On the zero slice of the sphere spectrum

Vladimir Voevodsky

December 2002

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Thom spaces of linear representations and their quotients … 2 3 Thick spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 Symmetric powers of T . . . . . . . . . . . . . . . . . . . . . . 9 5 Reformulation for smooth schemes . . . . . . . . . . . . . . . . 12 6 Motivic Eilenberg-MacLane spaces . . . . . . . . . . . . . . . 12

1 Introduction

In [3] we introduced the slice filtration on the motivic stable homotopy category which is a motivic analog of the filtration by the subcategories of n-connected spectra in topology. Since the inclusion functors between different terms of the filtration have right ajoints it makes sense to speak of the projection s_{n}(E) of a spectrum E to the n-th quotient of this filtration. This projection which is again an object of the motivic stable homotopy category is called the n-th slice of E. Its topological analog is the spectrum Σ^{n}H_{π_{n}(E)} where π_{n}(E) is the n-th stable homotopy group of E and H_{A} is the Eilenberg-MacLane spectrum corresponding to the abelian group A. In this sense slices provide a motivic replacement of the stable homotopy groups.

The goal of this paper is to prove over fields of characteristic zero the main conjecture of [3] which asserts that the zero slice of the sphere spectrum 1 is the motivic Eilenberg-MacLane spectrum H_{Z}. Using the analogy between slices and stable homotopy groups one may interpret this result as a motivic version of the statement that π_{0}^{s}(S^{0}) = Z. As an immediate corollary one gets that the slices of any spectrum are modules over H_{Z}.

We obtain our main result from an unstable statement about the motivic Eilenber-MacLane spaces K_{n}. We introduce the notion of a (homotopically) n-thick space and show that on the one hand the suspension spectrum of any n-thick space belongs to the n-th stage of the slice filtration and on the other hand that for n > 0 the cone of the natural map T^{n} → K_{n} is (n + 1)-thick.

The restriction on the characteristic of the base field k appears in our approach twice. Firstly, we use the model of K_{n} based on the symmetric powers of T^{n} = A^{n}/(A^{n}−{0}) which is only valid over fields of characteristic zero. Secondly, in order to analyze the structure of the d-th symmetric power we need invertibility of d! in k. We expect that the main result of the paper remains valid in any characteristic even though the intermediate steps we use do not.

The first draft of this paper (with some mistakes) was announced on the Great Lakes K-theory meeting in Evanston in 2001 and I would like to thank all the listeners for pointing out gaps in the original argument.

2 Thom spaces of linear representations and their quotients

We fix a base field k and let Sch/k denote the category of quasi-projective schemes over k. For a finite group scheme G over k we denote by G − Sch/k the category of G-objects in Sch/k. In [2] we defined an analog of the Nisnevich topology on G − Sch/k. We let (G − Sch/k)_{Nis} denote the corresponding site and Spc_{•}(G) denote the category of pointed simplicial sheaves on (G − Sch/k)_{Nis}. Following [2] one introduces the class of A^{1}- weak equivalences on Spc_{•}(G) and defines the corresponding A^{1}-homotopy category as the localization with respect to this class.

We let Σ_{s}^{1} and Σ_{T}^{1} denote the suspensions by the simplicial circle and the sphere T = A^{1}/(A^{1} − {0}) respectively, and Σ̃_{s} denote the unreduced s-suspension:

Σ̃_{s}F = cone(F_{+} → S^{0})

For a G-equivariant vector bundle E over X we let T h(E) denote the Thom space E/E − s_{0}(X) where s_{0} is the zero section. For X = Spec(k), equivariant vector bundles are linear representations V of G and T h(V) is the object V/V −{0}. Note that T h(V ⊕W) ≅ T h(V)∧T h(W) and T h(0) = S^{0}. Note also that since V is A^{1}-contractible as a G-scheme there is a natural A^{1}-equivalence of the form

T h(V) ≅ Σ̃_{s}(V − {0})

We let Quot_{G} : Spc_{•}(G) → Spc_{•} denote the functor which commutes with colimits and such that Quot(X_{+}) = (X/G)_{+} (see [2, Sec. 5.1]).

In this section we prove several general results about the structure of the quotients Quot_{G}(T h(V)). For simplicity, we assume in this section that G is a finite group. The following lemma is straightforward.

Lemma 2.1 Let V_{1}, V_{2} be linear representations of G_{1}, G_{2}. Then there is a natural isomorphism

Quot_{G_{1}×G_{2}}(T h(V_{1} × V_{2})) = Quot_{G_{1}}(V_{1}) ∧ Quot_{G_{2}}(V_{2})

In particular if O^{n} is the trivial representation of G of dimension n then

Quot_{G}(T h(V ⊕ O^{n})) = Σ_{T}^{n}Quot_{G}(T h(V))

For a subgroup H in G we let V^{≥H} denote the closed subset of H-invariant elements in V and by V^{>H} the subset of elements whose stabilizer is strictly greater than H. Note that for H_{1} ≠ H_{2} one has V^{≥H_{1}} ∩V^{≥H_{2}} ⊂ V^{>H_{1}} ∩V^{>H_{2}} and in particular

(V^{≥H_{1}} − V^{>H_{1}}) ∩ (V^{≥H_{2}} − V^{>H_{2}}) = ∅

For a closed subset Z of V we let GZ denote the orbit of Z i.e. the closed subset Im(Z ×G → V ×G → V) where the first map is the closed embedding and the second one is the action.

Lemma 2.2 Let H be a closed subgroup in G and N(H) the normalizer of H. Then one has

\begin{matrix}Quot_{G}((V − GV^{>H})/(V − GV^{≥H})) = \\ = Quot_{N(H)}((V − V^{>H})/(V − V^{≥H})) \end{matrix}  (2.1)

Proof: Let A = G/N(H) be the set of subgroups adjoint to H. The scheme GV^{≥H} − GV^{>H} is (non canonically) isomorphic to (V^{≥H} − V^{>H}) × A and there is a G-equivariant map GV^{≥H} − GV^{>H} → A. Consider the section of the projection (GV^{≥H} − GV^{>H}) × A → (GV^{≥H} − GV^{>H}) defined by this map. Since this projection is etale the image of this section is open and we may consider the closed complement C to this image. Consider the square:

(V − GV^{≥H}) × A ⟶ (V − GV^{>H}) × A − C
↓  ↓^{p}
V − GV^{≥H}  ⟶^{j}  V − GV^{>H}

where the vertical morphisms come from the obvious projections and the horizontal morphisms from the obvious embeddings. Let us show that this is an upper distinguished square in G − Sch/k. It is clearly a pull-back square, p is etale and j is an open embedding. It remains to check that p^{−1}(GV^{≥H} − GV^{>H}) → GV^{≥H} − GV^{>H} is an isomorphism which follows from our choice of C. We conclude that

(V − GV^{>H})/(V − GV^{≥H}) = (V − GV^{≥H}) × A/((V − GV^{>H}) × A − C)

and therefore

Quot_{G}((V − GV^{>H})/(V − GV^{≥H})) =
= Quot_{G}((V − GV^{≥H}) × A/((V − GV^{>H}) × A − C))

We further have:

Quot_{G}((V − GV^{≥H}) × A) = Quot_{N(H)}(V − GV^{≥H})

and

Quot_{G}((V − GV^{>H}) × A − C) = Quot_{N(H)}(V − GV^{>H} − (GV^{≥H} − V^{≥H}))

Since GV^{≥H} − GV^{>H} = (V^{≥H} − V^{>H}) ∐(GV^{≥H} − V^{≥H} − GV^{>H}) we have

(V − GV^{>H} − (GV^{≥H} − V^{≥H}))/(V − GV^{≥H}) = (V − V^{>H})/(V − V^{≥H})

and (2.1) follows.

Remark 2.3 An analog of Lemma 2.2 holds for any finite etale group scheme G and any X in G − Sch/k.

Let H be a normal subgroup of G. Then we may consider the relative analog of the Quot functor

Quot_{G,H} : Spc_{•}(G) → Spc_{•}(G/H)

which commutes with colimits and such that Quot(X_{+}) = (X/H)_{+} where X/H is considered with the natural action of G/H. One verifies easily that one has Quot_{G} = Quot_{G/H}Quot_{G,H}.

Lemma 2.4 If H is a normal subgroup of G of order prime to char(k) then one has:

Quot_{G}((V − V^{>H})/(V − V^{≥H})) =
= Quot_{G/H}((V^{≥H} − V^{>H})_{+} ∧ Quot_{G,H}(T h(V/V^{≥H})))

Proof: Since

Quot_{G} = Quot_{G/H}Quot_{G,H},

it is sufficient to show that Quot_{G,H}((V −V^{>H})/(V −V^{≥H})) is isomorphic to (V^{≥H} − V^{>H})_{+} ∧ Quot_{G,H}(T h(V/V^{≥H})) as a G/H-space. Since the order of H is prime to char(k) there is an isomorphism V = V^{≥H} ⊕ (V/V^{≥H}). Using this isomorphism we get an isomorphism

(V − V^{>H})/(V − V^{≥H}) = (V^{≥H} − V^{>H})_{+} ∧ T h(V/V^{≥H})

Since the action of H on V^{≥H} is trivial we get

Quot_{G,H}((V − V^{>H})/(V − V^{≥H})) = (V^{≥H} − V^{>H})_{+} ∧ Quot_{G,H}(T h(V/V^{≥H}))

Combining Lemmas 2.2 and 2.4 we get the following result.

Proposition 2.5 Let G be a finite group of order prime to char(k). Let further V be a linear representation of G over k such that V^{≥G} = 0. Then Quot_{G}(T h(V)) belongs to the smallest class which is closed under cones, finite coproducts and A^{1}-equivalences and contains the following objects:

1. Σ̃_{s}(Quot_{G}(V_{0})) where V_{0} is the open subscheme in V where G acts freely

2. Quot_{N(H)/H}((V^{≥H} − V^{>H})_{+} ∧ Quot_{N(H),H}(T h(V/V^{≥H}))) for all subgroups H in G such that H ≠ e, G.

Proof: Consider the sequence of open embeddings V_{0} → V_{1} → … → V_{N} = V where V_{i} is the subscheme of points with the stabilizer of order no more than i + 1 and N = |G| − 1. In particular V_{0} is the open subscheme where G acts freely. Under our assumption V_{N−1} = V −{0} and hence T h(V) = Σ̃_{s}(V_{N−1}).

Lemma 2.6 Let X_{0} → X_{1} → … → X_{m} be a sequence of monomorphisms in Spc_{•}(G). Then Σ̃_{s}(X_{m}) belongs to the smallest class which is closed under simplicial weak equivalences and cones and contains Σ̃_{s}(X_{0}) and X_{i}/X_{i−1} for i = 1, …, m.

Proof: Denote the smallest class satisfying the conditions of the lemma by B. let us show by induction on m that Σ̃_{s}(X_{m}) is in B. For m = 0 the statement is obvious. By induction we may assume that Σ̃_{s}X_{m−1} is in B. The digram

pt  ⟶  S^{0}  ⟶ S^{0}
↓  ↓  ↓
X_{m}/X_{m−1} ⟶ Σ_{s}X_{m−1,+} ⟶ Σ_{s}X_{m,+}
↓  ↓  ↓
X_{m}/X_{m−1} ⟶ Σ̃_{s}X_{m−1} ⟶ Σ̃_{s}X_{m}

whose rows and columns are cofibration sequences, shows that Σ̃_{s}X_{m} = cone(X_{m}/X_{m−1} → Σ̃_{s}X_{m−1}) and therefore Σ̃_{s}X_{m} is in B.

In view of Lemma 2.6 it is sufficient to show that the quotients V_{i}/V_{i−1} for i = 1, …, N − 1 belong to the class we consider. This follows directly from Lemmas 2.2 and 2.4.

3 Thick spaces

Definition 3.1 The class of n-thick objects is the smallest class A_{n} such that:

1. A_{n} is closed under A^{1}-equivalences

2. A_{n} is closed under filtering colimits

3. if (F_{i}) is a simplicial object in Δ^{op}Spc_{•}(G) such that F_{i} ∈ A_{n} for all i and Δ is the diagonal functor then Δ((F_{i})) ∈ A_{n}.

4. for any smooth X in G − Sch/k and Z closed in X everywhere of codimension ≥ n, X/(X − Z) is in A_{n}

We say that an object is thick if it is 1-thick.

Remark 3.2 Note that we constructed our definition in such a way that an object of Spc_{•}(G) is 0-thick if and only if it can be build out of smooth schemes by means of homotopy colimits. In particular, unless k has resolution of singularities, it is not clear that any object is 0-thick.

Lemma 3.3 The class A_{n} of n-thick objects has the following properties:

1. A_{n} is closed under coproducts

2. for a morphism f : X → Y where X, Y are in A_{n} one has cone(f) ∈ A_{n}

3. for X in A_{n} one has Σ_{T}^{1} X ∈ A_{n+1}

Proof: Since A_{n} is closed under filtering colimits and any coproduct is a filtering colimit of finite coproducts it is sufficient to show that if X and Y are in A_{n} then X ∨ Y is in A_{n}. Let us show first that for any space X in A_{n}, any smooth scheme U and any Z of codimension at least n + 1 in U, X ∨ U/(U − Z) is in A_{n}. Indeed the class of X for which this holds clearly satisfies the conditions of Definition 3.1 and therefore contains A_{n}. Similarly, the class of all Y such that X ∨ Y is in A_{n} clearly satisfies the first three conditions of Definition 3.1 and we have just shown that it satisfies the fourth. Hence it contains A_{n}.

The cone cone(f : X → Y) is the diagonal of a bisimplicial object with terms Y, Y ∨ X, Y ∨ X ∨ X etc. which one obtains if one writes the definition of the cone using second dimension for the simplicial interval Δ_{+}^{1}. Hence, any class closed under finite coproducts and diagonals is also closed under the cones.

To verify that last condition observe that the class of X such that Σ_{T}^{1} X is in A_{n+1} satisfies the first three conditions of Definition 3.1. It satisfies the fourth one since

Σ_{T}^{1} (X/(X − Z)) = X × A^{1}/(X × A^{1} − Z × {0})

and codim Z × {0} = codimZ + 1.

For a finite etale G-scheme W consider the functor F ↦ F^{∧W} introduced in [2].

Proposition 3.4 Let W be a finite etale G-scheme of degree d. Then the functor F ↦ F^{∧W} takes n-thick objects to nd-thick objects.

Proof: It is sufficient to show that the class of F such that F^{∧W} is nd-thick satisfies the conditions of Definition 3.1. The first condition follows from the fact that F ↦ F^{∧W} preserves A^{1}-equivalences (see [2, p.63, Prop. 5.2.11]). The second from the fact that F ↦ F^{∧W} commutes with filtering colimits. The third condition is obvious.

To see that it satisfies the fourth condition consider X in G − Sch/k and Z closed in X everywhere of codimension ≥ n. An easy generalization of [2, p.63, Rem. 5.2.8] shows that

(X/(X − Z))^{∧W} = X^{W} /(X^{W} − Z^{W}).

and it remains to note that if W is etale then codim(Z^{W}) = dcodim(Z).

Lemma 3.5 Let X be a scheme with a free action of G and F be a pointed solid G-sheaf which is n-thick. Then Quot_{G}(X_{+} ∧ F) is n-thick.

Proof: Let R : Spc_{•}(G) → Spc_{•}(G) be the resolution functor introduced in [2, Constr. 3.6.3, p.43] which takes a simplicial sheaf to a weakly equivalent one with terms being coproducts of representable sheaves. By [2, Prop. 5.1.4, p.57] functor Quot ∘ R respects A^{1}-equivalences and for a solid F the natural morphism Quot(R(F)) → Quot(F) is a weak equivalence. Hence it is sufficient to show that for F satisfying the conditions of the lemma, Quot(R(X_{+} ∧ F)) is n-thick.

Since F is assumed to be n-thick it is sufficient to check that the class of F such that Quot(R(X_{+} ∧F)) is n-thick satisfies the conditions of Definition 3.1. The first condition holds since the functor Quot(R(X_{+} ∧ −)) preserves A^{1}-equivalences. To see the second one recalls that R and Quot commute with filtering colimits. Similarly the third condition follows from the fact that both Quot and R commute with the diagonal functor.

To check the fourth condition we have to verify that Quot(R(X_{+}∧(Y/(Y − Z)))) is n-thick for any closed pair (Y, Z) where codim(Z) ≥ n. Using again [2, Prop. 5.1.4, p.57] we see that

Quot(R(X_{+} ∧ (Y/(Y − Z)))) → Quot(X_{+} ∧ (Y/(Y − Z)))

is an A^{1}-equivalence. We further have

Quot(X_{+} ∧ (Y/(Y − Z))) = Quot(X × Y)/(Quot(X × Y) − Quot(X × Z))

Since the action of G on X is free, the scheme Quot(X × Y) is smooth and since the projection X ×Y → Quot(X ×Y) is finite, codim(Quot_{G}(X ×Z)) = codim(X × Z) = codim(Z).

In the following two results G is the trivial group.

Proposition 3.6 Let X be an n-thick space over a perfect field k. Then the suspension spectrum Σ_{T}^{∞}(X) is in Σ_{T}^{n}SH^{eff}.

Lemma 3.7 Let U be a smooth rational scheme over k. Then Σ̃_{s}(U) is thick.

Proof: Since U is rational there is a dense open subscheme V in U such that V is an open subscheme in A^{n} for some n. Then Σ̃_{s}(V) ≅ A^{n}/V is thick and U/V is thick. We conclude by Lemma 2.6 and Lemma 3.3(2) that Σ̃_{s}U is thick.

4 Symmetric powers of T

Let S_{n} be the symmetric group. In this section we assume that char(k) = 0 or n < char(k). Consider the symmetric power

Symm^{n}(T^{m}) := Quot_{S_{n}}((T^{m})^{∧n})

Our goal is to prove the following result.

Theorem 4.1 For any n ≥ 2 and m ≥ 1 the space Symm^{n}(T^{m}) is (m + 1)- thick.

Consider the linear representations P (m, n) of S^{n} defined by the permutation action on (A^{m})^{n}. Then

Symm^{n}(T^{m}) = Quot_{S_{n}}(T h(P (m, n)))

Let V (m, n) be the reduced version of P (m, n)

V (m, n) = ker(p : (A^{m})^{n} → A^{m})

where p(x_{1}, …, x_{n}) = x_{1} +⋯+x_{m}. Under our assumptions on char(k) there is an isomorphism

P (m, n) ≅ V (m, n) × A^{m}

where the action on A^{m} is trivial. Therefore by Lemma 2.1

Symm^{n}(T^{m}) = Quot_{S_{n}}(T h(P (m, n))) = Σ_{T}^{m}Quot_{S_{n}}(T h(V (m, n)))

and Theorem 4.1 follows from Lemma 3.3(3) and Theorem 4.3 below.

Let (n) be the standard set of n-elements and j_{1}, …, j_{n} integers ≥ 0 such that

∑ ij_{i} = n.  (4.1)

Then there is a bijection

ϕ : (n) ≅ (1)^{j_{1}} ⨿ (2)^{j_{2}} ⨿ ⋯ ⨿ (n)^{j_{3}}  (4.2)

and any such bijection defines an embedding

e_{ϕ} : S_{1}^{j_{1}} × S_{2}^{j_{2}} × ⋯ × S_{n}^{j_{n}} → S_{n}

The stabilizer in S_{n} of a point x in P (m, n) is determined by the set of diagonals x_{i} = x_{j} which contain x and subgroups of the form Im(e_{ϕ}) are exactly the subgroups which occur as stabilizers of different points.

The ajunction class of a subgroup of the form Im(e_{ϕ}) is determined by the integers j_{1}, …, j_{n}. The case j_{1} = n and j_{i} = 0 for i ≠ 1 corresponds to a point with all components being different and the case j_{n} = 1 and j_{i} = 0 for i ≠ n to a point with all components being the same. We choose one isomorphism of the form (4.2) for each collection j_{1}, …, j_{n} satisfying (4.1) and let H_{j_{1},...,j_{n}} denote the corresponding subgroup.

The same classification of stabilizers applies to V (m, n) since V (m, n) is a subspace of P (m, n).

The normalizer of H = H_{j_{1},...,j_{n}} is of the form G_{j_{1},1} × ⋯ × G_{j_{n},n} where G_{j,i} is the semi-direct product of S_{j} and S_{i}^{j} with respect to the obvious permutational action. The quotient N(H)/H is of the form S_{j_{1}} × ⋯ × S_{j_{n}}.

Consider now the quotient V (m, n)/V (m, n)^{≥H}. One can easily see that it is isomorphic to the sum ⊕V (m, i)^{⊕j_{i}}, the action of N(H) on it is the direct product of actions of G_{j_{i},i} on V (m, i)^{⊕j_{i}} and the action of G_{j_{i},i} is given by the product of the standard actions of j_{i}-copies of S_{i} on V (m, i) and the permutation action of S_{j_{i}}. Summing this up we get the following result:

Lemma 4.2 One has an isomorphism of N(H)/H spaces of the form

Quot_{N(H),H}(T h(V (m, n)/V (m, n)^{≥H})) = ∧_{i=1}^{n}Quot_{S_{i}}(T h(V (m, i)))^{j_{i}}

Theorem 4.3 For n ≥ 2 and m > 0, the space Quot_{S_{n}}(T h(V (m, n))) is thick.

Example 4.4 For n = 2, V (m, n) is A^{m} with the sign action of S_{2} = Z/2. Over a field of odd characteristic one has

Quot_{S_{2}}(V (m, 2) − {0}) = O(−2)_{P^{m−1}} − z(P^{m−1})

where z is the sero section. In particular it is a rational variety and therefore

Quot_{S_{2}}(T h(V (m, 2))) = Σ̃_{s}(Quot_{S_{2}}(V (m, 2) − {0}))

is thick by Lemma 3.7.

Proof of Theorem 4.3: We proceed by induction on n ≥ 2. Let V_{0} be the open subscheme of V (m, n) where S_{n} acts freely. By Lemma 4.5 below Σ̃_{s}Quot_{S_{n}}(V_{0}) is thick. In view of Proposition 2.5 and Lemma 3.3(1,2) it remains to show that for H = H_{j_{1},...,j_{n}}, H ≠ e, S_{n} the space

Quot_{N(H)}((V^{≥H} − V^{>H})_{+} ∧ Quot_{N(H),H}(T h(V/V^{≥H})))

is thick. For n = 2 any subgroup of S_{n} is e or S_{n} and our statement is trivial. Hence we may assume inductively that n ≥ 3 and the theorem is proved for n − 1. Since the action of N(H) on V^{≥H} − V^{>H} is free and the space F = Quot_{N(H),H}(T h(V/V^{≥H})) is solid it is sufficient by Lemma 3.5 to see that F is thick as a N(H)/H-space. Since H ≠ e, S_{n} there exists n > i ≥ 2 such that j_{i} ≠ 0. By the inductive assumption Quot_{S_{i}}(T h(V (m, i))) is thick and therefore Quot_{S_{i}}(T h(V (m, i)))^{j_{i}} is thick as a S_{j_{i}}-space by Lemma 3.4. By Lemma 4.2, F is the smash product of the form F′ ∧ Quot_{S_{i}}(T h(V (m, i)))^{j_{i}} and N(H)/H = G×S_{j_{i}} where G acts on the first factor and S_{j_{i}} on the second and we conclude that F is thick.

Lemma 4.5 The object Quot_{S_{n}}(Σ̃_{s}V_{0}) is thick.

Proof: By Lemma 3.7 it is sufficient to show that Quot_{S_{n}}(V_{0}) is rational. Let V_{0,0} be the open subset in V_{0} which consists of x_{1}, …, x_{n} such that the first components of all x_{i} ∈ A^{m} are different. The projection to the first component

V_{0,0}(m, n) → V_{0}(1, n)  (4.3)

makes V_{0,0}(m, n) into an equivariant vector bundle over V_{0}(1, n). Since the action of S_{n} on V_{0}(1, n) is free, the map of quotient schemes defined by (4.3) is a vector bundle as well. Hence it is sufficient to show that Quot(V_{0}(1, n)) or, equivalently, Quot(V (1, n)) is rational.

The quotient Quot(P (1, n)) = A^{n}/S_{n} can be identified in the standard way with A^{n} where the first coordinate is given on P (1, n) by the sum of components. It follows that Quot(V (1, n)) = A^{n−1} and in particular that it is rational.

5 Reformulation for smooth schemes

Denote for a moment the (pointed, simplicial) sheaves on smooth schemes by Spc_{•}(Sm) and sheaves on all quasi-projective schemes by Spc_{•}(Sch). Let A_{n} be the class of n-thick spaces in Spc_{•}(Sch) and A_{n}^{Sm} the similarly defined class in Spc_{•}(Sm). The functor

π_{∗} : Spc_{•}(Sch) → Spc_{•}(Sm)

which takes a sheaf on Sch/k to its restriction on Sm/k respects limits, colimits and A^{1}-equivalences. This implies immediately that π_{∗} takes A_{n} to A_{n}^{Sm}. Therefore, all the results about thickness proved above remain valid if we work in the context of sheaves on smooth schemes.

6 Motivic Eilenberg-MacLane spaces

In this section k is a field of characteristic zero and we work in the context of sheaves on smooth schemes. Let K_{n} = Z_{tr}(A^{n})/Z_{tr}(A^{n} − {0}) be the nth motivic Eilenberg-Maclane space over k. Consider the obvious morphism T^{n} → K_{n} where T^{n} = T^{∧n} = A^{n}/(A^{n} − {0}). The goal of this section is to prove the following theorem.

Theorem 6.1 For n > 0, the space Σ_{s}cone(T^{n} → K_{n}) is (n + 1)-thick.

Example 6.2 Consider the case n = 1. It is easy to see using Lemma 6.3 below that K_{1} is A^{1}-equivalent to (P^{∞}, ∗) where ∗ is a rational point and the morphism T → K_{1} corresponds to the standard embedding (P^{1}, ∗) → (P^{∞}, ∗). The cone is given by P^{∞}/P^{1} and its s-suspension is 2-thick by the reduced analog of Lemma 2.6 and Lemma 3.3(2).

Consider the sheaf K_{n}^{eff} associated to the presheaf of the form

K_{n}^{eff} : U ↦ c^{eff}(U × A^{n}/U)/c^{eff}(U × (A^{n} − {0})/U)  (6.1)

where c^{eff}(X/U) is the monoid of effective finite cycles on X over U and the quotient on the right hand side means that two cycles Z_{1}, Z_{2} on U × A^{n} are identified if Z_{1} − Z_{2} is in U × (A^{n} − {0}).

Lemma 6.3 The obvious map K_{n}^{eff} → K_{n} is an A^{1}-equivalence for n ≥ 1.

Proof: Let F^{eff} be the presheaf (6.1) and let F be the presheaf given by

U ↦ Z_{tr}(A^{n})(U)/Z_{tr}(A^{n} − {0})(U)

such that the sheaf associated with F is K_{n}. Consider the singular simplicial presheaves C_{∗}(F^{eff}) and C_{∗}(F). The standard argument shows that the morphisms

K_{n}^{eff} = a_{Nis}F^{eff} → a_{Nis}C_{∗}(F^{eff})
K_{n} = a_{Nis}F → a_{Nis}C_{∗}(F)

are A^{1}-equivalences. For any smooth U the abelian group associated with the monoid F^{eff}(U) coincides with F (U). Therefore, a_{Nis}C_{∗}(F) is the sheaf of simplicial abelian groups associated with the sheaf of simplicial monoids a_{Nis}C_{∗}(F^{eff}). Let us show that the natural map

a_{Nis}C_{∗}(F^{eff}) → a_{Nis}C_{∗}(F)

is a local equivalence in the Nisnevich topology i.e. that for any henselian local S the map of simplicial sets

C_{∗}(F^{eff})(S) → C_{∗}(F)(S)

is a weak equivalence.

For any i, C_{i}(F^{eff})(S) = F^{eff}(A_{S}^{i}) is the free commutative monoid generated by irreducible closed subsets in Δ_{S}^{i} × A^{n} which are finite and equidimensional over Δ_{S}^{i} and have nontrivial intersection with Δ_{S}^{i} × {0}.

Therefore, by Lemma 6.4 below it is sufficient to show that

π_{0}(C_{∗}(F^{eff})(S)) = 0.

An element in C_{0}(F^{eff})(S) may be represented by a cycle on A_{S}^{n} of the form Z = ∑ n_{i}Z_{i} where Z_{i} are closed irreducible subsets of A_{S}^{n} and n_{i} > 0. Since Z_{i} are finite over S they are local (since S is henselian) and therefore we may assumed that the closed points of all Z_{i} lie in {0}_{S}. Consider now the cycle H on A_{S}^{1} × A^{n} obtained from Z by the pull-back with respect to the map (t, x) ↦ x − t. This cycle is finite over A_{S}^{1}. Its restriction to t = 0 is Z and the restriction to t = 1 has all the closed points in {1}_{S} and therefore lies in (A^{n} − {0})_{S}. Hence the image h of H in C_{1}(F^{eff})(S) has the property ∂_{0}h = Z and ∂_{1}h = 0 and we conclude that π_{0} = 0.

Lemma 6.4 Let M be a commutative simplicial monoid and f : M → M^{+} the canonical map from M to the associated simplicial group. If for any n the monoid M_{n} is a free commutative monoid and π_{0}(M) is a group then f is a weak equivalence.

Proof: If π_{0}(K) is a group the topological monoid |M| is a group-like associative H-space. For such H-spaces the canonical map |M| → ΩB(|M|) is a weak equivalence (follows from [1]). The same argument shows that |M^{+}| → ΩB(|M^{+}|) is a weak equivalence. It remains to show that B(|M|) → B(|M^{+}|) is a weak equivalence. By construction B(|M|) is the same as |B(M)| where B(M) is the diagonal of the bisimplicial object obtained by applying the classifying space construction to each term of M. Since the class of weak equivalences of simplicial sets is closed under taking diagonals it remains to show that for a free commutative monoid M the map B(M) → B(M^{+}) is a weak equivalence. For this case see [, ].

We are going now to analyze the structure of K_{n}^{eff}. Let K_{n}^{≤d} be the subsheaf in K_{n}^{eff} associated with the subpresheaf which takes U to the image in K_{n}^{eff}(U) of the set of finite cycles on A_{U}^{n}/U of degree everywhere ≤ d over U. Clearly, K_{n}^{eff} = colim_{d}K_{n}^{≤d}. The sheaf K_{n}^{≤1} coincides with T^{n} = A^{n}/(A^{n} − {0}) naturally embedded into K_{n}^{eff}. Consider the quotients K_{n}^{≤d}/K_{n}^{≤d−1}.

Lemma 6.5 Let k be a field of characteristic zero. Then there is an isomorphism

K_{n}^{≤d}/K_{n}^{≤d−1} ≅ Symm^{d}(T^{n})  (6.2)

where Symm^{d}(T^{n}) := Quot_{S_{d}}((T^{n})^{∧d}).

Proof: Let c^{≤d}(A^{n}) be the sheaf of finite cycles of degree ≤ d on A^{n}. By [, ], for k of characteristic zero this sheaf is represented by

∐_{i≤d} Symm^{i}(A^{n}) = ∐_{i≤d} Quot_{S_{i}}(A^{ni})

Since the right hand side of (6.2) is given by Quot_{S_{d}}(A^{nd}/(A^{nd} − {0})) it remains to show that the restriction of the map

c^{≤d}(A^{n}) → K_{n}^{≤d}/K_{n}^{≤d−1}

to Quot_{S_{d}}(A^{nd}) is surjective and that two sections of Quot_{S_{d}}(A^{nd}) over a henselian local scheme S project to the same section of K_{n}^{≤d}/K_{n}^{≤d−1} if and only if they project to the same section of Quot_{S_{d}}(A^{nd}/(A^{nd} − {0})).

The surjectivity is clear. Let Z = ∑ n_{i}Z_{i}, W = ∑ m_{j}W_{j} be two sections of c^{≤d}(A^{n}) over S of degree strictly d. By definition they project to the same section of K_{n}^{≤d}/K_{n}^{≤d−1} if both Z and W are equivalent modulo the components lying in A^{n} − {0} to cycle of degree ≤ d − 1. Since Z_{i} and W_{j} are local this means that not all of the closed points of the Z_{i}’s (resp. W_{j}’s) lie in {0}. This is equivalent to the condition that Z and W lie in Quot_{S_{d}}(A^{nd} − {0}).

Proof of Theorem 6.1: By Lemma 6.3, it is sufficient to show that the space Σ_{s}(K_{n}^{eff}/T^{n}) is (n + 1)-thick. Since T^{n} = K_{n}^{≤1} the space K_{n}^{eff}/T^{n} has a filtration which starts with K_{n}^{≤2}/K_{n}^{≤1} and has quotients of the form K_{n}^{≤d}/K_{n}^{≤d−1} for d ≥ 3. Since the class of thick objects is closed under filtering colimits it is sufficient by the obvious reduced analog of Lemma 2.6 and Lemma 3.3(2) to show that K_{n}^{≤d}/K_{n}^{≤d−1} are (n + 1)-thick for d ≥ 2. This follows immediately from Lemma 6.5 and Theorem 4.1.

As a corollary of Theorem 6.1 we can prove the main conjecture of [3] over fildes of characteristic zero.

Theorem 6.6 Let k be a field of characteristic zero. Then s_{0}(1) = H_{Z}.

Proof: Consider the unit map e : 1 → H_{Z}. Since s_{0}(H_{Z}) = H_{Z} it is sufficient to show that s_{0}(e) is an isomorphism i.e. that s_{0}(cone(e)) = 0. By definition of s_{0} this would follow if we can show that cone(e) belongs to Σ_{T}^{1} SH^{eff}. The map e is given on the level of spectra by the maps T^{n} → K_{n} considered above. In particular we have

cone(e) = hocolim_{n}Σ_{T}^{−n}Σ_{T}^{∞}(cone(T^{n} → K_{n}))

Since Σ_{T}^{1} SH^{eff} is closed under homotopy colimits it remains to check that

Σ_{T}^{−n}Σ_{T}^{∞}(cone(T^{n} → K_{n})) ∈ Σ_{T}^{1} SH^{eff}

i.e. that

Σ_{T}^{∞}(cone(T^{n} → K_{n})) ∈ Σ_{T}^{n+1}SH^{eff}

Since Σ_{T}^{n+1}SH^{eff} is stable under Σ_{s} it follows from Theorem 6.1 and Proposition 3.6.

References

[1] A. Dold and R. Lashof. Principal quasi-fibrations and fiber homotopy equivalence of bundles. Illinois J. Math., 3:285–305, 1959.

[2] Vladimir Voevodsky. Lectures on motivic cohomology 2000/2001 (written by Pierre Deligne). www.math.ias.edu/˜ vladimir/rear.html, 2000/2001.

[3] Vladimir Voevodsky. Open problems in the motivic stable homotopy theory, I. In Motives, Polylogarithms and Hodge Theory, Part I, pages 3–35. International Press, 2002.