On Additive Higher Chow Groups of Affine Schemes

Amalendu Krishna and Jinhyun Park

Received: July 5, 2015 Revised: December 18, 2015

Communicated by Takeshi Saito

Abstract. We show that the multivariate additive higher Chow groups of a smooth affine k-scheme Spec (R) essentially of finite type over a perfect field k of characteristic ≠ 2 form a differential graded module over the big de Rham-Witt complex 𝕎_{m}Ω_{R}^{•}. In the univariate case, we show that additive higher Chow groups of Spec (R) form a Witt-complex over R. We use these structures to prove an étale descent for multivariate additive higher Chow groups.

2010 Mathematics Subject Classification: Primary 14C25; Secondary 13F35, 19E15 Keywords and Phrases: algebraic cycle, additive higher Chow group, Witt vectors, de Rham-Witt complex

1. Introduction

The additive higher Chow groups TCH^{q}(X, n; m) emerged originally in [5] in part as an attempt to understand certain relative higher algebraic K-groups of schemes in terms of algebraic cycles. Since then, several papers [16], [17], [18], [19], [26], [27], [28] have studied various aspects of these groups. But lack of a suitable moving lemma for smooth affine varieties has been a hindrance in studies of their local behaviors. Its projective sibling was known by [17]. During the period of stagnation, the subject has evolved into the notion of ‘cycles with modulus’ CH^{q}(X|D, n) by Binda-Kerz-Saito in [1], [15] associated to pairs (X, D) of schemes and effective Cartier divisors D, setting a more flexible ground, while this desired moving lemma for the affine case was obtained by W. Kai [14] (See Theorem 4.1). The above developments now propel the authors to continue their program of realizing the relative K-theory K_{n}(X × Spec k[t]/(t^{m+1}), (t)) in terms of additive higher Chow groups. More specifically, one of the aims in the program considered in this paper is to understand via additive higher Chow groups, the part of the above relative K-groups which was proven in [2] to give the crystalline cohomology. This part turned out to be isomorphic to the de Rham-Witt complexes as seen in [12]. This article is the first of the authors’ papers that relate the additive higher Chow groups to the big de Rham-Witt complexes 𝕎_{m}Ω_{R}^{•} of [8] and to the crystalline cohomology theory. This gives a motivic description of the latter two objects. While the general notion of cycles with modulus for (X, D) provides a wider picture, the additive higher Chow groups still have a non-trivial operation not shared by the general case. One such is an analogue of the Pontryagin product on homology groups of Lie groups, which turns the additive higher Chow groups into a differential graded algebra (DGA). This product is induced by the structure of algebraic groups on 𝔸^{1} and 𝔾_{m} and their action on X×𝔸^{r} =: X[r] for r ≥ 1. The usefulness of such a product was already observed in the earliest papers on additive 0-cycles by Bloch-Esnault [5] and Rülling [28]. This product on higher dimensional additive higher Chow cycles was given in [19] for smooth projective varieties. In §5 of this paper, we extend this product structure in two directions: (1) toward multivariate additive higher Chow groups and (2) on smooth affine varieties. In doing so, we generalize some of the necessary tools, such as the following normalization theorem, proven as Theorem 3.2. Necessary definitions are recalled in §2.

Theorem 1.1. Let X be a smooth scheme which is either quasi-projective or essentially of finite type over a field k. Let D be an effective Cartier divisor on X. Then each cycle class in CH^{q}(X|D, n) has a representative, all of whose codimension 1 faces are trivial.

The above theorem for ordinary higher Chow groups was proven by Bloch and has been a useful tool in dealing with algebraic cycles. In this paper, we use the above theorem to construct the following structure of differential graded algebra and differential graded modules on the multivariate additive higher Chow groups, where Theorem 1.2 is proven in Theorems 7.1, 7.10, and 7.11, while Theorem 1.3 is proven in Theorem 6.13.

Theorem 1.2. Let X be a smooth scheme which is either affine essentially of finite type or projective over a perfect field k of characteristic ≠ 2

(1) The additive higher Chow groups {TCH^{q}(X, n; m)}_{q,n,m∈ℕ} has a functorial structure of a restricted Witt-complex over k.

(2) If X = Spec (R) is affine, then {TCH^{q}(X, n; m)}_{q,n,m∈ℕ} has a structure of a restricted Witt-complex over R.

(3) For X as in (2), there is a natural map of restricted Witt-complexes τ_{n,m}^{R} : 𝕎_{m}Ω_{R}^{n−1} → TCH^{n}(R, n; m).

Theorem 1.3. Let r ≥ 1. For a smooth scheme X which is either affine essentially of finite type or projective over a perfect field k of characteristic ≠ 2, the multivariate additive higher Chow groups \{CH^{q}(X[r]|D\underline{m}, n)\}_{q,n≥0} with modulus \underline{m} = (m_{1}, ⋯, m_{r}), where m_{i} ≥ 1, form a differential graded module over the DGA \{TCH^{q}(X, n; |\underline{m}| − 1)\}_{q,n≥1}, where |\underline{m}| = ∑_{i=1}^{r} m_{i}. In particular, each CH^{q}(X[r]|D\underline{m}, n) is a 𝕎_{(|\underline{m}|−1)}(R)-module, when X = Spec (R) is affine.

The above structures on the univariate and multivariate additive higher Chow groups suggest an expectation that these groups may describe the algebraic K-theory relative to nilpotent thickenings of the coordinate axes in an affine space over a smooth scheme. The calculations of such relative K-theory by Hesselholt in [9] and [10] show that any potential motivic cohomology which describes the above relative K-theory may have such a structure. As part of our program of connecting the additive higher Chow groups with the relative K-theory, we show in [22] that the above map τ_{n,m}^{R} is an isomorphism when X is semi-local in addition, and we show how one deduces crystalline cohomology from additive higher Chow groups. The results of this paper form a crucial part in the process. Recall that the higher Chow groups of Bloch and algebraic K-theory do not satisfy étale descent with integral coefficients. As an application of Theorem 1.3, we show that the étale descent is actually true for the multivariate additive higher Chow groups in the following setting:

Theorem 1.4. Let r ≥ 1 and let X be a smooth scheme which is either affine essentially of finite type or projective over a perfect field k of characteristic ≠ 2. Let G be a finite group of order prime to char(k), acting freely on X with the quotient f : X → X/G. Then for all q, n ≥ 0 and and \underline{m} = (m_{1}, ⋯, m_{r}) with m_{i} ≥ 1 for 1 ≤ i ≤ r, the pull-back map f^{∗} induces an isomorphism

CH^{q}(X/G[r]|D\underline{m}, n) ⟶^{≃} H^{0}(G, CH^{q}(X[r]|D\underline{m}, n)).

Note that the quotient X/G exists under the hypothesis on X. Since the corresponding descent is not yet known for the relative K-theory of nilpotent thickenings of the coordinate axes in an affine space over a smooth scheme, the above theorem suggests that this descent could be indeed true for the relative K-theory.

Conventions. In this paper, k will denote the base field which will be assumed to be perfect after §4. A k-scheme is a separated scheme of finite type over k. A k-variety is a reduced k-scheme. The product X × Y means usually X ×_{k} Y, unless said otherwise. We let Sch_{k} be the category of k-schemes, Sm_{k} of smooth k-schemes, and SmAff_{k} of smooth affine k-schemes. A scheme essentially of finite type is a scheme obtained by localizing at a finite subset (including ∅) of a finite type k-scheme. For C = Sch_{k}, Sm_{k}, SmAff_{k}, we let C^{ess} be the extension of the category C obtained by localizing at a finite subset (including ∅) of objects in C. We let SmLoc_{k} be the category of smooth semilocal k-schemes essentially of finite type over k. So, SmAff_{k}^{ess} = SmAff_{k} ∪ SmLoc_{k} for the objects. When we say a semi-local k-scheme, we always mean one that is essentially of finite type over k. Let SmProj_{k} be the category of smooth projective k-schemes.

2. Recollection of basic definitions

For ℙ^{1} = Proj_{k}(k[s_{0}, s_{1}]), we let y = s_{1}/s_{0} its coordinate. Let □ := ℙ^{1}\{1}. For n ≥ 1, let (y_{1}, ⋯, y_{n}) ∈ □^{n} be the coordinates. A face F ⊂ □^{n} means a closed subscheme defined by the set of equations of the form {y_{i_{1}} = ϵ_{1}, ⋯, y_{i_{s}} = ϵ_{s}} for an increasing sequence {i_{j}|1 ≤ j ≤ s} ⊂ {1, ⋯, n} and ϵ_{j} ∈ {0, ∞}. We allow s = 0, in which case F = □^{n}. Let \overline{□} := ℙ^{1}. A face of \overline{□}^{n} is the closure of a face in □^{n}. For 1 ≤ i ≤ n, let F_{n,i}^{1} ⊂ \overline{□}^{n} be the closed subscheme given by {y_{i} = 1}. Let F_{n}^{1} := ∑_{i=1}^{n} F_{n,i}^{1}, which is the cycle associated to the closed

subscheme \overline{□}^{n} \ □^{n}. Let □^{0} = \overline{□}^{0} := Spec (k). Let ι_{n,i,ϵ} : □^{n−1} ↪ □^{n} be the inclusion (y_{1}, ⋯, y_{n−1}) ↦ (y_{1}, ⋯, y_{i−1}, ϵ, y_{i}, ⋯, y_{n−1}).

2.1. Cycles with modulus. Let X ∈ Sch_{k}^{ess}. Recall ([21, §2]) that for effective Cartier divisors D_{1} and D_{2} on X, we say D_{1} ≤ D_{2} if D_{1} + D = D_{2} for some effective Cartier divisor D on X. A scheme with an effective divisor (sed) is a pair (X, D), where X ∈ Sch_{k}^{ess} and D an effective Cartier divisor. A morphism f : (Y, E) → (X, D) of seds is a morphism f : Y → X in Sch_{k}^{ess} such that f^{∗}(D) is defined as a Cartier divisor on Y and f^{∗}(D) ≤ E. In particular, f^{−1}(D) ⊂ E. If f : Y → X is a morphism of k-schemes, and (X, D) is a sed such that f^{−1}(D) = ∅, then f : (Y, ∅) → (X, D) is a morphism of seds.

Definition 2.1 ([1], [15]). Let (X, D) and (\overline{Y} , E) be schemes with effective divisors. Let Y = \overline{Y} \ E. Let V ⊂ X × Y be an integral closed subscheme with closure \overline{V} ⊂ X × \overline{Y}. We say V has modulus D (relative to E) if ν_{V}^{∗} (D × \overline{Y}) ≤ ν_{V}^{∗} (X ×E) on \overline{V}^{N}, where ν_{V} : \overline{V}^{N} → \overline{V} ↪ X ×\overline{Y} is the normalization followed by the closed immersion.

Recall the following containment lemma from [21, Proposition 2.4] (see also [1, Lemma 2.1] and [17, Proposition 2.4]):

Proposition 2.2. Let (X, D) and (\overline{Y} , E) be schemes with effective divisors and Y = \overline{Y} \ E. If V ⊂ X × Y is a closed subscheme with modulus D relative to E, then any closed subscheme W ⊂ V also has modulus D relative to E.

Definition 2.3 ([1], [15]). Let (X, D) be a scheme with an effective divisor. For s ∈ ℤ and n ≥ 0, let \underline{z}_{s}(X|D, n) be the free abelian group on integral closed subschemes V ⊂ X × □^{n} of dimension s + n satisfying the following conditions:

(1) (Face condition) for each face F ⊂ □^{n}, V intersects X × F properly.

(2) (Modulus condition) V has modulus D relative to F_{n}^{1} on X × □^{n}.

We usually drop the phrase “relative to F_{n}^{1}” for simplicity. A cycle in \underline{z}_{s}(X|D, n) is called an admissible cycle with modulus D. One checks that (n ↦ \underline{z}_{s}(X|D, n)) is a cubical abelian group. In particular, the groups \underline{z}_{s}(X|D, n) form a complex with the boundary map ∂ = ∑_{i=1}^{n}(−1)^{i}(∂_{i}^{∞} − ∂_{i}^{0}), where ∂_{i}^{ϵ} = ι_{n,i,ϵ}^{∗}.

Definition 2.4 ([1], [15]). The complex (z_{s}(X|D, •), ∂) is the nondegenerate complex associated to (n ↦ \underline{z}_{s}(X|D, n)), i.e., z_{s}(X|D, n) := \underline{z}_{s}(X|D, n)/\underline{z}_{s}(X|D, n)_{degn}. The homology CH_{s}(X|D, n) := H_{n}(z_{s}(X|D, •)) for n ≥ 0 is called higher Chow group of X with modulus D. If X is equidimensional of dimension d, for q ≥ 0, we write CH^{q}(X|D, n) = CH_{d−q}(X|D, n).

Here is a special case from [21]:

Definition 2.5. Let X ∈ Sch_{k}^{ess}. For r ≥ 1, let X[r] := X × 𝔸^{r}. When (t_{1}, ⋯, t_{r}) ∈ 𝔸^{r} are the coordinates, and m_{1}, ⋯, m_{r} ≥ 1 are integers, let D\underline{m} be the divisor on X[r] given by the equation {t_{1}^{m_{1}} ⋯ t_{r}^{m_{r}} = 0}. The groups CH^{q}(X[r]|D\underline{m}, n) are called multivariate additive higher Chow groups of X. For simplicity, we often say “a cycle with modulus \underline{m}” for “a cycle with modulus D\underline{m}.” For an r-tuple of integers \underline{m} = (m_{1}, ⋯, m_{r}), we write |\underline{m}| = ∑_{i=1}^{r} m_{i}. We shall say that \underline{m} ≥ p if m_{i} ≥ p for each i. When r = 1, we obtain additive higher Chow groups, and as in [19], we often use the older notations Tz^{q}(X, n + 1; m − 1) for z^{q}(X[1]|D_{m}, n) and TCH^{q}(X, n + 1; m − 1) for CH^{q}(X[1]|D_{m}, n). In such cases, note that the modulus m is shifted by 1 from the above sense.

Definition 2.6. Let W be a finite set of locally closed subsets of X and let e : W → ℤ_{≥0} be a set function. Let \underline{z}_{W,e}^{q}(X|D, n) be the subgroup generated by integral cycles Z ∈ \underline{z}^{q}(X|D, n) such that for each W ∈ W and each face F ⊂ □^{n}, we have codim_{W ×F} (Z ∩ (W × F)) ≥ q − e(W). They form a subcomplex \underline{z}_{W,e}^{q}(X|D, •) of \underline{z}^{q}(X|D, •). Modding out by degenerate cycles, we obtain the subcomplex z_{W,e}^{q}(X|D, •) ⊂ z^{q}(X|D, •). We write z_{W}^{q}(X|D, •) := z_{W,0}^{q}(X|D, •). For additive higher Chow cycles, we write \underline{Tz}_{W}^{q}(X, n; m) for \underline{z}_{W[1]}^{q}(X[1]|D_{m+1}, n − 1), where W[1] = \{W [1] | W ∈ W\}.

Here are some basic lemmas used in the paper:

Lemma 2.7 ([21, Lemma 2.2]). Let f : Y → X be a dominant map of normal integral k-schemes. Let D be a Cartier divisor on X such that the generic points of Supp(D) are contained in f(Y). Suppose that f^{∗}(D) ≥ 0 on Y. Then D ≥ 0 on X.

Lemma 2.8 ([21, Lemma 2.9]). Let f : Y → X be a proper morphism of quasiprojective k-varieties. Let D ⊂ X be an effective Cartier divisor such that f(Y) ⊄ D. Let Z ∈ z^{q}(Y |f^{∗}(D), n) be an irreducible cycle. Let W = f(Z) on X × □^{n}. Then W ∈ z^{s}(X|D, n), where s = codim_{X×□^{n}}(W).

Lemma 2.9. Let X be a k-scheme, and let {U_{i}}_{i∈I} be an open cover of X. Let Z ∈ z^{q}(X × □^{n}) and let Z_{U_{i}} be the flat pull-back to U_{i} × □^{n}. Then Z ∈ z^{q}(X|D, n) if and only if for each i ∈ I, we have Z_{U_{i}} ∈ z^{q}(U_{i}|D_{U_{i}}, n), where D_{U_{i}} is the restriction of D on U_{i}.

Proof. The direction (⇒) is obvious since flat pull-backs respect admissibility of cycles with modulus by [21, Proposition 2.12]. For the direction (⇐), we may assume Z is irreducible. In this case, it is easily checked that the face and the modulus conditions are both local on the base X. □

2.2. de Rham-Witt complexes.

2.2.1. Ring of big Witt-vectors. Let R be a commutative ring with unit. We recall the definition of the ring of big Witt-vectors of R (see [11, §4] or [28, Appendix A]). A truncation set S ⊂ ℕ is a non-empty subset such that if s ∈ S and t|s, then t ∈ S. As a set, let 𝕎_{S}(R) := R^{S} and define the map w : 𝕎_{S}(R) → R^{S} by sending a = (a_{s})_{s∈S} to w(a) = (w(a)_{s})_{s∈S}, where w(a)_{s} := ∑_{t|s} ta_{t}^{s/t}. When R^{S} on the target of w is given the component-wise ring structure, it is known that there is a unique functorial ring structure on 𝕎_{S}(R) such that w is a ring homomorphism (see [11, Proposition 1.2]). When S = {1, ⋯, m}, we write 𝕎_{m}(R) := 𝕎_{S}(R). There is another description. Let 𝕎(R) := 𝕎_{ℕ}(R). Consider the multiplicative group (1 + tR[[t]])^{×}, where t is an indeterminate. Then there is a natural bijection 𝕎(R) ≃ (1+tR[[t]])^{×}, where the addition in 𝕎(R) corresponds to the multiplication of formal power series. For a truncation set S, we can describe 𝕎_{S}(R) as the quotient of (1+tR[[t]])^{×} by a suitable subgroup I_{S}. See [28, A.7] for details. In case S = {1, ⋯, m}, we can write 𝕎_{m}(R) = (1 + tR[[t]])^{×}/(1 + t^{m+1}R[[t]])^{×} as an additive group. For a ∈ R, the Teichmüller lift [a] ∈ 𝕎_{S}(R) corresponds to the image of 1 − at ∈ (1 + tR[[t]])^{×}. This yields a multiplicative map [−] : R → 𝕎_{S}(R). The additive identity element of 𝕎_{m}(R) corresponds to the unit polynomial 1 and the multiplicative identity element corresponds to the polynomial 1 − t.

2.2.2. de Rham-Witt complex. Let p be an odd prime and R be a ℤ_{(p)}-algebra.^{1} For each truncation set S, there is a differential graded algebra 𝕎_{S}Ω_{R}^{•} called the big de Rham-Witt complex over R. This defines a contravariant functor on the category of truncation sets. This is an initial object in the category of V -complexes and in the category of Witt-complexes over R. For details, see [8] and [28, §1]. When S is a finite truncation set, we have 𝕎_{S}Ω_{R}^{•} = Ω_{𝕎_{S}(R)/ℤ}^{•}/N_{S}^{•}, where N_{S}^{•} is the differential graded ideal given by some generators ([28, Proposition 1.2]). In case S = {1, 2, ⋯, m}, we write 𝕎_{m}Ω_{R}^{•} for this object. Here is another relevant object for this paper from [8, Definition 1.1.1]; a restricted Witt-complex over R is a pro-system of differential graded ℤalgebras ((E_{m})_{m∈ℕ}, ℜ : E_{m+1} → E_{m}), with homomorphisms of graded rings (F_{r} : E_{rm+r−1} → E_{m})_{m,r∈ℕ} called the Frobenius maps, and homomorphisms of graded groups (V_{r} : E_{m} → E_{rm+r−1})_{m,r∈ℕ} called the Verschiebung maps, satisfying the following relations for all n, r, s ∈ ℕ:

(i) ℜF_{r} = F_{r}ℜ^{r}, ℜ^{r}V_{r} = V_{r}ℜ, F_{1} = V_{1} = Id, F_{r}F_{s} = F_{rs}, V_{r}V_{s} = V_{rs}; (ii) F_{r}V_{r} = r. When (r, s) = 1, F_{r}V_{s} = V_{s}F_{r} on E_{rm+r−1}; (iii) V_{r}(F_{r}(x)y) = xV_{r}(y) for all x ∈ E_{rm+r−1} and y ∈ E_{m}; (projection

formula) (iv) F_{r}dV_{r} = d, where d is the differential of the DGAs.

Furthermore, we require that there is a homomorphism of pro-rings (λ : 𝕎_{m}(R) → E_{m}^{0})_{m∈ℕ} that commutes with F_{r} and V_{r}, satisfying

1A definition of Witt-complex over a more general ring R can be found in [11, Definition 4.1].

(v) F_{r}dλ([a]) = λ([a]^{r−1})dλ([a]) for all a ∈ R and r ∈ ℕ.

The pro-system {𝕎_{m}Ω_{R}^{•}}_{m≥1} is the initial object in the category of restricted Witt-complexes over R (See [28, Proposition 1.15]).

3. Normalization theorem

Let k be any field. The aim of this section is to prove Theorem 3.2. Such results were known when D = ∅, or when X is replaced by X × 𝔸^{1} with D = {t^{m+1} = 0} for t ∈ 𝔸^{1}. We generalize it to higher Chow groups with modulus.

Definition 3.1. Let (X, D) be a scheme with an effective divisor. Let z_{N}^{q}(X|D, n) be the subgroup of cycles α ∈ z^{q}(X|D, n) such that ∂_{i}^{0}(α) = 0 for all 1 ≤ i ≤ n and ∂_{i}^{∞}(α) = 0 for 2 ≤ i ≤ n. One checks that ∂_{1}^{∞}∘∂_{1}^{∞} = 0. Writing ∂_{1}^{∞} as ∂^{N}, we obtain a subcomplex ι : (z_{N}^{q}(X|D, •), ∂^{N}) ↪ (z^{q}(X|D, •), ∂).

Theorem 3.2. Let X ∈ Sm_{k}^{ess} and let D ⊂ X be an effective Cartier divisor. Then ι : z_{N}^{q}(X|D, •) → z^{q}(X|D, •) is a quasi-isomorphism. In particular, every cycle class in CH^{q}(X|D, n) can be represented by a cycle α such that ∂_{i}^{ϵ}(α) = 0 for all 1 ≤ i ≤ n and ϵ = 0, ∞.

Let Cube be the standard category of cubes (see [24, §1]) so that a cubical abelian group is a functor Cube^{op} → (Ab). Recall also from loc.cit. that an extended cubical abelian is a functor ECube^{op} → (Ab), where ECube is the smallest symmetric monoidal subcategory of Sets containing Cube and the morphism μ : \underline{2} → \underline{1}. The essential point of the proof of Theorem 3.2 is

Theorem 3.3. Let X ∈ Sm_{k}^{ess} and D ⊂ X be an effective Cartier divisor. Then (\underline{n} ↦ z^{q}(X|D; n)) is an extended cubical abelian group.

If Theorem 3.3 holds, then [24, Lemma 1.6] implies Theorem 3.2. We suppose (X, D) is as in Theorem 3.2 in what follows. The idea is similar to that of [19, Appendix]. Let q_{1} : □^{2} → □ be the morphism (y_{1}, y_{2}) ↦ y_{1} + y_{2} − y_{1}y_{2} if y_{1}, y_{2} ≠ ∞, and (y_{1}, y_{2}) ↦ ∞ if y_{1} or y_{2} = ∞. Under the identification ψ : □ ≃ 𝔸^{1} given by y ↦ 1/(1 − y) (which sends {∞, 0} to {0, 1}), this map q_{1} is equivalent to q_{1,ψ} : 𝔸^{2} → 𝔸^{1} given by (y_{1}, y_{2}) ↦ y_{1}y_{2}. For our convenience, we use this □_{ψ} := (𝔸^{1}, {0, 1}) and cycles on X × □_{ψ}^{n}. The boundary operator is ∂ = ∑_{i=1}^{n}(−1)^{i}(∂_{i}^{0} − ∂_{i}^{1}), and we replace F_{n,i}^{1} by F_{n,i}^{∞} = {y_{i} = ∞}. We write F_{n}^{∞} = ∑_{i=1}^{n} F_{n,i}^{∞}. We write \overline{□}_{ψ} = (ℙ^{1}, \{0, 1\}). The group of admissible cycles is \underline{z}_{ψ}^{q}(X|D, n). Consider q_{n,ψ} : X ×□_{ψ}^{n+1} → X ×□_{ψ}^{n} given by (x, y_{1}, ⋯, y_{n+1}) ↦ (x, y_{1}, ⋯, y_{n−1}, y_{n}y_{n+1}).

Proposition 3.4. For Z ∈ z_{ψ}^{q}(X|D, n), we have q_{n,ψ}^{∗}(Z) ∈ z_{ψ}^{q}(X|D, n + 1).

The delicacy of its proof lies in that the product map q_{1,ψ} : 𝔸^{2} → 𝔸^{1} does not extend to a morphism (ℙ^{1})^{2} → ℙ^{1} of varieties so that checking the modulus condition becomes nontrivial. We use a correspondence instead. For n ≥ 1, let i_{n} : W_{n} ↪ X × □_{ψ}^{n+1} × \overline{□}_{ψ}^{1} be the closed subscheme defined by the equation

u_{0}y_{n}y_{n+1} = u_{1}, where (y_{1}, ⋯, y_{n+1}) ∈ □_{ψ}^{n+1} and (u_{0}; u_{1}) ∈ \overline{□}_{ψ}^{1} are the coordi-

nates. Let y := u_{1}/u_{0}. Its Zariski closure \overline{W}_{n} ↪ X ×\overline{□}_{ψ}^{n+1}×\overline{□}_{ψ}^{1} is given by the equation u_{0}u_{n,1}u_{n+1,1} = u_{1}u_{n,0}u_{n+1,0}, where (u_{1,0}, u_{1,1}), ⋯, (u_{n+1,0}, u_{n+1,1}) are the homogeneous coordinates of \overline{□}_{ψ}^{n+1} with y_{i} = u_{i,1}/u_{i,0}.

Consider θ_{n} : X × □_{ψ}^{n+1} × \overline{□}_{ψ}^{1} → X × □_{ψ}^{n} given by (x, y_{1}, ⋯, y_{n+1}, (u_{0}; u_{1})) ↦ (x, y_{1}, ⋯, y_{n−1}, y_{n}y_{n+1}), and let π_{n} := θ_{n}|_{W_{n}}. To extend this π_{n} to a morphism \overline{π}_{n} on \overline{W}_{n}, we use the projection \overline{θ}_{n} : X ×\overline{□}_{ψ}^{n+1} ×\overline{□}_{ψ}^{1} → X ×\overline{□}_{ψ}^{n−1} ×\overline{□}_{ψ}^{1}, that drops the coordinates (u_{n,0}; u_{n,1}) and (u_{n+1,0}; u_{n+1,1}), and the projection p_{n} : X × □_{ψ}^{n+1} × \overline{□}_{ψ}^{1} → X × □_{ψ}^{n+1}, that drops the last coordinate (u_{0}; u_{1}).

Lemma 3.5. (1) W_{n} ∩ {u_{0} = 0} = ∅, so that W_{n} ⊂ X × □_{ψ}^{n+1} × □_{ψ}^{1}. (2) \overline{θ}_{n}|_{W_{n}} = π_{n}. Thus, we define \overline{π}_{n} := \overline{θ}_{n}|_{\overline{W}_{n}}, which extends π_{n}. (3) The varieties W_{n} and \overline{W}_{n} are smooth. (4) Both π_{n} and \overline{π}_{n} are surjective flat morphisms of relative dimension 1.

Proof. Its proof is almost identical to that of [19, Lemma A.5]. Part (1) follows from the defining equation of W_{n}, and (2) holds by definition. Let ρ_{n} := p_{n}|_{W_{n}} : W_{n} → X×□_{ψ}^{n+1}. Since X is smooth, using Jacobian criterion we check that W_{n} is smooth. Furthermore, ρ_{n} is an isomorphism with the obvious inverse. Under this identification, the morphism π_{n} can also be regarded as the projection (x, y_{1}, ⋯, y_{n}, y) ↦ (x, y_{1}, ⋯, y_{n−1}, y) that drops y_{n}. In particular, π_{n} is a smooth and surjective of relative dimension 1. To check that \overline{W}_{n} is smooth, one can do it locally on each open set where each of u_{n,i}, u_{n+1,i}, u_{i} is nonzero for i = 0, 1. In each such open set, the equation for \overline{W}_{n} takes the same form as for W_{n}, so that it is smooth again by Jacobian criterion. Similarly as for π_{n}, one sees \overline{π}_{n} is of relative dimension 1. Since \overline{θ}_{n} is projective and π_{n} is surjective, the morphism \overline{π}_{n} is projective and surjective. So, since \overline{W}_{n} is smooth, the map \overline{π}_{n} is flat by [7, Exercise III-10.9, p.276]. Thus, we have (3) and (4). □

Lemma 3.6. Let n ≥ 1 and let Z ⊂ X ×□_{ψ}^{n} be a closed subscheme with modulus D. Then Z′ := (i_{n})_{∗}(π_{n}^{∗}(Z)) also has modulus D.

Proof. Let \overline{Z} and \overline{Z}′ be the Zariski closures of Z and Z′ in X × \overline{□}_{ψ}^{n} and

X × \overline{□}_{ψ}^{n+1}, respectively. By Lemma 3.5 and the projectivity of \overline{θ}_{n}, we see that \overline{θ}_{n}(\overline{Z}′) = \overline{Z}. Consider the commutative diagram

(3.1)  \xymatrix{\overline{Z}′^{N} \ar[d]^{f} \ar[r]^{g} \ar[rr]^{ν_{Z′}} & \overline{W}_{n} \ar@{^{(}->}[r]^{^{\overline{i}^{n}}} \ar[dr]^{_{\overline{π}_{n}}} & X × \overline{□}_{ψ}^{n+1} × \overline{□}_{ψ}^{1} \ar[d]^{_{\overline{θ}_{n}}} \\ _{\overline{Z}_{N}} \ar[rr]^{ν_{Z}} &  & X × \overline{□}_{ψ}^{n},}

Documenta Mathematica 21 (2016) 49–89

where f is induced by the surjection \overline{θ}_{n}|\overline{Z}′ : \overline{Z}′ → \overline{Z}, the maps g and ν_{Z} are normalizations of \overline{Z}′ and \overline{Z} composed with the closed immersions, and ν_{Z′} := \overline{i}_{n} ∘ g. By the definition of \overline{θ}_{n}, we have \overline{θ}_{n}^{∗}(D × \overline{□}_{ψ}^{n}) = D × \overline{□}_{ψ}^{n+2}, \overline{θ}_{n}^{∗}(F_{n,n}^{∞}) = F_{n+2,n+2}^{∞}, while \overline{θ}_{n}^{∗}(F_{n,i}^{∞}) = F_{n+2,i}^{∞} for 1 ≤ i ≤ n − 1. By the defining equation of \overline{W}_{n}, we have \overline{π}_{n}^{∗}F_{n,n}^{∞} = \overline{i}_{n}^{∗}F_{n+2,n+2}^{∞} = \overline{i}_{n}^{∗}\{u_{0} = 0\} ≤ \overline{i}_{n}^{∗}(\{u_{n,0} = 0\} + \{u_{n+1,0} = 0\}) = \overline{i}_{n}^{∗}(F_{n+2,n}^{∞} + F_{n+2,n+1}^{∞}). Thus, ν_{Z′}^{∗} \overline{θ}_{n}^{∗} ∑_{i=1}^{n} F_{n,i}^{∞} = ∑_{i=1}^{n−1} ν_{Z′}^{∗}F_{n+2,i}^{∞} + g^{∗}\overline{π}_{n}^{∗}F_{n,n}^{∞} ≤ ∑_{i=1}^{n−1} ν_{Z′}^{∗}F_{n+2,i}^{∞} + g^{∗}\overline{i}_{n}^{∗}(F_{n+2,n}^{∞} + F_{n+2,n+1}^{∞}) = ∑_{i=1}^{n+1} ν_{Z′}^{∗}F_{n+2,i}^{∞} ≤ ∑_{i=1}^{n+2} ν_{Z′}^{∗}F_{n+2,i}^{∞}. (In case n = 1, we just ignore the terms with ∑_{i=1}^{n−1} in the above.) That Z has modulus D means ν_{Z}^{∗}(D × \overline{□}_{ψ}^{n}) ≤ ∑_{i=1}^{n} ν_{Z}^{∗}F_{n,i}^{∞}. Applying f^{∗} and

using (3.1), we have ν_{Z′}^{∗}(D×\overline{□}_{ψ}^{n+2}) = ν_{Z′}^{∗} \overline{θ}_{n}^{∗}(D×\overline{□}_{ψ}^{n}) ≤ ν_{Z′}^{∗} \overline{θ}_{n}^{∗} ∑_{i=1}^{n} F_{n,i}^{∞}, which is bounded by ∑_{i=1}^{n+2} ν_{Z′}^{∗}F_{n+2,i}^{∞} as we saw above. This means Z′ has modulus D. □

Definition 3.7. For any closed subscheme Z ⊂ X × □_{ψ}^{n}, we define W_{n}(Z) := p_{n∗}i_{n∗}π_{n}^{∗}(Z), which is closed in X × □_{ψ}^{n+1}.

Lemma 3.8. Let n ≥ 1. If a closed subscheme Z ⊂ X × □_{ψ}^{n} intersects all faces properly, then W_{n}(Z) intersects all faces of X × □_{ψ}^{n+1} properly.

Proof. Our W_{n} is equal to τ^{∗}τ_{n}^{∗}τ_{n+1}^{∗}W_{n}^{X}, where W_{n}^{X} is that of [23, Lemma 4.1], and τ, τ_{n}, τ_{n+1} are the involutions (x ↦ 1 − x) for y, y_{n}, y_{n+1}, respectively. So, the lemma is a special case of loc.cit. □

Proof of Proposition 3.4. Consider the commutative diagram

\xymatrix{W_{n} \ar[d]_{π_{n}} \ar[dr]^{ρ_{n}=p_{n}|_{Wn}} \ar@{^{(}->}[r]^{i_{n}} & X × □_{ψ}^{n+1} × \overline{□}_{ψ} \ar[d]^{p_{n}} \\ X × □_{ψ}^{n} & X × □_{ψ}^{n+1}. \ar[l]_{q_{n,ψ}}}

By Lemma 3.5, ρ_{n} is an isomorphism so that ρ_{n∗}i_{n}^{∗}p_{n}^{∗} = Id. Hence, q_{n,ψ}^{∗}(Z) = ρ_{n∗}i_{n}^{∗}p_{n}^{∗}q_{n,ψ}^{∗}(Z) =^{†} ρ_{n∗}π_{n}^{∗}(Z) =^{‡} p_{n∗}i_{n∗}π_{n}^{∗}(Z) = W_{n}(Z), where †, ‡ are due to commutativity. So, we have reduced to showing that W_{n}(Z) ∈ z_{ψ}^{q}(X|D, n + 1). But, by Lemmas 3.6 and 3.8, we have i_{n∗}π_{n}^{∗}(Z) ∈ z_{ψ}^{q+1}(X × ℙ^{1}|D × ℙ^{1}, n + 1). Now, for the projection p_{n}, by Lemma 2.8, we have W_{n}(Z) = p_{n∗}i_{n∗}π_{n}^{∗}(Z) ∈ z_{ψ}^{q}(X|D, n + 1). This proves Proposition 3.4. □

Proof of Theorem 3.3. Since we know that (\underline{n} ↦ z^{q}(X|D; n)) is a cubical abelian group, every morphism h : \underline{r} → \underline{s} in Cube induces a morphism h : □^{r} → □^{s} which gives a homomorphism h^{∗} : z^{q}(X|D, s) → z^{q}(X|D, r). Furthermore, the morphism μ : \underline{2} → \underline{1} induces the morphism q_{1} : □^{2} → □^{1} of varieties, and for each Z ∈ z^{q}(X|D, 1), we have q_{1}^{∗}(Z) ∈ z^{q}(X|D, 2). Indeed, under the isomorphism ψ : □ ≃ 𝔸^{1}, y ↦ 1/(1 − y), this is equivalent to show that q_{1,ψ}^{∗} sends admissible cycles to admissible cycles, which we know by Proposition 3.4. So, it only remains to show the following “stability under products”: if h_{i} : \underline{r_{i}} → \underline{s_{i}}, i = 1, 2, are morphisms in ECube such that the corresponding morphisms h_{i} : □^{r_{i}} → □^{s_{i}} induce homomorphisms h_{i}^{∗} : z^{q}(X|D, s_{i}) → z^{q}(X|D, r_{i}), for i = 1, 2 and all q ≥ 0, then h := h_{1} × h_{2} : □^{r_{1}+r_{2}} → □^{s_{1}+s_{2}} induces a homomorphism h^{∗} : z^{q}(X|D, s) → z^{q}(X|D, r) for all q ≥ 0, where r = r_{1} + r_{2} and s = s_{1} + s_{2}. Since h = h_{1} × h_{2} = (Id_{r_{1}} × h_{2}) ∘ (h_{1} × Id_{r_{2}}), we reduce to prove it when h is either Id_{r_{1}} × h_{2} or h_{1} × Id_{r_{2}}. But the statement obviously holds for these cases. □

4. On moving lemmas

Let k be any field. In this section, we discuss some of moving lemmas on algebraic cycles with modulus conditions. By a ‘moving lemma’, we ask whether the inclusion z_{W}^{q}(Y |D, •) ⊂ z^{q}(Y |D, •) in Definition 2.6 is a quasi-isomorphism. It is known when Y is smooth quasi-projective and D = 0 (by [4]), and when Y = X × 𝔸^{1}, with X smooth projective, D = X × {t^{m+1} = 0}, and W consists of W × 𝔸^{1} for finitely many locally closed subsets W ⊂ X (by [17]). Recently, W. Kai [14] proved it when Y is smooth affine with a suitable condition. Kai’s cases include the above case of Y = X ×𝔸^{1}, where X is this time smooth affine. His proof applies to more general cases, possibly after Nisnevich sheafifications. In §4.1, we sketch the argument of Kai in the case of multivariate additive higher Chow groups of smooth affine k-variety. In §4.2, we generalize the moving lemma of [17] in the case of pairs (X × S, X × D) where X is smooth projective. In §4.3 and 4.4, we discuss the standard pull-back property and its consequences. In §4.5, we discuss a moving lemma for additive higher Chow groups of smooth semi-local k-schemes essentially of finite type.

4.1. Kai’s affine method for multivariate additive higher Chow groups. The moving lemma of W. Kai [14] is the first moving result that applies to cycle groups with a non-zero modulus over a smooth affine scheme. Since the work loc. cit. is at present not yet refereed, we give a detailed sketch the proof of the following special case on multivariate additive higher Chow groups. But, we emphasize that the most crucial part is due to Kai. Following Definition 2.5, we write X[r] := X × 𝔸^{r}.

Theorem 4.1 (W. Kai). Let X be a smooth affine variety over any field k. Let W be a finite set of locally closed subsets of X. Let W[r] := {W [r] | W ∈ W\}. Let \underline{m} = (m_{1}, ⋯, m_{r}) ≥ 1. Then the inclusion z_{W[r]}^{q}(X[r]|D\underline{m}, •) ↪ z^{q}(X[r]|D\underline{m}, •) is a quasi-isomorphism.

First recall some preparatory results:

Lemma 4.2 ([17, Lemma 4.5]). Let f : X → Y be a dominant morphism of normal varieties. Suppose that Y is integral with the generic point η ∈ Y, and let X_{η} be the fiber over η, with the inclusion j_{η} : X_{η} ↪ X. Let D be a Weil divisor on X such that j_{η}^{∗}(D) ≥ 0. Then there exists a non-empty open subset U ⊂ Y such that j_{U}^{∗}(D) ≥ 0, where j_{U} : f^{−1}(U) ↪ X is the inclusion.

The following generalizes [17, Proposition 4.7]:

Proposition 4.3 (Spreading lemma). Let k ⊂ K be a purely transcendental extension. Let (X, D) be a smooth quasi-projective k-scheme with an effective Cartier divisor, and let W be a finite collection of locally closed subsets of X. Let (X_{K}, D_{K}) and W_{K} be the base changes via Spec (K) → Spec (k). Let p_{K/k} : X_{K} → X_{k} be the base change map. Then the pull-back map

p_{K/k}^{∗} : \frac{z^{q}(X|D, •)}{z_{W}^{q}(X|D, •)} → \frac{z^{q}(X_{K}|D_{K}, •)}{z_{W_{K}}^{q} (X_{K}|D_{K}, •)}

is injective on homology.

Proof. It is similar to [17, Proposition 4.7]. We sketch its proof for the reader’s convenience. If k is finite, then we can use the standard pro-ℓ-extension argument to reduce the proof to the case when k is infinite, which we assume from now. We may also assume that tr.deg_{k}K < ∞ and furthermore that tr.deg_{k}K = 1, by induction. So, we have K = k(𝔸_{k}^{1}). Suppose Z ∈ z^{q}(X|D, n) is a cycle that satisfies ∂Z ∈ z_{W}^{q}(X|D, n − 1), and Z_{K} = ∂(B_{K}) + V_{K} for some B_{K} ∈ z^{q}(X_{K}|D_{K}, n + 1) and V_{K} ∈ z_{W_{K}}^{q} (X_{K}|D_{K}, n). Consider the inclusion z^{q}(X_{K}|D_{K}, •) ↪ z^{q}(X_{K}, •). Then there is a non-empty open U′ ⊂ 𝔸_{k}^{1} such that B_{K} = B_{U′}|_{η}, V_{K} = V_{U′}|_{η}, Z×U′ = ∂(B_{U′})+V_{U′} for some B_{U′} ∈ z^{q}(X×U′, n+1), V_{U′} ∈ z_{W×U′}^{q}(X×U′, n), where η is the generic point of U′. Let j_{η} : X × η → X × U′ be the inclusion, which is flat. Since B_{K}, V_{K} satisfy the modulus condition, we have j_{η}^{∗}(X×U′×F_{n+1}^{1}−D×U′×

\overline{□}^{n+1}) ≥ 0 on \overline{B}_{K}^{N} and similarly for \overline{V} _{K}^{N}. Furthermore, \overline{B}_{U′}^{N} → U′, \overline{V} _{U′}^{N} → U′ are dominant. Thus by Lemma 4.2, there is a non-empty open U ⊂ U′ such that j_{U}^{∗}(X × U′ × F_{n+1}^{1} − D × U′ × \overline{□}^{n+1}) ≥ 0 on \overline{B}_{U}^{N} and similarly for \overline{V} _{U}^{N}, for j_{U} : X × U ↪ X × U′. This proves that B_{U} and V_{U} have modulus D × U. Hence, B_{U} ∈ z^{q}(X × U|D × U, n + 1) and V_{U} ∈ z_{W×U}^{q}(X × U|D × U, n) with Z × U = ∂(B_{U}) + V_{U}. Since k is infinite, the set U(k) ↪ U is dense. We claim the following: Claim: There is a point u ∈ U(k) such that the pull-backs of B_{U} and V_{U} under the inclusion i_{u} : X × {u} ↪ X × U are both defined in z^{q}(X, n + 1) and z_{W}^{q}(X, n), respectively. Its proof requires the following elementary fact: Lemma: Let Y be any k-scheme. Let B ∈ z^{q}(Y × U) be a cycle. Then there exists a nonempty open subset U′′ ⊂ U such that for each u ∈ U′′(k), the closed subscheme Y × {u} intersects B properly on Y × U, thus it defines a cycle i_{u}^{∗}(B) ∈ z^{q}(Y), where Y is identified with Y × {u}. Note that for each u ∈ U(k), the subscheme Y × {u} ⊂ Y × U is an effective divisor, so its proper intersection with B is equivalent to that Y × {u} does not contain any irreducible component of B. If there exists a point u_{i} ∈ U(k) such that Y × {u_{i}} contains an irreducible component B_{i} of B, then for any other u ∈ U(k) \ {u_{i}}, we have (Y × {u}) ∩ B_{i} = ∅. So, for every irreducible component B_{i} of B, there exists at most one u_{i} ∈ U(k) such that Y × {u_{i}} contains B_{i}. Let S be the union of such points u_{i}, if they exist. There are only finitely many irreducible components of B, so |S| < ∞. Taking U′′ := U \ S, we have Lemma. We now prove Claim. Let F ⊂ □^{n+1} be any face, including the case F = □^{n+1}. Since B_{U} ∈ z^{q}(X ×U, n+1), by definition X ×U ×F and B_{U} intersect properly on X × U × □^{n+1}, so their intersection gives a cycle B_{U,F} ∈ z^{q}(X × U × F). By Lemma with Y = X × F, there exists a nonempty open subset U_{F} ⊂ U such that B_{U,F} defines a cycle in z^{q}(X × {u} × F) for every u ∈ U_{F} (k). Let U_{1} := ⋂_{F} U_{F}, where the intersection is taken over all faces F of □^{n+1}. This is a nonempty open subset of U. Similarly, let F ⊂ □^{n} be any face, including the case F = □^{n}. Here, V_{U} ∈ z_{W×U}^{q}(X × U, n), and repeating the above argument involving Lemma with Y = W ×F for W ∈ W, we get a nonempty open subset U_{W,F} ⊂ U such that we have an induced cycle in z^{q}(W × {u} × F) for every u ∈ U_{W,F} (k). Let U_{2} := ⋂_{W,F} U_{W,F}, where the intersection is taken over all pairs (W, F), with W ∈ W and a face F ⊂ □^{n}. Taking U := U_{1} ∩ U_{2}, which is a nonempty open subset of U, we now obtain Claim for every u ∈ U(k). Finally, for such a point u as in Claim, by the containment lemma (Proposition 2.2), i_{u}^{∗}(B_{U}) and i_{u}^{∗}(V_{U}) have modulus D. Hence, i_{u}^{∗}(B_{U}) ∈ z^{q}(X|D, n+1) and i_{u}^{∗}(V_{U}) ∈ z_{W}^{q}(X|D, n). This finishes the proof. □

Sketch of the proof of Theorem 4.1. Step 1. We first show it when X = 𝔸_{k}^{d}. Let K = k(𝔸_{k}^{d}) and let η ∈ X be the generic point. To facilitate the proof, as we did previously in §3, using the automorphism y ↦ 1/(1 − y) of ℙ^{1} we replace (□, {∞, 0}) by (𝔸^{1}, {0, 1}), and write □ = 𝔸^{1}. We use the homogeneous coordinates (u_{i,0}; u_{i,1}) ∈ \overline{□}^{1} = ℙ^{1}, where y_{i} = u_{i,1}/u_{i,0}, then the divisor F_{n,i}^{1} in the modulus condition is replaced by F_{n,i}^{∞} = {y_{i} = ∞} and F_{n}^{∞} = ∑_{i=1}^{n} F_{n,i}^{∞}. For any g ∈ 𝔸^{d} and an integer s > 0, define ϕ_{g,s} : 𝔸_{k(g)}^{d}[r] ×_{k(g)} □_{k(g)}^{1} → 𝔸_{k(g)}^{d}[r] by ϕ_{g,s}(\underline{x}, \underline{t}, y) := (\underline{x} + y(t_{1}^{m_{1}} ⋯ t_{r}^{m_{r}})^{s}g, \underline{t}), where k(g) is the residue field of g. (N.B. In terms of W. Kai’s homotopy, our g ∈ 𝔸^{d} corresponds to his v = (g, 0, ⋯, 0) ∈ 𝔸^{d}[r] = 𝔸^{d+r}.) For any cycle V ∈ z^{q}(X[r]|D\underline{m}, n), define H_{g,s}^{∗}(V) := (ϕ_{g,s}×Id_{□^{n}})^{∗}p_{k(g)/k}^{∗}(V), where p_{k(g)/k} : 𝔸_{k(g)}^{d}[r]×□^{n} → 𝔸_{k}^{d}[r]×□^{n} is the base change. Using [3, Lemma 1.2], one checks that H_{g,s}^{∗}(V) preserves the face condition for V. Moreover, if V ∈ z_{W}^{q}(X[r], n), then so does H_{g,s}^{∗}(V). When g = η, another application of [3, Lemma 1.2] shows that H_{g,s}^{∗}(V) intersects with all W [r] × F properly, where W ∈ W and a F ⊂ □^{n} is a face. The argument for proving these face conditions follows the same steps as that of the proof of [17, Lemma 5.5, Case 2] though the present case is slightly different so that we use [3, Lemma 1.2] instead of [3, Lemma 1.1] (see [14, Lemma 3.5] for more detail). On the other hand, we have the following crucial and central assertion due to W. Kai (cf. [14, Proposition 3.3]):

Claim: For each irreducible V ∈ z^{q}(𝔸_{k}^{d}[r]|D\underline{m}, n), there is s(V) ∈ ℤ_{≥0} such that for any s > s(V) and for any g ∈ 𝔸^{d}, the cycle H_{g,s}^{∗}(V) has modulus D\underline{m}. Once it is proven, call the smallest such integer s(V), the threshold of V, for simplicity. Here, instead of translations by g ∈ 𝔸^{d} used in usual higher Chow groups of 𝔸^{d} (which correspond to s = 0), Kai uses adjusted translations as in the definition of ϕ_{g,s}, so that near the divisors {t_{i} = 0}, the effect of adjusted translation is also small, while away from the divisors {t_{i} = 0}, the effect of adjusted translation gets larger, so that for a sufficiently large s, this imbues the desired modulus condition into cycles. Note the following elementary fact (cf. [14, Lemma 3.2]), which amounts to rewriting the definitions: Let A be a commutative ring with unity, 𝔭 ⊂ A a prime ideal, ζ ∈ A, and u ∈ A \ 𝔭. Then the element ζ/u of κ(𝔭) is integral over A/𝔭 if and only if there is a homogeneous polynomial E(a, b) ∈ A[a, b], which is monic in the variable a, with E(ζ, b) ∈ 𝔭 in A. For each I ⊂ \{1, ⋯, n\}, consider the open subset U_{I} ⊂ 𝔸_{k}^{d} × \overline{□}^{n} given by the conditions u_{i,0} ≠ 0 for i ∈ I and u_{i,1} ≠ 0 for i ∉ I. For i ∉ I, we let \overline{y}_{i} = u_{i,0}/u_{i,1} = y_{i}^{−1}. Hence, U_{I} = Spec (R_{I}), where R_{I} := k[\underline{x}, \underline{t}, \{y_{i}\}_{i∈I}, \{ȳ_{i}\}_{i∉I}], where \underline{x} = (x_{1}, ⋯, x_{d}) and \underline{t} = (t_{1}, ⋯, t_{r}). On U_{I}, the divisor F_{n}^{∞} used in the definition of the modulus condition is given by the polynomial ∏_{i∉I} \overline{y}_{i}. For an irreducible V ∈ z^{q}(𝔸_{k}^{d}[r]|D\underline{m}, n), let \overline{V} be its Zariski closure in 𝔸_{k}^{d}[r] × \overline{□}^{n}. For a given I, the restriction \overline{V} ∩ (𝔸_{k}^{d}[r] × U_{I}) is given by an ideal of R_{I}, say, generated by a finite set of polynomials f_{λ}^{I}(\underline{x}, \underline{t}, \{y_{i}\}_{i∈I}, \{\overline{y}_{i}\}_{i∉I}) ∈ R_{I} for λ ∈ Λ_{I}. By the above fact and the assumption that V has the modulus condition, there is a polynomial E_{I}(a, b) = E_{I}(\underline{x}, \underline{t}, \{y_{i}\}_{i∈I}, \{\overline{y}_{i}\}_{i∉I}, a, b) ∈ R_{I}[a, b], homogeneous in a, b and monic in a, satisfying the condition inside the ring R_{I}:

(4.1)  E_{I}(∏_{i∉I} \overline{y}_{i}, t\underline{m}) ∈ ∑_{λ∈Λ_{I}}(f_{λ}^{I}), where t\underline{m} = t_{1}^{m_{1}} ⋯ t_{r}^{m_{r}}.

If necessary, by multiplying a power of a to E_{I}, we may assume deg E_{I} ≥ deg\underline{x} f_{λ}^{I}, where deg is the homogeneous degree of E_{I} in the variables a, b and deg\underline{x} is the total degree with respect to \underline{x}. In doing so, we may further assume that deg E_{I} is the same for all subset I ⊂ {1, ⋯, n}. For this choice of degrees, we let s(V) = deg E_{I}. If V is not irreducible, then take the maximum of s(V_{i}) over all irreducible components V_{i} of V to define s(V). The heart of the proof is to show that this number satisfies the assertions of Claim, which we do now. We may assume V is irreducible. For any fixed s > s(V) and g ∈ 𝔸^{d}, let V′ be an irreducible component of H_{g,s}^{∗}(V) and let \overline{V}′ be its Zariski closure in

𝔸_{κ}^{d}[r] × \overline{□}^{n+1}, where κ = k(g). We use the coordinates (y, y_{1}, ⋯, y_{n}) ∈ \overline{□}^{n+1}, and for the first \overline{□} = ℙ^{1}, use the homogeneous coordinate (u_{0}; u_{1}) so that

y = u_{1}/u_{0} and \overline{y} := u_{0}/u_{1} = y^{−1}. Let ν : \overline{V}′^{N} → \overline{V} be the normalization.

Note that whether a divisor is effective or not on \overline{V}′^{N} is a Zariski local question on \overline{V}′^{N} (thus on \overline{V}′), so we may check the modulus condition Zariski locally near any point P ∈ \overline{V}′. Fix a point P. Let I ⊂ \{1, ⋯, n\} be the set points i such that P does not map to ∞ ∈ ℙ_{κ}^{1} of the (i + 1)-th projection \overline{V}′ ↪ 𝔸_{κ}^{d}[r] × \overline{□}^{n+1} → \overline{□}_{κ} = ℙ_{κ}^{1}. There are two possibilities. In the first case P ∈ 𝔸_{κ}^{d}[r]×𝔸^{1}×\overline{□}^{n}, i.e. P does not map to ∞ ∈ ℙ^{1} for the first projection to \overline{□}_{κ}, the morphism p_{κ/k} ∘(ϕ_{g,s} ×Id_{□}^{n}) : 𝔸_{κ}^{d}[r]×𝔸^{1}×□^{n} → 𝔸_{k}^{d}[r]×□^{n} extends uniquely to 𝔸_{κ}^{d}[r]×𝔸^{1}×\overline{□}^{n} → 𝔸_{k}^{d}[r]×\overline{□}^{n}. Thus, by pulling-back the relation (4.1), we obtain in the ring R_{I}[y],

E_{I}(\underline{x} + y(t\underline{m})^{s}g, \underline{t}, \{y_{i}\}_{i∈I}, \{\overline{y}_{i}\}_{i∉I}, ∏_{i∉I} \overline{y}_{i}, t\underline{m}) ∈ (4.2)

∈ ∑_{λ∈Λ_{I}}(f_{λ}^{I}(\underline{x} + y(t\underline{m})^{s}g, \underline{t}, \{y_{i}\}_{i∈I}, \{\overline{y}_{i}\}_{i∉I})).

Here, the polynomials f_{λ}^{I}(\underline{x} + y(t\underline{m})^{s}g, \{y_{i}\}_{i∈I}, \{\overline{y}_{i}\}_{i∉I}) over λ ∈ Λ_{I} define the underlying closed subscheme of the Zariski closure of H_{g,s}^{∗}(V) restricted on the region Spec (R_{I}[y]). Due to the choice of the degrees of E_{I} and f_{λ}^{I}, the relation (4.2) implies that the rational function ∏_{i∉I} \overline{y}_{i}/t\underline{m} is integral using fact. In particular, V′ satisfies the modulus condition in a neighborhood of P. In the remaining case P ∉ 𝔸_{κ}^{d}[r] × 𝔸^{1} × \overline{□}^{n}, i.e. P does map to ∞ ∈ ℙ^{1} for the first projection to \overline{□}_{κ}, we use the affine open chart Spec (R_{I}[\overline{y}]) where u_{1} ≠ 0. The defining ideal of \overline{V}′ ∩ Spec (R_{I}[\overline{y}]) in the ring R_{I}[\overline{y}] contains the polynomials ϕ_{λ}^{I}(\underline{x}, \underline{t}, \overline{y}, \{y_{i}\}_{i∈I}, \{\overline{y}_{i}\}_{i∉I}) := f_{λ}^{I}(\underline{x} + \frac{1}{\overline{y}} (t\underline{m})^{s}g, \underline{t}, \{y_{i}\}_{i∈I}, \{\overline{y}_{i}\}_{i∉I}) ·

\overline{y}^{deg\underline{x}(f_{λ}^{I})}, where λ ∈ Λ_{I}. By expanding the definition of ϕ_{λ}^{I}, we see that it is of the form

(4.3)  ϕ_{λ}^{I} = \overline{y}^{deg\underline{x}(f_{λ}^{I})}f_{λ}^{I}(\underline{x}, \underline{t}, \{y_{i}\}_{i∈I}, \{\overline{y}_{i}\}_{i∉I}) + (t\underline{m})^{s}h, h ∈ R_{I}[\overline{y}].

Express (4.1) as E_{I}(∏_{i∉I} \overline{y}_{i}, t\underline{m}) = ∑_{λ∈Λ_{I}} b_{λ}f_{λ}^{I} for some b_{λ} ∈ R_{I}. Let c_{λ} :=

\overline{y}^{s(V)−deg\underline{x}(f_{λ}^{I})} · b_{λ} (which is in R_{I} because s(V) ≥ deg\underline{x}(f_{λ}^{I})). Then from (4.3),

(4.4)  ∑_{λ∈Λ_{I}} c_{λ}ϕ_{λ}^{I} = \overline{y}^{s(V)} · E_{I}(∏_{i∉I} \overline{y}_{i}, t\underline{m}) + (t\underline{m})^{s}g,

where (keep in mind that s ≥ s(V)) the right hand side becomes (\overline{y} ∏_{i∉I} \overline{y}_{i})^{s(V)} + e_{1}\overline{y}(\overline{y} ∏_{i∉I} \overline{y}_{i})^{s(V)−1}t\underline{m} + ⋯ + (e_{s(v)}\overline{y}^{s(V)} + (t\underline{m})^{s−s(V)}h) · (t\underline{m})^{s(V)}, which we write as E′(\overline{y} ∏_{i∉I} \overline{y}_{i}, t\underline{m}) for a polynomial E′(a, b) ∈ R_{I}[\overline{y}][a, b], homogeneous in a, b and monic in a. Thus (4.4) is ∑_{λ∈Λ_{I}} c_{λ}ϕ_{λ}^{I} = E′(\overline{y} ∏_{i∉I} \overline{y}_{i}, t\underline{m}), which implies that the rational function \overline{y} ∏_{i∉I} \overline{y}_{i}/t\underline{m} is in-

tegral on \overline{V}′ ∩ Spec (R_{I}[\overline{y}]) using fact. Thus V′ also satisfies the modulus condition near P. Combining these two cases, we have now proven Claim. Now consider the subgroup z_{W[r],e}^{q}(X[r]|D\underline{m}, n)^{≤s} ⊂ z_{W[r],e}^{q}(X[r]|D\underline{m}, n) for s > 0, consisting of cycles V with its threshold s(V) ≤ s (cf. [14, §3.4]). We deduce

\frac{z_{W[r],e}^{q}(X[r]|D\underline{m}, n)}{z_{W[r]}^{q}(X[r]|D\underline{m}, n)} = lim_{→s} \frac{z_{W[r],e}^{q}(X[r]|D\underline{m}, n)^{≤s}}{z_{W[r]}^{q}(X[r]|D\underline{m}, n)^{≤s}} .

Then one has the induced map from H_{η,s}^{∗},

H_{η,s}^{∗} : \frac{z_{W[r],e}^{q}(X[r]|D\underline{m}, n)^{≤s}}{z_{W[r]}^{q}(X[r]|D\underline{m}, n)^{≤s}} → \frac{z_{W[r],e}^{q}(X_{K}[r]|D\underline{m}, n + 1)}{z_{W[r]}^{q}(X_{K}[r]|D\underline{m}, n + 1)},

which gives a homotopy between the base change p_{K/k}^{∗} and H_{η,s}^{∗}|_{y=1}. However, H_{η,s}^{∗}|_{y=1} is zero on the quotient, while p_{K/k}^{∗} is injective on homology by Proposition 4.3, after taking s → ∞, so that the map p_{K/k}^{∗} is in fact zero on homology. This means, the quotient z_{W[r],e}^{q}(X[r]|D\underline{m}, n)/z_{W[r]}^{q}(X[r]|D\underline{m}, n) is acyclic, proving the moving lemma for X = 𝔸_{k}^{d}. Step 2. If X is a general smooth affine k-variety of dimension d, we use the standard generic linear projection trick. We choose a closed immersion X ↪ 𝔸^{N} for some N ≫ d and run the steps of §6 of [17] (with ℙ^{n} replaced by 𝔸^{N} everywhere) mutatis mutandis to conclude the proof of the moving lemma for X from that of affine spaces. We leave the details for the reader. □

4.2. Projective method for multivariate additive higher Chow groups. The following theorem generalizes the moving lemma for additive higher Chow groups of smooth projective schemes [17, Theorem 4.1] to a general setting which includes the multivariate additive higher Chow groups.

Theorem 4.4. Let (S, D) be a smooth quasi-projective k-variety with an effective Cartier divisor. Let X be a smooth projective k-variety. Let W be a finite collection of locally closed subsets of X. We let W × S := {W × S|W ∈ W}. Then the inclusion z_{W×S}^{q}(X × S|X × D, •) ↪ z^{q}(X × S|X × D, •) is a quasi-isomorphism. In particular, when \underline{m} = (m_{1}, ⋯, m_{r}) ≥ 1, and (S, D) = (𝔸^{r}, D\underline{m}), the moving lemma holds for multivariate additive higher Chow groups of smooth projective varieties over k.

Proof. Most arguments of [17, Theorem 4.1] work with minor changes, so we sketch the proof. Step 1. We first prove the theorem when X = ℙ_{k}^{d}. The algebraic group SL_{d+1,k} acts on ℙ^{d}. Let K = k(SL_{d+1,k}). Then there is a K-morphism ϕ : □_{K}^{1} → SL_{d+1,K} such that ϕ(0) = Id, and ϕ(∞) = η, where η is the generic point of SL_{d+1,k}. See [17, Lemma 5.4]. For such ϕ, consider the composition H_{n} of morphisms

ℙ^{d} × S × □_{K}^{n+1} →^{μ_{ϕ}} ℙ^{d} × S × □_{K}^{n+1} →^{pr′_{K}} ℙ^{d} × S × □_{K}^{n} →^{p_{K/k}} ℙ^{d} × S × □_{k}^{n},

where μ_{ϕ}(\underline{x}, s, y_{1}, ⋯, y_{n+1}) = (ϕ(y_{1})\underline{x}, s, y_{1}, ⋯, y_{n+1}), pr′_{K} is the projection dropping y_{1}, and p_{K/k} is the base change. We claim that H_{n}^{∗} carries z_{W×S}^{q}(ℙ^{d}× S|ℙ^{d} × D, n) to z_{W×S}^{q}(ℙ_{K}^{d} × S|ℙ_{K}^{d} × D, n + 1), i.e., for an irreducible cycle Z ∈ z_{W×S}^{q}(ℙ^{d} × S|ℙ^{d} × S, n), we show that Z′ := H_{n}^{∗}(Z) ∈ z_{W×S}^{q}(ℙ_{K}^{d} × S|ℙ_{K}^{d} × D, n + 1).

To do so, we first claim that Z′ intersects with W × S × F_{K} properly for each W ∈ W and each face F ⊂ □^{n+1}.

(1) In case F = {0} × F′ for some face F′ ⊂ □^{n}, because ϕ(0) = Id, we have Z′ ∩ (W × S × F_{K}) ≃ Z_{K} ∩ (W × S × F′_{K}). Note that dim(W × S × F_{K}) = dim(W × S × F′_{K}). Hence, codim_{W ×S×F_{K}} (Z′ ∩ (W × S × F_{K})) = dim(W × S × F_{K})−dim(Z′∩(W ×S ×F_{K})) = dim(W ×S ×F′_{K})−dim(Z_{K} ∩(W ×S ×F′_{K})) = dim(W ×S×F′)−dim(Z∩(W ×S×F′)) = codim_{W ×S×F′}(Z∩(W ×S×F′)) ≥ q, because Z ∈ z_{W×S}^{q}(ℙ^{d} × S|ℙ^{d} × D, n).

(2) In case F = {∞} × F′ for some face F′ ⊂ □^{n}, dim(W × S × F_{K}) = dim(W ×S×F′_{K}) and Z′∩(W ×S×F_{K}) ≃ η·(Z_{K})∩(W ×S×F′_{K}), where SL_{d+1,k} acts on ℙ^{d}×S×F′, naturally on ℙ^{d} and trivially on S×F′. Let A := W ×S×F′ and B := Z ∩ (ℙ^{d} × S × F′). Thus, codim_{W ×S×F_{K}} (Z′ ∩ (W × S × F_{K})) = dim(W ×S×F_{K})−dim(Z′∩(W ×S×F_{K})) = dim(W ×S×F′_{K})−dim(η·(Z_{K})∩ (W ×S ×F′_{K})) =^{†} dim(A_{K})−dim(η·B_{K} ∩A_{K}) = codim_{A_{K}} (η·B_{K} ∩A_{K}), where † holds because Z ∩ A = B ∩ A. By applying [3, Lemma 1.1] to G = SL_{d+1,k}, and the above A, B on X := ℙ^{d} × S × F′, there is a non-empty open subset U ⊂ G such that for all g ∈ U, the intersection (g · A) ∩ B is proper on X. By shrinking U, we may assume U is invariant under inverse map, so g = η^{−1} ∈ U. Thus, codim_{A_{K}} ((η · B_{K}) ∩ A_{K}) ≥ codim_{X_{K}} (η · B_{K}). Since codim_{X_{K}} (η · B_{K}) = codim_{X_{K}} B_{K} and codim_{X_{K}} B_{K} = q, we get codim_{W ×S×F_{K}} (Z′∩(W ×S×F_{K})) = codim_{A_{K}} ((η · B_{K}) ∩ A_{K}) ≥ codim_{X_{K}} B_{K} = q.

(3) In case F = □ × F′ for some face F′ ⊂ □^{n}, the projection Z′ ∩ (W × S × □ × F′_{K}) → □_{K} is flat, being a dominant map to a curve, so dim(Z′ ∩ (W × S × □ × F′_{K})) = dim(Z′ ∩ (W × S × {∞} × F′_{K})) + 1. We also have dim(W × S × □ × F′_{K}) = dim(W × S × {∞} × F′_{K}) + 1. Hence, we deduce codim_{W ×S×F_{K}} (Z′ ∩ (W × S × F_{K})) = dim(W × S × □ × F_{K}) − dim(Z′ ∩ (W × S × □ × F′_{K})) = codim_{W ×S×{∞}×F′_{K}} (Z′ ∩ (W × S × {∞} × F′_{K})) ≥^{†} q, where † follows from case (2). This shows Z′ intersects all faces properly. Now we show that Z′ has modulus ℙ^{d} × D. We drop all exchange of the factors, for simplicity. For p : ℙ^{d} → Spec (k), we take V = p(Z) on S × □^{n}. Because Z ⊂ p^{−1}(p(Z)) = ℙ^{r} × V, we have Z′ = μ_{ϕ}^{∗}(□_{K}^{1} × Z) ⊂ μ_{ϕ}^{∗}(ℙ^{d} × □_{K}^{1} × V) = ℙ^{d} × □_{K}^{1} × V := Z_{1}. By Lemma 2.8, V is admissible on S × □^{n}. So, p^{∗}[V] = ℙ^{d} × V is admissible on ℙ^{d} × S × □^{n}. In particular, ℙ^{d} × V has modulus ℙ^{d} × D. Hence, Z_{1} = ℙ^{d} × □_{K}^{1} × V also has modulus ℙ_{K}^{d} × D. Now, Z′ ⊂ Z_{1} shows that Z′ has modulus ℙ_{K}^{d} × D by Proposition 2.2. Thus, we proved Z′ ∈ z_{W×S}^{q}(ℙ_{K}^{d} × S|ℙ_{K}^{d} × D, n + 1). Going back to the proof, one checks that H_{•}^{∗} : z^{q}(ℙ^{d} × S|ℙ^{d} × D, •) → z^{q}(ℙ_{K}^{d} × S|ℙ^{d} × D, • + 1) is a chain homotopy satisfying ∂H^{∗}(Z) + H^{∗}∂(Z) = Z_{K} − η · (Z_{K}), and the same holds for z_{W×S} by a straightforward computation (see [17, Lemma 5.6]). Furthermore, for each admissible Z, we have η · Z_{K} ∈ z_{W_{K}×S}^{q}(ℙ_{K}^{d} × S|ℙ_{K}^{d} × D, n), by the above proof of proper intersection of Z′ with W × S × F_{K}, where F = {∞} × F′ for a face F′ ⊂ □^{n}. Hence, the base change p_{K/k}^{∗} : z^{q}(ℙ_{k}^{d} × S|ℙ_{k}^{d} × D, •)/z_{W×S}^{q}(ℙ_{k}^{d} × S|ℙ_{k}^{d} × D, •) → z^{q}(ℙ_{K}^{d} × S|ℙ_{K}^{d} × D, •)/z_{W_{K}×S}^{q}(ℙ_{K}^{d} × S|ℙ_{K}^{d} × D, •) is homotopic to η · p_{K/k}^{∗}, which is zero on the quotient. That is, p_{K/k}^{∗} on the above quotient complex is zero on homology. However, by the spreading argument (Proposition 4.3), p_{K/k}^{∗} is injective on homology. (N.B. We used here an elementary fact that k(SL_{d+1,k}) is purely transcendental over k. To check this fact, first note that by definition k[SL_{d+1,k}] ≃ k[{T_{i,j}|1 ≤ i, j ≤ d + 1}]/(det(M) − 1) for the (d + 1, d + 1)- matrix M = [T_{ij}] consisting of indeterminates T_{i,j} for 1 ≤ i, j ≤ d + 1. Here by Cramer’s rule we can write det(M) − 1 = αT_{d+1,d+1} − β − 1, where α = det(M_{d+1,d+1}), β = ∑_{1≤j≤d}(−1)^{d+1+j} det(M_{d+1,j}) and M_{ij} is the (i, j)-minor of M. Here both α and β do not have T_{d+1,d+1}. Hence k[SL_{d+1,k}] ≃ k[{T_{ij}|1 ≤ i, j ≤ d + 1, (i, j) ≠ (d + 1, d + 1)\}, \frac{β+1}{α}]. Thus, k(SL_{d+1,k}) ≃ k(\{T_{ij}|1 ≤ i, j ≤ d + 1, (i, j) ≠ (d + 1, d + 1)}), which is purely transcendental over k.) Hence, the quotient complex z^{q}(ℙ^{d} × S|ℙ^{d} × D, •)/z_{W×S}^{q}(ℙ^{d} × S|ℙ^{d} × D, •) is acyclic, i.e., the moving lemma holds for (ℙ^{d} × S, ℙ^{d} × D), finishing Step 1. Step 2. Now let X be a general smooth projective variety of dimension d. In this case, we choose a closed immersion X ↪ ℙ^{N} for some N ≫ d. We now run the linear projection argument of [17, §6] again without any extra argument to deduce the proof of the moving lemma for X from that of the projective spaces. We leave out the details. □

4.3. Contravariant functoriality. The following contravariant functoriality of multivariate additive higher Chow groups is an immediate application of the moving lemma and the proof is identical to that of [17, Theorem 7.1].

Theorem 4.5. Let f : X → Y be a morphism of k-varieties, with Y smooth affine or smooth projective. Let r ≥ 1 and \underline{m} = (m_{1}, ⋯, m_{r}) ≥ 1. Then there exists a pull-back f^{∗} : CH^{q}(Y [r]|D\underline{m}, n) → CH^{q}(X[r]|D\underline{m}, n). If g : Y → Z is another morphism with Z smooth affine or smooth projective, then we have (g ∘ f)^{∗} = f^{∗} ∘ g^{∗}.

Remark 4.6. As a special case, when r = 1, we have the pull-back map f^{∗} : TCH^{q}(Y, n; m) → TCH^{q}(X, n; m).

4.4. The presheaf T CH. For the rest of the section, we concentrate on additive higher Chow groups. Let m ≥ 0. By Theorem 4.5, we see that T_{n,m}^{q} := TCH^{q}(−, n; m) is a presheaf of abelian groups on the category SmAff_{k}, but we do not know if it is a presheaf on the categories Sm_{k} or Sch_{k}. However, we can exploit Theorem 4.5 further to define a new presheaf on Sm_{k} and Sch_{k}. The idea of this detour occurred to the authors while working on [20]. We do it for somewhat more general circumstances. Let C be a category and D be a full subcategory. Let F be a presheaf of abelian groups on D, i.e. F : D^{op} → (Ab) is a functor to the category of abelian groups. For each object X ∈ C, let (X ↓ D) be the category whose objects are the morphisms X → A in C, with A ∈ D, and a morphism from h_{1} : X → A to h_{2} : X → B, with A, B ∈ D, is given by a morphism g : A → B in C such that g ∘ h_{1} = h_{2}. The functor F : D^{op} → (Ab) induces the functor (X ↓ D)^{op} → (Ab) given by (X →^{h} A) ↦ F (A), also denoted by F.

Definition 4.7. Suppose that for each X ∈ C, the category (X ↓ D) is cofiltered. Then define F(X) := colim_{(X↓D)^{op}}F.

In particular, when C = Sch_{k} and D = SmAff_{k}, one checks that (X ↓ SmAff_{k}) is cofiltered, and for X ∈ Sch_{k}, we define T CH^{q}(X, n; m) := colim_{(X↓SmAff_{k})^{op}}T_{n,m}^{q}.

Proposition 4.8. Let C be a category and D be a full subcategory such that for each X ∈ C, the category (X ↓ D) is cofiltered. Let F be a presheaf of abelian groups on D and let F be as in Definition 4.7. Let f : X → Y be a morphism in C. Then for X ∈ C, the association X ↦ F(X) satisfies the following properties:

(1) There is a canonical homomorphism α_{X} : F(X) → F (X).

(2) If X ∈ D, then α_{X} is an isomorphism, and α : F → F defines an isomorphism of presheaves on D.

(3) There is a canonical pull-back f^{∗} : F(Y) → F(X). If g : Y → Z is another morphism in C, then we have (g∘f)^{∗} = f^{∗}∘g^{∗}. So, F is a presheaf of abelian groups on C. In particular, T CH^{q}(−, n; m) is a presheaf of abelian groups on Sch_{k}, which is isomorphic to TCH^{q}(−, n; m) on SmAff_{k}.

Proof. (1) Let (X ⟶^{h} A) ∈ (X ↓ D)^{op}. By the given assumption, we have the pull-back h^{∗} : F (A) → F (X). Regarding F (X) as a constant functor on (X ↓ D)^{op}, this gives a morphism of functors F → F (X). Taking the colimits over all h, we obtain F(X) → F (X), where α_{X} = colim_{h} h^{∗}.

(2) When X ∈ D, the category (X ↓ D)^{op} has the terminal object Id_{X} : X → X. Hence, the colimit F(X) is just F (X).

(3) A morphism f : X → Y in C defines a functor f^{♯} : (Y ↓ D)^{op} → (X ↓ D)^{op} given by (Y →^{h} A) ↦ (X →^{f} Y →^{h} A). Thus, taking the colimits of the functors induced by F, we obtain f^{∗} : F(Y) → F(X). For another morphism g : Y → Z, that (g ∘ f)^{∗} = f^{∗} ∘ g^{∗} can be checked easily using the universal property of the colimits. In the special case when C = Sch_{k} and D = SmAff_{k} with F = TCH^{q}(−, n; m), by Theorem 4.5 we know that F is a presheaf on SmAff_{k}. So, the above general discussion holds. □

Remark 4.9. Since additive higher Chow groups have pull-backs for flat maps (see [16, Lemma 4.7]), it follows that for X ∈ Sm_{k}, α_{(−)} defines a map of presheaves T CH^{q}(−, n; m) → TCH^{q}(−, n; m) on the small Zariski site X_{Zar} of X. Proposition 4.8(2) says that this map is an isomorphism for affine open subsets of X. Thus, this map of presheaves on X_{Zar} induces an isomorphism of their Zariski sheafifications.

4.5. Moving lemma for smooth semi-local schemes. One remaining objective in Section 4 is to prove the following semi-local variation of Theorem 4.1:

Theorem 4.10. Let Y ∈ SmLoc_{k}. Let W be a finite set of locally closed subsets of Y. Then the inclusion Tz_{W}^{q}(Y, •; m) ↪ Tz^{q}(Y, •; m) is a quasiisomorphism.

We begin with some basic results related to cycles over semi-local schemes. Recall that when A is a ring and Σ = {p_{1}, ⋯, p_{N}} is a finite subset of Spec (A), the localization at Σ is the localization A → S^{−1}A, where S = ⋂_{i=1}^{N}(A \ p_{i}). For a quasi-projective k-scheme X and a finite subset Σ of (not necessarily closed) points of X, the localization X_{Σ} is defined by reducing it to the case when X is affine by the following elementary fact (see [25, Proposition 3.3.36]) that we use often.

Lemma 4.11. Let X be a quasi-projective k-scheme. Given any finite subset Σ ⊂ X and an open subset U ⊂ X containing Σ, there exists an affine open subset V ⊂ U containing Σ.

For X ∈ Sch_{k} and a point x ∈ X, the open neighborhoods of x form a cofiltered category and we have functorial flat pull-back maps (j_{U}^{V})^{∗} : \underline{Tz}^{q}(V, n; m) → \underline{Tz}^{q}(U, n; m) for j_{U}^{V} : U ↪ V in this category.

Lemma 4.12. Let X ∈ Sch_{k} and let x ∈ X be a scheme point. Let Y = Spec (O_{X,x}). Then we have colim_{x∈U} \underline{Tz}^{q}(U, n; m) ⟶^{≃} \underline{Tz}^{q}(Y, n; m), where the colimit is taken over all open neighborhoods U of x.

Proof. Replacing x by an affine open neighborhood of x ∈ X, we may assume that X is affine and write X = Spec (A). Let 𝔭_{x} ⊂ A be the prime ideal that corresponds to the point x and let S := A \ 𝔭_{x}, so that Y = Spec (S^{−1}A). To facilitate our proof, using the automorphism y ↦ 1/(1 − y) of ℙ^{1}, we identify □ with 𝔸^{1} and take {0, 1} ⊂ 𝔸^{1} as the faces. So, X × B_{n} = X × 𝔸^{1} × 𝔸^{n−1} = Spec (A[t, y_{1}, ⋯, y_{n−1}]). Let α ∈ \underline{Tz}^{q}(Y, n; m). We need to find an open subset U ⊂ X containing x such that the closure of α in U × 𝔸^{1} × 𝔸^{n−1} is admissible. For this, we may assume α is irreducible, i.e., it is a closed irreducible subscheme Z ⊂ Y × 𝔸^{1} × 𝔸^{n−1}. Let \overline{Z} be its Zariski closure in X × 𝔸^{1} × 𝔸^{n−1}. Let 𝔭 be the prime ideal of B := A[t, y_{1}, ⋯, y_{n−1}] such that V (𝔭) = \overline{Z}. For the proper intersection with faces, let 𝔮 ⊂ B be the prime ideal (y_{i_{1}} − ϵ_{1}, ⋯, y_{i_{s}} − ϵ_{s}), where 1 ≤ i_{1} < ⋯ < i_{s} ≤ n − 1 and ϵ_{j} ∈ {0, 1}. Let 𝔓 be a minimal prime of 𝔭 + 𝔮. One checks immediately from the behavior of prime ideals under localizations that there is a ∈ S such that either 𝔓B[a^{−1}] = B[a^{−1}] or ht(𝔓B[a^{−1}]) ≥ q + s. This means, over U_{𝔮} := Spec (A[a^{−1}]), either the intersection of \overline{Z}_{U_{𝔮}} with V (𝔮) is empty, or has codimension ≥ q + s. Applying this argument to all faces, we can take U_{1} := ⋂_{𝔮} U_{𝔮}. Then \overline{Z}_{U_{1}} intersects all faces of U_{1} × 𝔸^{1} × 𝔸^{n−1} properly. For the modulus condition, let ν : Ẑ^{N} → Ẑ ↪ X × ℙ^{1} × (ℙ^{1})^{n−1} be the normalization composed with the closed immersion of the further Zariski closure Ẑ of \overline{Z}. Let F_{n}^{∞} = ∑_{i=1}^{n−1}\{y_{i} = ∞\} be the divisor at infinity. For an open set j : U ↪ X, the modulus condition of \overline{Z}_{U} means (m + 1)[j^{∗}ν^{∗}\{t = 0\}] ≤ [j^{∗}ν^{∗}(F_{n}^{∞})] on Ẑ_{U}^{N}. Note that there exist only finitely many prime Weil divisors P_{1}, ⋯, P_{ℓ} on Ẑ^{N} such that ord_{P_{i}}(ν^{∗}(F_{n}^{∞}) − (m + 1)ν^{∗}{t = 0}) < 0. Their images Q_{i} under the normalization map Ẑ^{N} → Ẑ are still irreducible proper closed subsets of Ẑ, thus of X × ℙ^{1} × (ℙ^{1})^{n−1}. Since Z = \overline{Z}_{Y} has the modulus condition on Y × B_{n} by the given assumption, we have (Y × B̂_{n}) ∩ Q_{i} = ∅ for each 1 ≤ i ≤ ℓ. Thus, there is an affine open subset U_{2} ⊂ X containing x such that (U_{2} × B̂_{n}) ∩ Q_{i} = ∅ for each 1 ≤ i ≤ ℓ. Now, by construction, \overline{Z}_{U_{2}} on U_{2} × B_{n} satisfies the modulus condition. So, taking an affine open subset U ⊂ U_{1} ∩ U_{2} containing x, we have \overline{Z}_{U} ∈ \underline{Tz}^{q}(U, n; m). That (\overline{Z}_{U})_{Y} = Z is obvious. □

We can extend this colimit description to semi-local schemes:

Lemma 4.13. Let Y be a semi-local k-scheme obtained by localizing at a finite set Σ of scheme points of a quasi-projective k-variety X. For a cycle Z on Y × B_{n}, let \overline{Z} be its Zariski closure in X × B_{n}. Then Z ∈ \underline{Tz}^{q}(Y, n; m) if and only if there exists an affine open subset U ⊂ X containing Σ, such that \overline{Z}_{U} ∈ \underline{Tz}^{q}(U, n; m), where \overline{Z}_{U} is the pull-back of \overline{Z} via the open immersion U → X.

Proof. The direction (⇐) is obvious by pulling back via the flat morphism Y ↪ U. For the direction (⇒), by Lemma 4.12, for each x ∈ Σ we have an affine open neighborhood U_{x} ⊂ X of x such that \overline{Z}_{U_{x}} ∈ \underline{Tz}^{q}(U_{x}, n; m). Take W = ⋃_{x∈Σ} U_{x}. This is an open subset of X containing Σ. By Lemma 2.9, we have \overline{Z}_{W} ∈ \underline{Tz}^{q}(W, n; m). On the other hand, by Lemma 4.11, there exists an affine open subset U ⊂ W containing Σ. By taking the flat pull-back via the open immersion U ↪ W, we get \overline{Z}_{U} ∈ \underline{Tz}^{q}(U, n; m). □

Lemma 4.14. Let Y be a semi-local integral k-scheme obtained by localizing at a finite set Σ of scheme points of an integral quasi-projective k-scheme X. Let Z ∈ Tz^{q}(Y, n; m), W ∈ Tz^{q}(Y, n+1; m), and let \overline{Z}, \overline{W} be their Zariski closures in X × B_{n} and X × B_{n+1}, respectively. For every open subset U ⊂ X, the subscript U means the pull-back to U. Then we have the following:

(1) If ∂Z = 0, we can find an affine open subset U ⊂ X containing Σ such that \overline{Z}_{U} ∈ Tz^{q}(U, n; m) and ∂\overline{Z}_{U} = 0.

(2) If Z = ∂W, we can find an affine open subset U ⊂ X containing Σ such that \overline{Z}_{U} ∈ Tz^{q}(U, n; m), \overline{W}_{U} ∈ Tz^{q}(U, n + 1; m) and \overline{Z}_{U} = ∂\overline{W}_{U}.

Proof. Note that (1) is a special case of (2), so we prove (2) only. Let Z′ := \overline{Z} − ∂\overline{W} ∈ z^{q}(X × B_{n}). If Z′ is 0 as a cycle, then take U_{0} = X. If not, let Z′_{1}, ⋯, Z′_{s} be the irreducible components of Z′. Since Z = ∂W, each component Z′_{i} has empty intersection with Y × B_{n}. So, each π((Z′_{i})^{c}) is a nonempty open subset of X containing Σ, where π : X ×B_{n} → X is the projection, which is open. Take U_{0} = ⋂_{i=1}^{s} π((Z′_{i})^{c}). On the other hand, Lemma 4.13 implies that there exist open sets U_{1}, U_{2} ⊂ X containing Σ such that \overline{Z}_{U_{1}} ∈ Tz^{q}(U_{1}, n; m) and \overline{W}_{U_{2}} ∈ Tz^{q}(U_{2}, n + 1; m).

Choose an affine open subset U ⊂ U_{0} ∩ U_{1} ∩ U_{2} containing Σ, using Lemma 4.11. Then part (2) holds over U by construction. □

Proof of Theorem 4.10. We show that the chain map Tz_{W}^{q}(Y, •; m) ↪ Tz^{q}(Y, •; m) is a quasi-isomorphism. Let X be a smooth affine k-variety with a finite subset Σ ⊂ X such that Y = Spec (O_{X,Σ}). For surjectivity on homology, let Z ∈ \underline{Tz}^{q}(Y, n; m) be such that ∂Z = 0. Let \overline{Z} be the Zariski closure of Z in X × B_{n}. Here, ∂\overline{Z} may not be zero, but by Lemma 4.14(1), there exists an affine open subset U ⊂ X containing Σ such that we have ∂\overline{Z}_{U} = 0, where \overline{Z}_{U} is the pull-back of \overline{Z} to U. Let W_{U} = {W_{U}|W ∈ W}, where W_{U} is the Zariski closure of W in U. Then the quasi-isomorphism Tz_{W_{U}}^{q} (U, •; m) ↪ Tz^{q}(U, •; m) of Theorem 4.1 shows that there are some C ∈ Tz^{q}(U, n + 1; m) and Z′_{U} ∈ Tz_{W_{U}}^{q} (U, n; m) such that ∂C = \overline{Z}_{U} − Z′_{U}. Let ι : Y ↪ U be the inclusion. So, by applying the flat pull-back ι^{∗} (which is equivariant with respect to taking faces), we obtain ∂(ι^{∗}C) = Z − ι^{∗}Z′_{U}, and here ι^{∗}Z′_{U} ∈ Tz_{W}^{q}(Y, n; m), i.e., Z is equivalent to a member in Tz_{W}^{q}(Y, n; m). For injectivity on homology, let Z ∈ Tz_{W}^{q}(Y, n; m) be such that Z = ∂Z′ for some Z′ ∈ Tz^{q}(Y, n + 1; m). Let \overline{Z} and \overline{Z}′ be the Zariski closures of Z and Z′ on X × B_{n} and X × B_{n+1}, respectively. Then by Lemma 4.14(2), there exists a nonempty open affine subset U ⊂ X containing Σ such that \overline{Z}_{U} = ∂\overline{Z}′_{U}. Then the quasi-isomorphism Tz_{W_{U}}^{q} (U, •; m) ↪ Tz^{q}(U, •; m) of Theorem 4.1 shows that there exists Z′′ ∈ Tz_{W_{U}}^{q} (U, n + 1; m) such that \overline{Z}_{U} = ∂Z′′. Pulling back via ι : Y ↪ U then shows Z = ∂(ι^{∗}Z′′), with ι^{∗}Z′′ ∈ Tz_{W}^{q}(Y, n + 1; m). □

Using an argument identical to Theorem 4.5 (see [17, Theorem 7.1]), we get:

Corollary 4.15. Let f : Y_{1} → Y_{2} be a morphism in Sch_{k}^{ess}, where Y_{2} ∈ SmLoc_{k}. Then there is a natural pull-back f^{∗} : TCH^{q}(Y_{2}, n; m) → TCH^{q}(Y_{1}, n; m).

5. The Pontryagin product

Let R be a commutative ring and let (A, d_{A}) be a differential graded algebra over R. Recall that (left) differential graded module M over A is a left Amodule M with a grading M = ⊕_{n∈ℤ}M_{n} and a differential d_{M} such that A_{m}M_{n} ⊂ M_{m+n}, d_{M}(M_{n}) ⊂ M_{n+1} and d_{M}(ax) = d_{A}(a)x + (−1)^{n}ad_{M}(x) for a ∈ A_{n} and x ∈ M. A homomorphism of differential graded modules f : M → N over A is an A-module map which is compatible with gradings and differentials. In this section, we show that the multivariate additive higher Chow groups have a product structure that resembles the Pontryagin product. We construct a differential operator on these groups in the next section and show that the product and the differential operator together turn multivariate additive higher Chow groups groups into a differential graded module over 𝕎_{m}Ω_{R}^{•} for suitable m, when X = Spec (R) is in SmAff_{k}^{ess}. This generalizes the DGA-structure on additive higher Chow groups of smooth projective varieties in [19]. The base field k is perfect in this section.

5.1. Some cycle computations. We generalize some of [19, §3.2.1, 3.2.2, 3.3]. Let (X, D) be a k-scheme with an effective divisor. Recall that a permutation σ ∈ 𝔖_{n} acts naturally on □^{n} via σ(y_{1}, ⋯, y_{n}) := (y_{σ(1)}, ⋯, y_{σ(n)}). This action extends to cycles on X × □^{n} and X × \overline{□}^{n}. Let n, r ≥ 1 be given. Consider the finite morphism χ_{n,r} : X × □^{n} → X × □^{n} given by (x, y_{1}, ⋯, y_{n}) ↦ (x, y_{1}^{r}, y_{2}, ⋯, y_{n}). Given an irreducible cycle Z ⊂ X × □^{n}, define Z{r} := (χ_{n,r})_{∗}([Z]) = [k(Z) : k(χ_{n,r}(Z))] · [χ_{n,r}(Z)]. We extend it ℤ-linearly.

Lemma 5.1. If Z is an admissible cycle with modulus D, then so is Z{r}.

Proof. The proof is almost identical to that of [19, Lemma 3.11], except that the divisor (m + 1)\{t = 0\} there should be replaced by D × \overline{□}^{n}. We give its argument for the reader’s convenience. We may assume Z is irreducible. It is enough to show that χ_{n,r}(Z) is admissible with modulus D. We first check that it satisfies the face condition of Definition 2.3. When n = 1, the proper faces of □ are of codimension 1, and for ϵ ∈ {0, ∞}, we have ∂_{1}^{ϵ}(χ_{n,r}(Z)) = r∂_{1}^{ϵ}(Z). When n ≥ 2, for ϵ ∈ {0, ∞}, we have ∂_{1}^{ϵ}(χ_{n,r}(Z)) = r∂_{1}^{ϵ}(Z) and ∂_{i}^{ϵ}(χ_{n,r}(Z)) = χ_{n−1,r}(∂_{i}^{ϵ}(Z)) if i ≥ 2. For faces F ⊂ □^{n} of higher codimensions, we consequently have F ·(χ_{n,r}(Z)) = r(F ·Z) if F involves the equations {y_{1} = ϵ}, and F ·(χ_{n,r}(Z)) = χ_{n−c,r}(F ·Z), otherwise, where c is the codimension of F. Since the intersection F · Z is proper, so is χ_{n−c,r}(F · Z) by induction on the codimension of faces. This shows χ_{n,r}(Z) satisfies the face condition. To show that W := χ_{n,r}(Z) has modulus D, consider the commutative diagram

\xymatrix{\overline{Z}^{N} \ar[r]^{ν_{Z}} \ar[d]^{_{\overline{χ}{N \atop n,r}}} & \overline{Z} \ar[d]^{_{\overline{χ}_{n,r}}} \ar@{^{(}->}[r]^{ι_{Z}} & X × \overline{□}^{n} \ar[d]^{χ_{n,r}} \\ _{\overline{W}_{N}} \ar[r]^{ν_{W}} & \overline{W} \ar@{^{(}->}[r]^{ι_{W}} & X × \overline{□}^{n},}

where \overline{Z}, \overline{W} are the Zariski closures of Z and W in X × \overline{□}^{n} and ν_{Z}, ν_{W} are the respective normalizations. The morphisms χ_{n,r}, \overline{χ}_{n,r} are the natural induced maps, and \overline{χ}_{n,r}^{N} is induced by the universal property of normalization. Since Z has modulus D, we have the inequality

(5.1)  [ν_{Z}^{∗}ι_{Z}^{∗}(D × \overline{□}^{n})] ≤ ∑_{i=1}^{n}[ν_{Z}^{∗}ι_{Z}^{∗}\{y_{i} = 1\}].

By the definition of χ_{n,r}, we have χ_{n,r}^{∗}(D × \overline{□}^{n}) = D × \overline{□}^{n}, χ_{n,r}^{∗}\{y_{1} = 1\} ≥ {y_{1} = 1}, and χ_{n,r}^{∗}{y_{i} = 1} = {y_{i} = 1} for i ≥ 2. Hence (5.1) implies that [ν_{Z}^{∗}ι_{Z}^{∗}χ_{n,r}^{∗}(D × \overline{□}^{n})] ≤ ∑_{i=1}^{n}[ν_{Z}^{∗}ι_{Z}^{∗}χ_{n,r}^{∗}\{y_{i} = 1\}]. By the commutativity of the

diagram, this implies that \overline{χ}_{n,r}^{N}^{∗} (∑_{i=1}^{n} ν_{W}^{∗} ι_{W}^{∗} \{y_{i} = 1\} − ν_{W}^{∗} ι_{W}^{∗} (D × \overline{□}^{n})) ≥ 0.

By Lemma 2.7, this implies ∑_{i=1}^{n} ν_{W}^{∗} ι_{W}^{∗} \{y_{i} = 1\} − ν_{W}^{∗} ι_{W}^{∗} (D × \overline{□}^{n}) ≥ 0, which means W has modulus D. This completes the proof. □

Let n, i ≥ 1. Suppose X is smooth quasi-projective essentially of finite type over k. Let (x, y_{1}, ⋯, y_{n}, y, λ) be the coordinates of X × \overline{□}^{n+2}. Consider the closed subschemes V_{X}^{i} on X ×□^{n+2} given by the equation (1−y)(1−λ) = 1−y_{1} if i = 1, and (1−y)(1−λ) = (1−y_{1})(1+y_{1} +⋯+y_{1}^{i−1} −λ(1+y_{1} +⋯+y_{1}^{i−2})) if i ≥ 2. Let V̂_{X}^{i} be the Zariski closure of V_{X}^{i} in X × \overline{□}^{n+2}. Let π_{1} : X × \overline{□}^{n+2} → X × \overline{□}^{n+1} be the projection that drops y_{1}, and let π′_{1} := π_{1}|_{V_{X}^{i}}. As in [19, Lemma 3.12], one sees that π′_{1} is proper surjective. For an irreducible cycle Z ⊂ X × □^{n}, define (see [19, Definition 3.13]) γ_{Z}^{i} := π′_{1}_{∗}(V_{X}^{i} · (Z × □^{2})) as an abstract algebraic cycle. One checks that it is also the Zariski closure of ν^{i}(Z × □), where ν^{i} : X × □^{n} × □ → X × □^{n+1} is the rational map given by ν^{i}(x, y_{1}, ⋯, y_{n}, y) = (x, y_{2}, y_{3}, ⋯, y_{n}, y, \frac{y−y_{1}^{i}}{y−y_{1}^{i−1}} ). We extend the definition of

γ_{Z}^{i} ℤ-linearly.

Lemma 5.2. Let Z ∈ z^{q}(X|D, n). Then γ_{Z}^{i} ∈ z^{q}(X|D, n + 1).

Proof. Once we have Lemma 5.1, the proof of Lemma 5.2 is very similar to that of [19, Lemma 3.15], except we replace (m + 1)\{t = 0\} by D × \overline{□}^{n+1}. We give its argument for the reader’s convenience. We may assume Z is irreducible. To keep track of n, we write γ_{Z,n}^{i} = γ_{Z}^{i}. We first check that it satisfies the face condition of Definition 2.3. Let ϵ ∈ {0, ∞}. Let F ⊂ □^{n+1} be a face. If F involves the equation {y_{j} = ϵ} for j = n, n + 1, then by direction computations, we see that ∂_{n}^{0}(γ_{Z,n}^{i}) = σ · Z, ∂_{n+1}^{0}(γ_{Z,n}^{i}) = σ · (Z{i}) for the cyclic permutation σ = (1, 2, ⋯, n), and ∂_{n}^{∞}(γ_{Z,n}^{i}) = 0, ∂_{n+1}^{∞}(γ_{Z,n}^{i}) = σ · (Z{i − 1}). Since Z is admissible with modulus D, so are Z{i} and Z{i − 1} by Lemma 5.1. In particular, all of σ · Z, σ · (Z{i}), and σ · (Z{i − 1}) intersect all faces properly. Hence γ_{Z,n}^{i} intersects F properly. In case F does not involve the equations {y_{j} = ϵ} for j = n, n + 1, we prove it by induction on n ≥ 1. By direction calculations, for j < n, we have ∂_{j}^{ϵ}(γ_{Z,n}^{i}) = γ_{∂_{j}^{ϵ}Z,n−1}^{i} so that the dimension of ∂_{j}^{i}(γ_{Z,n}^{i}) is at least one less by the induction hypothesis. Repeated applications of this argument for all other defining equations of F then give the result. It remains to show that γ_{Z}^{i} has modulus D. Every irreducible component of γ_{Z}^{i} is of the form W′ = π′_{1}(Z′), where Z′ is an irreducible component of V_{X}^{i} · (Z × □^{2}). We prove W′ has modulus D. Consider the following commutative diagram

\xymatrix{\overline{Z}′^{N} \ar[d]^{π_{1}^{N}} \ar[r]^{ν_{Z′}} & V̂_{X}^{i} \ar@{^{(}->}[r]^{ι} \ar[d]^{_{\overline{π}{′ \atop 1}}} \ar[dr]^{π′_{1}} & X × \overline{□}^{n+2} \ar[d]^{π_{1}} \\ _{\overline{W}_{′}}_{N} \ar[r]^{ν} & _{\overline{W}_{′}} \ar@{^{(}->}[r]^{ι_{W′}} & X × \overline{□}^{n+1},}

where ν_{Z′} is the normalization of the Zariski closure \overline{Z}′ of Z′ in V̂_{X}^{i}, ν is the normalization of the Zariski closure \overline{W}′ of W′ in X × \overline{□}^{n+1}, and π_{1}^{N}, \overline{π}′_{1} are the induced morphisms. We use (x, y_{1}, ⋯, y_{n}, y, λ) ∈ X × \overline{□}^{n+2} and (x, y_{2}, ⋯, y_{n}, y, λ) ∈ X × \overline{□}^{n+1} as the coordinates. From the modulus D condition of Z, we deduce

(5.2)  ν_{Z′}^{∗}ι^{∗}(D × \overline{□}^{n+2}) ≤ ∑_{j=1}^{n} ν_{Z′}^{∗}ι^{∗}\{y_{j} = 1\}.

Note that the above does not involve the divisors {y = 1} and {λ = 1}. Since V_{X}^{i} is an effective divisor on X × □^{n+2} defined by the equation (1 − y_{1})(∗) = (1 − y)(1 − λ) for some polynomial (∗), we have [ν_{Z′}^{∗}ι^{∗}{y_{1} = 1}] ≤ [ν_{Z′}^{∗}ι^{∗}{y = 1}] + [ν_{Z′}^{∗}ι^{∗}{λ = 1}]. Since the above diagram commutes, from (5.2) we deduce π_{1}^{N}^{∗}ν^{∗}ι_{W′}^{∗}(D × \overline{□}^{n+1}) ≤ π_{1}^{N}^{∗} (∑_{j=2}^{n} ν^{∗}ι_{W′}^{∗}\{y_{j} = 1\} + \{y = 1\} + \{λ = 1\}). Hence by Lemma

2.7, we deduce ν^{∗}ι_{W′}^{∗}(D × \overline{□}^{n+1}) ≤ ∑_{j=2}^{n} ν^{∗}ι_{W′}^{∗}\{y_{j} = 1\} + \{y = 1\} + \{λ = 1\}, which means W′ has modulus D. This finishes the proof. □

Lemma 5.3. Let n ≥ 2 and let Z ∈ z^{q}(X|D, n) such that ∂_{i}^{ϵ}(Z) = 0 for all 1 ≤ i ≤ n and ϵ ∈ {0, ∞}. Let σ ∈ 𝔖_{n}. Then there exists γ_{Z}^{σ} ∈ z^{q}(X|D, n + 1) such that Z = (sgn(σ))(σ · Z) + ∂(γ_{Z}^{σ}).

Proof. Its proof is almost identical to that of [19, Lemma 3.16], except that we use Lemma 5.2 instead of [19, Lemma 3.15]. We give its argument for the reader’s convenience. First consider the case when σ is the transposition τ = (p, p + 1) for 1 ≤ p ≤ n − 1. We do it for p = 1 only, i.e. τ = (1, 2). Other cases of τ are similar. Let ξ be the unique permutation such that ξ · (x, y_{1}, ⋯, y_{n+1}) = (x, y_{n}, y_{1}, y_{n+1}, y_{2}, ⋯, y_{n−1}). Consider the cycle γ_{Z}^{τ} := ξ · γ_{Z}^{1}, where γ_{Z}^{1} is as in Lemma 5.2. Being a permutation of an admissible cycle, so is this cycle γ_{Z}^{ξ}. Furthermore, by direction calculations, we have ∂_{1}^{∞}(γ_{Z}^{τ}) = 0, ∂_{1}^{0}(γ_{Z}^{τ}) = τ · Z, ∂_{3}^{∞}(γ_{Z}^{τ}) = 0 and ∂_{3}^{0}(γ_{Z}^{τ}) = Z. On the other hand, for ϵ ∈ {0, ∞}, ∂_{2}^{ϵ}(γ_{Z}^{τ}) is a cycle obtained from γ_{∂_{2}^{ϵ}(Z)}^{1} by a permutation action. So, it is 0 because ∂_{2}^{ϵ}(Z) = 0 by the given assumptions. Similarly for j ≥ 4, we have ∂_{j}^{ϵ}(γ_{Z}^{τ}) = 0. Hence ∂(γ_{Z}^{τ}) = Z + τ · Z, as desired. Now let σ ∈ 𝔖_{n} be any. By a basic result from group theory, we can express σ = τ_{r}τ_{r−1} ⋯ τ_{2}τ_{1}, where each τ_{i} is a transposition of the form (p, p + 1) as considered before. Let σ_{0} := Id and σ_{ℓ} := τ_{ℓ}τ_{ℓ−1} ⋯ τ_{1} for 1 ≤ ℓ ≤ r. For each such ℓ, by the previous case considered, we have (−1)^{ℓ−1}σ_{ℓ−1} · Z + (−1)^{ℓ−1}τ_{ℓ} · σ_{ℓ−1} · Z = ∂((−1)^{ℓ−1}γ_{σ_{ℓ−1}·Z}^{τ_{ℓ}}). Since τ_{ℓ} · σ_{ℓ−1} = σ_{ℓ}, by taking the sum of the above equations over all 1 ≤ ℓ ≤ r, after cancellations, we obtain Z + (−1)^{r−1}σ · Z = ∂(γ_{Z}^{σ}), where γ_{Z}^{σ} := ∑_{ℓ=1}^{r}(−1)^{ℓ−1}γ_{σ_{ℓ−1}·Z}^{τ_{ℓ}}. Since (−1)^{r} = sgn(σ), we obtain the desired result. □

5.2. Pontryagin product. Let X ∈ Sch_{k}^{ess} be an equidimensional scheme. For \underline{m} = (m_{1}, ⋯, m_{r}) ≥ 1, let CH(X[r]|D\underline{m}) := ⊕_{q,n}CH^{q}(X[r]|D\underline{m}, n). For m ≥ 1, we let TCH(X; m) = ⊕_{q,n}TCH^{q}(X, n; m) = ⊕_{q,n}CH^{q}(X[1]|D_{m+1}, n − 1). The objective of §5.2 is to prove the following result which generalizes [19, §3].

Theorem 5.4. Let k be a perfect field. Let m ≥ 0 and let \underline{m} = (m_{1}, ⋯, m_{r}) ≥ 1. Let X, Y be both either in SmAff_{k}^{ess} or in SmProj_{k}. Then we have the following:

(1) TCH(X; m) is a graded commutative algebra with respect to a product ∧_{X}.

(2) CH(X[r]|D\underline{m}) is a graded module over TCH(X; |\underline{m}| − 1).

(3) For f : Y → X with d = dimY − dimX, f^{∗} : CH(X[r]|D\underline{m}) → CH(Y [r]|D\underline{m}) and f_{∗} : CH(Y [r]|D\underline{m}) → CH(X[r]|D\underline{m})[−d] (if f is proper in addition) are morphisms of graded TCH(X; |\underline{m}|−1)-modules.

The proof requires a series of results and will be over after Lemma 5.13.

Lemma 5.5. Let X_{1}, X_{2} ∈ Sch_{k}^{ess}. For i = 1, 2 and r_{i} ≥ 1, let V_{i} be a cycle on X_{i} × 𝔸^{r_{i}} × □^{n_{i}} with modulus \underline{m}_{i} = (m_{i1}, ⋯, m_{ir_{i}}), respectively. Then V_{1} × V_{2}, regarded as a cycle on X_{1} × X_{2} × 𝔸^{r_{1}+r_{2}} × □^{n_{1}+n_{2}} after a suitable exchange of factors, has modulus (\underline{m}_{1}, \underline{m}_{2}).

Proof. We may assume that V_{1} and V_{2} are irreducible. It is enough to show that each irreducible component W ⊂ V_{1} × V_{2} has modulus (\underline{m}_{1}, \underline{m}_{2}). Let ι_{i} : \overline{V}_{i} ↪ X_{i} ×𝔸^{r_{i}} ×\overline{□}^{n_{i}} be the Zariski closure of V_{i}, and let ν_{\overline{V}_{i}} : \overline{V} _{i}^{N} → \overline{V}_{i} be the normalization for i = 1, 2. Since k is perfect, [16, Lemma 3.1] says that the morphism ν := ν_{\overline{V}_{1}} ×ν_{\overline{V}_{2}} : \overline{V} _{1}^{N} ×\overline{V} _{2}^{N} → \overline{V}_{1}×\overline{V}_{2} = \overline{V_{1} × V_{2}} is the normalization.

Hence, the composite \overline{W}^{N} →^{ν_{W}} \overline{W} ↪^{ι} \overline{V}_{1} ×\overline{V}_{2}, where \overline{W} is the Zariski closure of

W and ν_{W} is the normalization of \overline{W}, factors into \overline{W}^{N} →^{ι^{N}} \overline{V} _{1}^{N} ×\overline{V} _{2}^{N} →^{ν} \overline{V}_{1}×\overline{V}_{2}, where ι^{N} is the natural inclusion. Let (t_{1}, ⋯, t_{r_{1}}, t′_{1}, ⋯, t′_{r_{2}}, y_{1}, ⋯, y_{n_{1}+n_{2}}) ∈ 𝔸^{r_{1}+r_{2}} × \overline{□}^{n_{1}+n_{2}} be the coordinates. Consider two divisors D^{1} := ∑_{i=1}^{n_{1}}{y_{i} = 1} − ∑_{j=1}^{r_{1}} m_{1j}{t_{j} = 0}, D^{2} := ∑_{i=n_{1}+1}^{n_{1}+n_{2}}{y_{i} = 1} − ∑_{j=1}^{r_{2}} m_{2j}{t′_{j} = 0}. By the modulus conditions satisfied by V_{1} and V_{2}, we have ((ι_{1} × 1) ∘ (ν_{\overline{V}_{1}} × 1))^{∗}D^{1} ≥ 0 and ((1 × ι_{2}) ∘ (1 × ν_{\overline{V}_{2}}))^{∗}D^{2} ≥ 0. Thus, we have ν^{∗}(ι_{1} × ι_{2})^{∗}(D^{1} + D^{2}) ≥ 0 on

\overline{V} _{1}^{N} ×\overline{V} _{2}^{N} so that (ι^{N})^{∗}ν^{∗}(ι_{1} ×ι_{2})^{∗}(D^{1} +D^{2}) ≥ 0 on \overline{W}^{N}. Since ι∘ν_{W} = ν ∘ι^{N}, this is equivalent to ν_{W}^{∗} ι^{∗}(ι_{1} × ι_{2})^{∗}(D^{1} + D^{2}) ≥ 0, which shows W has modulus (\underline{m}_{1}, \underline{m}_{2}). □

Definition 5.6. Let r ≥ 1 be an integer and define μ : X_{1} × 𝔸^{1} × □^{n_{1}} × X_{2} × 𝔸^{r} × □^{n_{2}} → X_{1} × X_{2} × 𝔸^{r} × □^{n_{1}+n_{2}} by (x_{1}, t, {y_{j}}) × (x_{2}, {t_{i}}, {y′_{j}}) ↦ (x_{1}, x_{2}, {tt_{i}}, {y_{j}}, {y′_{j}}).

The map μ is flat, but not proper. But, the following generalization of [19, Lemma 3.4] gives a way to take a push-forward:

Proposition 5.7. Let V_{1} ⊂ X_{1} × 𝔸^{1} × □^{n_{1}} and V_{2} ⊂ X_{2} × 𝔸^{r} × □^{n_{2}} be closed subschemes with moduli m and \underline{m} ≥ 1, respectively. Then μ|_{V_{1}×V_{2}} is finite.

Proof. Since μ is an affine morphism, the proposition is equivalent to show that μ|_{V_{1}×V_{2}} is projective. Set X = X_{1} × X_{2} × □^{n_{1}+n_{2}}. Let Γ ↪ X_{1} × X_{2} × 𝔸^{1} × 𝔸^{r} × 𝔸^{r} × □^{n_{1}+n_{2}} = X × 𝔸^{1} × 𝔸^{r} × 𝔸^{r} denote the graph of the morphism μ and let \overline{Γ} ↪ X × ℙ^{1} × (ℙ^{1})^{r} × (ℙ^{1})^{r} = X × P_{1} × P_{2} × P_{3} be its closure, where P_{1} = ℙ^{1} and P_{2} = P_{3} = (ℙ^{1})^{r}. Let p_{i} be the projection of X × ℙ^{1} × (ℙ^{1})^{r} × (ℙ^{1})^{r} to X × P_{i} for 1 ≤ i ≤ 3. Set \overline{Γ}^{0} = p_{3}^{−1}(X × 𝔸^{r}). Then p_{3} : \overline{Γ}^{0} → X × 𝔸^{r} is projective. Using the homogeneous coordinates of P_{1} × P_{2} × P_{3}, one checks easily that Z := \overline{Γ}^{0} \Γ ⊂ E ∪(⋃_{i=1}^{r} E_{i}) (the union is taken inside X ×P_{1} ×P_{2} ×P_{3}), where E = X ×{∞}×({0})^{r} ×𝔸^{r} and E_{i} = X ×{0}×((ℙ^{1})^{i−1} ×{∞}×(ℙ^{1})^{r−i})×𝔸^{r}. Let V = V_{1}×V_{2}. Let Γ_{V} be the graph Γ restricted to V and let \overline{Γ}_{V} be its Zariski closure in X × P_{1} × P_{2} × P_{3}. Since p_{3} : \overline{Γ}^{0} → X × 𝔸^{r} is projective, so is the map \overline{Γ}_{V}^{0} := \overline{Γ}_{V} ∩ \overline{Γ}^{0} → X × 𝔸^{r}. So, if we show \overline{Γ}_{V}^{0} ∩ Z = ∅, then V ≃ Γ_{V} = \overline{Γ}_{V}^{0} is projective over X × 𝔸^{r}, which is the assertion of the proposition. To show \overline{Γ}_{V}^{0} ∩ Z = ∅, consider the projections X × P_{1} × P_{2} × P_{3} →^{p_{1}} X × P_{1} →^{π_{1}} X_{1} × P_{1} × □^{n_{1}}. Since the closure \overline{V}_{1} has modulus m ≥ 1 on X_{1} × P_{1} × □^{n_{1}}, we have \overline{V}_{1} ∩ (X_{1} × \{0\} × □^{n_{1}}) = ∅. In particular, \overline{Γ}_{V} ∩ E_{i} ↪ (π_{1} ∘ p_{1})^{−1}(\overline{V}_{1} ∩ (X_{1} × {0} × □^{n_{1}})) = ∅ for 1 ≤ i ≤ r. To show that \overline{Γ}_{V}^{0} ∩ E = ∅, consider the projections X × P_{1} × P_{2} × P_{3} →^{p_{2}} X × P_{2} →^{π_{2}} X_{2} × P_{2} × □^{n_{2}}. Since the closure \overline{V}_{2} has modulus \underline{m} ≥ 1 on X_{2} ×P_{2} ×□^{n_{2}}, we have \overline{V}_{2} ∩(X_{2} ×(\{0\})^{r} ×□^{n_{2}}) = ∅. In particular, \overline{Γ}_{V} ∩E ↪ (π_{2} ∘ p_{2})^{−1}(\overline{V}_{2} ∩ (X_{2} × (\{0\})^{r} × □^{n_{2}})) = ∅. This finishes the proof. □

Lemma 5.8. Let X ∈ Sch_{k}^{ess} and let V be a cycle on X × 𝔸^{1} × 𝔸^{r} × □^{n} with modulus (|\underline{m}|, \underline{m}), where \underline{m} = (m_{1}, ⋯, m_{r}) ≥ 1. Suppose μ|_{V} is finite. Then the closed subscheme μ(V) on X × 𝔸^{r} × □^{n} has modulus \underline{m}.

Proof. This is a straightforward generalization of [19, Proposition 3.8] and is a simple application of Lemma 2.7. We skip the detail. We only remark that it is crucial for the proof that the 𝔸^{1}-component of the modulus is at least |\underline{m}|. □

Definition 5.9. For any irreducible closed subscheme V ⊂ X × 𝔸^{1} × 𝔸^{r} × □^{n} such that μ|_{V} : V → μ(V) is finite, where μ is as in Definition 5.6, define μ_{∗}(V) as the push-forward μ_{∗}(V) = deg(μ|_{V}) · [μ(V)]. Extend it ℤ-linearly. If V_{1} is a cycle on X_{1} × 𝔸^{1} × □^{n_{1}} and V_{2} is a cycle on X_{2} × 𝔸^{r} × □^{n_{2}} such that μ|_{V_{1}×V_{2}} is finite, we define the external product V_{1} ×_{μ} V_{2} := μ_{∗}(V_{1} × V_{2}). If p_{i} = dim V_{i}, then dim(V_{1} ×_{μ} V_{2}) = p_{1} + p_{2}. If X_{1} × X_{2} is equidimensional and if q_{i} is the codimension of V_{i}, then V_{1} ×_{μ} V_{2} has codimension q_{1} + q_{2} − 1.

Lemma 5.10. Let V_{1} ∈ z^{q_{1}}(X_{1}[1]|D_{m}, n_{1}) and V_{2} ∈ z^{q_{2}}(X_{2}[r]|D\underline{m}, n_{2}) with X_{1}, X_{2} ∈ Sch_{k}^{ess} and m, \underline{m} ≥ 1. Then V_{1} ×_{μ} V_{2} intersects all faces of X_{1} × X_{2} × 𝔸^{r} × □^{n_{1}+n_{2}} properly.

Proof. We may assume that V_{1} and V_{2} are irreducible. V_{1}×V_{2} clearly intersects all faces of X_{1}×X_{2}×𝔸^{1}×𝔸^{r} ×□^{n_{1}+n_{2}} properly. It follows from Proposition 5.7 that μ|_{V_{1}×V_{2}} is finite. In this case, the proper intersection property of μ(V_{1} ×_{μ} V_{2}) follows exactly like that of the finite push-forwards of Bloch’s higher Chow cycles. □

Corollary 5.11. Let X_{1}, X_{2}, X_{3} ∈ Sch_{k}^{ess} be equidimensional and let \underline{m} ≥ 1. Then there is a product

×_{μ} : z^{q_{1}}(X_{1}[1]|D_{|\underline{m}|}, n_{1}) ⊗ z^{q_{2}}(X_{2}[r]|D\underline{m}, n_{2}) →

→ z^{q_{1}+q_{2}−1}((X_{1} × X_{2})[r]|D\underline{m}, n_{1} + n_{2})

which satisfies the relation ∂(ξ ×_{μ} η) = ∂(ξ) ×_{μ} η + (−1)^{n_{1}}ξ ×_{μ} ∂(η). It is associative in the sense that (α_{1} ×_{μ} α_{2}) ×_{μ} β = α_{1} ×_{μ} (α_{2} ×_{μ} β) for α_{i} ∈ z^{q_{i}}(X_{i}[1]|D_{|\underline{m}|}, n_{i}) for i = 1, 2 and β ∈ z^{q_{3}}(X_{3}[r]|D\underline{m}, n_{3}). In particular, it induces operations ×_{μ} : CH^{q_{1}}(X_{1}[1]|D_{|\underline{m}|}, n_{1}) ⊗ CH^{q_{2}}(X_{2}[r]|D\underline{m}, n_{2}) → CH^{q_{1}+q_{2}−1}((X_{1} × X_{2})[r]|D\underline{m}, n_{1} + n_{2}).

Proof. The existence of ×_{μ} on the level of cycle complexes follows from the combination of Proposition 5.7, Lemma 5.8 and Lemma 5.10. The associativity follows from that of the Cartesian product × and the product μ : 𝔸^{1}×𝔸^{1} → 𝔸^{1}. By definition, one checks ∂(ξ × η) = ∂(ξ) × η + (−1)^{n_{1}}ξ × ∂(η). So, by applying μ_{∗}, we get the required relation. That ×_{μ} descends to the homology follows. □

Definition 5.12. Let \underline{m} = (m_{1}, ⋯, m_{r}) ≥ 1 and let X be in SmAff_{k}^{ess} or in SmProj_{k}. For cycle classes α_{1} ∈ CH^{q_{1}}(X[1]|D_{|\underline{m}|}, n_{1}) and α_{2} ∈ CH^{q_{2}}(X[r]|D\underline{m}, n_{2}), define the internal product α_{1} ∧_{X} α_{2} to be Δ_{X}^{∗}(α_{1} ×_{μ} α_{2}) via the diagonal pull-back Δ_{X}^{∗} : CH^{q_{1}+q_{2}−1}((X × X)[r]|D\underline{m}, n_{1} + n_{2}) → CH^{q_{1}+q_{2}−1}(X[r]|D\underline{m}, n_{1} + n_{2}). This map exists by Theorem 4.5 and Corollary 4.15.

Lemma 5.13. ∧_{X} is associative in the sense that (α_{1} ∧_{X} α_{2}) ∧_{X} β = α_{1} ∧_{X} (α_{2} ∧_{X} β) for α_{1}, α_{2} ∈ CH(X[1]|D_{|\underline{m}|}) and β ∈ CH(X[r]|D\underline{m}). ∧_{X} is also graded-commutative on CH(X[1]|D_{|\underline{m}|}).

Proof. The associativity holds by Corollary 5.11. For the gradedcommutativity, first note by Theorem 3.2 that we can find representatives α_{1} and α_{2} of the given cycle classes whose codimension 1 faces are all trivial. Let σ be the permutation that sends (1, ⋯, n_{1}, n_{1} + 1, ⋯, n_{1} + n_{2}) to (n_{1} + 1, ⋯, n_{1} + n_{2}, 1, ⋯, n_{1}) so that sgn(σ) = (−1)^{n_{1}+n_{2}}. It follows from Lemma 5.3 that α_{1} ∧_{X} α_{2} = (−1)^{n_{1}+n_{2}}α_{2} ∧_{X} α_{1} + ∂(W) for some admissible cycle W, as desired. □

Proof of Theorem 5.4. The proof of (1) and (2) is just a combination of the above discussion under the observation that TCH^{q}(X, n; m) = CH^{q}(X[1]|D_{m+1}, n − 1) for m ≥ 0 and n ≥ 1. To prove (3) for f^{∗}, consider the commutative diagram

(5.3)  \xymatrix{Y [r] × □^{n} \ar[d]^{f} \ar[r]^{Δ_{Y}} & (Y × Y)[r] × □^{n} \ar[d]^{f×f} & (Y × Y)[r + 1] × □^{n} \ar[d]^{f×f} \ar[l]_{μ_{Y}} \\ X[r] × □^{n} \ar[r]^{Δ_{X}} & (X × X)[r] × □^{n} & (X × X)[r + 1] × □^{n}. \ar[l]_{μ_{X}}}

There is a finite set W of locally closed subsets of X such that f^{∗} : z_{W}^{q_{1}}(X[1]|D_{|\underline{m}|}, •) → z^{q_{1}}(Y [1]|D_{|\underline{m}|}, •) and f^{∗} : z_{W}^{q_{2}}(X[r]|D\underline{m}, •) → z^{q_{2}}(Y [r]|D\underline{m}, •) can be defined as taking cycles associated to the inverse images. Moreover, it is enough to consider the product of cycles in z_{W}^{q_{1}}(X[1]|D_{|\underline{m}|}, •) and z_{W}^{q_{2}}(X[r]|D\underline{m}, •) by the moving lemmas Theorems 4.1 and 4.4. For irreducible cycles V_{1} ∈ z^{q_{1}}(X[1]|D_{|\underline{m}|}, n_{1}) and V_{2} ∈ z^{q_{2}}(X[r]|D\underline{m}, n_{2}), the map μ_{Y} is finite when restricted to f^{∗}(V_{1}) × f^{∗}(V_{2}) by Lemma 5.7. In particular, μ_{Y} (f^{∗}(V_{1}) × f^{∗}(V_{2})) ∈ z^{q_{1}+q_{2}−1}((Y × Y)[r]|D\underline{m}, n_{1} + n_{2}). Since the right square in the diagram (5.3) is transverse, it follows that f^{∗}(μ_{X}(V_{2} × V_{2})) = μ_{Y} (f^{∗}(V_{1}) × f^{∗}(V_{2})) as cycles. The desired commutativity of the product with f^{∗} now follows from the commutativity of the left square in (5.3) and the composition law of Theorem 4.5. The proof of (3) for f_{∗} is just the projection formula, whose proof is identical to the one given in [19, Theorem 3.19] in the case when X_{1}, X_{2} ∈ SmProj_{k}. □

As applications, we obtain:

Corollary 5.14. Let X be in SmAff_{k}^{ess} or in SmProj_{k}. Then for q, n ≥ 0 and \underline{m} ≥ 1, the group CH^{q}(X[r]|D\underline{m}, n) is a 𝕎_{(|\underline{m}|−1)}(k)-module.

Proof. Applying Theorem 5.4 to X and the structure map X → Spec (k), it follows that CH(X[r]|D\underline{m}) is a graded module over TCH(k; |\underline{m}| − 1). By Corollary 5.11, this yields a TCH^{1}(k, 1; |\underline{m}| − 1)-module structure on each CH^{q}(X[r]|D\underline{m}, n). The corollary now follows from the fact that there is a ring isomorphism 𝕎_{m}(k) →^{∼} TCH^{1}(k, 1; m) for every m ≥ 1 by [28, Corollary 3.7]. □

We can explain the homotopy invariance of the groups CH^{q}(X, n) in terms of additive higher Chow groups as follows.

Corollary 5.15. For X ∈ Sch_{k}^{ess} which is equidimensional and for q, n ≥ 0, we have CH^{q}(X[1]|D_{1}, n) = 0.

Proof. By Corollary 5.11, we have a map ×_{μ} : CH^{1}(pt[1]|D_{1}, 0) ⊗ CH^{q}(X[1]|D_{1}, n) → CH^{q}(X[1]|D_{1}, n) and it follows from the definition of ×_{μ} that [1] ×_{μ} α = α for every α ∈ CH^{q}(X[1]|D_{1}, n), where [1] ∈ CH^{1}(pt[1]|D_{1}, 0) is the cycle given by the closed point 1 ∈ 𝔸^{1}(k). It therefore suffices to show that the homology class of 1 is zero. To do so, we may use the identification (□, {∞, 0}) ≃ (𝔸^{1}, {0, 1}) given by y ↦ 1/(1 − y) again. Then the cycle C ⊂ 𝔸^{2} given by {(t, y) ∈ 𝔸^{2}|ty = 1} is an admissible cycle in z^{1}(pt[1]|D_{1}, 1) such that ∂_{1}([C]) = [1] and ∂_{0}([C]) = 0. □

6. The structure of differential graded modules

In this section, we construct a differential operator on the graded module of §5 of multivariate additive higher Chow groups over the univariate additive higher Chow groups, generalizing [19, §4]. We assume that k is perfect and char(k) ≠ 2.

6.1. Differential. Let X be a smooth quasi-projective scheme essentially of finite type over k. Let r ≥ 1 and let \underline{m} = (m_{1}, ⋯, m_{r}) ≥ 1. Let (𝔾_{m}^{r})^{×} := {(t_{1}, ⋯, t_{r}) ∈ 𝔾_{m}^{r} | t_{1} ⋯ t_{r} ≠ 1}. Consider the morphism δ_{n} : (𝔾_{m}^{r})^{×} × □^{n} → 𝔾_{m}^{r} × □^{n+1}, (t_{1}, ⋯, t_{r}, y_{1}, ⋯, y_{n}) ↦ (t_{1}, ⋯, t_{r}, \frac{1}{t_{1}···t_{r}} , y_{1}, ⋯, y_{n}). It induces δ_{n} : X × (𝔾_{m}^{r})^{×} × □^{n} → X × 𝔾_{m}^{r} × □^{n+1}. Recall a closed subscheme Z ⊂ X ×𝔸^{r} ×□^{n} with modulus \underline{m} does not intersect the divisor {t_{1} ⋯ t_{r} = 0}. So, it is closed in X × 𝔾_{m}^{r} × □^{n}. For such Z, we define Z^{×} := Z|_{X×(𝔾_{m}^{r})^{×}×□^{n}}.

Lemma 6.1. For a closed subscheme Z ⊂ X × 𝔸^{r} × □^{n} with modulus \underline{m}, the image δ_{n}(Z^{×}) is closed in X × 𝔾_{m}^{r} × □^{n+1}.

Proof. It is enough to show that δ_{n} : X × (𝔾_{m}^{r})^{×} × □^{n} → X × 𝔾_{m}^{r} × □^{n+1} is a closed immersion. It reduces to show that the map (𝔾_{m}^{r})^{×} → 𝔾_{m}^{r} × (ℙ^{1} \ {1}) given by (t_{1}, ⋯, t_{r}) ↦ (t_{1}, ⋯, t_{r}, 1/(t_{1} ⋯ t_{r})) is a closed immersion. This is obvious because the image coincides with the closed subscheme given by the equation t_{1} ⋯ t_{r}y = 1, where (t_{1}, ⋯, t_{r}, y) ∈ 𝔾_{m}^{r} × □ are the coordinates. □

Definition 6.2 (cf. [19, Definition 4.3]). For a closed subscheme Z ⊂ X × 𝔸^{r} × □^{n} with modulus \underline{m}, we write δ_{n}(Z) := δ_{n}(Z^{×}). If Z is a cycle, we define δ_{n}(Z) by extending it ℤ-linearly. We may often write δ(Z) if no confusion arises.

Lemma 6.3. Let Z be a cycle on X × 𝔸^{r} × □^{n} with modulus \underline{m}. Then δ_{n}(Z) is a cycle on X × 𝔸^{r} × □^{n+1} with modulus \underline{m}.

Proof. We may suppose that Z is irreducible. Let V = δ_{n}(Z), which is a priori closed in X ×𝔾_{m}^{r} ×□^{n+1}. If the closure V′ of V in X ×𝔸^{r} ×□^{n+1} has modulus \underline{m}, then it does not intersect the divisor \{t_{1} ⋯ t_{r} = 0\} of X × 𝔸^{r} × □^{n+1}, so V = V′, and V is closed in X × 𝔸^{r} × □^{n+1} with modulus \underline{m}. So, we reduce to show that V′ has modulus \underline{m}. Let \overline{Z} and \overline{V} be the Zariski closures of Z and V′ in X × 𝔸^{r} × \overline{□}^{n} and X × 𝔸^{r} × \overline{□}^{n+1}, respectively. Observe that δ_{n} extends to \overline{δ}_{n} : X × 𝔸^{r} × \overline{□}^{n} → X × 𝔸^{r} × \overline{□}^{n+1}, which is induced from 𝔸^{r} →^{Γ} 𝔸^{r} × \overline{□} →^{Id×σ} 𝔸^{r} × \overline{□}, where Γ is the graph morphism of the composite 𝔸^{r}→𝔸^{1} ↪ \overline{□} of the product map followed by the open inclusion, (t_{1}, ⋯, t_{r}) ↦ (t_{1} ⋯ t_{r}) ↦ (t_{1} ⋯ t_{r}; 1), while σ : \overline{□} → \overline{□} is the antipodal automorphism (a; b) ↦ (b; a), where (a; b) ∈ \overline{□} = ℙ^{1} are the homogeneous coordinates. Since Γ is a closed immersion and Id × σ is an isomorphism, the morphism \overline{δ}_{n} is projective. Hence, the dominant map δ_{n}|_{Z^{×}} : Z^{×} → V induces \overline{δ}_{n}|\overline{Z} : \overline{Z} → \overline{V}. In particular, we have a commutative diagram

(6.1)  \xymatrix{\overline{Z}^{N} \ar[d]^{δ̃_{n}} \ar[r]^{ν_{Z}} & \overline{Z} \ar[d]^{\overline{δ}_{n}|\overline{Z}} \ar@{^{(}->}[r]^{ι_{Z}} & X × 𝔸^{r} × \overline{□}^{n} \ar[d]^{_{\overline{δ}_{n}}} \\ _{\overline{V}_{N}} \ar[r]^{ν_{V}} & \overline{V} \ar@{^{(}->}[r]^{ι_{V}} & X × 𝔸^{r} × \overline{□}^{n+1},}

where ι_{Z}, ι_{V} are the closed immersions, ν_{Z}, ν_{V} are normalizations, and δ̃_{n} is given by the universal property of normalization for dominant maps. By definition, \overline{δ}_{n}^{∗}\{t_{j} = 0\} = \{t_{j} = 0\} for 1 ≤ j ≤ r. First consider the case n ≥ 1. Then \overline{δ}_{n}^{∗}F_{n+1,i}^{1} = F_{n,i−1}^{1} for 2 ≤ i ≤ n + 1. Now, δ̃_{n}^{∗}ν_{V}^{∗} ι_{V}^{∗} (∑_{i=1}^{n+1} F_{n+1,i}^{1} − ∑_{j=1}^{r} m_{j}{t_{j} = 0}) ≥ δ̃_{n}^{∗}ν_{V}^{∗} ι_{V}^{∗} (∑_{i=2}^{n+1} F_{n+1,i}^{1} − ∑_{j=1}^{r} m_{j}\{t_{j} = 0\}) =^{†} ν_{Z}^{∗}ι_{Z}^{∗} \overline{δ}_{n}^{∗}(∑_{i=2}^{n+1} F_{n+1,i}^{1} − ∑_{j=1}^{r} m_{j}\{t_{j} = 0\}) = ν_{Z}^{∗}ι_{Z}^{∗}(∑_{i=2}^{n+1} F_{n,i−1}^{1} − ∑_{j=1}^{r} m_{j}{t_{j} = 0}) = ν_{Z}^{∗}ι_{Z}^{∗}(∑_{i=1}^{n} F_{n,i}^{1} − ∑_{j=1}^{r} m_{j}{t_{j} = 0}) ≥^{‡} 0, where † holds by the commutativity of (6.1) and ‡ holds as Z has modulus \underline{m}. Using Lemma 2.7, we can drop δ̃_{n}^{∗}, i.e., V′ has modulus \underline{m}. When n = 0, we have for 1 ≤ j ≤ r, δ̃_{0}^{∗}ν_{V}^{∗} ι_{V}^{∗} \{t_{j} = 0\} = ν_{Z}^{∗}ι_{Z}^{∗} \overline{δ}_{0}^{∗}\{t_{j} = 0\} = ν_{Z}^{∗}ι_{Z}^{∗}\{t_{j} = 0\}, which is 0 because \overline{Z} ∩ \{t_{j} = 0\} = ∅. Hence, δ̃_{0}^{∗}ν_{V}^{∗} ι_{V}^{∗} (F_{1,1}^{1} − ∑_{j=1}^{r} m_{j}{t_{j} = 0}) = δ̃_{0}^{∗}ν_{V}^{∗} ι_{V}^{∗} F_{1,1}^{1} ≥ 0. Dropping δ̃_{0}^{∗}, we get V′ has modulus \underline{m}. □

Proposition 6.4. Let Z ∈ z^{q}(X[r]|D\underline{m}, n). Then δ(Z) ∈ z^{q+1}(X[r]|D\underline{m}, n+1). Furthermore, δ and ∂ satisfy the equality δ∂ + ∂δ = 0.

Proof. We may assume that Z is an irreducible cycle. Let ∂_{n,i}^{ϵ} be the boundary given by the face F_{n,i}^{ϵ} on X × 𝔸^{r} × □^{n}, for 1 ≤ i ≤ n and ϵ = 0, ∞. Claim: For ϵ = 0, ∞, (i) ∂_{n+1,1}^{ϵ} ∘ δ_{n} = 0, (ii) ∂_{n+1,i}^{ϵ} ∘ δ_{n} = δ_{n−1} ∘ ∂_{n,i−1}^{ϵ} for 2 ≤ i ≤ n + 1. For (i), we show that δ_{n}(Z) ∩ {y_{1} = ϵ} = ∅ for ϵ = 0, ∞. Since δ_{n}(Z) ⊂ V (t_{1} ⋯ t_{r}y_{1} = 1), we have δ_{n}(Z) ∩ {y_{1} = 0} = ∅. On the other hand, if δ_{n}(Z) intersects {y_{1} = ∞}, then some t_{i} must be zero on Z, i.e., Z intersects {t_{i} = 0} for some 1 ≤ i ≤ r. However, since Z has modulus \underline{m}, this can not happen. Thus, δ_{n}(Z) ∩ {y_{1} = ∞} = ∅. This shows (i). For (ii), by the definition of δ_{n}, the diagram

\xymatrix{(𝔾_{m}^{r})^{×} × □^{n−1} \ar[r]^{ι_{i−1}^{ϵ}} \ar[d]^{δ_{n−1}} & (𝔾_{m}^{r})^{×} × □^{n} \ar[d]^{δ_{n}} \\ 𝔾_{m}^{r} × □^{n} \ar[r]^{ι_{i}^{ϵ}} & 𝔾_{m}^{r} × □^{n+1}}

is Cartesian. Thus, δ_{n−1}((ι_{i−1}^{∗}(Z)) = (ι_{i}^{ϵ})^{∗}(δ_{n}(Z)) by [6, Proposition 1.7], i.e., (ii) holds. This proves the claim. By Lemma 6.3, we know δ_{n}(Z) has modulus \underline{m}. Since Z intersects all faces properly, so does δ_{n}(Z) by applying (i) and (ii) of the above claim repeatedly. For ∂δ+δ∂ = 0, note that ∂δ_{n}(Z) = ∑_{i=1}^{n+1}(−1)^{i}(∂_{n+1,i}^{∞}δ_{n}(Z)−∂_{n+1,i}^{0}δ_{n}(Z)) =^{†} ∑_{i=2}^{n+1}(−1)^{i}(δ_{n−1}∂_{n,i−1}^{∞}(Z) − δ_{n−1}∂_{n,i−1}^{0}(Z)) = − ∑_{i=1}^{n}(−1)^{i}(δ_{n−1}∂_{n,i}^{∞}(Z) − δ_{n−1}∂_{n,i−1}^{0}(Z)) = −δ_{n−1} ∑_{i=1}^{n}(−1)^{i}(∂_{n,i}^{∞}(Z) − ∂_{n,i}^{0}(Z)) = −δ_{n−1} ∘ ∂(Z), where † holds by the claim. □

Lemma 6.5 and Corollary 6.6 below, which generalize [19, §4.2], have much simpler proofs than loc.cit.

Lemma 6.5. Let Z ∈ z^{q}(X[r]|D\underline{m}, n) be such that ∂_{i}^{ϵ}(Z) = 0 for 1 ≤ i ≤ n and ϵ = 0, ∞. Then 2δ^{2}(Z) is the boundary of an admissible cycle with modulus \underline{m}.

Proof. Note that δ^{2}(Z) is an admissible cycle on X × 𝔸^{r} × □^{n+2} with modulus \underline{m}, by Proposition 6.4. For the transposition τ = (1, 2) on the set \{1, ⋯, n+2\}, we have τ · δ^{2}(Z) = δ^{2}(Z), by the definition of δ. On the other hand, we have τ · δ^{2}(Z) = −δ^{2}(Z) + ∂(γ) for some admissible cycle γ, by Lemma 5.3. Hence, we have −δ^{2}(Z) + ∂(γ) = δ^{2}(Z), i.e., 2δ^{2}(Z) = ∂(γ), as desired. □

Corollary 6.6. Let k be a perfect field of characteristic ≠ 2 and let X be in SmAff_{k}^{ess} or in SmProj_{k}. Let \underline{m} ≥ 1. Then δ^{2} = 0 on CH^{q}(X[r]|D\underline{m}, n).

Proof. If r = \underline{m} = 1, by Corollary 5.15, there is nothing to prove. So, suppose either r ≥ 2 or |\underline{m}| ≥ 2. But, if r ≥ 2, then we automatically have |\underline{m}| ≥ 2, so we just consider the latter case. Given α ∈ CH^{q}(X[r]|D\underline{m}, n), by Theorem 3.2, we can find a representative Z ∈ z^{q}(X[r]|D\underline{m}, n) such that ∂_{i}^{ϵ}(Z) = 0 for 1 ≤ i ≤ n and ϵ = 0, ∞. Then by Lemma 6.5, we have 2δ^{2}(α) = 0. On the other hand, by Corollary 5.14, the group CH^{q}(X[r]|D\underline{m}, n) is a 𝕎_{(|\underline{m}|−1)}(k)-module. As |\underline{m}| ≥ 2 and char(k) ≠ 2, it follows that 2 ∈ (𝕎_{(|\underline{m}|−1)}(k))^{×}. In particular, δ^{2}(α) = 0. □

6.2. Leibniz rule. We now discuss the Leibniz rule, generalizing [19, §4.3]. Let X ∈ Sch_{k}^{ess}. Let (x, t, t_{1}, ⋯, t_{r}, y_{1}, ⋯, y_{n+2}) ∈ X × 𝔸^{r+1} × □^{n+2} be the coordinates. Let T ⊂ X × 𝔸^{r+1} × □^{n+2} be the closed subscheme defined by the equation ty_{n+1} = y_{n+2}(tt_{1} ⋯ t_{r}y_{n+1} − 1).

Definition 6.7 (cf. [19, Definition 4.9]). Given a closed subscheme Z ⊂ X × 𝔸^{r+1} × □^{n}, define C_{Z} := T · (Z × □^{2}) on X × 𝔸^{r+1} × □^{n+2}. This is extended ℤ-linearly to cycles.

Lemma 6.8. Let Z be a cycle on X × 𝔸^{r+1} × □^{n} with modulus \underline{m} = (m_{1}, ⋯, m_{r+1}). Then C_{Z} has modulus \underline{m} on X × 𝔸^{r+1} × □^{n+2}.

Proof. We may assume Z is irreducible. We show that each irreducible component V ⊂ C_{Z} has modulus \underline{m}. Let \overline{Z} and \overline{V} be the Zariski closures of Z and V in X × 𝔸^{r+1} × \overline{□}^{n} and X × 𝔸^{r+1} × \overline{□}^{n+2}, respectively. The projection pr : X × 𝔸^{r+1} × \overline{□}^{n+2} → X × 𝔸^{r+1} × \overline{□}^{n} that ignores the last two \overline{□}^{2} is projective, while its restriction to X × 𝔸^{r+1} × □^{n+2} maps V into Z. So, pr maps \overline{V} to \overline{Z}, giving a commutative diagram

(6.2)  \xymatrix{\overline{V}^{N} \ar[r]^{ν_{V}} \ar[d]^{_{pr_{N}}} & \overline{V} \ar@{^{(}->}[r]^{ι_{V}} \ar[d]^{pr|\overline{V}} & X × 𝔸^{r+1} × \overline{□}^{n+2} \ar[d]^{pr} \\ _{\overline{Z}_{N}} \ar[r]^{ν_{Z}} & \overline{Z} \ar@{^{(}->}[r]^{ι_{Z}} & X × 𝔸^{r+1} × \overline{□}^{n},}

where ι_{V} and ι_{Z} are the closed immersions, ν_{V} and ν_{Z} are normalizations, and pr^{N} is induced by the universal property of normalization for dominant maps. The modulus condition for V is now easily verified using the pull-back of the modulus condition for Z on \overline{Z}^{N} and the fact that pr^{∗}\{t_{j} = 0\} = \{t_{j} = 0\} for all j and pr^{∗}F_{n,i}^{1} = F_{n+2,i}^{1} for all i. □

Corollary 6.9. Let X_{1}, X_{2} ∈ Sch_{k}^{ess}. Let V_{1} ⊂ X_{1} × 𝔸^{1} × □^{n_{1}} and V_{2} ⊂ X_{2} × 𝔸^{r} × □^{n_{2}} be closed subschemes with moduli |\underline{m}| and \underline{m}, respectively with \underline{m} ≥ 1. Under the exchange of factors X_{1}×𝔸^{1}×□^{n_{1}} ×X_{2}×𝔸^{r}×□^{n_{2}} ≃ X_{1}×X_{2}×𝔸^{r+1}× □^{n}, where n = n_{1} + n_{2}, consider the cycle C_{V_{1}×V_{2}} on X_{1} × X_{2} × 𝔸^{r+1} × □^{n+2}. Then μ|_{C_{V_{1}×V_{2}}} is finite. In particular, μ_{∗}(C_{V_{1}×V_{2}}) as in Definition 5.9 is welldefined, and has modulus \underline{m}.

Proof. We set V = V_{1} × V_{2}. From the definition of μ, the map μ : V × □^{2} → X_{1} × X_{2} × 𝔸^{r} × □^{n+2} is of the form μ|_{V} × Id_{□^{2}}. By Proposition 5.7, the map μ|_{V} is finite, thus so is μ|_{V} × Id_{□^{2}} : V × □^{2} → X_{1} × X_{2} × 𝔸^{r} × □^{n+2}. Hence, its restriction to C_{V} = T · (V × □^{2}) is also finite. The modulus condition for μ_{∗}(C_{V}) follows from Lemmas 5.8 and 6.8. □

Definition 6.10 (cf. [19, Definition 4.12]). Let V_{1} ∈ z^{q_{1}}(X_{1}[1]|D_{|\underline{m}|}, n_{1}) and V_{2} ∈ z^{q_{2}}(X_{2}[r]|D\underline{m}, n_{2}) with X_{1}, X_{2} ∈ Sch_{k}^{ess}. Let n = n_{1} + n_{2} and define V_{1} ×_{μ′} V_{2} be the cycle σ · μ_{∗}(C_{V_{1}×V_{2}}), where σ = (n + 2, n + 1, ⋯, 1)^{2} ∈ 𝔖_{n+2}.

Lemma 6.11. Let V_{1}, V_{2} be as in Definition 6.10. Then V_{1} ×_{μ′} V_{2} ∈ z^{q_{1}+q_{2}−1}((X_{1} × X_{2})[r]|D\underline{m}, n_{1} + n_{2} + 2).

Proof. By Corollary 6.9, the cycle μ_{∗}(C_{V_{1}×V_{2}}) has modulus \underline{m}, thus so does W := V_{1} ×_{μ′} V_{2}. It remains to prove that W intersects all faces properly. Let σ_{n_{1}} = (n_{1} + 1, n_{1}, ⋯, 1) ∈ 𝔖_{n+1}. Then by direct calculations, we have (6.3)

\{ \begin{matrix}∂_{1}^{∞}W = σ_{n_{1}}(V_{1} ×_{μ} δ(V_{2})), ∂_{1}^{0}W = 0, ∂_{2}^{∞}W = δ(V_{1} ×_{μ} V_{2}), \\ ∂_{2}^{0}W = δ(V_{1}) ×_{μ} V_{2}, \\ ∂_{i}^{ϵ}W = \{ \begin{matrix}∂_{i−2}^{ϵ}(V_{1}) ×_{μ′} V_{2}, & for 3 ≤ i ≤ n_{1} + 2, \\ V_{1} ×_{μ′} ∂_{i−n_{1}−2}^{ϵ}(V_{2}), & for n_{1} + 3 ≤ i ≤ n + 2, \end{matrix} ϵ ∈ \{0, ∞\}. \end{matrix}

Since each V_{i} is admissible, using (6.3), Lemma 5.10, Proposition 6.4 and induction on the codimension of faces, we deduce that W intersects all faces properly. □

Proposition 6.12. Let X_{1}, X_{2} ∈ Sm_{k}^{ess}. Let ξ ∈ z^{q_{1}}(X_{1}[1]|D_{|\underline{m}|}, n_{1}) and η ∈ z^{q_{2}}(X_{2}[r]|D\underline{m}, n_{2}). Let n = n_{1} + n_{2} and q = q_{1} + q_{2}. Suppose that all codimension one faces of ξ and η vanish. Then in the group z^{q−1}((X_{1} × X_{2})[r]|D\underline{m}, n+1), the cycle δ(ξ ×_{μ} η)−δξ ×_{μ} η −(−1)^{n_{1}}ξ ×_{μ} δη is the boundary of an admissible cycle.

Proof. By (6.3), for 3 ≤ i ≤ n_{1} + 2, we have ∂_{i}^{ϵ}(ξ ×_{μ′} η) = ∂_{i−2}^{ϵ}(ξ) ×_{μ′} η = 0, while for n_{1} +3 ≤ i ≤ n+2, we have ∂_{i}^{ϵ}(ξ ×_{μ′} η) = ξ ×_{μ′} ∂_{i−n_{1}−2}^{ϵ}(η) = 0. Hence, ∂(ξ×_{μ′} η) = ∑_{i=1}^{n+2}(−1)^{i}(∂_{i}^{∞}−∂_{i}^{0})(ξ×_{μ′} η) = δ(ξ×_{μ}η)−{σ_{n_{1}} ·(ξ×_{μ}δη)+δξ×_{μ}η} by (6.3) for i = 1, 2. Equivalently,

(6.4)  δ(ξ ×_{μ} η) − δξ ×_{μ} η − σ_{n_{1}} · (ξ ×_{μ} δη) = ∂(ξ ×_{μ′} η).

But, for ξ ×_{μ} δη, notice that

(6.5) ∂_{i}^{ϵ}(ξ ×_{μ} δη) = \{ \begin{matrix}∂_{i}^{ϵ}ξ ×_{μ} δη = 0, & for 1 ≤ i ≤ n_{1}, \\ ξ ×_{μ} ∂_{i−n_{1}}^{ϵ}(δη), & for n_{1} + 1 ≤ i ≤ n + 1, \end{matrix} ϵ ∈ \{0, ∞\}.

We have ∂_{1}^{ϵ}(δη) = 0 when i = n_{1} + 1 by Claim (i) of Proposition 6.4, and ∂_{i−n_{1}}^{ϵ}(δη) = δ(∂_{i−n_{1}−1}^{ϵ}η) = δ(0) = 0 when n_{1} + 2 ≤ i ≤ n + 1 by Claim (ii) of Proposition 6.4. Hence, ξ ×_{μ} δη is a cycle with trivial codimension 1 faces, so, by Lemma 5.3, for some admissible cycle γ, we have σ_{n_{1}} · (ξ ×_{μ} δη) = sgn(σ_{n_{1}})(ξ ×_{μ} δη)+∂(γ) = (−1)^{n_{1}}ξ ×_{μ} δη +∂(γ). Putting this back in (6.4), we obtain δ(ξ ×_{μ} η) − δξ ×_{μ} η − (−1)^{n_{1}}ξ ×_{μ} δη = ∂(ξ ×_{μ′} η) − ∂(γ), as desired. □

The above discussion summarizes as follows:

Theorem 6.13. Let X be in SmAff_{k}^{ess} or in SmProj_{k} over a perfect field k with char(k) ≠ 2. Let r ≥ 1 and \underline{m} = (m_{1}, ⋯, m_{r}) ≥ 1. Then the following hold:

(1) (CH(X[1]|D_{|\underline{m}|}), ∧_{X}, δ) forms a commutative differential graded 𝕎_{(|\underline{m}|−1)}Ω_{k}^{•}-algebra.

(2) (CH(X[r]|D\underline{m}), δ) forms a differential graded (CH(X[1]|D_{|\underline{m}|}), ∧_{X}, δ)- module.

In particular, (CH(X[r]|D\underline{m}), δ) is a differential graded 𝕎_{(|\underline{m}|−1)}Ω_{k}^{•}-module.

Proof. The commutative differential graded algebra structure on CH(X[1]|D_{|\underline{m}|}) and the differential graded module structure on CH(X[r]|D\underline{m}) over CH(X[1]|D_{|\underline{m}|}) follows by combining Theorem 5.4, Corollary 6.6 and Proposition 6.12 using Theorem 3.2. The structure map p : X → Spec (k) turns (CH(X[1]|D_{|\underline{m}|}), ∧_{X}, δ) into a differential graded algebra over (CH(pt[1]|D_{|\underline{m}|}), ∧_{pt}, δ) via p^{∗}. Since ⊕_{n≥0}CH^{n+1}(pt[1]|D_{|\underline{m}|}, n) forms a differential graded sub-algebra of (CH(pt[1]|D_{|\underline{m}|}), ∧_{pt}, δ). The map of commutative differential graded algebras 𝕎_{(|\underline{m}|−1)}Ω_{k}^{•} → ⊕_{n≥0}CH^{n+1}(pt[1]|D_{|\underline{m}|}, n) (see [28]) finishes the proof of the theorem. □

As a consequence of Theorem 6.13 (use Corollary 5.15 when |\underline{m}| = 1), we obtain the following property of multivariate additive higher Chow groups.

Corollary 6.14. Let r ≥ 1 and \underline{m} ≥ 1 and let X be in SmAff_{k}^{ess} or in SmProj_{k}. Then each CH^{q}(X[r]|D\underline{m}, n) is a k-vector space provided char(k) = 0.

7. Witt-complex structure on additive higher Chow groups

Let k be a perfect field of characteristic ≠ 2. In this section, a smooth affine k-scheme means an object in SmAff_{k}^{ess}, i.e., an object of either SmAff_{k} or SmLoc_{k}. Rülling proved in [28] that the additive higher Chow groups of 0-cycles over Spec (k) form a restricted Witt-complex over k. When X is a smooth projective variety over k, it was proven in [19] that additive higher Chow groups of X form a restricted Witt-complex over k. Our objective is to prove the stronger assertion that the additive higher Chow groups of Spec (R) ∈ SmAff_{k}^{ess} have the structure of a restricted Witt-complex over R. Since we exclusively use the case r = 1 only, we use the older notations Tz^{q}(X, n; m) and TCH^{q}(X, n; m) instead of z^{q}(X[1]|D_{m+1}, n − 1) and CH^{q}(X[1]|D_{m+1}, n − 1). For X ∈ Sch_{k}^{ess}, we let TCH(X; m) := ⊕_{n,q}TCH^{q}(X, n; m) and TCH^{M}(X; m) := ⊕_{n}TCH^{n}(X, n; m). The superscript M is for Milnor. Let TCH(X) := ⊕_{m}TCH(X; m) and TCH^{M}(X) := ⊕_{m}TCH^{M}(X; m). We similarly define T CH(X; m), T CH^{M}(X; m), T CH(X), and T CH^{M}(X) for X ∈ Sch_{k} using Definition 4.7.

7.1. Witt-complex structure over k. In this section, we show that the additive higher Chow groups for an object of SmAff_{k}^{ess} form a functorial restricted Witt-complex over k. For r ≥ 1, let ϕ_{r} : 𝔸^{1} → 𝔸^{1} be the morphism x ↦ x^{r}, which induces ϕ_{r} : Spec (R) × B_{n} → Spec (R) × B_{n}. By [19, §5.1, 5.2], we have the Frobenius F_{r} : TCH^{q}(R, n; rm+r −1) → TCH^{q}(R, n; m) and the Verschiebung V_{r} : TCH^{q}(R, n; m) → TCH^{q}(R, n; rm+r−1) given by F_{r} = ϕ_{r∗} and V_{r} = ϕ_{r}^{∗}. We also have a natural inclusion ℜ : Tz^{q}(R, •; m + 1) → Tz^{q}(R, •; m) for any m ≥ 1, which induces ℜ : TCH^{q}(R, n; m + 1) → TCH^{q}(R, n; m), called the restriction. Finally, by Theorem 6.13, there is a differential δ : Tz^{q}(R, •; m) → Tz^{q}(R, • + 1; m), which induces δ : TCH^{q}(R, n; m) → TCH^{q}(R, n + 1; m).

Theorem 7.1. Let X ∈ SmAff_{k}^{ess} and m ≥ 1. Then TCH(X; m) is a DGA and TCH^{M}(X; m) is its sub-DGA. Furthermore, with respect to the operations δ, ℜ, F_{r}, V_{r} in the above together with λ = f^{∗} : 𝕎_{m}(k) = TCH^{1}(k, 1; m) → TCH^{1}(X, 1; m) for the structure morphism f : X → Spec (k), TCH(X) is a restricted Witt-complex over k and TCH^{M}(X) is a restricted sub-Witt-complex over k. These structures are functorial.

Proof. In [19, Theorem 1.1, Scholium 1.2], it was stated that TCH(X; m) and TCH^{M}(X; m) are DGAs, and that TCH(X) and TCH^{M}(X) are restricted Witt-complexes over k with respect to the above δ, ℜ, F_{r}, V_{r}, provided the moving lemma holds for X. But this is now shown in Theorems 4.1 and 4.10. We give a very brief sketch of this structure and its functoriality.

The functoriality of the restriction operator ℜ recalled above, was stated in [19, Corollary 5.19], which we easily check here: let f : X → Y be a morphism in SmAff_{k}^{ess} and consider the following commutative diagram:

\xymatrix{Tz_{W}^{q}(Y, •; m + 1) \ar[r]^{f^{∗}} \ar@{^{(}->}[d] & Tz^{q}(X, •; m + 1) \ar@{^{(}->}[d] \\ Tz_{W}^{q}(Y, •; m) \ar[r]^{f^{∗}} & Tz^{q}(X, •; m),}

where W is a finite set of locally closed subsets of Y, and the horizontal maps are chain maps given by the inverse images as in the proof of Theorem 4.5 and Corollary 4.15. The diagram and Theorems 4.1 and 4.10 imply that f^{∗}ℜ = ℜf^{∗} because the vertical inclusions induce ℜ by definition. For each r ≥ 1, the Frobenius F_{r} and Verschiebung V_{r} recalled in the above are functorial as proven in [19, Lemmas 5.4, 5.9], and that F_{r} is a graded ring homomorphism is proven in [19, Corollary 5.6]. Finally, the properties (i), (ii), (iii), (iv), (v) in Section 2.2.2, are all proven in [19, Theorem 5.13], where none requires the projectivity assumption. □

Corollary 7.2. Let m ≥ 1 be an integer. Then T CH(−; m) and T CH^{M}(−; m) define presheaves of DGAs on Sch_{k}, and the pro-systems T CH(−) and T CH^{M}(−) define presheaves of restricted Witt-complexes over k on Sch_{k}.

Proof. Let X ∈ Sch_{k}. By definition, T CH(X; m) is the colimit over all (X → A) ∈ (X ↓ SmAff_{k})^{op} of TCH(A; m). But the category of DGAs is closed under filtered colimits (see [13]) so that T CH(X; m) is a DGA. For each morphism f : X → Y in Sch_{k}, one checks f^{∗} : T CH(Y; m) → T CH(X; m) is a morphism of DGAs. The other assertions follow easily using Theorem 7.1. □

Before we discuss Witt-complexes over R, we state the following behavior of various operators under finite push-forward maps.

Proposition 7.3. Let f : X → Y be a finite map in SmAff_{k}^{ess}. Then for r ≥ 1, we have: (a) f_{∗}ℜ = ℜf_{∗}; (b) f_{∗}δ = δf_{∗}; (c) f_{∗}F_{r} = F_{r}f_{∗}; (d) f_{∗}V_{r} = V_{r}f_{∗}.

Proof. The item (a) is obvious and (b) and (c) follow at once from the fact that these operators are defined as push-forward under closed immersion and finite maps and they preserve the faces. For (d), we consider the commutative diagram

(7.1)  \xymatrix{X × 𝔸^{1} \ar[r]^{Id×ϕ_{r}} \ar[d]^{f×Id} & X × 𝔸^{1} \ar[d]^{f×Id} \\ Y × 𝔸^{1} \ar[r]^{Id×ϕ_{r}} & Y × 𝔸^{1}.}

Since this diagram is Cartesian and f as well as ϕ preserve the faces, we conclude from [6, Proposition 1.7] that f_{∗} ∘ ϕ_{r}^{∗} = ϕ_{r}^{∗} ∘ f_{∗}. □

7.2. Witt-complex structure over R. Let X = Spec (R) ∈ SmAff_{k}^{ess}. The objective of this section is to strengthen Theorem 7.1 by showing that TCH(X) is a restricted Witt-complex over R.

Let m ≥ 1 be an integer. We first define a group homomorphism τ_{R} : 𝕎_{m}(R) → TCH^{1}(R, 1; m) for any k-algebra R. Recall that the underlying abelian group of 𝕎_{m}(R) identifies with the multiplicative group (1+tR[[t]])^{×}/(1+t^{m+1}R[[t]])^{×}. For each polynomial p(t) ∈ (1 + R[[t]])^{×}, consider the closed subscheme of Spec (R[t]) given by the ideal (p(t)), and let Γ_{(p(t))} be its associated cycle. By definition, Γ_{(p(t))} ∩ {t = 0} = ∅ so that Γ_{(p(t))} ∈ Tz^{1}(R, 1; m). We set Γ_{a,n} = Γ_{(1−at^{n})} for n ≥ 1 and a ∈ R.

Lemma 7.4. Let f(t), g(t) be polynomials in R[t], and let h(t) ∈ R[t] be the unique polynomial such that (1−tf(t))(1−tg(t)) = 1−th(t). Then Γ_{(1−th(t))} = Γ_{(1−tf(t))} + Γ_{(1−tg(t))} in Tz^{1}(R, 1; m).

Proof. This is obvious by (1 − tf(t))(1 − tg(t)) = 1 − th(t). □

Lemma 7.5. For n ≥ m + 1, we have Γ_{(1−t^{n}f(t))} ≡ 0 in TCH^{1}(R, 1; m).

Proof. Consider the closed subscheme Γ ⊂ X ×𝔸^{1} ×□ given by y_{1} = 1−t^{n}f(t). Let ν : \overline{Γ}^{N} → \overline{Γ} ↪ X ×𝔸^{1} ×ℙ^{1} be the normalization of the Zariski closure \overline{Γ} in X × 𝔸^{1} × ℙ^{1}. Since f(t)t^{n} = 1 − y_{1} on \overline{Γ}, we see that nν^{∗}\{t = 0\} ≤ ν^{∗}\{y_{1} = 1\} on \overline{Γ}^{N}. Since n ≥ m + 1, this shows that Γ satisfies the modulus m condition. Since ∂_{1}^{∞}(Γ) = 0 and ∂_{1}^{0}(Γ) = Γ_{(1−t^{n}f(t))} (which is of codimension 1), the cycle Γ is an admissible cycle in Tz^{1}(R, 2; m) such that ∂Γ = Γ_{(1−t^{n}f(t))}. This shows that Γ_{(1−t^{n}f(t))} ≡ 0 in TCH^{1}(R, 1; m). □

Proposition 7.6. Let R be a k-algebra. Then the map τ_{R} : (1 + tR[t]) → Tz^{1}(R, 1; m) that sends a polynomial 1 − tf(t) to Γ_{(1−tf(t))}, defines a group homomorphism τ_{R} : 𝕎_{m}(R) → TCH^{1}(R, 1; m).

Proof. Every element p(t) ∈ (1 + tR[[t]])^{×} has a unique expression p(t) = ∏_{n≥1}(1 − a_{n}t^{n}) for a_{n} ∈ R. For any such p(t), set p^{≤m}(t) = ∏_{n=1}^{m}(1 − a_{n}t^{n}). We define τ_{R}(p(t)) = Γ_{(p^{≤m}(t))}. It follows from Lemmas 7.4 and 7.5 that this map descends to a group homomorphism from 𝕎_{m}(R). □

Recall from [28, Appendix A] that for each r ≥ 1, we have the Frobenius F_{r} : 𝕎_{rm+r−1}(R) → 𝕎_{m}(R) and the Verschiebung V_{r} : 𝕎_{m}(R) → 𝕎_{rm+r−1}(R). They are given by F_{r}(1 − at^{n}) = (1 − a^{\frac{r}{s}} t^{\frac{n}{s}})^{s}, where s = gcd(r, n) and V_{r}(1 − at^{n}) = 1 − at^{rn}. On the other hand, as seen in Section 7.1, we have operations F_{r} and V_{r} on {TCH^{1}(R, 1; m)}_{m∈ℕ}.

Lemma 7.7. Let R be a k-algebra. Then the maps τ_{R} : 𝕎_{m}(R) → TCH^{1}(R, 1; m) of Proposition 7.6 commute with the F_{r} and V_{r} operators on both sides.

Proof. That τ_{R}V_{r} = V_{r}τ_{R}, is easy: we have V_{r}(τ_{R}(1−at^{n})) = V_{r}(Γ_{a,n}) = Γ_{a,rn}, while τ_{R}(V_{r}(1 − at^{n})) = Γ_{(1−at^{rn})} = Γ_{a,rn}. That τ_{R}F_{r} = F_{r}τ_{R}, is slightly more involved. Recall that F_{r}(1 − at^{n}) = (1−a^{\frac{r}{s}} t^{\frac{n}{s}})^{s}, where s = gcd(r, n). Write n = n′s and r = r′s, where 1 = (r′, n′). Hence, we have τ_{R}F_{r}(1 − at^{n}) = sΓ_{a^{\frac{r}{s}} , \frac{n}{s}} = sV\frac{n}{s} (Γ_{a^{\frac{r}{s}} ,1}) = sV_{n′}(Γ_{a^{r′} ,1}) =: ♣, while F_{r}τ_{R}(1 − at^{n}) = F_{r}Γ_{a,n} = F_{r}V_{n}(Γ_{a,1}) =: ♡. First observe that when n = 1, we have s = 1, r = r′, n = n′ = 1, and we have ♡ = F_{r}(Γ_{a,1}) = Γ_{a^{r},1} = ♣, so that τ_{R}F_{r}(1 − at) = F_{r}τ_{R}(1 − at), indeed. For a general n ≥ 1, we have F_{r}V_{n} = F_{r′}F_{s}V_{s}V_{n′} = F_{r′} ∘ (s · Id) ∘ V_{n′} = sF_{r′}V_{n′} =^{†} sV_{n′}F_{r′}, where † holds because (r′, n′) = 1. Since F_{r′}(Γ_{a,1}) = Γ_{a^{r′} ,1} (by the first case), we have ♡ = F_{r}V_{n}(Γ_{a,1}) = sV_{n′}F_{r′}(Γ_{a,1}) = sV_{n′}(Γ_{a^{r′} ,1}) = ♣. This shows τ_{R}F_{r} = F_{r}τ_{R}. □

Remark 7.8. In the proof of Lemma 7.7, we saw that for s = (r, n),

(7.2)  F_{r}(Γ_{a,n}) = sΓ_{a^{\frac{r}{s}} , \frac{n}{s}} , V_{r}(Γ_{a,n}) = Γ_{a,rn}.

Proposition 7.9. For X = Spec (R) ∈ SmAff_{k}^{ess}, the maps τ_{R} : 𝕎_{m}(R) → TCH^{1}(R, 1; m) form a morphism of pro-rings that commutes with F_{r} and V_{r} for r ≥ 1.

Proof. It is clear from the definition of τ_{R} in Proposition 7.6 that it commutes with ℜ. We saw that τ_{R} commutes with F_{r} and V_{r} in Lemma 7.7. So, we only need to show that τ_{R} respects the products. By [2, Proposition (1.1)], it is enough to prove that for a, b ∈ R and u, v ≥ 1,

(7.3)  Γ_{a,u} ∧ Γ_{b,v} = wΓ_{a^{\frac{v}{w}} b^{\frac{u}{w}} , \frac{uv}{w}} in TCH^{1}(R, 1; m),

where w = gcd(u, v) and ∧ = ∧_{X} is the product structure on the ring TCH^{1}(R, 1; m) as in Theorem 7.1. Step 1. First, consider the case when u = v = 1, i.e., we prove Γ_{a,1} ∧ Γ_{b,1} = Γ_{ab,1}. Recall that ∧ is defined as the composition Δ^{∗} ∘ μ_{∗} ∘ × in

X × 𝔸^{1} × X × 𝔸^{1} →^{μ} X × X × 𝔸^{1} ←^{Δ} X × 𝔸^{1}.

Under the identification X × X ≃ Spec (R ⊗_{k} R), we have μ_{∗}(Γ_{a,1} × Γ_{b,1}) = Γ_{(a⊗1)(1⊗b),1}, and Δ^{∗}(Γ_{(a⊗1)(1⊗b),1}) = Γ_{ab,1}, because Δ is given by the multiplication R ⊗_{k} R → R. This proves (7.3) for Step 1. For the following remaining two steps, we use the projection formula: x ∧ V_{s}(y) = V_{s}(F_{s}(x) ∧ y), which we can use by Theorem 7.1. Step 2. Consider the case when v = 1, but u ≥ 1 is any integer. We apply the projection formula to x = Γ_{b,1} and y = Γ_{a,1} with s = u. Since TCH^{1}(R, 1; m) is a commutative ring, by the projection formula, we get V_{u}(Γ_{a,1}) ∧ Γ_{b,1} = V_{u}(Γ_{a,1} ∧ F_{u}(Γ_{b,1})). Here, the left hand side is Γ_{a,u} ∧ Γ_{b,1} by eqn:FV identity, while the right hand side is =^{1} V_{u}(Γ_{a,1} ∧ Γ_{b^{u},1}) =^{2} V_{u}(Γ_{ab^{u},1}) =^{3} Γ_{ab^{u},u}, where =^{1} and =^{3} hold by (7.2) and =^{2} holds by Step 1. This proves (7.3) for Step 2. Step 3. Finally, let u, v ≥ 1 be any integers. Let w = gcd(u, v). We again apply the projection formula to x = V_{u}(Γ_{a,1}), y = Γ_{b,1}, s = v, so that V_{u}(Γ_{a,1})∧ V_{v}(Γ_{b,1}) = V_{v}(F_{v}(V_{u}(Γ_{a,1}))∧Γ_{b,1}). Its left hand side coincides with that of (7.3) by (7.2). Its right hand side is =^{1} V_{v}(F_{v}(Γ_{a,u}) ∧ Γ_{b,1}) =^{2} V_{v}(wΓ_{a^{\frac{v}{w}} , \frac{u}{w}} ∧ Γ_{b,1}),

where =^{1} and =^{2} hold by (7.2). But, Step 2 says that Γ_{a^{\frac{v}{w}} , \frac{u}{w}} ∧Γ_{b,1} = Γ_{a^{\frac{v}{w}} b^{\frac{u}{w}} , \frac{u}{w}}

so that V_{v}(wΓ_{a^{\frac{v}{w}} , \frac{u}{w}} ∧ Γ_{b,1}) = wV_{v}(Γ_{a^{\frac{v}{w}} b^{\frac{u}{w}} , \frac{u}{w}}) =^{†} wΓ_{a^{\frac{v}{w}} b^{\frac{u}{w}} , \frac{uv}{w}}, where = † holds by (7.2). This last expression is the right hand side of (7.3). Thus, we obtain the equality (7.3) and this finishes the proof. □

Theorem 7.10. For Spec (R) ∈ SmAff_{k}^{ess}, TCH(R) is a restricted Wittcomplex over R, and its sub-pro-system TCH^{M}(R) is a restricted sub-Witt-complex over R.

Proof. As saw in the proof of Theorem 7.1, we already have the restriction ℜ, the differential δ, the Frobenius F_{r} and the Verschiebung V_{r} defined by the same formulas. Furthermore, by Proposition 7.9, now we have ring homomorphisms λ = τ_{R} : 𝕎_{m}(R) → TCH^{1}(R, 1; m) for m ≥ 1. The properties (i), (ii), (iii), (iv) in Section 2.2.2 are independent of the choice of the ring, so that what we checked in Theorem 7.1 still work. To prove the theorem, the only thing left to be checked is the property (v) that for all a ∈ R and r ≥ 1,

(7.4)  F_{r}δτ_{R}([a]) = τ_{R}([a]^{r−1})δτ_{R}([a]),

where we have shrunk the product notation ∧ and taken the ring homomorphism λ to be τ_{R}. To check this, we identify 𝕎_{m}(R) with (1 + tR[[t]])^{×}/(1 + t^{m+1}R[[t]])^{×}. If a = 0, then τ_{R}([a]) = Γ_{(1−0·t)} = ∅. So, both sides of (7.4) are zero. If a = 1, then τ_{R}([a]) = τ_{R}(1 − t) = Γ_{(1−t)}. But, in our definition of δ, to compute it, we should first restrict the cycle Γ_{(1−t)} ⊂ Spec (R) × 𝔾_{m} onto Spec (R) × (𝔾_{m} \ {1}), which becomes empty. Hence, δτ_{R}([a]) = δΓ_{(1−t)} = 0, so again both sides of (7.4) are zero. Let a ∈ R \ {0, 1}. Then τ_{R}([a]) = Γ_{(1−at)} ⊂ Spec (R) × 𝔸^{1}, and δτ_{R}([a]) is given by the ideal (1 − at, 1 − ty_{1}) in R[t, y_{1}]. Since t is not a zero-divisor in R[t, y_{1}], we have (1 − at, 1 − ty_{1}) = (1 − at, y_{1} − a) as ideals. Hence, F_{r}δτ_{R}([a]) is given by the ideal (1 − a^{r}t, y_{1} − a) in R[t, y_{1}]. On the other hand,

(7.5)

τ_{R}([a]^{r−1})δτ_{R}([a]) = Γ_{(1−a^{r−1}t)} ∧ Spec (\frac{R[t,y_{1}]}{(1−at,y_{1}−a)})
= Δ^{∗} (\frac{(R⊗_{k}R)[t,y_{1}]}{(1−(a^{r−1}⊗1)(1⊗a),y_{1}−(1⊗a))}) =^{†} Spec (\frac{R[t,y_{1}]}{(1−a^{r}t,y_{1}−a)}) ,

where † holds because Δ is induced by the product homomorphism R⊗_{k}R → R. Hence, both hand sides of (7.4) coincide. This completes the proof. □

Theorem 7.11. For Spec (R) ∈ SmAff_{k}^{ess} and n, m ≥ 1, there is a unique homomorphism τ_{n,m}^{R} : 𝕎_{m}Ω_{R}^{n−1} → TCH^{n}(R, n; m) that defines a morphism of restricted Witt-complexes over R, {τ_{•,m}^{R} : 𝕎_{m}Ω_{R}^{•−1} → TCH^{•}(R, •; m)}_{m}, such that τ_{1,m}^{R} = τ_{R}.

Proof. The theorem follows from Theorem 7.10 and [28, Proposition 1.15]. We have τ_{1,m}^{R} = τ_{R} because the map λ of §2.2.2 is given by τ_{R} in Theorem 7.10. □

We have shown in Propositions 7.6 and 7.9 that τ_{R} is a group homomorphism for any k-algebra R and is a ring homomorphism if R is smooth. Here, we provide the following information on τ_{R}.

Theorem 7.12. Let R be an integral domain which is an essentially of finite type k-algebra. Then τ_{R} is injective. It is an isomorphism if R is a UFD.

Proof. Let K := Frac(R) and ι : R ↪ K be the inclusion. This induces a commutative diagram

\xymatrix{𝕎_{m}(R) \ar[r]^{𝕎_{m}(ι)} \ar[d]_{τ_{R}} & 𝕎_{m}(K) \ar[d]^{≃}_{τ_{K}} \\ TCH^{1}(R, 1; m) \ar[r] & TCH^{1}(K, 1; m),}

where the bottom map is the flat pull-back via Spec (K) → Spec (R), and τ_{K} is the isomorphism by [28, Corollary 3.7]. Since 𝕎_{m}(ι) is clearly injective (see [28, Properties A.1.(i)]), it follows that τ_{R} is injective, too. Suppose now R is a UFD and V is an irreducible admissible cycle in Tz^{1}(R, 1; m). Then we must have (I(V), t) = R[t], where I(V) is the ideal of V. Since R[t] is a UFD, using basic commutative algebra, one checks that I(V) = (1 − tf(t)) for some non-zero polynomial f(t) ∈ R[t]. In particular, the map τ_{R} is surjective and hence an isomorphism. □

7.3. Étale descent. Finally:

Proof of Theorem 1.4. By Corollary 5.15, we can assume |\underline{m}| ≥ 2. We set Y = X/G, λ = |G| and consider the diagram

(7.6)  \xymatrix{G × X \ar[r]^{γ} \ar[d]_{p} & X \ar[d]^{f} \\ X \ar[r]_{f} & Y,}

where γ is the action map and p is the projection. Since G acts freely on X, this square is Cartesian and f is étale of degree λ. By [6, Proposition 1.7], we have f^{∗} ∘ f_{∗} = p_{∗} ∘ γ^{∗} : CH^{q}(X[r]|D\underline{m}, n) → CH^{q}(X[r]|D\underline{m}, n). Since f is G-equivariant with respect to the trivial G-action on Y, we see that f^{∗} induces a map f^{∗} : CH^{q}(Y [r]|D\underline{m}, n) → CH^{q}(X[r]|D\underline{m}, n)^{G}. Moreover, it follows from [21, Theorem 3.12] that f_{∗} ∘ f^{∗} is multiplication by λ. On the other hand, it follows easily from the action map γ that p_{∗} ∘ γ^{∗}(α) = ∑_{g∈G} g^{∗}(α). In particular, p_{∗} ∘ γ^{∗}(α) = λ · α if α ∈ CH^{q}(X[r]|D\underline{m}, n)^{G}.

Since λ ∈ k^{×} and the Teichmüller map is multiplicative with |\underline{m}| ≥ 2, we see that λ ∈ (𝕎_{(|\underline{m}|−1)}(k))^{×}. We conclude from Theorem 5.4(3) and Corol-

lary 5.14 that the composite CH^{q}(Y [r]|D\underline{m}, n) ⟶^{f^{∗}} CH^{q}(X[r]|D\underline{m}, n)^{G} ⟶^{λ^{−1}f_{∗}} CH^{q}(Y [r]|D\underline{m}, n) yields the desired isomorphism. □

Acknowledgments. The authors thank Wataru Kai for letting them know about his moving lemma for affine schemes. The authors also feel very grateful to the referee whose careful and detailed comments had improved the paper and the editor of the Documenta Mathematica for various kind suggestions. During this research, JP was partially supported by the National Research Foundation of Korea (NRF) grants No. 2013042157 and No. 2015R1A2A2A01004120, Korea Institute for Advanced Study (KIAS) grant, all funded by the Korean government (MSIP), and TJ Park Junior Faculty Fellowship funded by POSCO TJ Park Foundation.

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Amalendu Krishna School of Mathematics Tata Institute of Fundamental Research 1 Homi Bhabha Road Colaba, Mumbai, India amal@math.tifr.res.in Jinhyun Park Department of Mathematical Sciences KAIST 291 Daehak-ro Yuseong-gu Daejeon 34141 Republic of Korea (South) jinhyun@mathsci.kaist.ac.kr jinhyun@kaist.edu 90

Documenta Mathematica 21 (2016)