for all F and p < 0 we are done, since cohomology commutes with direct limits in the sheaf variable.

For any n, L_{𝔸^{1}}^{(n)}(C) is the total complex of a cubical diagram of chain complexes. For n = 1, it is the total complex of the diagram

\xymatrix{Hom(I, C) \ar[r] & C,}

where I = ℤ(𝔸_{•}^{1}). Similarly, for n = 2, it is the total complex of the following diagram:

\xymatrix{Hom(I ⊗ I, C) \ar[r] \ar[d] & Hom(I, C) \ar[d] \\ Hom(I, C) \ar[r] & C}

(The arrows in the diagram are induced by the map ℤ → I.)

For general n, there is a similar (n − 1)-cubical diagram. In order to prove the result, then, it suffices to show that H_{p}(Hom(I^{⊗k}, C))(F) = 0 for p < −k.

However, this group is ℍ^{−p} ((𝔸_{F}^{1} )^{∧k}, C). There is a hypercohomology spectral sequence

H^{r} ((𝔸_{F}^{1} )^{∧k}, H_{s}(C)) ⇒ ℍ^{r−s}((𝔸_{F}^{1} )^{∧k}, C).

The groups H^{r} vanish for −r > k because of dimension reasons; therefore, ℍ^{p} vanishes for −p > k. As a consequence of the connectivity theorem, we get a t-structure on the derived category D_{𝔸^{1}}(Ab(V)) as follows.

The derived category D(Ab(V)) has truncation operations giving a t-structure, as follows. There are functors τ_{≤p} and τ^{>p} such for any C, there is a natural exact triangle

τ_{≥p}C → C → τ^{<p},

where the left-hand map is an isomorphism on H_{k} for k ≥ p, and the right-hand map is an isomorphism on H_{k} for k < p. This gives a t-structure on D(Ab(V)). Specifically, we have two subcategories

D_{≥0} = {C : H_{p}C = 0 for p < 0}

and

D_{<0} = {C : H_{p}C = 0 for p ≥ 0}.

The “heart” of this t-structure is the abelian category of sheaves.

Theorem. If C is 𝔸^{1}-local and fibrant, then τ_{≥p}C and τ^{<p}C are 𝔸^{1}-local.

As a result of the theorem, we can define a t-structure on D_{𝔸^{1}} by

(D_{𝔸^{1}})_{≥0} = D_{𝔸^{1}} ∩ D_{≥0}

and

(D_{𝔸^{1}})_{<0} = D_{𝔸^{1}} ∩ D_{<0}.

This t-structure has heart equal to the category of strictly 𝔸^{1}-invariant sheaves.

Corollary. A fibrant object C is 𝔸^{1}-local if and only if the sheaves H_{p}C are strictly 𝔸^{1}-invariant.

Corollary. Strictly 𝔸^{1}-invariant sheaves form an abelian category. (!)

Proof of theorem. Suppose C is 𝔸^{1}-local. We have the following maps.

\xymatrix{τ_{≥p}C \ar[r] \ar[d] & C \ar[d]^{∼} \\ L_{𝔸^{1}}τ_{≥p}C \ar[r] & L_{𝔸^{1}}C}

However, τ_{≥p}C is universal for maps into C from (p − 1)-connected objects, so there is a splitting L_{𝔸^{1}}τ_{≥p}C → τ_{≥p}C. Therefore, τ_{≥p}C is a summand of an 𝔸^{1}-local object, hence is 𝔸^{1}-local itself. The corresponding statement for τ^{<p} follows from the natural exact triangle.