Cancellation theorem

Vladimir Voevodsky

January 2002

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Finite correspondences . . . . . . . . . . . . . . . . . . . . . . 2 3 Intersecting relative cycles with divisors . . . . . . . . . . . . . 5 4 Cancellation theorem . . . . . . . . . . . . . . . . . . . . . . . 9

1 Introduction

Let SmCor(k) be the category of smooth finite correspondences over a field k. Denote by G_{m} the scheme A^{1} − {0}. One defines the sheaf with transfers S_{t}^{1} by the condition that Z_{tr}(G_{m}) = S_{t}^{1} ⊕ Z where Z is split off by the projection to the point and the point 1. For any scheme X consider the sheaf with transfers

F_{X} : Y ↦ Hom(S_{t}^{1} ⊗ Z_{tr}(Y), S_{t}^{1} ⊗ Z_{tr}(X))

We prove that the obvious map Z_{tr}(X) → F_{X} defines a quasi-isomorphism of singular simplicial complexes

C_{∗}(Z_{tr}(X)) → C_{∗}(F_{X})

as complexes of presheaves i.e. for any Y the map of complexes of abelain groups

C_{∗}(Z_{tr}(X))(Y) → C_{∗}(F_{X})(Y)

is a quasi-isomorphism. We then deduce from this result the cancellation theorem which asserts that if k is a perfect field then for any K, L in DM_{−}^{eff}(k) the map

Hom(K, K′) → Hom(K(1), K′(1))

is bijective.

This result was previously known in two particular situations. For varieties over a field k with resolution of singularities it was proved in [6]. For K′ being the motivic complex Z(n)[m] and any field k it was proved in [4]. Both proofs are very long.

The main part of our argument does not use the assumption that we work with smooth schemes over a field and we give it for separated schemes of finite type over a noetherian base. To be able to do it we define in the first section the category of finite correspondences for separated schemes of finite type over a base. The definition is a straightforward generalization of the definition for schemes over a field based on the constructions of [3] and can be skipped. In the second section we define intersection of relative cycles with Cartier divisors and prove the properties of this construction which we need. In the third we prove our main theorem 4.6 and deduce from it the cancellation theorem over perfect fileds 4.10.

In this paper we say “a relative cycle” instead of “an equidimensional relative cycle”. All schemes are separated. The letter S is typically reserved for the base scheme which is assumed to be noetherian. All the standard schemes P^{1}, A^{1} etc. are over S. When no confusion is possible we write XY instead of X ×_{S} Y.

2 Finite correspondences

For a scheme X of finite type over a noetherian scheme S we denote by c(X/S) the group of finite relative cycles on X over S. In [3] this group was denoted by c_{equi}(X/S, 0). If S is regular or if S is normal and the characteristic of X is zero, c(X/S) is the free abelian group generated by closed irreducible subsets of X which are finite over S and surjective over a connected component of S. For the general definition see [3, after Lemma 3.3.9]. A morphism f : S′ → S defines the pull-back homomorphism c(X/S) → c(XS′/S′) which we denote by cycl(f).

For two schemes X, Y of finite type over S we define the group c(X, Y) of finite correspondences from X to Y as c(XY/X).

Let us recall the following construction from [3, §3.7]. Let X′ → X → S be morphisms of finite type, W a relative cycle on X′ over X and Z a relative cycle on X over S. Then one defines a cycle Cor(W, Z) on X′ as follows. Let Z_{i} be the components of the support of Z present with multiplicites n_{i} and e_{i} : Z_{i} → X the corresponding closed embeddings. Let e′_{i} : Z_{i} ×_{X} X′ → X′ denote the projections. We set

Cor(W, Z) = ∑_{i} n_{i}(e′_{i})_{∗}cycl(e_{i})(W)

where (e′_{i})_{∗} is the (proper) push-forward on cycles.

Let X, Y be schemes of finite type over S and

f ∈ c(X, Y) = c(XY/X)
g ∈ c(Y, Z) = c(Y Z/Y)

finite correspondences. Let

p_{X} : XY → Y
p_{Y} : XY Z → XZ

be the projections. We define the composition g ∘ f by the formula:

g ∘ f = (p_{Y})_{∗}Cor(cycl(p_{X})(g), f)  (2.1)

This operation is linear in both arguments and thus defines a homomorphism of abelian groups

c(X, Y) ⊗ c(Y, Z) → c(X, Z)

The lemma below follows immediately from the definition of Cor(−, −) and the fact that the (proper) push-forward commutes with the cycl(−) homomorphisms ([3, Prop. 3.6.2]).

Lemma 2.1 Let Y → X → S be a sequence of morphisms of finite type, p : Y → Y′ a morphism over X, Y ∈ Cycl(Y/X, r)⊗Q and X ∈ Cycl(X/S, s)⊗ Q. Assume that p is proper on the support of Y. Then

p_{∗}Cor(Y, X) = Cor(p_{∗}(Y, X)).

Lemma 2.2 For any f ∈ c(X, Y), g ∈ c(Y, Z), h ∈ c(Z, T) one has

(h ∘ g) ∘ f = h ∘ (g ∘ f).

Proof: Consider the following diagram

XT ⟵^{4} XY T ⟶ Y T
7↑  ^{8}↑  ^{2}↑
XZT ⟵^{9} XY ZT ⟶ Y ZT ⟶ ZT ⟶ T
↓  ↓  ↓  ↓
XZ ⟵^{5} XY Z ⟶^{9} Y Z ⟶^{1} Z
↓  ↓
XY ⟶^{3} Y
↓
X

where the morphisms are the obvious projections. Note that all the squares are cartesian. We will also use the projection 6 : XZ → Z.

We have f ∈ c(XY/X), g ∈ c(Y Z/Y) and h ∈ c(ZT/Z). The compositions are given by:

g ∘ f = 5_{∗}Cor(cycl(3)(g), f)
h ∘ g = 2_{∗}Cor(cycl(1)(h), g)

(h∘g)∘f = 4_{∗}Cor(cycl(3)(h∘g), f) = 4_{∗}Cor(cycl(3)(2_{∗}Cor(cycl(1)(h), g)), f)

h∘(g∘f) = 7_{∗}Cor(cycl(6)(h), g∘f) = 7_{∗}Cor(cycl(6)(h), 5_{∗}Cor(cycl(3)(g), f))

We have:

4_{∗}Cor(cycl(3)(2_{∗}Cor(cycl(1)(h), g)), f) =
= 4_{∗}Cor(8_{∗}cycl(3)Cor(cycl(1)(h), g), f) =
= 4_{∗}8_{∗}Cor(cycl(3)Cor(cycl(1)(h), g), f) =
= 4_{∗}8_{∗}Cor(Cor(cycl(1 ∘ 9)(h), cycl(3)(g)), f)

where the first equality holds by [3, Prop. 3.6.2], the second by Lemma 2.1 and the third by [3, Th. 3.7.3]. We also have:

7_{∗}Cor(cycl(6)(h), 5_{∗}Cor(cycl(3)(g), f)) =
= 7_{∗}9_{∗}Cor(cycl(6 ∘ 5)(h), Cor(cycl(3)(g), f))

by [3, Lemma 3.7.1]. We conclude that (h ∘ g) ∘ f = h ∘ (g ∘ f) by [3, Prop. 3.7.7].

We denote by Cor(S) the category of finite correspondences whose objects are schemes of finite type over S, morphisms are finite correspondences and the composition of morphisms is defined by (2.1).

For a morphism of schemes f : X → Y let Γ_{f} be its graph considered as an element of c(XY/X). One verifies easily that Γ_{gf} = Γ_{g} ∘ Γ_{f} and we get a functor Sch/S → Cor(S). Below we use the same symbol for a morphism of schemes and its graph considered as a finite correspondence.

The external product of cycles defines pairings

c(X, Y) ⊗ c(X′, Y′) → c(XX′, Y Y′)

and one verifies easily using the results of [3] that this pairing extends to a tensor structure on Cor(S) with X ⊗ Y := XY.

3 Intersecting relative cycles with divisors

Let X be a noetherian scheme and D a Cartier divisor on X i.e. a global section of the sheaf M^{∗}/O^{∗}. One defines the cycle cycl(D) associated with D as follows. Let U_{i} be an open covering of X such that D_{U_{i}} is of the form f_{i,+}/f_{i,−} ∈ M^{∗}(U_{i}). Then cycl(D) is determined by the property that

cycl(D)_{|U_{i}} = cycl(f_{i,+}^{−1}(0)) − cycl(f_{i,−}^{−1}(0))

where on the right hand side one considers the cycles associated with closed subschemes ([3, ]). One defines the support of D as the closed subset supp(D) := supp(cycl(D)).

We say that a cycle Z = ∑ n_{i}z_{i} on X intersects D properly if the points z_{i} do not belong to supp(D). Let Z_{i} be the closure of z_{i} considered as a reduced closed subscheme and e_{i} : Z_{i} → X the closed embedding. If Z and D intersect properly we define their intersection (Z, D) as the cycle

(Z, D) := ∑ n_{i}(e_{i})_{∗}(cycl(e_{i}^{∗}(D)))

If p : X → S is a morphism of finite type and Z is a relative cycle of relative dimension d over S, we say that D intersects Z properly relative to p (or properly over S) if the dimension of fibers of supp(D) ∩ supp(Z) over S is ≤ d − 1. This clearly implies that Z intersects D properly and (Z, D) is defined.

Proposition 3.1 Assume that S is reduced and D and Z intersect properly over S. Then:

1. (Z, D) is a relative cycle of relative dimension d − 1 over S

2. let p : S′ → S be a morphism and p′ : XS′ → X the projection, assume that (p′)^{∗}(D) is defined, then one has

p^{∗}(cycl(Z), D) = (p^{∗}(cycl(Z)), (p′)^{∗}(D)).  (3.1)

Proof: One can easily see that D intersects Z properly relative to p if and only if for any i and any point x of X there exists a neighborhood U of x such that D_{|U} = f_{+}/f_{−} where f_{+} and f_{−} are not zero divisors on the fibers of the maps Z_{i} → S. Since the problem is local in X we may assume that D = D(f) where f is a regular function on X which is not a zero divisor on fibers of the maps Z_{i} → S. We write (Z, f) instead of (Z, D(f)).

To prove both statements of the proposition it is clearly sufficient to show that for any fat point Spec(k) →^{x_{0}} Spec(R) →^{x_{1}} S one has

(x_{0}, x_{1})^{∗}((Z, f)) = ((x_{0}, x_{1})^{∗}(Z), fx′_{1}x′_{0})

where x′_{0}, x′_{1} are the projections

XSpec(k) →^{x′_{0}} XSpec(R) →^{x′_{1}} X

Let j : Spec(K(R)) → Spec(R) be the generic point of R and

pr : Spec(K(R))X → X

the projection. Since S is reduced, Z_{i} are flat over a dense open subset of S and (x_{1}j)^{∗}((Z, f)) = ∑ m_{j}w_{j} is defined and by Lemma 3.3 we have

(x_{1}j)^{∗}((Z, f)) = ((x_{1}j)^{∗}(Z), f ∘ pr)

For a cycle W = ∑ m_{j}w_{j} over the generic point of Spec(R) denote by cl W the same cycle considered over Spec(R). Since cl(x_{1}j)^{∗}(Z) is a formal linear combination of cycles of flat subschemes, Lemma 3.3 applies and we conclude that:

(x_{0}, x_{1})^{∗}((Z, f)) = x_{0}^{∗}(cl(x_{1}j)^{∗}((Z, f))) = x_{0}^{∗}(cl(x_{1}j)^{∗}(Z), fx′_{1}) =
= (x_{0}^{∗}cl(x_{1}j)^{∗}(Z), fx′_{1}x′_{0}) = ((x_{0}, x_{1})^{∗}(Z), fx′_{1}x′_{0})

Lemma 3.2 Let Z be a noetherian scheme and f a regular function on Z which is not a zero divisor. Let further e_{i} : Z_{i} → Z be the connected components of Z and n_{i} their multiplicities. Then one has:

cycl(f^{−1}(0)) = ∑ n_{i}(e_{i})_{∗}(cycl(f_{|Z_{i}}^{−1}(0)))  (3.2)

Proof: (P.Deligne) Replacing Z by the local scheme of a generic point of f^{−1}(0) we may assume that Z = Spec(A) is local of dimension 1. By definition of the cycle associated to a closed subscheme (3.2) is equivalent to

l(A/(f)) = ∑ l(A/p_{i})l(A/(p_{i} + (f)))

where p_{i} are the minimal ideals of A and l(M) is the length of an A-module. The function M → l(M) extends to an additive function on the K_{0} of the abelain category of A-modules of finite length. Consider the function l_{f} : M ↦ l(cone(M →^{f} M)). Equivalently we have l_{f}(M) = l(M ⊗^{L} A/f) which shows that it is an additive function on K_{0}(A − mod). In K_{0}(A − mod) one has

A = (∑ l(A/p_{i})A/p_{i}) + M

where M is a module of finite length. This implies:

l(A/f) = l_{f}(A) = ∑ l(A/p_{i})l_{f}(A/p_{i}) = ∑ l(A/p_{i})l(A/(p_{i} + (f))).

Lemma 3.3 Let Z be a closed subscheme of X which is flat and equidimensional over S and f be a regular function on Z such that f^{−1}(0) ∩ Z is equidimensional over S. Then (cycl(Z), f) is a relative cycle and for any morphism p : S′ → S one has:

p^{∗}(cycl(Z), f) = (p^{∗}(cycl(Z)), fp′)

where p′ : XS′ → X.

Proof: Lemma 3.2 implies that

(cycl(Z), f) = cycl(f_{|Z}^{−1}(0)).

Our condition on f implies that, for any s ∈ S, f restricted to the fiber Z_{s} is not a zero divisor. Since Z is flat over S this implies that f_{|Z}^{−1}(0) is flat over S (see e.g. [1, ]). Since Z is equidimensional of relative dimension d, f_{|Z}^{−1}(0) is equidimensional of relative dimension d − 1. We conclude by [3, ] that cycl(f_{|Z}^{−1}(0)) is a relative cycle.

For p; S′ → S we have:

p^{∗}((cycl(Z), f)) = p^{∗}(cycl(f_{|Z}^{−1}(0))) = cycl((fp′)_{|ZS′}^{−1}(0)) =
= (cycl(ZS′), fp′) = (p^{∗}(cycl(Z)), fp′).

Corollary 3.4 Let X′ →^{f} X → S be morphisms of finite type, Z a relative cycle on X over S and W a relative cycle on X′ over X of dimension 0. Let further D be a Cartier divisor on X′ which intersects W properly over X. Then D intersects Cor(W, Z) properly over S and one has:

(Cor(W, Z), D) = Cor((W, D), Z)  (3.3)

Proof: It is a straightforward corollary of the definition of Cor(−, −) and (3.1).

Lemma 3.5 Let f : X′ → X be a morphism of schemes of finite type over S, Z a relative cycle on X′ such that f is proper on supp(Z) and D a Cartier divisor on X. Assume that f^{∗}(D) is defined and Z intersects f^{∗}(D) properly over S. Then f_{∗}(Z) intersects D properly over S and one has:

f_{∗}(Z, f^{∗}(D)) = (f_{∗}(Z), D)  (3.4)

Proof: Let d be the relative dimension of Z over S. To see that f_{∗}(Z) intersects D properly over S we need to check that the dimension of the fibers of supp(D) ∩ supp(f_{∗}(Z)) over S is ≤ d − 1. This follows from our assumption and the inclusion

supp(D) ∩ supp(f_{∗}(Z)) ⊂ supp(D) ∩ f(supp(Z)) =
= f(f^{−1}(supp(D)) ∩ supp(Z)) = f(supp(f^{∗}(D)) ∩ supp(Z))

To verify (3.4) it is sufficient to consider the situation locally around the generic points of f(supp(f^{∗}(D)) ∩ supp(Z)). Therefore we may assume that D = D(g) is the divisor of a regular function g and Z = z is just one point with the closure Z. Replacing X′ by Z and X by f(Z) we may assume that X, X′ are integral, f is surjective and X is local of dimension 1. Let A = O(X), B = O(X′). As in the proof of 3.2, consider the function l_{g} : M ↦ l_{A}(M ⊗^{L} A/g) on K_{0}(A−mod). This function vanishes on modules with the support in the closed point which implies that

l_{g}(B) = deg(f)l_{g}(A) = deg(f)l_{A}(A/g)

On the other hand l_{g}(A) = l_{A}(B/(f^{∗}(g))). Let x′_{i} be the closed points of X′, k′_{i} their residue fields and k the residue field of the closed point of X. Let further M_{i} be the part of B/(f^{∗}(g)) supported in x′_{i}. One can easily see that l_{A}(B/(f^{∗}(g))) = ∑_{i}[k′_{i} : k]l_{B}(M_{i}). Combining our equalities we get:

deg(f)l_{A}(A/g) = ∑_{i}[k′_{i} : k]l_{B}(M_{i})  (3.5)

which is equivalent to (3.4).

4 Cancellation theorem

Consider a finite correspondence

Z ∈ c(G_{m}X, G_{m}Y) = c(G_{m}XG_{m}Y/G_{m}X).

Let f_{1}, f_{2} be the projections to the first and the second copy of G_{m} respectively and let g_{n} denote the rational function (f_{1}^{n+1} − 1)/(f_{1}^{n+1} − f_{2}) on G_{m}XG_{m}Y.

Lemma 4.1 For any Z there exists N such that for all n ≥ N the divisor of g_{n} intersects Z properly over X and the cycle (Z, D(g_{n})) is finite over X.

Proof: Let f̄_{1}×q̄: C̄ → P^{1}X be a finite morphism which extends the projection supp(Z) → G_{m}X. Let N be an integer such that the rational function f̄_{1}^{N}/f_{2} is regular in a neighborhood of f̄_{1}^{−1}(0) and the rational function f_{2}/f̄_{1}^{N} is regular in a neighborhood of f̄_{1}^{−1}(∞). Then for any n ≥ N one has:

1. the restriction of g_{n}f_{2} to supp(Z) is regular on a neighborhood of f̄_{1}^{−1}(0) and equals 1 on f̄_{1}^{−1}(0)

2. the restriction of g_{n} to supp(Z) is regular a neighborhood of f̄_{1}^{−1}(∞) and equals 1 on f̄_{1}^{−1}(∞)

Conditions (1),(2) imply that the divisor of g_{n} intersects Z properly over X and that the relative cycle (Z, D(g_{n})) is finite over X.

If (Z, D(g_{n})) is defined as a finite relative cycle we let ρ_{n}(Z) ∈ c(X, Y) denote the projection of (Z, D(g_{n})) to XY.

Remark 4.2 Note that we can define a finite correspondence ρ_{g}(Z) : X → Y for any function g satisfying the conditions (1),(2) in the same way as we defined ρ_{n} = ρ_{g_{n}}. In particular, if n and m are large enough then the function tg_{n} + (1 − t)g_{m} defines a finite correspondence h = h_{n,m} : XA^{1} → Y such that h_{|X×{0}} = ρ_{m}(Z) and h_{|X×{1}} = ρ_{n}(Z), i.e. we get a canonical A^{1}-homotopy from ρ_{m}(Z) to ρ_{n}(Z).

Lemma 4.3 (i) For a finite correspondence W : X → Y and any n ≥ 1 one has ρ_{n}(Id_{G_{m}} ⊗ W) = W

(ii) Let e_{X} be the composition G_{m}X ⟶^{pr} X ⟶^{{1}×Id} G_{m}X. Then ρ_{n}(e_{X}) = 0 for any n ≥ 0.

Proof: The cycle on G_{m}XG_{m}Y over G_{m}X which represents Id_{G_{m}} ⊗ W is Δ_{∗}(G_{m} × W) where Δ is the diagonal embedding G_{m}XY → G_{m}XG_{m}Y. The cycle (Δ_{∗}(G_{m} × W), g_{n}) is Δ_{∗}(D ⊗ W) where D is the divisor of the function (t^{n+1} − 1)/(t^{n+1} − t) on G_{m}. The push-forward of Δ_{∗}(D ⊗ W) to XY is the cycle deg(D)W. Since deg(D) = 1 we get the first statement of the lemma.

The cycle Z on G_{m}XG_{m}X representing e_{X} is the image of the embedding G_{m}X → G_{m}XG_{m}X which is diagonal on X and of the form t ↦ (t, 1) on G_{m}. This shows that the restriction of g_{n} to supp(Z) equals 1 and (Z, D(g_{n})) = 0.

Lemma 4.4 Let Z : G_{m}X → G_{m}Y be a finite correspondence such that ρ_{n}(Z) is defined. Then for any finite correspondence W : X′ → X, ρ_{n}(Z ∘ (Id_{G_{m}} ⊗ W)) is defined and one has

ρ_{n}(Z ∘ (Id_{G_{m}} ⊗ W)) = ρ_{n}(Z) ∘ W  (4.1)

Proof: Let us show that (4.1) holds. In the process it will become clear that the left hand side is defined. We can write ρ_{n}(Z) ∘ W as the composition

X′ ⟶^{W} X ⟶^{(Z,D(g_{n}))} G_{m}G_{m}Y ⟶^{pr} Y

and ρ_{n}(Z ∘ (Id_{G_{m}} ⊗ W)) as the composition

X′ ⟶^{Y} G_{m}G_{m}Y ⟶^{pr} Y

where Y = (Z ∘ (Id_{G_{m}} ⊗ W), D(g_{n})). Consider the diagram

G_{m}X′G_{m}Y ⟵^{p_{1}} G_{m}X′XG_{m}Y ⟶ G_{m}XG_{m}Y
↓  ↓
X′X  ⟶^{p_{2}}  X
↓
X′

where the arrows are the obvious projections. If we consider Z as a cycle of dimension 1 over X then the cycle Z ∘ (Id_{G_{m}} ⊗ W), considered as a cycle over X′, is (p_{1})_{∗}Cor(cycl(p_{2})(Z), W) and we have

((p_{1})_{∗}Cor(cycl(p_{2})(Z), W), D(g_{n})) =

= (p_{1})_{∗}(Cor(cycl(p_{2})(Z), W), D(g_{n})) = (p_{1})_{∗}Cor((cycl(p_{2})(Z), D(g_{n})), W) =

= (p_{1})_{∗}Cor(cycl(p_{2})(Z, D(g_{n})), W)

where the first equality holds by (3.4), the second by (3.3) and the third by (3.1).

The last expression represents the composition W ∘ (Z, D(g_{n})) and we conclude that

ρ_{n}(Z) ∘ W = ρ_{n}(Z ∘ (Id_{G_{m}} ⊗ W))

Lemma 4.5 Let Z : G_{m}X → G_{m}Y be a finite correspondence such that ρ_{n}(Z) is defined. Then for any morphism of schemes f : X′ → Y′, ρ_{n}(Z ⊗f) is defined and one has

ρ_{n}(Z ⊗ f) = ρ_{n}(Z) ⊗ f  (4.2)

Proof: Consider the diagram

G_{m}XX′G_{m}Y Y′ ⟵^{p_{1}} G_{m}XX′G_{m}Y ⟶ G_{m}XG_{m}Y
↓  ↓
XX′  ⟶^{p_{2}}  X

where p_{1} is defined by the embedding X′ ⟶^{f×Id} X′Y′ and the rest of the morphisms are the obvious projections. Consider Z as a cycle over X. Then ρ_{n}(Z ⊗ f) is given by the composition

G_{m}XX′ ⟶^{Y_{1}} G_{m}G_{m}Y ⟶^{pr} Y Y′

where Y_{1} = ((p_{1})_{∗}cycl(p_{2})(Z), g_{n}) and ρ_{n}(Z) ⊗ f by the composition

G_{m}XX′ ⟶^{Y_{2}} G_{m}G_{m}Y ⟶^{pr} Y Y′

where Y_{2} = (p_{1})_{∗}(cycl(p_{2})((Z, g_{n}))). The equality Y_{1} = Y_{2} follows from (3.4) and (3.1).

For our next result we need to use presheaves with transfers. A presheaf with transfers on Sch/S is an additive contravariant functor from Cor(S) to the category of abelian groups. For X in Sch/S we let Z_{tr}(X) denote the functor represented by X on Cor(S). One defines tensor product of presheaves with transfers in the usual way such that Z_{tr}(X) ⊗ Z_{tr}(Y) = Z_{tr}(X × Y). To simplify notations we will write X instead of Z_{tr}(X) and identify morphisms Z_{tr}(X) → Z_{tr}(Y) with finite correspondences X → Y. Note in particular that G_{m} denotes the presheaf with transfers Z_{tr}(G_{m}) not the presheaf with transfers represented by G_{m} as a scheme. To preserve compatibility with the notation XY for the product of X and Y we write F G for the tensor product of presheaves with transfers F and G.

Let S_{t}^{1} denote the presheaf with transfers ker(G_{m} → S). We consider it as a direct summand of G_{m} with respect to the projection Id − e where e is defined by the composition G_{m} → S ⟶^{1} G_{m}. In the following theorem we let f ≅ g denote that the morphisms f and g are A^{1}-homotopic.

Theorem 4.6 Let F be a presheaf with transfers such that there is an epimorphism X → F for a scheme X. Let ϕ : S_{t}^{1} ⊗ F → S_{t}^{1}Y be a morphism. Then there exists a unique up to an A^{1}-homotopy morphism ρ(ϕ) : F → Y such that Id_{S_{t}^{1}} ⊗ ρ(ϕ) ≅ ϕ.

Proof: Let us fix an epimorphism p : X → F. Then the morphism ϕ defines a finite correspondence Z : G_{m}X → G_{m}Y and for n sufficiently large we may consider ρ_{n}(Z) : X → Y. Lemma 4.4 implies immediately that ρ_{n}(Z) vanishes on ker(p) and therefore it defines a morphism ρ_{n}(ϕ) : F → X.

Consider a morphism ϕ of the form Id_{S_{t}^{1}} ⊗ ψ. Then Z is of the form (Id_{G_{m}} − e) ⊗ W where W : X → Y corresponds to ψ. By Lemma 4.3 we have ρ_{n}(Z) = W and therefore ρ_{n}(Id_{S_{t}^{1}} ⊗ ψ) = ψ for any n ≥ 1. If ρ, ρ′ are two morphims such that Id_{S_{t}^{1}} ⊗ρ ≅ ϕ and Id_{S_{t}^{1}} ⊗ρ′ ≅ ϕ then for a sufficiently large n we have

ρ = ρ_{n}(Id_{S_{t}^{1}} ⊗ ρ) ≅ ρ_{n}(Id_{S_{t}^{1}} ⊗ ρ′) = ρ′

This implies the uniqueness part of the theorem.

To prove the existence let us show that for a sufficiently large n one has Id_{S_{t}^{1}} ⊗ ρ_{n}(ϕ) ≅ ϕ. Let ϕ̃ be the morphism G_{m}F → G_{m}Y defined by ϕ and let

ϕ̃^{∗} : F G_{m} → Y G_{m}

be the morphism obtained from ϕ̃ by the obvious permutation.

Lemma 4.7 The morphisms ϕ̃ ⊗ (Id_{G_{m}} − e) and (Id_{G_{m}} − e) ⊗ ϕ̃^{∗} are A^{1}- homotopic.

Proof: One can easily see that these two morphisms are obtained from the morphisms

ϕ ⊗ Id_{S_{t}^{1}}, Id_{S_{t}^{1}} ⊗ ϕ^{∗} : S_{t}^{1}F S_{t}^{1} → S_{t}^{1}Y S_{t}^{1}

by using the standard direct sum decomposition. One can see further that ϕ ⊗ Id_{S_{t}^{1}} = σ_{Y} (Id_{S_{t}^{1}} ⊗ ϕ^{∗})σ_{F} where σ_{F} and σ_{Y} are the permutations of the two copies of S_{t}^{1} in S_{t}^{1}F S_{t}^{1} and S_{t}^{1}Y S_{t}^{1} respectively. Lemma 4.8 below implies now that ϕ ⊗ Id_{S_{t}^{1}} ≅ Id_{S_{t}^{1}} ⊗ ϕ^{∗}.

Lemma 4.8 The permutation on S_{t}^{1}S_{t}^{1} is A^{1}-homotopic to {−1}Id ⊗ Id where {−1} : S_{t}^{1} → S_{t}^{1} is defined by the morphism G_{m} ⟶^{x↦x^{−1}} G_{m}.

Proof: The same arguments as the ones used in [2, p.142] show that for any scheme X and any pair of invertible functions f, g on X the morphism X ⟶^{f⊗g} S_{t}^{1}S_{t}^{1} is A^{1}-homotopic to the morphism g ⊗ f^{−1}. This implies immediately that the permutation on S_{t}^{1}S_{t}^{1} is A^{1}-homotopic to the morphism Id ⊗ ({−1}Id) where {−1}Id : S_{t}^{1} → S_{t}^{1} is the morphism defined by the map

G_{m} ⟶^{x↦x^{−1}} G_{m}.

For a sufficiently large n we have

ρ_{n}(ϕ ⊗ (Id_{G_{m}} − e)) = ρ_{n}(ϕ) ⊗ (Id_{G_{m}} − e)

by Lemma 4.5. On the other hand

ρ_{n}((Id_{G_{m}} − e) ⊗ ϕ^{∗}) = ϕ^{∗}

by Lemma 4.3. By Lemma 4.7 we conclude that

ϕ^{∗} ≅ ρ_{n}(ϕ) ⊗ (Id_{G_{m}} − e)

which is equivalent to Id_{S_{t}^{1}} ⊗ ρ_{n}(ϕ) ≅ ϕ. Theorem 4.6 is proved.

Corollary 4.9 Denote by F_{Y} the presheaf

X ↦ Hom(S_{t}^{1}X, S_{t}^{1}Y)

and consider the obvious map Y → F_{Y}. Then for any X the corresponding map of complexes of abelian groups

C_{∗}(Y)(X) → C_{∗}(F_{Y})(X)

is a quasi-isomorphism

Proof: Let Δ^{n} ≅ A^{n} be the standard algebraic simplex and ∂Δ^{n} the subpresheaf in Δ^{n} which is the union of the images of the face maps Δ^{n−1} → Δ^{n}. Then the n-th homology group of the complex C_{∗}(F)(X) for any F is the group of homotopy classes of maps from X ⊗ (Δ^{n}/∂Δ^{n}) to F. Our result now follows directly from 4.6.

Corollary 4.10 Let k be a perfect field. Then for any K, L in DM_{−}^{eff}(k) the map Hom(K, L) → Hom(K(1), L(1)) is a bijection.

Proof: Since DM_{−}^{eff} is generated by objects of the form X it is enough to check that for smooth schemes X, Y over k and n ∈ Z one has

Hom(S_{t}^{1}X, S_{t}^{1}Y [n]) = Hom(X, Y [n])

By Corollary 4.9 we know that the map

Y → F_{Y} = \underline{Hom}(S_{t}^{1}, S_{t}^{1}Y)

is an isomorphism in DM. Let us show now that for any sheaf with transfers F and any X one has

Hom_{DM}(S_{t}^{1}X, F [n]) = Hom_{DM}(X, \underline{Hom}(S_{t}^{1}, F)[n])  (4.3)

The left hand side of (4.3) is the hypercohomology group H^{n}(G_{m}X, C_{∗}(F)) modulo the subgroup H^{n}(X, C_{∗}(F)). The right hand side is the hypercohomology group H^{n}(X, C_{∗}\underline{Hom}(G_{m}, F)) modulo similar subgroup. Let p : G_{m}X → X be the projection. It is easy to see that (4.3) asserts that Rp_{∗}(C_{∗}(F)) ≅ C_{∗}(p_{∗}(F)). There is a spectral sequence which converges to the cohomology sheaves of Rp_{∗}(C_{∗}(F)) and starts with the higher direct images R^{i}p_{∗}(\underline{H}^{j}(C_{∗}(F))). We need to verify that R^{i}p_{∗}(\underline{H}^{j}(C_{∗}(F))) = 0 for i > 0 and that p_{∗}(\underline{H}^{j}(C_{∗}(F))) = \underline{H}^{j}(C_{∗}(p_{∗}(F))). Both statements follow from [5, Prop. 4.34, p.124] and the comparison of Zariski and Nisnevich cohomology for homotopy invariant presheaves with transfers.

References

[1] J.S. Milne. Etale Cohomology. Princeton Univ. Press, Princeton, NJ, 1980.

[2] Andrei Suslin and Vladimir Voevodsky. Bloch-Kato conjecture and motivic cohomology with finite coefficients. In The arithmetic and geometry of algebraic cycles, pages 117–189. Kluwer, 2000.

[3] Andrei Suslin and Vladimir Voevodsky. Relative cycles and Chow sheaves. In Cycles, transfers and motivic homology theories, Annals of Math Studies, pages 10–86. Princeton Univ. Press, 2000.

[4] Vladimir Voevodsky. Motivic cohomology are isomorphic to higher Chow groups. www.math.uiuc.edu/K-theory/378, 1999.

[5] Vladimir Voevodsky. Cohomological theory of presheaves with transfers. In Cycles, transfers and motivic homology theories, Annals of Math Studies, pages 87–137. Princeton Univ. Press, 2000.

[6] Vladimir Voevodsky. Triangulated categories of motives over a field. In Cycles, transfers and motivic homology theories, Annals of Math Studies, pages 188–238. Princeton Univ. Press, 2000.