CHAPTER I BASIC DEFINITIONS AND PROPERTIES

1. Introduction

This chapter contains the basic definitions and properties of the higher chow groups. The focus will be entirely on cycles; any mention of K-theory will be purely (co)incidental. We fix a ground field, usually denoted k, and scheme will usually mean quasi-projective algebraic k-scheme. A variety is a reduced, irreducible algebraic k-scheme. Occasionally, we may work with schemes like Spec(R), where R is the local ring at a point on an algebraic k-scheme.

1.1. An algebraic cycle on a scheme X is a finite, formal linear combination ∑ n_{V} [V] of subvarieties V ⊂ X with integer coefficients. When X is an equidimension algebraic k-scheme, we write Z^{r}(X) for the group of all codimension r algebraic cycles on X. Our reference for foundational questions involving cycles is [FULTON]. Let D : f = 0 be a principal divisor on X, and let V ⊂ X be an irreducible subvariety. Assume V ⊄ D. We can then define an intersection cycle D · V as in [FULTON, 1.3].We extend by linearity to define D · ( ∑ n_{V} [V]). More generally, if W ⊂ X is a complete intersection, W : f_{1} = … = f_{n} = 0, one says that W and V meet properly if W ∩ V has codimension ≥ n in V. When W and V meet properly, the above procedure can be iterated to define a cycle W · V. Again, W · ( ∑ n_{V} [V]) is defined by linearity. Because it is done one divisor at a time, this intersection is clearly associative. It is commutative, i.e. W_{1}▪W_{2}▪V = W_{2}▪W_{1}▪V by [FULTON, 2.4].

1.2. We write Δ^{n} for an affine simplex of the form

Δ^{n} = Spec (k[t_{0}, …, t_{n}]/(∑ t_{i} − 1))

Let Δ▪ be the cosimplicial variety associated to the Δ^{n}. In other words, given ρ : {0, …, m} → {0, …, n} a (weakly) increasing map, define ρ̃: Δ^{m} → Δ^{n} by

ρ̃^{∗}(t_{i}) = \{ \begin{matrix}0 & if ρ^{−1}(i) = ∅ \\ ∑_{ρ(j)=i} t_{j} & otherwise \end{matrix}

If ρ is injective, ρ̃ is called a face map and ρ̃(Δ^{m}) is a face. Maps ρ̃ for ρ surjective are degeneracies. Face maps are closed immersions, and degeneracies are flat. The notions of faces, face maps, and degeneracies extend in an obvious way to schemes X × Δ^{n}.

1.3. Let X be an equidimension algebraic k-scheme, n, r ≥ 0 integers.

Typeset by AMS-TEX

Definition 1.3.1. Z^{r}(X, n) ⊂ Z^{r}(X × Δ^{n}) is the free abelian group on irreducible subvarieties V ⊂ X × Δ^{n} of codimension r such that V meets all faces of X × Δ^{n} properly.

Proposition 1.3.2. Z^{r}(X, ▪) is a simplicial abelian group.

Proof. The assertion is that one can functorially associate to a weakly increasing map ρ : {0, …, m} → {0, …, n} a pullback map

ρ^{∗} : Z^{r}(X, n) → Z^{r}(X, m).

Suppose first that ρ is surjective. Then ρ̃ : X × Δ^{m} → X × Δ^{n} is flat, so by [FULTON, 1.7], one has defined a pullback ρ^{∗} : Z^{r}(X × Δ^{n}) → Z^{r}(X × Δ^{m}). Note that ρ^{∗}(V) has support in ρ̃^{−1}(V). For ρ injective, ρ̃(X × Δ^{m}) ⊂ X × Δ^{n} is a face, and, for V meeting faces properly, we define ρ^{∗}(V) = ρ̃(X × Δ^{m})▪V. A general ρ can be written uniquely as ρ = ρ_{1}^{∘}ρ_{2} with ρ_{1} injective and ρ_{2} surjective. We define ρ^{∗}(V) = ρ_{2}^{∗}^{∘}ρ_{1}^{∗}(V). Another way to understand this is to notice that, given T ⊂ X × Δ^{m} a closed subvariety of codimension r and V ⊂ X × Δ^{n} a closed subvariety of codimension r meeting faces properly, the multiplicity of T in ρ^{∗}(V) coincides with the length of ρ̃^{−1}(V) at the generic point of T. This description of multiplicities also makes functoriality of pullback clear, providing that at each stage the cycle meets faces properly.

Finally, for V ⊂ X ×Δ^{n} meeting faces properly, we must show that ρ^{∗}(V) meets faces properly in X ×Δ^{m}. We argue by induction on m. The assertion is immediate for ρ injective. In general, one sees easily that ρ̃^{−1}(V) has pure codimension r in X × Δ^{m}, i.e. ρ^{∗}(V) meets the codimension 0 face properly. Let σ ⊂ Δ^{m} be a proper face. Then ρ̃|σ corresponds to a map τ : {0, …, p} → {0, …, n} for some p < m. By induction, τ^{∗}(V) meets faces properly on X × τ. We must check that τ^{∗}(V) = ρ^{∗}(V)▪(X × σ), but this follows from the above description of the multiplicities of components of τ^{∗}(V).

We will write

∂_{i} : Z^{r}(X, n) → Z^{r}(X, n − 1)

for the pullback map on cycles associated to the inclusion Δ^{n−1} ⊂ Δ^{n} defined by t_{i} = 0.

1.4 Higher Chow groups.

Definition 1.4.1. Let X be an equidimensional algebraic k-scheme as above. The higher Chow groups are defined to be the homotopy groups of the simplicial abelian group Z^{∗}(X, ▪):

CH^{r}(X, n) := π_{n}(Z^{r}(X, ▪)).

Let δ = ∑(−1)^{i}∂_{i} : Z^{r}(X, n) → Z^{r}(X, n − 1). One checks easily that δ^{∘}δ = 0, so (Z^{r}(X, ▪), δ) is a chain complex. On the other hand, one defines the normalized complex

Z^{r}(X, n)_{norm} = ⋂_{i=0}^{i=n−1} ker(∂_{i}) ⊂ Z^{r}(X, n)

with boundary maps ∂_{n} : Z^{r}(X, n)_{norm} → Z^{r}(X, n − 1)_{norm}.

Proposition 1.4.2. We have

CH^{r}(X, n) = π_{n}(Z^{r}(X, ▪)) ≅ H_{n}(Z^{r}(X, ▪), δ) ≅ H_{n}(Z^{r}(X, ▪)_{norm}).

Proof. This is standard from the structure of simplicial abelian groups, cf. [MAY, §22].

Proposition 1.4.3. CH^{r}(X, 0) ≅ CH^{r}(X), the usual Chow group of algebraic cycles on X modulo rational equivalence.

Proof. By definition, the Chow group coincides with the cokernel of Z(X, 1)\overrightarrow{→}Z(X, 0). QED

2. Functoriality

Proposition 2.1. The complex Z(X, ▪) is covariant functorial for proper maps and contravariant functorial for flat maps.

Proof. Recall Z(X, n) ⊂ Z(X × △^{n}) is generated by subvarieties meeting all the faces X × △^{m} ⊂ X × △^{n} properly. Let f : X → Y be proper. Then f × 1 : X × △^{n} → Y × △^{n} is also proper, so there is a map on cycles f_{∗} : Z(X × △^{n}) → Z(Y × △^{n}). For Z ⊂ X × △^{n} and △^{m} ⊂ △^{n} one has

f(Z) ∩ (Y × △^{m}) = f(Z ∩ (X × △^{m}))
dim(f(Z) ∩ Y × △^{m}) ≤ dim(Z ∩ (X × △^{m})).

It follows that the arrows in the square

Z(X, n) ⟶^{pullback} Z(X, m)
↓^{f_{∗}}  ↓^{f_{∗}}
Z(Y, n) ⟶^{pullback} Z(Y, m)

are defined. The fact that the square commutes is e.g. ([**], Th. 6.2(a)). Actually, in the reference, Fulton works with cycles mod rational equivalence, but we can replace X × △^{n}, Y × △^{n}, X × △^{m}, Y × △^{m} with Z, f(Z), Z ∩ (X × △^{m}), f(Z ∩ (X × △^{m})) and assume we are working with cycles of codimension 0 where rational equivalence changes nothing.

The assertion of contravariant functoriality for flat maps is ([**], 1.7). QED

Corollary 2.1.1. Let π : Spec(k′) → Spec(k) be a finite extension of fields. Then one has

Z(X_{k}, ▪) ⟶^{π^{∗}} Z(X_{k′}, ▪) ⟶^{π_{∗}} Z(X_{k}, ▪).

If π has degree n, the composition

π_{∗}π^{∗} = multiplication by n.

Proof. Immediate. QED

2.2. The following lemma is the prototype for an argument which will be used repeatedly in the sequel.

Lemma 2.2.1. Let k ⊂ K be a purely transcendental extension. Then the pullback π^{∗} : Z(X_{k}, ▪) → Z(X_{K}, ▪) is injective on homology.

Proof. Let Z ∈ Z(X_{k}, n) with ∂Z = 0 in Z(X_{k}, n − 1). Let W ∈ Z(X_{K}, n + 1) satisfy ∂W = π^{∗}Z. Clearly, W is defined over a finitely generated purely transcendental extension k(t_{1}, …, t_{p}) ⊂ K. Standard “spreading out” results in algebraic geometry imply there exists U an open subvariety of A^{p} and a cycle W in Z(X ×_{k} U, n + 1) such that

(2.2.1.1)  ∂W = Z ×_{k} U.

Again by standard results in algebraic geometry, there exists a closed subvariety V ⫋ U such that if u ∈ U − V is a k-point, then the intersection W_{u} = W · X × {u} is defined and ∂W_{u} = Z × {u}. If k is infinite, then such a k-point u exists, and the class of Z in H_{n}(Z(X_{k}, ▪)) is zero.

If k is a finite field, for any prime ℓ there exists an infinite pro-ℓ extension k_{ℓ} of k. By (2.1.1), the kernel of the map CH(X_{k}, n) → CH(X_{k_{ℓ}}, n) is ℓ-torsion. By the above argument, since k_{ℓ} is infinite, the kernel of CH(X_{k}, n) → CH(X_{K}, n) is ℓ-torsion. Since ℓ is arbitrary, this map is injective. QED

Here is a variant on (2.2.1) needed in the next section. Let Y be a smooth k-variety, W ⊂ Y a closed algebraic subset. Let

(2.2.2)  Z_{W} (Y, ▪) ⊂ Z(Y, ▪)

be generated by irreducible subvarieties of X × △^{n} meeting W × △^{m} properly for all faces △^{m} ⊂ △^{n}. More generally, if W = {W_{1}, …, W_{q}} is a collection of closed algebraic subsets of Y, we define Z_{W}(Y, ▪) ⊂ Z(Y, ▪) to be generated by irreducible cycles meeting all W_{i} × △^{m} properly. Returning to the case of a single closed set, write C_{∗} for the quotient complex Z(Y, ▪)/Z_{W} (Y, ▪). Let K ⊃ k be a purely transcendental extension, and write C_{K,∗} for the corresponding quotient for W_{K} ⊂ Y_{K}.

Lemma 2.2.3. The pullback map ι : C_{∗} → C_{K,∗} induces an injection on homology.

Proof. The proof is precisely analogous to (2.2.1) and is left for the reader.

The interest in the complex Z_{W} (Y, ▪) arises from the following lemma:

Lemma 2.2.4. With notation as above, assume either that Y is smooth or W ⊂ Y is a local complete intersection. Then pullback of cycles induces a map of complexes

Z_{W} (Y, ▪) → Z(W, ▪).

More generally, if W = {W_{1}, …, W_{q}} with W = W_{1} ⊂ Y as above and W_{i} ⊂ W_{1} for i ≥ 2 then pullback induces a map of complexes

Z_{W}(Y, ▪) → Z_{{W_{2},... ,W_{q}}}(W_{1}, ▪).

Proof. This follows from commutativity of the intersection product [FULTON, 6.4]. QED

2.3 Cones. The purpose of this section and the next is to prove what we are tempted to call the “easy” moving lemma:

Proposition 2.3.1. Let Y be a smooth k-variety, W = {W_{1}, …, W_{r}} a collection of closed algebraic subsets of Y. Assume Y is either affine or projective. Then with notation as in (2.2.2), the inclusion Z_{W}(Y, ▪) ⊂ Z(Y, ▪) is a quasi-isomorphism.

Proof. In this section, we will assume (2.3.1) for Y = 𝔸^{n} and Y = ℙ^{n} and prove the general result. The result for 𝔸^{n} and ℙ^{n} will be proved in the following section by a different method.

To fix ideas, suppose Y is projective. Let K be a large, purely transcendental extension of k as above, and define C_{∗} and C_{K,∗} as in (2.2). By (2.2.3) it will suffice to prove

Lemma 2.3.1.1. The map ι : C_{∗} → C_{K,∗} induces the zero map on homology.

Proof of 2.3.1.1. Using (2.2.3), we may enlarge K if we want. Let dim(Y) = d. Fix an embedding Y ↪ ℙ^{N} for N >> 0. Let L ⊂ ℙ^{N} be a linear space of codimension d + 1 which is k-generic and defined over K. We have an inclusion j : Y ↪ ℙ^{N} − L. Projecting from L defines a map

(2.3.1.2)  π_{L} : ℙ^{N} − L → ℙ^{d}.

The composition ρ = π ∘ j : Y → ℙ^{d} is finite so there is a map of complexes

ρ_{∗} : Z^{∗}(Y, ▪) → Z^{∗}(ℙ^{d}, ▪).

Let Q be the blowup of ℙ^{N} along L. There is an evident diagram

Q
f ↙  ↘ g
ℙ^{d}  ℙ^{N}

with f flat and g proper. We define the cone map

(2.3.1.3)  cone_{L} : Z^{r}(Y, ▪) → Z^{r}(ℙ_{K}^{N}, ▪)

to be the composition cone_{L} = g_{∗} ∘f^{∗} ∘ρ_{∗}. In fact, the cones in question meet faces on Y properly. More generally, for W ⊂ Y closed, if a cycle Z in Z^{r}(Y, n) meets W × △^{m} properly for a face △^{m} ⊂ △^{n} it is easy to check that cone_{L}(Z) does as well. We get maps of complexes

cone_{L} : Z^{∗}(Y, ▪) → Z_{Y}^{∗} (ℙ_{K}^{N}, ▪)
cone_{L} : Z_{W}^{∗}(Y, ▪) → Z_{Y,W_{1},... ,W_{r}}^{∗}(ℙ_{K}^{N}, ▪)

Using (2.2.4), we have a composed map

C_{∗} ⟶^{cone_{L}} Z_{Y}^{∗} (ℙ_{K}^{N}, ▪)/Z_{Y,W_{1},... ,W_{r}}^{∗}(ℙ_{K}^{N}, ▪) ⟶^{j^{∗}} C_{K,∗}.

We will prove in the next section that the quotient complex in the middle here is acyclic, so j^{∗} ∘ cone_{L} is zero on homology. On the other hand, given z ∈ H_{n}(C_{∗}), the standard construction for moving cycles ([**]) implies that for N, p >> 0 (depending on n), there exists a chain of purely transcendental extensions k ⊂ K_{1} ⊂ K_{2} … ⊂ K_{p} and a chain of codimension d + 1 linear spaces L_{i} defined over K_{i} and independent generic over K_{i−1} such that the composition (with obvious notation)

(ι_{p} − j^{∗} ∘ cone_{L_{p}}) ∘ … ∘ (ι_{1} − j^{∗} ∘ cone_{L_{1}})

is zero on z. (We remark that the usual moving lemma is for cycles on Y, while our application is to cycles on Y × △^{n}. To deal with this we must apply the classical argument separately to Y × Spec(k(t)) as t runs through the generic points of all faces of △^{n}. This explains the possible dependence of N and p on n.)

It follows that ι is zero on homology. This completes the proof of (2.3.1.1) and (2.3.1) for Y projective, modulo the proof of the corresponding assertions for Y = ℙ^{N}. QED

For the case Y affine, we consider 𝔸_{k}^{N} ⊂ ℙ_{k}^{N} and we fix a linear space L of codimension d in the hyperplane at ∞, ℙ^{N} − 𝔸^{N}. Let G be the Euclidean group (semi-direct product of 𝔸^{N} and GL(N).) We take a k-embedding Y ↪ 𝔸^{N} which we compose with g ∈ G, where we think of K as containing the function field k(G) and take g to be the k-generic K-point of G. In this way we obtain a K-embedding of Y in 𝔸_{K}^{N}. Projection from L preserves cycles “at ∞” and so induces a coning map Z^{∗}(Y, ▪) → Z^{∗}(𝔸_{K}^{N}, ▪). The rest of the argument parallels the projective case. QED

2.4 Contravariant Functoriality for Projective and Affine Space.

In this section, we verify (2.3.1) for the cases Y = 𝔸^{N} and Y = ℙ^{N}.

Lemma 2.4.1. Let X be an algebraic k-scheme and G a connected algebraic kgroup acting on X. Let A, B ⊂ X be closed subsets, and assume the fibres of the map

G × A → X (g, a) ↦ g · a

all have the same dimension, and that this map is dominant. (Note both conditions hold when G acts transitively on X.) Then there exists a non-empty open set U ⊂ G such that for g ∈ U the intersection g(A) ∩ B is proper.

Proof. Consider the diagram

G ← G × A → X
↖  ∪  □ ∪
C  → B

where C is the fibre product. Our hypotheses imply

dim G + dim A + dim B − dim X = dim C.

We may take for U the open set in G where the fibres of C → G have smallest dimension. QED

Lemma 2.4.2. Let hypotheses be as in (2.4.1). Assume, moreover, we are given an overfield K ⊃ k and a K-morphism ψ : X_{K} → G_{K}. Let V ⊂ X be non-empty open such that for every x ∈ V_{K} a scheme point, we have

tr. deg_{k} k(φ ∘ ψ(x), π(x)) ≥ dim G,

where π : X_{K} → X_{k} and φ : G_{K} → G_{k}. (In other words, we assume that the subfield of K(x) generated over k by the residue fields at x and ψ(x) has transcendence degree over k at least equal to the dimension of G.) Define

ϕ : X_{K} → X_{K} ϕ(x) = ψ(x) · x.

Assume ϕ is an isomorphism. Then the intersection ϕ(A ∩ V) ∩ B is proper.

Proof. the map A_{K} → A_{k}. We have the diagram

(V ∩ A)_{K}  ⟶^{(ψ,π)} G ×_{k} A ⟶ X
∪  ∪  ∪
(V ∩ A)_{K} ∩ C ⟶  C  ⟶ B

As in (2.4.1), dim C = dim G+dim A+dim B −dim X. Also (V ∩A)_{K} ∩C coincides under ϕ with ϕ(V ∩A)∩B. It therefore suffices to show the intersection C ∩(V ∩A) is proper on G × A. We can replace A by V ∩ A and ignore V.

We regard this as an intersection problem on G × A for C arbitrary. Replacing C by a hyperplane section, we may assume C ∩ A is zero-dimensional. For the intersection to be improper we must have then dim C < dim G. Let a ∈ A be a scheme point such that (ψ(a), π(a)) ∈ C. We must have tr. deg_{k} k(ψ(a), a) < dim G, contradicting the hypotheses. QED

Remark 2.4.2.1. In the application, X = Y ×𝔸^{1}, with G acting trivially on 𝔸^{1}. We are given ψ_{0} : 𝔸^{1} → G_{K} and an open U_{k} ⊂ 𝔸_{k}^{1} such that for all scheme points w ∈ U_{K}, tr. deg_{k} ψ_{0}(w) ≥ dim G − 1 and tr. deg_{k} ψ_{0}(w) = dim G if w is algebraic over k. Then the map ψ = ψ_{0} ∘ projection : X → G satisfies the hypotheses of (2.4.2). Indeed, the map ϕ : X = Y × 𝔸^{1} → X is ϕ(x, a) = (ψ_{0}(a)x, a) and is thus an isomorphism.

For the case Y = 𝔸^{N} (resp. Y = ℙ^{N}) we take G = 𝔸^{N} acting by translation (resp. G = SL(N + 1) acting in the usual way). In both cases, k(G) ⊂ K with K purely transcendental over k, and we can find a map ψ_{0} : 𝔸_{K}^{1} → G_{K} with ψ_{0}(0) = identity and ψ_{0}(1) k-generic in G. (Use the fact that SL(N + 1) is generated by transvections.)

Proposition 2.4.3. With notation as above, let W be a finite collection of closed sets in Y. Then C_{∗} = Z(Y, ▪)/Z_{W}(Y, ▪) is acyclic.

Proof. We begin by fixing triangulations for △^{n} × △^{1} for n = 0, 1, 2, . . .. Vertices in △^{n} × △^{1} will be pairs (v, w) with v a vertex of △^{n} and w a vertex of △^{1}. For S = ((v_{0}, w_{0}), …, (v_{r}, w_{r})) a collection of r + 1 vertices, we define

θ_{S} : △^{r} → △^{n} × △^{1} (t_{0}, …, t_{r}) ↦ ∑ t_{i}(v_{i}, w_{i})

. We can consider formal linear combinations of maps between products of simplices. There is an obvious notion of composition of such linear combinations. For example

∂_{△^{r}} = ∑(−1)^{i}∂_{i} : △^{r−1} → △^{r}.

A triangulation T is a collection S_{n}_{n=0,1,...}, where each S_{n} is itself a collection of sets S of n + 1 vertices in △^{n−1} × △^{1} together with signs σ(S) such that writing

T_{n} = ∑_{S∈S_{n}} σ(S)θ_{S} : △^{n} → △^{n−1} × △^{1}

we have as formal linear combinations of maps △^{n} → △^{n} × △^{1}

T_{n+1} ∘ ∂_{△^{n+1}} − ((∂_{△^{n−1}}) × id_{△^{1}}) ∘ T_{n} = id_{△^{n}} ×ι_{1} − id ×ι_{0} : △^{n} → △^{n} × △^{1}.

We fix such a triangulation T = T_{n}. For example, T_{1} might look like

fig. 1

with signs arranged so the diagonal faces on T_{1} ∘ ∂ cancel. We would like to think that pulling back along T. would define a homotopy

T^{∗} : Z(Y × 𝔸^{1}, ▪) → Z(Y, ▪ + 1).

Unfortunately, for example, with reference to fig. 1 above, taking t ≠ 0, 1 the cycle Y × {t} × {t} ∈ Z(Y × 𝔸^{1}, 1) does not meet the diagonal faces properly.

Lemma 2.4.3.1. Consider the composition

Y × 𝔸_{K}^{1} ⟶^{ϕ} Y × 𝔸_{K}^{1} ⟶^{pr_{1}} Y_{K} ⟶^{π} Y_{k}.

The mapping

(2.4.3.2)  ρ^{∗} = T^{∗} ∘ ϕ^{∗} ∘ pr_{1}^{∗} ∘π^{∗} : Z(Y_{k}, ▪) → Z(Y_{K}, ▪ + 1)

is well defined. This same map carries Z_{W}(Y_{k}, ▪) → Z_{W_{K}} (Y_{K}, ▪ + 1).

Proof of 2.4.3.1. We prove the last assertion. Let Z represent a cycle in Z_{W}(Y_{k}, n) and let F ⊂ △^{n} × △^{1} be a face of some △^{n+1} entering into the triangularization. We must show the intersection

ϕ^{∗} pr_{1}^{∗} π^{∗}(Z) ∩ (W × F)

is proper for W ∈ W. Replacing ψ_{0} by its image under composition with the inverse map in G_{K}, we might as well consider

pr_{1}^{∗}π^{∗}(Z) ∩ ϕ^{∗}(W × F).

If F ⊂ △^{n} × {0} this intersection is proper since ψ_{0}(0) = id and Z ∈ Z_{W}(Y, n). Otherwise, we apply (2.4.2) with B = W × F, A = pr_{1}^{∗}(Z) ∩ (Y × F). (Note the intersection in the definition of A is proper.) This proves (2.4.3.1). QED end of proof of 2.4.3. Note the homotopy ρ^{∗} in (2.4.3.2) satisfies

∂ ∘ ρ_{n}^{∗} − ρ_{n−1}^{∗} ∘ ∂ = ψ_{0}(1)^{∗} − ψ_{0}(0)^{∗} : Z(Y_{k}, n) → Z(Y_{K}, ▪).

Next we claim

ψ_{0}(1)^{∗} : Z(Y_{k}, ▪) → Z_{W}(Y_{k}, ▪).

We apply (2.4.2) with ψ the constant map with image ψ_{0}(1). For ambiant space we take Y ×△^{m} for some face △^{m} ⊂ △^{n}. We take A = W ×△^{m}, and B = z∩(Y ×△^{m}) with z ∈ Z(Y_{k}, n).

In conclusion, we see the map C_{∗} → C_{K,∗} induces 0 on homology. Again by the argument in (2.2.3), C_{∗} is acyclic, and (2.4.3) follows. QED

Remark 2.4.3.3. Let Y and W = {W_{1}, …, W_{n}} be as in (2.4.3) and let X be a k-scheme. Write X × W = {X × W_{1}, …, X × W_{n}}. The same argument as in the proof of (2.4.3) shows that the inclusion Z_{X×W}(X × Y_{k}, ▪) ⊂ Z(X × Y_{k}, ▪) is a quasi-isomorphism.

2.5 Contravariant Functoriality; Applications.

Proposition 2.5.1. Let f : X → Y be a morphism of algebraic k-schemes. Assume Y is smooth and either affine or projective. Then there is defined a pullback map

(2.5.1.1)  f^{∗} : CH^{∗}(Y, n) → CH^{∗}(X, n).

Given a composition W ⟶^{f} X ⟶^{g} Y with X and Y smooth and either affine or projective, we have

(g^{∘}f)^{∗} = f^{∗}^{∘}g^{∗} : CH^{∗}(Y, n) → CH^{∗}(W, n).

Proof. There exists a collection T = {T_{1}, …, T_{p}} of closed subsets of Y such that for any n and any cycle Z ∈ Z_{T} (Y, n), the pullback f^{−1}(Z) is defined in Z(X, n). Indeed, the diagonal △_{Y} ⊂ Y ×_{k} Y is a local complete intersection since Y is smooth. Hence, the graph Γ_{f} = (f × 1_{Y})^{−1}△_{Y} ⊂ X × Y is also a local complete intersection. Let T_{i} be the set over which the fibres of f have dimension ≥ dim X −dim Y +i. If Z meets T_{i}×△^{n} properly for all i, then X ×Z ⊂ X ×Y ×△^{n} meets Γ_{f} × △^{n} properly, so the intersection is defined. Under the identification Γ_{f} ≅ X, this is the pullback. Since H_{∗}(Z_{T} (Y, ▪)) ≅ H_{∗}(Z(Y, ▪)), the pullback (2.5.1.1) is defined. The remaining argument is similar and is left for the reader. QED

The hypothesis that Y be either projective or affine in (2.3.1) and (2.5.1) can be removed using the “strong” moving lemma to be proved in (**).

Proposition 2.5.2. Let Y be smooth quasi-projective, and let W = {W_{1}, …, W_{p}} be closed subsets of Y. Assume the strong moving lemma (**) holds. Then the inclusion Z_{W}(Y, ▪) ⊂ Z(Y, ▪) is a quasi-isomorphism.

Proof. Induction on d = dim Y. For d ≤ 1, Y is affine or projective and the assertion is (2.3.1). In general, let Ȳ be a projective closure of Y with Z = Ȳ − Y.

Let X̄ be a hypersurface section of large degree on Ȳ which contains Z but which is otherwise generic. Let X = X̄ −Z. Then X is smooth quasi-projective of dimension d − 1, and Y − X is affine. Write W ∩ X = {W_{1} ∩ X, …, W_{p} ∩ X}. We have a diagram

Z_{W∩X}(X, ▪) ⟶ Z_{W}(Y, ▪) ⟶ Z_{W}(Y − X, ▪)
↓  ↓  ↓
Z(X, ▪) ⟶ Z(Y, ▪) ⟶ Z(Y − X, ▪)

By induction, the outside vertical arrows are quasi-isomorphisms. The strong moving lemma (**) and (**) implies the two horizontal sequences give distinguished triangles in the derived category. It follows from the five-lemma that the middle vertical arrow is a quasi-isomorphism. QED

3. The Homotopy Theorem

Theorem 3.1. Let X be a quasi-projective k-variety. The pullback map p^{∗} : Z(X, ▪) → Z(X × 𝔸^{n}, ▪) is a quasi-isomorphism.

Proof. One reduces by induction to the case n = 1. Let K = k(𝔸^{1}) be the function field of 𝔸^{1}. Let

(1_{X} × ρ)^{∗} : Z(X ×_{k} 𝔸^{1}, ▪) → Z((X ×_{k} 𝔸^{1})_{K}, ▪ + 1)

be defined as in (2.4.3.2). The first point is that this map is defined, and that

(1_{X} × ρ)^{∗}(Z_{X×{0,1}}(X ×_{k} 𝔸^{1}, ▪)) ⊂ Z_{X×{0,1}}((X ×_{k} 𝔸^{1})_{K}, ▪ + 1).

The arguments here are the same as in (2.4) (with Y = 𝔸^{1}), and are left to the reader. In particular, the inclusion Z_{X×{0,1}}(X × 𝔸^{1}, ▪) ⊂ Z(X × 𝔸^{1}, ▪) is a quasiisomorphism, and, writing θ^{∗} for the pullback on cycles along the map translation by the k-generic point t, we have that θ^{∗} is homotopic to the natural pullback π^{∗} mapping Z(X × 𝔸_{k}^{1}, ▪) → Z(X × 𝔸_{K}^{1}, ▪) with K = k(t). Also,

θ^{∗}(Z(X × 𝔸_{k}^{1}, ▪)) ⊂ Z_{X×{0,1}}(X × 𝔸_{K}^{1}, ▪).

Writing T for a triangulation, the composition

Z_{X×{0,1}}(X × 𝔸_{k}^{1}, ▪) ⟶^{1_{X}×θ^{∗}} Z_{X×{0,1}}(X × 𝔸_{K}^{1}, ▪) ⟶^{T^{∗}} Z(X_{K}, ▪ + 1)

is defined. Since T is a triangulation, we have

∂^{∘}T^{∗}^{∘}θ^{∗} − T^{∗}^{∘}θ^{∗}^{∘}∂ = (∂^{∘}T^{∗} − T^{∗}^{∘}∂)^{∘}θ^{∗} = (ι_{1}^{∗} − ι_{0}^{∗})^{∘}θ^{∗}.

If we view these as maps Z_{X×{0,1}}(X × 𝔸_{k}^{1}, ▪) → Z(X_{K}, ▪) and consider the effect on homology we have

0 = H_{∗}((ι_{1}^{∗} − ι_{0}^{∗})^{∘}θ^{∗}) = H_{∗}((ι_{1}^{∗} − ι_{0}^{∗})^{∘}π^{∗}) = H_{∗}(π^{∗}^{∘}(ι_{1}^{∗} − ι_{0}^{∗})).

Since π^{∗} is injective on homology, we conclude

H_{∗}(ι_{0}^{∗}) = H_{∗}(ι_{1}^{∗}) : H_{∗}(Z_{X×{0,1}}(X × 𝔸_{k}^{1}, ▪)) → H_{∗}(Z(X, ▪)).

Let

τ : 𝔸^{1} × 𝔸^{1} → 𝔸^{1} τ(a, b) = ab.

The map τ is flat, so we have a pullback (2.1) 1_{X} × τ^{∗} : Z(X × 𝔸^{1}, ▪) → Z(X × 𝔸^{1} × 𝔸^{1}, ▪). For Z ⊂ X × 𝔸^{1} × △^{n} one has

(1_{X} × τ)^{−1}(Z) ∩ (X × 𝔸^{1} × {0} × △^{n}) ⊂ (1_{X} × τ)^{−1}(Z ∩ (X × {0} × △^{n})).

It follows easily, again using flatness, that

1_{X} × τ^{∗}(Z_{X×{0}}(X × 𝔸^{1}, ▪)) ⊂ Z_{X×𝔸^{1}×{0,1}}(X × 𝔸^{1} × 𝔸^{1}, ▪)).

The compositions

ι_{j}^{∗}^{∘}τ^{∗} : Z_{X×{0}}(X × 𝔸^{1}, ▪) → Z(X × 𝔸^{1}, ▪)

are defined for j = 0, 1 and agree on homology. But ι_{1}^{∗}^{∘}τ^{∗} = identity while ι_{0}^{∗}^{∘}τ^{∗} = (τ^{∘}ι_{0})^{∗} = (ι_{0}^{∘}p)^{∗}. It follows that

p^{∗} : H_{∗}(Z(X, ▪)) → H_{∗}(Z(X × 𝔸^{1}, ▪))

is surjective. Since p^{∗}(Z(X, ▪)) ⊂ Z_{X×{0}}(X × 𝔸^{1}, ▪), we have identity = ι_{0}^{∗}^{∘}p^{∗} defined so p^{∗} is injective. this completes the proof of (3.1). QED

4. Cubical Chow Groups

In this section we show how the higher Chow groups can be computed using cubes rather than simplices. The use of cubes simplifies both the product structure (**) and the proof of the strong moving lemma (**).

4.1. Let □^{n} = 𝔸_{X}^{n}. (The variety X will be fixed throughout the discussion and will frequently be omitted from the notation.)

δ_{i}^{j} : □^{n} ↪ □^{n+1} (4.1.1)

δ_{i}^{j}(x_{1}, …, x_{n}) = (x_{1}, …, x_{i−1}, j, x_{i}, …, x_{n}) 1 ≤ i ≤ n, j = 0, 1

and

p_{i} : □^{n} → □^{n−1} (4.1.2)

p_{i}(x_{1}, …, x_{n}) = (x_{1}, …, x̂_{i}, …, x_{n})

Codimension 1 faces of □^{n} are defined by setting a coordinate = 0, 1. Codimension r faces are intersections of r codimension 1 faces. Let

(4.1.3)  Z_{c}(n) = Z_{c}(X, n)

be the free abelian group on irreducible subvarieties of □^{n} meeting all faces properly (**). We have pullback maps

d_{i}^{j} = (δ_{i}^{j})^{∗} : Z_{c}(n) → Z_{c}(n − 1) (4.1.4) π_{i} = p_{i}^{∗} : Z_{c}(n − 1) → Z_{c}(n) (4.1.5)

These satisfy the following identities for ν < μ and j, k = 0, 1

d_{μ−1}^{k}^{∘}d_{ν}^{j} = d_{ν}^{j}^{∘}d_{μ}^{k} (4.1.5) π_{ν}^{∘}π_{μ−1} = π_{μ}^{∘}π_{ν} (4.1.6)

π_{ν}^{∘}d_{μ−1}^{j} = d_{μ}^{j}^{∘}π_{ν} (4.1.7)

π_{μ−1}^{∘}d_{ν}^{j} = d_{ν}^{j}^{∘}π_{μ} (4.1.8)

Finally, we define

(4.1.8)  d = ∑_{i=1}^{n}(−1)^{i}(d_{i}^{0} − d_{i}^{1}) : Z_{c}(n) → Z_{c}(n − 1).

It follows from 4.1.5 that d^{2} = 0, so Z_{c}(∗) is a complex.

4.2. Define the degenerate cubical cycles

(4.2.1)  D(n) := ∑_{i=1}^{n} π_{i}(Z_{c}(n − 1)) ⊂ Z_{c}(n).

It follows from 4.1.7 and 4.1.8 that D(∗) is a subcomplex of Z_{c}(∗).

The reduced cubical cycles are given by

(4.2.2)  Z_{c}(n)_{0} := ⋂_{i=1}^{n} ker(d_{i}^{0}) ⊂ Z_{c}(n).

It follows from 4.1.5 that Z_{c}(∗)_{0} is a subcomplex of Z_{c}(∗). The differential is

d = ∑(−1)^{i−1}d_{i}^{1} : Z_{c}(n)_{0} → Z_{c}(n − 1)_{0}.

Lemma 4.2.3. Z_{c}(n) = D(n) ⨁ Z_{c}(n)_{0}.

Proof. Let z ∈ Z_{c}(n) and assume d_{r+1}^{0}(z) = … = d_{n}^{0}(z) = 0. It follows from 4.1.5 that z′ = z − π_{r}^{∘}d_{r}^{0}(z) satisfies d_{r}^{0}(z′) = … = d_{n}^{0}(z′) = 0. By induction we conclude that Z_{c}(n) = D(n) + Z_{c}(n)_{0}. To see the sum is direct, let r be minimal such that there exists 0 ≠ z ∈ Z_{c}(n)_{0} with z = ∑_{i=1}^{r} π_{i}w_{i}. Using 4.1.7 and 4.1.8 togher with d_{r}^{0}z = 0, we get

w_{r} = − ∑_{i=1}^{r−1} d_{r}^{0}π_{i}w_{i} = − ∑_{i=1}^{r−1} π_{i}d_{r−1}^{0}w_{i}

so by 4.1.6

z = ∑_{i=1}^{r−1} π_{i}(w_{i} − π_{r−1}d_{r−1}^{0}w_{i}),

contradicting the minimality of r. QED Definition 4.2.4. The cubical cycle groups, CH_{c}^{∗}(X, p) := H_{p}(Z_{c}(∗)_{0}).

Remark 4.2.5. Note the full cubical complex Z_{c}(∗) does not give the correct groups. For example, one checks easily that CH_{c}^{0}(X, p) = ℤ for all p ≥ 0.

Theorem 4.3. CH_{c}^{∗}(X, p) ≅ CH^{∗}(X, p)

Lemma 4.3.1. Let W = {W_{1}, …, W_{q}} be closed subsets of 𝔸^{n}. Then the inclusion Z_{c,X×W}(X × 𝔸^{n}, ∗)_{0} ⊂ Z_{c}(X × 𝔸^{n}, ∗)_{0} is a quasi-isomorphism.

Proof of 4.3.1. The proof is analogous to the proof of 2.4.3 and 2.4.3.3 and is left for the reader. QED

Lemma 4.3.2. CH_{c}^{∗}(X, p) ≅ CH_{c}^{∗}(X × 𝔸^{1}, p)

Proof of Lemma 4.3.2. The proof is analogous to, but easier than, the proof of (**). One simply notes that the identification X × 𝔸^{1} × □^{n} = X × □^{n+1} gives a homotopy T^{∗} : Z_{c,X×{0,1}}(X × 𝔸^{1}, n)_{0} → Z_{c}(X, n + 1)_{0}. Indeed, for Z on X × 𝔸^{1} × □^{n}, write Z̄ for the corresponding cycle on X × □^{n+1} under the obvious identification. Define T^{∗}Z = Z̄ − π_{1}^{∘}ι_{0}^{∗}(Z), where ι_{i}^{∗} : Z_{c,X×{0,1}}(X × 𝔸^{1}, ∗) → Z_{c}(X, ∗) is the pullback, i = 0, 1.. One has dT^{∗} − T^{∗}d = ι_{1}^{∗} − ι_{0}^{∗}. The rest of the proof is just as in (**), using 4.3.1 in place of 2.4.3.3. QED

Proof of Theorem 4.3. Define Z_{c}(p, q)_{0} to be cycles on X ×Δ^{p} ×□^{q} meeting all (intersections of simplicial and cubical) faces properly and trivial on faces defined by setting cubical coordinates = 0. One has boundary maps d′ : Z_{c}(p, q)_{0} → Z_{c}(p−1, q)_{0} (simplicial) and d′′ : Z_{c}(p, q)_{0} → Z_{c}(p, q−1)_{0} (cubical). It is immediate from 4.3.1 and 4.3.2 that

′′H_{q}(Z_{c}(p, ∗)_{0}) ≅ CH_{c}(X × Δ^{p}, q) ≅ CH_{c}(X, q).

In particular the boundary map ′′H_{q}(Z_{c}(p, ∗)_{0}) → ′′H_{q}(Z_{c}(p − 1, ∗)_{0}) is zero for p odd and an isomorphism for p even. We have

E_{q,p}^{2} = ′H_{p}( ′′H_{q}(Z_{c}(∗, ∗)_{0})) = \{ \begin{matrix}0, & p > 0 \\ CH_{c}(X, q), & p = 0 \end{matrix}

so H_{p}(Z_{c}(∗, ∗)_{0}) ≅ CH_{c}(X, p).

On the other hand, if we drop the condition that cycles be trivial on faces defined by setting cubical coordinates = 0, we get a double complex Z_{c}(∗, ∗) ⊃ Z_{c}(∗, ∗)_{0}, and by the homotopy theorem 3.1 for the simplicial groups:

′H_{p}(Z_{c}(∗, q)) ≅ CH(X × □^{q}, p) ≅ CH(X × □^{0}, p) ≅ ′H_{p}(Z_{c}(∗, 0)).

It follows that the map restriction to the point with cubical coordinates (0, …, 0) induces a quasi-isomorphism Z_{c}(∗, q) → Z_{c}(∗, 0). Since Z_{c}(∗, q)_{0} ⊂ Z_{c}(∗, q) is a direct summand (mapping to 0 in Z_{c}(∗, 0) if q > 0) we conclude

′H_{p}(Z_{c}(∗, q)_{0}) = \{ \begin{matrix}0, & q > 0 \\ CH(X, p), & q = 0. \end{matrix}

The spectral sequence E_{p,q}^{2} = ′′H_{q}(′H_{p}(Z_{c}(∗, ∗)_{0})) therefore degenerates, so H_{p}(Z_{c}(∗, ∗)_{0}) ≅ CH(X, p). We conclude CH(X, p) ≅ CH_{c}(X, p). QED

4.4. Finally, for the hard moving lemma, it will be useful to work with a complex of cubical cycles which are trivial on all but one face.

Definition 4.4.1. Z_{c}(∗)_{0i} = {x ∈ Z_{c}(∗)_{0} | d_{j}^{1}x = 0 for j ≥ i + 2.}.

For example, d_{1}^{1} is the only non-trivial face map on Z_{c}(∗)_{00}. It follows from 4.1.5 that Z_{c}(∗)_{0i} ⊂ Z_{c}(∗)_{0} is a subcomplex.

Theorem 4.4.2. The inclusion Z_{c}(∗)_{00} ⊂ Z_{c}(∗)_{0} is a homotopy equivalence.

Proof. Given integers ℓ ≤ n − 1 define h^{ℓ} : □^{n+1} → □^{n} by

h^{ℓ}(x_{1}, …, x_{n+1}) = (x_{1}, …, x_{n−ℓ−1}, x_{n−ℓ} · x_{n−ℓ+1}, x_{n−ℓ+2}, …, x_{n+1}).

For Z a cycle on □^{n}, define H^{ℓ}(Z) = (−1)^{n−ℓ}(h^{ℓ})^{−1}(Z) on □^{n+1}. For ℓ ≥ n define H^{ℓ}(Z) = 0. It is easy to check that H^{ℓ} : Z_{c}(∗)_{0} → Z_{c}(∗ + 1)_{0}. Consider the composition of endomorphisms of Z_{c}(∗)_{0}:

φ := ⋯ (id −(d^{∘}H^{ℓ} + H^{ℓ}^{∘}d))^{∘}(id −(d^{∘}H^{ℓ−1} + H^{ℓ−1}^{∘}d))^{∘} ⋯^{∘}(id −(d^{∘}H^{0} + H^{0}^{∘}d)).

Although this is an infinite composition, it stabilizes in any degree and so defines an endomorphsim φ : Z_{c}(∗)_{0} → Z_{c}(∗)_{0} which is homotopic to the identity.

It will be convenient to write an element z ∈ Z_{c}(n)_{0} as z(x_{1}, …, x_{n}). We have, for example, d_{i}^{1}z(x_{1}, …, x_{n}) = z(x_{1}, …, x_{i−1}, 1, x_{i}, …, x_{n−1}). Thus,

dH^{ℓ}z = (−1)^{n−ℓ}z(1, x_{1}, …, x_{n−ℓ−2}, x_{n−ℓ−1} · x_{n−ℓ}, x_{n−ℓ+1}, …, x_{n})
− … + z(x_{1}, …, x_{n−ℓ−2}, 1, x_{n−ℓ−1} · x_{n−ℓ}, x_{n−ℓ+1}, …, x_{n})
− z(x_{1}, …, x_{n−ℓ−1}, x_{n−ℓ} · x_{n−ℓ+1}, 1, x_{n−ℓ+2}, …, x_{n})
+ … + (−1)^{ℓ}z(x_{1}, …, x_{n−ℓ−1}, x_{n−ℓ} · x_{n−ℓ+1}, x_{n−ℓ+2}, …, x_{n}, 1).

Also

H^{ℓ}dz = (−1)^{n−ℓ−1}z(1, x_{1}, …, x_{n−ℓ−2}, x_{n−ℓ−1} · x_{n−ℓ}, x_{n−ℓ+1}, …, x_{n})
− … + (−1)^{ℓ}z(x_{1}, …, x_{n−ℓ−2}, x_{n−ℓ−1} · x_{n−ℓ}, x_{n−ℓ+1}, …, x_{n}, 1)

It follows that

dH^{ℓ}z + H^{ℓ}dz = − z(x_{1}, …, x_{n−ℓ−1}, x_{n−ℓ} · x_{n−ℓ+1}, 1, x_{n−ℓ+2}, …, x_{n})

+ … + (−1)^{ℓ}z(x_{1}, …, x_{n−ℓ−1}, x_{n−ℓ} · x_{n−ℓ+1}, x_{n−ℓ+2}, …, x_{n}, 1)

+ z(x_{1}, …, x_{n−ℓ−2}, x_{n−ℓ−1} · x_{n−ℓ}, 1, x_{n−ℓ+1}, …, x_{n})

− … + (−1)^{ℓ}z(x_{1}, …, x_{n−ℓ−2}, x_{n−ℓ−1} · x_{n−ℓ}, x_{n−ℓ+1}, …, x_{n}, 1)

Suppose now z ∈ Z_{c}(n)_{0i} for some i ≤ n − 1. If ℓ ≤ n − i − 2, we see from the above that dH^{ℓ}z + H^{ℓ}dz = 0. Thus the first few factors of φ may act as the identity on z, but eventually we get to ℓ = n − i − 1. At this point,

z′ = z − (dH^{ℓ}z + H^{ℓ}dz)
= z(x_{1}, …, x_{n}) − z(x_{1}, …, x_{n−ℓ−2}, x_{n−ℓ−1} · x_{n−ℓ}, 1, x_{n−ℓ+1}, …, x_{n}).

Note that d_{j}^{1}z′ = 0 for j ≥ i + 1, i.e. z′ ∈ Z_{c}(n)_{0,i−1}. The next factor of φ is ℓ = n − i = n − (i − 1) − 1, so z′ is mapped to some z′′ ∈ Z_{c}(n)_{0,i−2}, etc. Once z ∈ Z_{c}(n)_{00}, it follows again from the above that dH^{ℓ}z + H^{ℓ}dz = 0 so factors of φ are the identity. Thus, φ : Z_{c}(∗)_{0} → Z_{c}(∗)_{00}, φ is the identity on Z_{c}(∗)_{00}, and φ composed with the inclusion is homotopic to the identity on Z_{c}(∗)_{0}. QED