KAIST 수리과학과 | - DGA-structure on additive higher Chow ...mathsci.kaist.ac.kr/~jinhyun/papers/CDGA.pdfMumbai, India and 2Department of Mathematical Sciences, KAIST, 291 Daehak-ro,

Krishna, A., and J. Park. (20xx) “DGA-structure on additive higher Chow groups,”International Mathematics Research Notices, Vol. 20xx, Article ID rmn???, 31 pages.doi:10.1093/imrn/rmn???

DGA-structure on additive higher Chow groups

Amalendu Krishna1 and Jinhyun Park2

1School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Colaba,Mumbai, India and 2Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu,Daejeon, 305-701, Republic of Korea (South)

Correspondence to be sent to: [email protected]

We construct a graded-commutative differential graded algebra structure on additive higher Chow groups of a smooth

projective variety over a perfect field. We show that these groups are equipped with Frobenius and Verschiebung

operators, that turn the collection into a Witt-complex.

1 Introduction

The objective of this paper is to show that additive higher Chow groups of modulus m ≥ 1 of a smooth projectivevariety form a graded-commutative differential graded algebra (DGA), and that the projective system of theDGAs over all m ≥ 1 forms a Witt-complex over k. Its zero-dimensional case was studied by [Rülling, 2007].

The additive higher Chow theory can be seen as an attempt to understand motivic cohomology of non-reduced schemes. Even when the underlying reduced spaces are smooth, such schemes generally have non-trivialrelative Quillen K-groups that the usual higher Chow groups in [Bloch, 1986] cannot capture. This theory hopesto describe such relative K-groups in terms of algebraic cycles. The first attempt was the simplicial additivehigher Chow group in [Bloch and Esnault, 2003a], where it is shown that the additive higher Chow groups ofzero-cycles for the reduced space Spec (k) form the absolute Kähler differentials Ω∗k/Z. A cubical version was

proposed in [Bloch and Esnault, 2003b] for zero-cycles over the reduced space Spec (k) with a certain moduluscondition for which one had the same result, and Rülling computed the group of zero-cycles with higher modulusm ≥ 1 in [Rülling, 2007] in terms of the big de Rham-Witt forms. The higher dimensional cases were subsequentlydefined and studied in [Park, 2007], [Krishna and Levine, 2008], [Park, 2009], [Krishna and Park, 2012a], and[Krishna and Park, 2012b].

An interesting point is that, all of the above three calculations of zero-cycles in [Bloch and Esnault, 2003a],[Bloch and Esnault, 2003b], [Rülling, 2007] gave collections of cycle groups that are always certain differentialforms. In fact, the K-theory of the infinitesimal deformations of the base field k is expressed in terms of themodules of absolute Kähler differentials and the absolute de Rham-Witt complex of Hesselholt-Madsen, as shownfor example in [Hesselholt, 2005]. These de Rham-Witt complexes have the structure of a graded-commutativeDGA and they are initial objects in the category of so called Witt-complexes (see [Hesselholt and Madsen, 2001],[Rülling, 2007]).

This paper is along the line of extending these results to higher dimensional cases, and the following resultssummarize the outcomes of our attempts:

Theorem 1.0.1. Let X be a smooth projective variety over a perfect field k of characteristic 6= 2. Then, for eachinteger m ≥ 1, the additive higher Chow groups TCH(X;m) := (TCHq(X,n;m),∧, δ)q,n≥1 of X of modulus mform a DGA with the following operators: the graded-commutative associative product

∧ : TCHq1(X,n1;m)⊗Z TCHq2(X,n2;m)→ TCHq(X,n;m), (1.0.1)

where q = q1 + q2 − 1, n = n1 + n2 − 1 and the graded derivation δ for the product ∧ satisfying the Leibnizrule: for ξ ∈ TCHq1(X,n1, ;m) and η ∈ TCHq2(X,n2;m),{

ξ ∧ η = (−1)(n1−1)(n2−1)η ∧ ξ,δ(ξ ∧ η) = δξ ∧ η + (−1)n1−1ξ ∧ δη.

Received d m y; Revised d m y; Accepted d m yCommunicated by

c© The Author 20xx. Published by Oxford University Press. All rights reserved. For permissions,please e-mail: [email protected].

2 A. Krishna and J. Park

The product and derivation commute with the pull-back maps of additive higher Chow groups, and thepush-forward maps satisfy a projection formula. Furthermore, the projective system {TCH(X;m)}m≥1 of DGAsis equipped with certain operators λ,R, Fr, Vr (r ≥ 1), and they form a restricted Witt-complex over k. ThisWitt-complex structure is natural with respect to pull-back maps.

If we let TCH(X;m)0 = ⊕n≥1 TCHn(X,n;m), then it follows that (TCH(X;m)0,∧, δ) is a differentialgraded sub-algebra of TCH(X;m) and the projective system {TCH(X;m)0}m≥1 is a sub-Witt-complex. IfX = Spec (k), this coincides with the de Rham-Witt complex description in [Rülling, 2007].

The product ∧ in (1.0.1) is given by the formula ∆∗ ◦ µ ◦�; here, � is the concatenation of two cycleson X ×Gm ×�ni−1, i = 1, 2, and the concatenated cycle lies on X ×X ×Gm ×Gm ×�n−1. We apply the“product” µ induced by the multiplication µ : Gm ×Gm → Gm, and take the pull-back via the diagonal∆ : X → X ×X. Showing that these operations µ and ∆∗ make sense on additive higher Chow groups is apart of the goals of this paper, and by definition on the level of cycles this product pairing ∧ is nontrivial, andit isn’t obtained trivially by “extension by zero”. For more discussion on nontriviality, see Remark 3.5.2.

The entire paper is devoted to achieving this goal. Section 2 recalls the definition of additive higher Chowcycles and discusses some results on modulus conditions, and Section 3 defines the product structure ∧. Section4 defines the differential operator δ and proves the Leibniz rule. The Witt-complex structure is discussed inSection 5. Establishing Theorem 1.0.1 required us to introduce numerous concrete additive higher Chow cycles,and we hope they play useful roles for other future works in the subject. Some calculations require a technicalresult, that we call the normalization theorem (see Theorem 1.0.3 below).

We couldn’t yet extend Theorem 1.0.1 to smooth affine (or, more generally quasi-projective) varieties, dueto the current absence of the moving lemma as in [Krishna and Park, 2012a, Theorem 4.1] for such varieties(See Remarks 3.0.3, 4.3.10). We leave a memorandum that follows from the arguments of this paper:

Scholium 1.0.2. If the moving lemma for additive higher Chow groups holds for smooth affine (or, quasi-projective) varieties, too, then Theorem 1.0.1 extends to them, as well.

The Appendix (Section 6) proves the normalization theorem used in the proof of Theorem 1.0.1:

Theorem 1.0.3. Let X be a smooth quasi-projective variety. Then, the natural inclusion TzqN (X, •;m) ↪→Tzq(X, •;m) of complexes is a quasi-isomorphism, where Tzq(X, •;m) (TzqN (X, •;m)) is the (normalized)additive cycle complex in Definition 2.2.1 (Definition 6.1.1, respectively).

For higher Chow complexes, a similar result was known by [Bloch, online note, Theorem 4.4.2] and suchresults help us in proving the moving lemma for higher Chow groups of smooth quasi-projective varieties as in[Bloch, 1994]. We hope Theorem 1.0.3 will lead us to an eventual resolution of Scholium 1.0.2 in the future.

Conventions: In this paper, k is always a fixed perfect field of characteristic 6= 2. A k-scheme, or a schemeover k, is a separated scheme of finite type over k. A k-variety is a reduced k-scheme. We frequently work withequidimensional k-schemes to use codimensions of cycles, but one can work with more general k-schemes byusing dimensions of cycles. All products × of k-schemes are assumed to be ×k, unless specified otherwise. Foreach k-scheme X, we denote by XN the normalization of Xred.

2 Additive higher Chow groups

We recall some definitions regarding additive higher Chow complexes from [Krishna and Park, 2012a]. Let Xbe an equidimensional k-scheme. Let A1 = Spec k[t], Gm = Spec k[t, t−1], P1 = Proj k[Y0, Y1], and �n = (P1 \{1})n. For n ≥ 1, let Bn = Gm ×�n−1, Bn = A1 × (P1)n−1 and B̂n = P1 × (P1)n−1 ⊃ Bn. Let (t, y1, · · · , yn−1)be the coordinates on B̂n.

On B̂n, define the Cartier divisors F1n,i := {yi = 1} for 1 ≤ i ≤ n− 1, Fn,0 := {t = 0} (which is contained

in Bn), and let F1n :=

∑n−1i=1 F

1n,i. A face of Bn is a closed subscheme defined by a set of equations of the

form yi1 = �1, . . . , yis = �s where �j ∈ {0,∞}. For � = 0,∞, and i = 1, · · · , n− 1, let ιn,i,� : Bn−1 → Bn be theinclusion (t, y1, . . . , yn−2) 7→ (t, y1, . . . , yi−1, �, yi, . . . , yn−2). Its image is identified with a codimension one face.

Recall that the (cubical) higher Chow complex (see [Bloch, 1986] for the simplicial version), from which theadditive one is derived, is defined as follows: let zq(X,n) be the subgroup of zq(X ×�n) of cycles that intersectall faces of �n properly, where faces are defined in the same way as the above except we ignore the Gm-factors.The group zq(X,n) is zq(X,n) modulo the degenerate cycles. This gives a cubical object, and the associatedcomplex is the higher Chow complex. Its homology groups are the higher Chow groups CHn(X,n). See [Totaro,1992].

The additive higher Chow complex is defined similarly using the spaces Bn instead of �n, but we imposean additional condition called the modulus conditions, that control how the cycles should behave at “infinity”.

3

2.1 Modulus conditions

We recall all modulus conditions that have been used in the literature in studying additive higher Chow groups.

Definition 2.1.1. Let X be a k-scheme, and let V be an integral closed subscheme of X ×Bn. Let V denotethe Zariski closure of V in X × B̂n and let ν : V

N → V ⊂ X × B̂n be the normalization of V . Fix an integerm ≥ 1 and let n ≥ 1. When n = 1, the notations F 11 and F 11,i below are regarded as the zero divisors.

(i) We say that V satisfies the modulus m condition Msum on X ×Bn, if as Weil divisors on VN

we have(m+ 1)[ν∗(Fn,0)] ≤ [ν∗(F 1n)].

(ii) We say that V satisfies the modulus m condition Msup on X ×Bn, if as Weil divisors on VN

we have(m+ 1)[ν∗(Fn,0)] ≤ sup1≤i≤n−1[ν∗(F 1n,i)].

(iii) We say that V satisfies the modulus m condition Mssup on X ×Bn if there exists an integer 1 ≤ i ≤ n− 1such that as Weil divisors on V

N, we have (m+ 1)[ν∗(Fn,0)] ≤ [ν∗(F 1n,i)].

If V is a cycle on X ×Bn, we say that V satisfies the modulus condition if all of its irreducible componentssatisfy the modulus condition.

When no confusion arises, we simply say that V satisfies the modulus condition M without mentioning theinteger m. Note that since V is contained in X ×Bn, its closure V intersects all the Cartier divisors Fn,0 andF 1n,i (1 ≤ i ≤ n− 1) properly in X × B̂n. In particular, their pull-backs of Fn,0 and F 1n,i are all effective Cartierdivisors on V

N.

Remark 2.1.2. For n ≥ 1, let V ⊂ X ×Bn be an irreducible closed subvariety satisfying a modulus conditionin Definition 2.1.1. The modulus condition says that the closure of V in X × B̂n can intersect {t = 0} only alongthe divisor F 1n . In particular, the closure of V in X ×A1 ×�n−1 does not intersect {t = 0}. Furthermore, forn = 1, that the closure of V in X ×A1 does not intersect {t = 0} is equivalent to that V satisfies any one of theabove modulus conditions.

Here, we present another property that shows how nontrivial the modulus conditions are.

Lemma 2.1.3. Let V ⊂ X ×Bn be an irreducible closed subvariety satisfying a modulus condition above.Then,

(i) V does not contain a nonempty subset of the form Gm ×W , where W ⊂ X ×�n−1.(ii) The composition V ↪→ X ×Bn

pr→ X ×�n−1 is quasi-finite.

Proof . (i) If V ⊃ Gm ×W , then the Zariski closure V in X ×A1 ×�n−1 contains A1 ×W so that V ∩ {t =0} 6= ∅. Contradiction. (ii) Given a point p, the fiber V ∩ pr−1(p) is a proper closed subscheme of Gm × {p} by(1). Hence, it is finite.

Remark 2.1.4. The modulus condition Msum was used in [Bloch and Esnault, 2003b] and [Rülling, 2007]. Themodulus condition Msup was used in [Krishna and Levine, 2008] and [Park, 2009]. The modulus condition Mssupwas recently introduced in [Krishna and Park, 2012a]. However, some experts seem to think that Mssup maynot be as suitable as Msum owing to its stringent behavior.

2.2 Additive cycle complexes

We recall the definition of additive cycle complexes based on the above modulus conditions from [Krishna andPark, 2012a, Definition 2.5]. (N.B. We correct a minor error in loc.cit. on part (0) below.)

Definition 2.2.1. Let M be a modulus condition in Section 2.1. Let X be a k-scheme, and let r,m, n be integerswith m,n ≥ 1.

(0) Tzr(X, 1;m)M is the free abelian group on integral closed subschemes Z of X ×Gm of dimension r,satisfying the modulus condition.

For n > 1, Tzr(X,n;m)M is the free abelian group on integral closed subschemes Z of X ×Bn of dimensionr + n− 1 such that:

(1) For each face F of Bn, Z intersects X × F properly on X ×Bn.


(2) Z satisfies the modulus m condition M on X ×Bn.For each i = 1, · · · , n− 1 and � = 0,∞, let ∂�i (Z) := [(IdX × ιn,i,�)∗(Z)]. The proper intersection with facesensures that ∂�i (Z) are well-defined.If M is the modulus condition Msup, then we further suppose that

(3) ∂�i (Z) lies in Tzr(X,n− 1;m)M .

Algebraic cycles on X ×Bn that belong to Tzr(X,n;m)M are called the admissible cycles (additive higher Chowcycles, or additive cycles) with respect to the modulus condition M . When the scheme X is equidimensional ofdimension d over k, we write for q ≥ 0, Tzq(X,n;m)M := Tzd+1−q(X,n;m)M .

From the above, we have the cubical abelian group (n 7→ Tzq(X,n;m)M ) in the sense of [Krishna andLevine, 2008, §1.1]. As seen in [Krishna and Park, 2012a], using the containment lemma [Krishna and Park,2012a, Proposition 2.4], for Msum and Mssup, the condition (3) is implied from (1) and (2). For Msup this isn’tnecessarily the case.

Definition 2.2.2. Let M be a modulus condition. The additive higher Chow complex, or just the additive cyclecomplex, Tzq(X, •;m)M of X in codimension q and with the modulus m condition M is the nondegeneratecomplex associated to the cubical abelian group (n 7→ Tzq(X,n;m)M ), i.e.,

Tzq(X,n;m)M := Tzq(X,n;m)M/Tz

q(X,n;m)M,degn.

The boundary map of this complex at level n is given by ∂ :=∑n−1

i=1 (−1)i(∂∞i − ∂0i ), and it satisfies ∂2 = 0.The homology TCHq(X,n;m)M := Hn(Tz

q(X, •;m)M ) for n ≥ 1 is the additive higher Chow group of X withmodulus m condition M .

Remark 2.2.3. The modulus conditions in Section 2.1 satisfy the implications Mssup ⇒Msup ⇒Msum, sothat we have the natural homomorphisms of abelian groups TCHq(X,n;m)Mssup → TCH

q(X,n;m)Msup →TCHq(X,n;m)Msum . Whether these additive Chow groups are identical is not known in general. However forX = Spec (k), where char(k) 6= 2, the additive higher Chow groups of zero-cycles are all isomorphic using thearguments in [Rülling, 2007] without modification.

Remark 2.2.4. Earlier works used a bit different notations. The following shows how they are related to thenotation of this paper:

(i) THnM (k, n) in [Bloch and Esnault, 2003b] ' TCHn(Spec (k), n; 1)Msum ,

(ii) CHn(Ak(m+ 1), n− 1) in [Rülling, 2007] ' TCHn(Spec (k), n;m)Msum ,(iii) ACHq(X,n− 1;m+ 1) in [Park, 2009] ' TCHq(X,n;m)Msup ,(iv) TCHq(X,n;m) in [Krishna and Levine, 2008] ' TCHq(X,n;m)Msup ,(v) THq(X,n;m) in [Krishna and Park, 2012a] ' TCHq(X,n;m).

Conventions. We study the additive higher Chow groups based on only the modulus condition Msum in thispaper. Henceforth, we drop the subscript M from the notations of additive higher Chow groups.

2.3 Double modulus conditions

The modulus condition in Definition 2.1.1 have multivariate analogues. Their principal usages will appear in aforthcoming paper [Krishna and Park, in progress] that studies some multivariate analogues of de Rham-Wittcomplexes in terms of algebraic cycles. Although this paper is about the “univariate” case, some considerations ofthe multiple modulus conditions simplify our discussions. Below, we consider only the double modulus conditionfor Msum:

Definition 2.3.1. Let n ≥ 2 and let X be a k-scheme.

(i) Let B2,n := (Gm)2 ×�n−2 and B2,n = (A1)2 × (P1)n−2.(ii) Let (t1, t2, y1, · · · , yn−2) ∈ Bn be the coordinates. The faces of B2,n are similarly defined by a sequence of

equations of the form yi = � with � ∈ {0,∞}.

5

(iii) Let V be an integral closed subscheme of X ×B2,n. Let V denote the Zariski closure of V in X ×B2,n,and let ν : V

N → V ⊂ X ×B2,n be the normalization of V . Let m1,m2 ≥ 1 be integers.We say that V satisfies the modulus (m1,m2) condition on X ×B2,n, if as Weil divisors on V

Nwe have

2∑i=1

(mi + 1)[ν∗{ti = 0})] ≤

n−2∑i=1

[ν∗{yi = 1}]. (2.3.1)

Here, when n = 2 we interpret the right hand side to be zero so that we have [ν∗{t1 = 0}] = [ν∗{t2 =0}] = 0, in particular. We say that a cycle on X ×B2,n satisfies the modulus (m1,m2) condition if eachirreducible component of V satisfies it.

Remark 2.3.2. Since (mi + 1)[ν∗{ti = 0}] ≤

∑2i=1(mi + 1)[ν

∗{ti = 0})], we deduce that V satisfies themodulus mi conditions on X ×Gm ×Bn−1.

Lemma 2.3.3. Let X and Y be k-schemes, and let V1 and V2 be cycles satisfying the modulus m1 and m2conditions on X ×Bn1 and Y ×Bn2 , respectively. Then, V1 × V2 satisfies the modulus (m1,m2) condition onX × Y ×B2,n1+n2 .

Proof . We may assume that V1, V2 are irreducible. Let W ⊂ V1 × V2 be an irreducible component. It is enoughto show that W satisfies the modulus (m1,m2) condition. Let ι1 : V 1 ⊂ X ×Bn1 and ι2 : V 2 ⊂ Y ×Bn2 be theZariski closures of V1 and V2, respectively. Take their normalizations νV i : V

N

i → V i for i = 1, 2. By [Krishnaand Levine, 2008, Lemma 3.1], the product of two reduced normal k-schemes is again normal over perfect

fields. Thus, the morphism ν = νV 1 × νV 2 : VN

1 × VN

2 → V 1 × V 2 = V1 × V2 is the normalization under theidentification X × Y ×Bn1+n2 ' X ×Bn1 × Y ×Bn2 . Thus, we regard V

N

1 × VN

2 as V1 × V2N

. This gives acommutative diagram

VN

1 × VN

2

ν

%%

WN νW //

ιN::

W � ι // V 1 × V 2 �

ι1×ι2 // X × Y ×A2 × (P1)n−1.

Here W is the Zariski closure of W , νW is the normalizations of W , and ιN is the natural inclusion. Here,

n = n1 + n2 − 1.Let (t1, t2, y1, · · · , yn−1) ∈ A2 ×

(P1)n−1

be the coordinates. Then, for D1 :=∑n1−1

i=1 {yi = 1} − (m1 +1){t1 = 0}, D2 :=

∑n−1i=n1{yi = 1} − (m2 + 1){t2 = 0}, and n− 1 = (n1 − 1) + (n2 − 1), we have ((ι1 × 1) ◦

(νV 1 × 1))∗D1 ≥ 0 and ((1× ι2) ◦ (1× νV 2))

∗D2 ≥ 0. These hold by the modulus conditions of V1 × Y ×Bn2and X ×Bn1 × V2, regarding V1 × Y ×Bn2 as a cycle in Tz(X × Y ×Bn2 , n1;m), and similarly for X ×Bn1 ×V2. Thus, we have ν

∗(ι1 × ι2)∗(D1 +D2) ≥ 0 on VN

1 × VN

2 so that (ιN )∗ν∗(ι1 × ι2)∗(D1 +D2) ≥ 0 on W

N. By

the commutativity, this means ν∗W ι∗(ι1 × ι2)∗(D1 +D2) ≥ 0 on W

N, which is the desired modulus (m1,m2)

condition of W .

2.4 A remark on effective divisors

The following result, that refines [Krishna and Levine, 2008, Lemma 3.2] and [Krishna and Park, 2012a, Lemma2.2], is used often in this paper:

Lemma 2.4.1. Let f : Y → X be a dominant map of normal integral k-schemes. Let D be a Cartier divisor onX such that the generic points of Supp(D) are contained in f(Y ). Suppose that f∗(D) ≥ 0 on Y . Then, D ≥ 0on X.

Proof . Note that f(Y ) is a constructible (by Chevalley) dense subset of X, so that by [Hartshorne, 1977,Exercise II-3.18(b)], there is a nonempty open subset U ⊂ X contained in f(Y ). Since the generic points ofSupp(D) are all contained in f(Y ), we may assume that U also contains the generic points of Supp(D). Then,D|U ≥ 0 on U if and only if D ≥ 0 on X. Thus, we may replace X by U , Y by f−1(U), and D by D|U , i.e., wemay assume that f : Y → X is surjective. Then, by [Krishna and Park, 2012a, Lemma 2.2] we are done.


3 Product structure

In this section, we construct a graded-commutative associative algebra structure on the additive higher Chowgroups of a smooth projective variety, which is compatible with the module structure of [Krishna and Levine,2008, Theorem 4.10] on these groups over the ordinary Chow ring of the variety. We deduce the algebra structurein Theorem 3.5.1 from the following result, whose proof will occupy most of this section.

Proposition 3.0.2. Let X and Y be smooth quasi-projective varieties over k. Then there exists an externalproduct on the additive higher Chow groups

×µ : TCHq1(X,n1;m)⊗Z TCHq2(Y, n2;m)→ TCHq(X × Y, n;m), (3.0.1)

where q = q1 + q2 − 1, n = n1 + n2 − 1, and qi, ni,m ≥ 1 for i = 1, 2. Furthermore, one has

ξ×µη = (−1)(n1−1)(n2−1)η ×µ ξ (3.0.2)

for all classes ξ ∈ TCHq1(X,n1;m) and η ∈ TCHq2(Y, n2;m).

Once we have the above external product, in case X is smooth projective, the internal product structureessentially comes via the diagonal pull-back ∆∗X : TCH

q(X ×X,n;m)→ TCHq(X,n;m) which exists due tothe general functoriality in [Krishna and Park, 2012a, Theorem 7.1], as in the classical intersection theory. Thisfunctoriality result is based on the moving lemma of [Krishna and Park, 2012a, Theorem 4.1].

Remark 3.0.3. If one can extend [Krishna and Park, 2012a, Theorem 4.1] to smooth quasi -projective varieties,then the functoriality result [Krishna and Park, 2012a, Theorem 7.1] also extends, and we have the internalproduct for all smooth quasi-projective varieties. At present, this argument works only for the smooth projectivecase.

3.1 External product

Let X and Y be equidimensional k-schemes. The external product ×µ is based on the product mapµ : Gm ×Gm → Gm. Consider the diagram, where n = n1 + n2 − 1:

X ×Gm ×�n1−1 × Y ×Gm ×�n2−1τ //

µ

++

X × Y ×Gm ×Gm ×�n−1

1×1×µ×1��

X × Y ×Gm ×�n−1.

(3.1.1)

Here, τ is the transposition map (x, t, y, x′, t′, y′) 7→ (x, x′, t, t′, y, y′). We denote the composite map also by µ.For a given pair of admissible cycles V1 ∈ Tzq1(X,n1;m) and V2 ∈ Tzq2(Y, n2;m), we want to define an

admissible cycle V1 ×µ V2.

Lemma 3.1.1. The map µ in (3.1.1) restricted to V1 × V2 is quasi-finite. In particular, codimX×Y×Bn(µ(V1 ×V2)) = q, or, dim(µ(V1 × V2)) = dim(V1 × V2).

Proof . Regard V1 as a subvariety of Gm ×X ×�n−1. By Lemma 2.1.3, each fiber of the compositionπ1 : V1 ↪→ Gm ×X ×�n1−1

pr→ X ×�n1−1 is finite, where pr is the projection. Similarly, the fibers of π2 : V2 ↪→Gm × Y ×�n2−1 → Y ×�n2−1 are also finite. Then, the lemma follows from :Claim: Let πi : Vi ↪→ Yi ×Xi

pr→ Xi be the compositions for i = 1, 2, that are quasi-finite. Let f : Y1 × Y2 → Y3be any morphism. Then, the composition

π : V1 × V2 ↪→ Y1 ×X1 × Y2 ×X2 ' Y1 × Y2 ×X1 ×X2f̄→ Y3 ×X1 ×X2

is quasi-finite, too, where f̄ = f × IdX1×X2 .Indeed, for each point p ∈ Y3 ×X1 ×X2, let xi = pri(p) for projections pri : Y3 ×X1 ×X2 → Xi. Then,

since π−1(p) ⊂ π−11 (x1)× π−12 (x2) and the latter set is finite by assumption, the fiber π

−1(p) is finite. Applythis to X1 = X ×�n1−1, X2 = Y ×�n2−1, Y1 = Y2 = Y3 = Gm, and f = µ. From this, µ : V1 × V2 → µ(V1 × V2)is surjective with finite fibers. Hence, dim(V1 × V2) = dim(µ(V1 × V2)).

A form of the following lemma was suggested by an anonymous referee:

Lemma 3.1.2. The map µ in (3.1.1) restricted on V1 × V2 is a projective (and hence finite) morphism.

7

Proof . We show that µ|V1×V2 is projective. Let Γµ ⊂ (Gm)2 ×Gm be the graph of µ : Gm ×Gm → Gm, andlet Γµ be its Zariski closure in (P1)2 × P1. The projection onto the third factor pr3 : Γµ → P1 is projective, sothat its pull-back along the natural inclusion Gm ↪→ P1

Γ◦µ : = Γµ ×P1 Gm = Γµ ∪ ({0} × {∞} ×Gm) ∪ ({∞} × {0} ×Gm)→ Gm

is also projective.Let V ⊂ X × Y × Γµ ×�n−1 be the pull-back of V1 × V2 along the isomorphism Γµ ' Gm ×Gm (up

to a suitable permutation of factors), and let V be its Zariski closure in X × Y × Γµ ×�n−1. The mapV → X × Y × P1 ×�n−1 induced by pr3 is projective so that its pull-back along the inclusion Gm ↪→ P1

V ∩(X × Y × Γ◦µ ×�n−1

)→ X × Y ×Gm ×�n−1 (3.1.2)

is projective. Let V 1 be the closure of V1 in X × P1 ×�n1−1. Let

π1 : X × Y × Γµ ×�n−1 → X × P1 ×�n1−1

be the composition of the projection to X × Γµ ×�n1−1 with the map pr1 : Γµ → P1, which is the restrictionof the projection (P1)2 → P1 to the first factor. Then, π−11 (V ) is closed and it contains V . Hence, itcontains V . On the other hand, the modulus condition for V1 implies that V 1 ∩ (X × {0} ×�n1−1) = ∅.Hence, V ∩ (X × Y × ({0} × {∞} ×Gm)×�n1−1) = ∅. The same argument for V2 shows that V ∩ (X × Y ×({∞} × {0} ×Gm)×�n−1) = ∅. Hence, by (3.1.2), V ∩ (X × Y × Γµ ×�n−1) = V ' V1 × V2 is projective overX × Y ×Gm ×�n−1 via µ. This shows that the map µ|V1×V2 is projective. The map µ|V1×V2 is also quasi-finiteby Lemma 3.1.1 and we deduce that it is finite.

Definition 3.1.3. (i) For any irreducible closed subvariety V ⊂ X × Y ×Gm ×Gm ×�n−1 such thatµ|V : V → µ(V ) is finite, define µ∗(V ) as the push-forward deg(V/µ(V )) · [µ(V )]. This is a cycle onX × Y ×Gm ×�n−1. We can extend it Z-linearly to the subgroup of cycles spanned by such cycles.

(ii) For V1 and V2 as before, we define the external product as V1 ×µ V2 : = µ∗(V1 × V2), where V1 × V2 is seenas the cycle associated to the scheme V1 × V2. It is a sum over irreducible components of V1 × V2, and itis a well-defined codimension q1 + q2 − 1 cycle on X × Y ×Gm ×�n−1. We can extend it Z-bilinearly forall admissible cycles V1 and V2.

Remark 3.1.4. When X = Spec (k) and V1, V2 are 0-cycles, this definition coincides with [Rülling, 2007,Definition-Proposition 3.9 (ii)].

Lemma 3.1.5. Let m1,m2 ≥ 1. Let V1 ∈ Tzq1(X,n1;m1), V2 ∈ Tzq2(Y, n2;m2). Then, the cycle V1 ×µ V2intersects properly with all faces of X × Y ×Bn.

Proof . We may assume that V1, V2 are irreducible. Let F ⊂ Bn be a face. We need to prove that dim(Z ∩(X × Y × F )) ≤ dimZ − codimX×Y×Bn(X × Y × F ). Since ∂�i (V1 ×µ V2) = ∂�i (V1)×µ V2 if 1 ≤ i ≤ n1 − 1, and= V1 ×µ ∂�i (V2) if n1 ≤ i ≤ n− 1, by induction on the codimension of X × Y × F , it is enough to show the lemmawhen F are codimension one faces. We do it for 1 ≤ i ≤ n1 − 1 as the other case n1 ≤ i ≤ n− 1 is exactlythe same. In this case, dim(∂�i (V1 ×µ V2)) = dim(∂�i (V1)×µ V2) =† dim(∂�i (V1)× V2) ≤ dimV1 − 1 + dimV2 =‡dim(V1 ×µ V2)− 1, where † and ‡ hold by Lemmas 2.3.3 and 3.1.1. The inequality holds because V1 intersectswith the faces properly. This finishes the proof.

Proposition 3.1.6. Let m1,m2 ≥ 1. For any cycle V on X ×B2,n+1 satisfying the modulus (m1,m2) conditionsuch that V → µ∗(V ) is finite, the cycle µ∗(V ) satisfies the modulus min{m1,m2} condition on X ×Bn.

Proof . Let m := min{m1,m2}. We may assume that V is irreducible. Since µ∗(V ) = deg(V/µ(V )) · [µ(V )], itis enough to show that µ(V ) satisfies the modulus m condition. Consider the commutative diagram

VN νV //

µN

��

V � ι //

µ

��

X ×A2 × (P1)n−1

µ

��

µ(V )N νµ(V )

// µ(V ) � ιµ(V )

// X ×A1 × (P1)n−1.


Here, V and µ(V ) are Zariski closures of V and µ(V ), and νV and νµ(V ) are the normalizations of the closures.The map µ̄N is given by the universal property of normalization.

Let (t1, t2, y1, · · · , yn−1) ∈ A2 ×(P1)n−1

and (w, y1, · · · , yn−1) ∈ A1 ×(P1)n−1

be the coordinates. Let

D :=∑n−1

i=1 {yi = 1} −∑2

j=1(mj + 1){tj = 0} onX × Y ×A2 × (P1)n−1. Since V satisfies the modulus (m1,m2)condition, we have ν∗V ι

∗(D) ≥ 0.On the other hand, consider the Cartier divisor D′ :=

∑n−1i=1 {yi = 1} − (m+ 1){w = 0} on X ×A1 ×(

P1)n−1

. On X ×A2 ×(P1)n−1

, we have µ∗D′ =∑n−1

i=1 {yi = 1} −∑2

j=1(m+ 1){tj = 0} ≥ D. Thus, by pulling-back via ι ◦ νV , we have ν∗V ι∗µ∗D′ ≥ ν∗V ι∗(D) ≥ 0. This inequality is equivalent to (µN )∗ν∗µ(V )ι

∗µ(V )D

′ ≥ 0 bythe commutativity of the diagram. Applying Lemma 2.4.1 to µN , we get ν∗µ(V )ι

∗µ(V )D

′ ≥ 0. This is the modulusm condition for µ(V ).

Corollary 3.1.7. Let X and Y be equidimensional k-schemes. Let V1 ∈ Tzq1(X,n1;m) and V2 ∈ Tzq2(Y, n2;m).Then µ∗(V1 × V2) = V1 ×µ V2 is in Tzq(X × Y, n;m).

From the associativity of µ we immediately have:

Corollary 3.1.8. The external product ×µ is associative on the level of cycles.

3.2 Some cycle computations

Before we proceed further, we collect some concrete cycle calculations that are used in several parts ofthis paper. Let σ be a permutation on {1, 2, · · · , n− 1} acting naturally on �n−1 of X ×Gm ×�n−1 viaσ · (x, t, y1, · · · , yn−1) := (x, t, yσ(1), · · · , yσ(n−1)), where X is a k-scheme. This action on the space naturallyinduces an action on the group of admissible cycles. Note that if Z is an admissible cycle, then for any permutationσ, the cycle σ · Z is admissible.

3.2.1 The cycles Z{r}For r ≥ 1 and n ≥ 2, consider the morphism χn,r : X ×Gm ×�n−1 → X ×Gm ×�n−1 given by(x, t, y1, · · · , yn−1) 7→ (x, t, yr1, y2, · · · , yn−1). Given an irreducible closed subvariety Z ⊂ X ×Gm ×�n−1, letχn,r(Z) denote its image under the finite (and flat) map χn,r and define Z{r} to be the cycle

Z{r} := (χn,r)∗([Z]) = [k(Z), k(χn,r(Z))] · [χn,r(Z)]. (3.2.1)

Notice that χn,r(Z) ( X ×Gm ×�n−1 is closed as χn,r is a finite morphism. We can extend this definitionZ-linearly so that Z{r} is defined for all cycles Z.Lemma 3.2.1. Let Z ∈ Tzq(X,n;m). Then, χn,r(Z) (and hence Z{r}) is also in Tzq(X,n;m).

Proof . We may assume that Z is irreducible. We first show that χn,r(Z) intersects properly with all faces.We do it by induction on n ≥ 1. For codimension 1 faces, note that ∂�1(χn,r(Z)) = r∂�1(Z), and ∂�i (χn,r(Z)) =(∂�i (χn,r(Z)) for i ≥ 2. Consequently, for general faces F , we have F · (χn,r(Z)) = r(F · Z) if F involves {y1 = �},and F.(Zχn,r(Z)) = (χn,r(F · Z) otherwise. Since the intersection F.Z is proper, we deduce that so doesχn,r(F · Z) by the induction hypothesis. This shows that the intersection F · χn,r(Z) is proper.

Now, we show that χn,r(Z) satisfies the modulus condition. For simplicity, let W : = χn,r(Z). Let Z and W

be the Zariski closures of Z and W in X × B̂n and let νZ : ZN → Z and νW : W

N →W be their normalizations.Thus, we have the commutative diagram

ZN

χ̄Nn,r��

νZ // Z

χ̄n,r

��

ιZ // X × B̂n

χn,r

��

WN νW // W

ιW // X × B̂n.

Here, we have

(m+ 1)[ν∗Zι∗Z(Fn,0)] ≤†

n−1∑i=1

[ν∗Zι∗Z(F

1n,i)] ≤ r[ν∗Zι∗Z(F 1n,1)] +

n−1∑i=2

[ν∗Zι∗Z(F

1n,i)], (3.2.2)

where the inequality † follows from the modulus condition of Z. By the definition of χn,r, note thatχ∗n,rFn,0 = Fn,0, χ

∗n,rF

1n,1 = rF

1n,1, and χ

∗n,rF

1n,i = F

1n,i for i ≥ 2. Hence, the inequality (3.2.2) is equiva-

lent to (m+ 1)[ν∗Zι∗Zχ∗n,r(Fn,0)] ≤

∑n−1i=1 [ν

∗Zι∗Zχ∗n,r(F

1n,i)] = [ν

∗Zι∗Zχ∗n,r(F

1n)]. By the commutativity of the dia-

gram, this is [(χ̄Nn,r)∗(ν∗W ι

∗W (F

1n)− (m+ 1)ν∗W ι∗W (Fn,0)] ≥ 0. By Lemma 2.4.1, this implies [ν∗W ι∗W (F 1n)− (m+

1)ν∗W ι∗W (Fn,0)] ≥ 0, which is the modulus condition for W = χn,r(Z). Hence, χn,r(Z) is an admissible cycle.

9

3.2.2 The cycles γiZ

We consider somewhat mutated analogues V iX for i ≥ 1 of WXn in [Levine, 1994, Lemma 4.1]. The case i = 1 isused in Section 3.3 to prove Lemma 3.3.1, and the general cases i ≥ 1 are used in Section 5.4 in the discussionof Witt-complex structures.

Let X be a smooth quasi-projective k-variety. Let n ≥ 2. Let (x, t, y1, · · · , yn−1, y, λ) be the coordinates ofX × B̂n+2. Consider the closed subscheme ι : V iX ↪→ X ×Bn+2 given by the equation

(1− y)(1− λ) ={

1− y1, if i = 1,(1− y1)(1 + y1 + · · ·+ yi−11 − λ(1 + y1 + · · ·+ y

i−21 )), if i ≥ 2.

(3.2.3)

Let V̂ iX be the Zariski-closure of ViX in X × B̂n+2. Let π1 : X × B̂n+2 → X × B̂n+1 be the projection that ignores

the coordinate y1, and let π′1 be the restriction π1|V̂ iX .

Lemma 3.2.2. The map π′1 : ViX → X ×Bn+1 is an isomorphism for i = 1, and a surjection for i ≥ 2.

Proof . When i = 1, we define the inverse θ : X ×Bn+1 → V 1X , given by

(x, t, y2, · · · , yn−1, y, λ) 7→ (x, t,−(1− y)(1− λ) + 1, y2, · · · , yn−1, y, λ).

That this is inverse to π′1 is immediate. For i ≥ 2, that π′1 is a surjection is obvious.

Definition 3.2.3. For i ≥ 1 and n ≥ 2, and an irreducible closed subvariety Z ⊂ X ×Gm ×�n−1, defineγiZ := π

′1(V

iX · (Z ×�2)) as an abstract algebraic cycle. For a general cycle Z, we can define γiZ by extending

the above Z-linearly.

Remark 3.2.4. The cycle γiZ can be described as follows, too : for i ≥ 1 and n ≥ 2, consider the rational mapsνi : X ×Gm ×�n−1 ×�→ X ×Gm ×�n given by

νi : (x, t, y1, · · · , yn−1)× y 7→ (x, t, y2, y3, · · · , yn−1, y,y − yi1y − yi−11

). (3.2.4)

Then, γiZ is the Zariski closure of νi(Z ×�). Indeed, if we let (y − yi1)/(y − yi−11 ) = λ, we get

(1− y)(1− λ) = (1− yi1)− λ(1− yi−11 ).

After pulling out the factor (1− y1) from the right hand side, we obtain the defining equation (3.2.3) of V iX .

Lemma 3.2.5. Let Z ∈ Tzq(X,n;m). Then, γiZ is in Tzq(X,n+ 1;m).

Proof . We may assume that Z is irreducible. We first prove that γiZ intersects properly with the faces ofX ×Bn+1. Write γiZ,n = γiZ to keep track of n. Let � ∈ {0,∞}.

If a given face F of X ×Bn+1 involves a face {yj = �} for j ∈ {n− 1, n}, then γiZ,n intersects Fproperly. Indeed, (see (3.2.4)) we see that ∂0n−1(γ

iZ,n) = σ · Z, ∂0n(γiZ,n) = σ · (Z{i}) for the cyclic permutation

σ = (1, 2, · · · , n), and ∂∞n−1(γiZ,n) = 0, ∂∞n (γiZ,n) = σ · (Z{i− 1}). Since Z is admissible, so are Z{i} and Z{i− 1}by Lemma 3.2.1, thus, in particular all of σ · Z, σ · (Z{i}), and σ · (Z{i− 1}) intersect properly with all faces.In particular γiZ,n intersects F properly.

We now show that γiZ,n intersects properly with all faces F that do not involve {yj = �} for j ∈ {n− 1, n}.This follows by induction on n ≥ 2. Indeed, from (3.2.4), for j < n and � = 0,∞, we have ∂�j(γiZ,n) = γi∂�jZ,n−1so that the dimension of ∂�j(γ

iZ,n) is at least one less by the induction hypothesis. Repeatedly applying this

argument for the remaining defining equations {yj = �} of the face F , we are done.We now show that γiZ satisfies the modulus condition. Let Z

′ be an irreducible component of V iX · (Z ×�2)and let W ′ = π′1(Z

′). Any irreducible component of γiZ is of this form by Definition 3.2.3. Thus, it is enough toshow that W ′ satisfies the modulus condition. Since Z ×�2 satisfies the modulus condition, by the containmentlemma in [Krishna and Park, 2012a, Proposition 2.4], Z ′ also satisfies the modulus condition. In fact, it satisfies

a stronger condition (3.2.5) below: let Z′

be the Zariski closure of Z ′ in X × B̂n+2 and let νZ′ : Z′N → Z ′ ↪→ V̂ iX


be the normalization composed with the inclusion. Similarly, W′

is the Zariski closure of W ′ and ν : W′N →W ′

is the normalization. Consider the diagram

Z′N νZ′ //

πN1��

V̂ iX� ι //

π′1

##

π̄′1��

X × B̂n+2

π1

��

W′N ν // W

′ � ιW ′ // X × B̂n+1.

We use (x, t, y1, · · · , yn−1, y, λ) and (x, t, y2, · · · , yn−1, y, λ) for the coordinates of X × B̂n+2 and X × B̂n+1,respectively. From the modulus condition of Z, we have

(m+ 1)[ν∗Z′ι∗{t = 0}] ≤ [ν∗Z′ι∗

n−1∑j=1

{yj = 1}]. (3.2.5)

Note that we did not include the divisors {y = 1} and {λ = 1}. But, V iX is an effective divisor onX ×Bn+2 given by the equation (1− y1)(∗) = (1− y)(1− λ) for some polynomial (∗) as in (3.2.3)so that [ν∗Z′ι

∗{y1 = 1}] ≤ [ν∗Z′ι∗{y = 1}] + [ν∗Z′ι∗{λ = 1}] (The equality holds if i = 1.). By the com-mutativity of the diagram, we thus deduce from (3.2.5) the inequality (m+ 1)[(πN1 )

∗ν∗ι∗W ′{t = 0}] ≤[(πN1 )

∗ν∗ι∗W ′(∑n−1

j=2 {yj = 1}+ {y = 1}+ {λ = 1})

]. Since πN1 is dominant by Lemma 3.2.2, using Lemma 2.4.1

we deduce

(m+ 1)[ν∗ι∗W ′{t = 0}] ≤ [ν∗ι∗W ′

(n−1∑j=2

{yj = 1}+ {y = 1}+ {λ = 1}

)].

But, this is the modulus condition for W ′. Hence γiZ satisfies the modulus condition.

3.3 Graded commutativity

We now come back to the algebra structure on additive higher Chow groups. We use cycles considered in Section3.2.2 for the case of i = 1.

Lemma 3.3.1. Let X be a smooth quasi-projective k-variety. Let n ≥ 3. Let Z ∈ Tzq(X,n;m) be a cycle suchthat ∂�i (Z) = 0 for all 1 ≤ i ≤ n− 1 and � ∈ {0,∞}. Let σ be a permutation on {1, 2, · · · , n− 1}. Then, thereexists a cycle γσZ ∈ Tz

q(X,n+ 1;m) such that

Z = (sgn(σ))(σ · Z) + ∂(γσZ). (3.3.1)

Proof . Case 1. We first consider the case when σ is the transposition τ = (p, p+ 1) for 1 ≤ p ≤ n− 2. We do itfor τ = (1, 2) only, for the other cases are similar. For the unique permutation ξ on {1, · · · , n} determined by ξ ·(x, t, y1, y2, · · · , yn) = (x, t, yn−1, y1, yn, y2, · · · , yn−2), (more precisely, ξ = (2, 3)(n, n− 1, · · · , 2, 1)2), considerthe cycle γτZ := ξ · γ1Z , where γ1Z is as in Definition 3.2.3 and (3.2.4). (N.B. The components of this cycle can bedescribed also as follows: consider the morphism ν : X ×Gm ×�n−1 ×�→ X ×Gm ×�n given by

ν(x, t, y1, · · · , yn−1, y) = (x, t, y, y2,y − y1y − 1

, y3, · · · , yn−1). (3.3.2)

This is obtained from (3.2.4) with i = 1 by permuting the coordinates of �n so that set-theoretically γτZ is theclosure of ν(Z ×�).) Being a permutation of the cycle in Lemma 3.2.5, this cycle γτZ is also admissible. Here,we have ∂∞1 (γ

τZ) = 0, ∂

01(γ

τZ) = τ · Z and ∂∞3 (γτZ) = 0, ∂03(γτZ) = Z. On the other hand, for � ∈ {0,∞}, ∂�2(γτZ)

is a permutation of γ1∂�2(Z), which is 0 because ∂�2(Z) = 0 by the given assumptions. Similarly, for j ≥ 4 and

� ∈ {0,∞}, we have ∂�j(γτZ) is a permutation of γ1∂�j−1(Z), which is again 0. In particular, ∂(γτZ) = Z + τ · Z, so

we are done for this case.Case 2. Now let σ be any permutation on {1, · · · , n− 1}. By a basic fact from group theory, we can write

σ as a product σ = τrτr−1 · · · τ2τ1, where each τi is a transposition of the form (p, p+ 1) considered in Case 1.Let σ0 := Id and σl := τlτl−1 · · · τ1 for 1 ≤ l ≤ r. Then, for each 1 ≤ l ≤ r, by Case 1, we have

(−1)l−1σl−1 · Z + (−1)l−1τl · σl−1 · Z = ∂(

(−1)l−1γτlσl−1·Z). (3.3.3)

Since τl · σl−1 = σl, by taking the sum of (3.3.3) over 1 ≤ l ≤ r, after cancellations we have Z + (−1)r−1σ · Z =∂(γσZ), where γ

σZ :=

∑rl=1(−1)l−1γ

τlσl−1·Z . Since (−1)

r = sgn(σ), we now obtain (3.3.1).

11

Lemma 3.3.2. Let X and Y be smooth quasi-projective k-varieties. For two cycles ξ ∈ Tzq1(X,n1;m) andη ∈ Tzq2(Y, n2;m) all of whose codimension 1 faces are trivial, we have

ξ ×µ η = (−1)(n1−1)(n2−1)η ×µ ξ + ∂(γr,s) (3.3.4)

for an admissible cycle γr,s, where r = n1 − 1, s = n2 − 1.

Proof . Let n := n1 + n2 − 1. By the given assumptions, all codimension one faces of ξ ×µ η are also trivial.Thus, the lemma follows from Lemma 3.3.1 applied to ξ ×µ η ∈ Tzq(X × Y, n;m) by taking γr,s = γσξ×µη, whereσ is the permutation on {1, · · · , r + s} given by σ(i) = r + i for i ≤ s and σ(i) = i− s for i ≥ s+ 1.

3.4 External product on homology

We now show that ×µ on the cycle complexes descends onto the homology groups. Note first that from thedefinition of ∂, we immediately have:

Lemma 3.4.1. Let X,Y be equidimensional k-schemes. For two cycles ξ ∈ Tzq1(X,n1;m) and η ∈Tzq2(Y, n2;m), we have

∂(ξ ×µ η) = ∂ξ ×µ η + (−1)n1−1ξ ×µ ∂η. (3.4.1)

Proof . (of Proposition 3.0.2) The external product in Definition 3.1.3 is

×µ : Tzq1(X,n1;m)⊗Z Tzq2(Y, n2;m)→ Tzq(X × Y, n;m). (3.4.2)

Suppose that ξ ∈ Tzq1(X,n1;m) and η ∈ Tzq2(Y, n2;m). If ∂ξ = 0 and ∂η = 0, then by Lemma 3.4.1, wehave ∂(ξ ×µ η) = 0. Hence, ×µ maps a pair of cocycles to a cocycle. If ∂ξ = 0, then ξ ×µ ∂η = (−1)n1−1∂(ξ ×µ η).Similarly, if ∂η = 0, then ∂ξ ×µ η = ∂(ξ ×µ η). Hence, ×µ maps a pair consisting of a cocycle and a boundaryto a boundary. Hence, the map ×µ of (3.4.2) descends to

×µ : TCHq1(X,n1;m)⊗Z TCHq2(Y, n2;m)→ TCHq(X × Y, n;m).

The associativity comes from Corollary 3.1.8. To prove the graded-commutativity, by the normalization theorem(Theorem 6.1.2 in Appendix I), we know that every pair of classes ξ ∈ TCHq1(X,n1;m) and η ∈ TCHq2(Y, n2;m)can be represented by a pair of cycles (also denoted by ξ, η) all of whose codimension one faces are trivial. Then,by Lemma 3.3.2, we have (3.3.4). This completes the proof of Proposition 3.0.2.

3.5 Internal product

We now prove the first main result of the paper:

Theorem 3.5.1. Let X be a smooth projective variety over k. Then there exists an internal product on theadditive higher Chow groups of X

∧X : TCHq1(X,n1;m)⊗Z TCHq2(X,n2;m)→ TCHq(X,n;m), (3.5.1)

where qi, ni,m ≥ 1 for i = 1, 2 and q = q1 + q2 − 1, n = n1 + n2 − 1. This is associative and satisfies the equation

ξ∧Xη = (−1)(n1−1)(n2−1)η∧Xξ, (3.5.2)

for all classes ξ ∈ TCHq1(X,n1;m) and η ∈ TCHq2(X,n2;m).This internal product is natural with respect to the pull-back maps of additive higher Chow groups and

satisfies the projection formula

f∗ (a∧Xf∗(b)) = f∗(a)∧Y b (3.5.3)

for a morphism f : X → Y of smooth projective varieties.


Proof . Define ∧X := ∆∗X ◦ ×µ via TCHq(X ×X,n;m), where ×µ is well-defined by Proposition 3.0.2, and the

pull-back ∆∗X exists for additive higher Chow groups by [Krishna and Park, 2012a, Theorem 7.1], which requiresthat X is smooth projective. Associativity follows from Corollary 3.1.8, and (3.5.2) follows from Lemma 3.3.2.

The rest of the proof concerns the naturality and the projection formula. Observe that if Xf−→ Y g−→ Z are

morphisms of smooth projective varieties, then from the contravariance functoriality of the additive higher Chowgroups [Krishna and Park, 2012a, Theorem 7.1], one deduces the naturality of the internal products easily. Notealso that if the projection formulas for f and g hold, then it holds for g ◦ f :

(g ◦ f)∗[a∧X(g ◦ f)∗(b)] = g∗[f∗{a∧Xf∗(g∗(b))}] = g∗[f∗(a)∧Y g∗(b)]

= (g∗ ◦ f∗)(a)∧Zb = (g ◦ f)∗(a)∧Zb.

We remark that the projection formula means

f∗[∆∗X {µ∗(a× f∗(b))}] = ∆∗Y [µ∗ {f∗(a)× b}]. (3.5.4)

Since any given morphism f : X → Y factors as the composite X ↪→ X × Y p2→ Y , it only remains to prove theprojection formula (3.5.4) for closed embeddings and projections, separately.

For any closed embedding f : X → Y , by the moving lemma [Krishna and Park, 2012a, Theorem 1.1], it isenough to prove (3.5.4) for the cycles a ∈ Tzq1(X,n1;m), b ∈ Tzq2(Y, n2;m) for which all pull-backs below makesense. Indeed,

f∗[∆∗X{µ∗(a× ([X] · b))}] =† ∆∗Y [(f × f)∗{µ∗(a× ([X] · b))}]

= ∆∗Y [µ∗{(f × f)∗(a× ([X] · b))}] =‡ ∆∗Y [µ∗{f∗(a)× b}],

where ‡ follows from [Serre, 1965, V-30, (11)] and † follows from the left Cartesian diagram of below:

X∆X−−−−→ X ×Xyf yf×f

Y∆Y−−−−→ Y × Y

X × Y IdX×∆Y−−−−−−→ X × Y × Yyf yf×IdYY

∆Y−−−−→ Y × Y

(3.5.5)

Now, consider the projection f : X × Y → Y . Let p : X × Y ×X × Y → X × Y × Y be the obviousprojection. Let Z := X × Y . Then for cycles a ∈ Tzq1(Z, n1;m) and b ∈ Tzq2(Y, n2;m), whenever all pull-backsbelow make sense, we have

f∗[∆∗Z{µ∗(a× f∗(b))}] = f∗[∆∗Z{µ∗(a× (X × b))}] = f∗[∆∗Z{p∗ (µ∗(a× b))}]

= f∗[(IdX ×∆Y )∗ (µ∗(a× b))]=†∆∗Y [(f × IdY )∗ (µ∗(a× b))]= ∆∗Y [µ∗{(f × IdY )∗(a× b)}] = ∆

∗Y [µ∗{f∗(a)× b)}],

where † follows from the right Cartesian square of (3.5.5) above. By the moving lemma again, the equality thenholds for any additive higher Chow classes. This completes the proof of the projection formula (3.5.4).

Remark 3.5.2. One may question if this product structure ∧X is indeed nontrivial at the level of additive higherChow groups, although may not be apparent from its definition. Here is a way to see it using the projectionformula, just proven in Theorem 3.5.1: for a given smooth projective variety X, choose a closed immersionι : X ↪→ PN for some N > 0. Then, the projection formula ι∗(a ∧X ι∗b) = ι∗(a) ∧PN b reduces the question ofnontriviality to the case of projective spaces. But, for projective spaces, these groups satisfy the projectivebundle formula (see [Krishna and Park, 2012a, Theorem 3.2] cf. [Krishna and Levine, 2008, Theorem 5.6]) sothat the question further reduces to Spec (k), where we know the product is not trivial by [Rülling, 2007]. Anexplicit example on how this works is provided below.

Example 3.1. Let X be a smooth projective curve over a perfect field k of characteristic different from two.We show that the internal product ∧ : TCH1(X, 1; 2)⊗ TCH2(X, 1; 2)→ TCH2(X, 1; 2) is nontrivial.

There is a finite and flat map f : X → P1k. Since the map CH1(X)→ CH1(P1) ' Z is the degree map, it

is surjective. One can now check easily from the projection formula (cf. [Krishna and Levine, 2008, Theorem4.10]) and the projective bundle formula that the map f∗ : TCH

2(X, 1; 2)→ TCH2(P1, 1; 2) is surjective.Now, let π : X → Spec (k) be the structure map. Since the push-forward map TCH2(P1, 1; 2)→

TCH1(k, 1; 2) is surjective (use the projective bundle formula), we conclude that the map π∗ : TCH2(X, 1; 2)→

TCH1(k, 1; 2) is surjective. Using this, Theorem 3.5.1 and the isomorphism k'−→ TCH1(k, 1; 2), one deduces

easily that the pairing TCH1(X, 1; 2)⊗ TCH2(X, 1; 2)→ TCH2(X, 1; 2) is nontrivial.

13

For a smooth projective variety X over k, let TCHn(X;m) =⊕

q≥1 TCHq(X,n+ 1;m) and let

TCH(X;m) =⊕

n≥0 TCHn(X;m). Let CH(X) =⊕

q CHq(X).

Corollary 3.5.3. For a smooth projective variety X, there is an internal product structure∧ : TCH(X;m)⊗CH(X)TCH(X;m)→ TCH(X;m), that makes TCH(X;m) a graded-commutative ring and ‘∧’is bilinear with respect to the CH(X)-module structure on TCH(X;m). The unit of the ring structure is[X × {1}] ∈ TCH1(X, 1;m).

Proof . This follows from Theorem 3.5.1 once we know that the internal product ∧ of Theorem 3.5.1 is bilinearover the ring CH(X), where the CH(X)-module structure on TCH(X;m) is given by [Krishna and Levine, 2008,§4]. For a (Chow) cycle α, the action α · (−) is ∆∗X(α× (−)), if the pull-back is defined. (By the moving lemma,it is defined on a quasi-isomorphic subcomplex.) Since ∧ = ∆∗X ◦ ×µ, where ×µ is as in Definition 3.1.3, to provethe bilinearity α · (ξ ∧ η) = (α · ξ) ∧ η = ξ ∧ (α · η) for a Chow cycle α on X, we need to show that

∆∗X [α×∆∗X{µ∗(ξ × η)}] = ∆∗X [µ∗{∆∗X(α× ξ)× η}] = ∆∗X [µ∗{ξ ×∆∗X(α× η)}], (3.5.6)

whenever the pull-backs ∆∗X are defined on the level of cycles (again, which is true on quasi-isomorphicsubcomplexes of cycles by the moving lemma).

Note that from the commutative diagram

X ×X ×Gmµ←−−−− X ×X ×Gm ×Gmx∆X x∆X

X ×Gmµ←−−−− X ×Gm ×Gm,

(3.5.7)

we have the equality ∆∗X ◦ µ∗ = µ∗ ◦∆∗X . Thus, the second term of (3.5.6) is

∆∗X(µ∗((∆X × Id)∗(α× ξ × η))) = µ∗(∆∗X((∆X × Id)∗(α× ξ × η))), (3.5.8)

and the third term of (3.5.6) is

∆∗X(µ∗((Id×∆X)∗(ξ × α× η))) = µ∗(∆∗X((Id×∆X)∗(ξ × α× η))). (3.5.9)

Thus, the second equality of (3.5.6) holds via the transposition α× ξ × η 7→ ξ × α× η.Treating ξ × η as a higher Chow cycle on X ×X ×Gm ×Gm and pulling back via ∆X , the first term of

(3.5.6) is ∆∗X(µ∗(α×∆∗X(ξ × η))). Hence, by ∆∗X ◦ µ∗ = µ∗ ◦∆∗X , this first term of (3.5.6) is

µ∗(∆∗X((Id×∆X)∗(α× ξ × η))). (3.5.10)

Comparing (3.5.8) with (3.5.10), to prove the first equality of (3.5.6), it is enough to check that ∆∗X ◦ (Id×∆X)

∗ = ∆∗X ◦ (∆X × Id)∗ as compositions of pull-backs of higher Chow cycles. This follows immediately fromthe diagram

X ×X Id×∆X−−−−−→ X ×X ×Xx∆X x∆X×IdX

∆X−−−−→ X ×X.This finishes the proof of (3.5.6), thus the corollary.

As another consequence of Theorem 3.5.1, we get the following result :

Corollary 3.5.4. Let X be a smooth projective k-variety with char(k) 6= 2. Then for any q, n,m ≥ 1, thegroup TCHq(X,n;m) is a Wm(k)-module, where Wm(k) is the big Witt ring of k of length m. In particular,TCHq(X,n;m) is a k-vector space if char(k) = 0.

Proof . This follows from Theorem 3.5.1 by considering the composite map

TCH1(k, 1;m)⊗Z TCHq(X,n;m)f∗⊗Id

//

++

TCH1(X, 1;m)⊗Z TCHq(X,n;m)

∧��

TCHq(X,n;m),

where f : X → Spec (k) is the structure map, and using the isomorphism Wm(k)'−→ TCH1(k, 1;m) of [Rülling,

2007]. That this gives a module structure also follows from Theorem 3.5.1. Finally, in characteristic zero, Wm(k)is itself a k-vector space.


Remark 3.5.5. In characteristic zero, the Witt ring Wm(k) has several k-vector structures so thatTCHq(X,n;m) also inherits several k-vector space structures.

4 Differential operator

We saw in Section 3 that the additive higher Chow groups of a smooth projective variety form a graded-commutative algebra. Our goal in this section is to show (as Theorem 4.3.9) that this algebra is equipped witha differential operator, which turns this algebra into a differential graded algebra (DGA).

4.1 The operator δ

Let X be an equidimensional k-scheme. Let G×m denote the variety Gm\{1}. We have natural inclusions of opensets G×m ↪→ � ↪→ P1. For n ≥ 1, define the map

δn : X ×G×m ×�n−1 → X ×Gm ×�n, (x, t, y1, · · · , yn−1) 7→ (x, t, t−1, y1, · · · , yn−1). (4.1.1)

For any irreducible cycle Z ⊂ X ×Gm ×�n−1, let Z× denote its restriction to the open set X ×G×m ×�n−1.

Lemma 4.1.1. For any irreducible admissible cycle Z ∈ Tzq(X,n;m), δn(Z×) is closed in X ×Gm ×�n.

Proof . It is enough to show that δn is a closed immersion. For this, it is enough to show that the mapG×m → Gm ×�, given by t 7→ (t, t−1), is a closed immersion. But this is obvious because its image is given bythe equation ty = 1, where (t, y) ∈ Gm ×�.

Lemma 4.1.2. For Z as in Lemma 4.1.1, the closed subvariety V := δn(Z×) satisfies the modulus condition.

Proof . Consider the following commutative diagram:

ZN

//

δ̃n��

f++

Z �

//

��

X × P1 × (P1)n−1

δn

��

VN

//

g

33V� // X × P1 × (P1)n,

(4.1.2)

where Z and V are the Zariski closures of Z and V , ZN → Z and V N → V are their normalizations, and f

and g are the compositions. The map δn is induced by the composition P1diag↪→ P1 × (P1) Id×σ→ P1 × (P1), where

σ : P1 → P1 is the isomorphism induced by k[Y0, Y1]→ k[Y0, Y1], Y0 7→ Y1, Y1 7→ Y0. This δn is projective. Here,the dominant map Z× → V gives Z → V , and the induced dominant map δ̃n is consequently projective andsurjective.

From the definition of δn, we have δ∗n(Fn+1,0) = Fn,0. First consider the case n ≥ 2. Then, δ

∗n(F

1n+1,i) =

F 1n,i−1 for i ≥ 2. Thus,

[δ̃∗ng∗(F 1n+1 − (m+ 1)Fn+1,0)] ≥

n∑i=1

[δ̃∗ng∗(F 1n+1,i)]− [δ̃∗ng∗((m+ 1)Fn+1,0)]

=

n∑i=2

[f∗δ∗n(F

1n+1,i)]− [f∗δ

∗n((m+ 1)Fn+1,0)] = [f

∗(F 1n − (m+ 1)Fn,0)] ≥† 0,

where ≥† follows by the modulus condition for Z. We deduce from Lemma 2.4.1 that [g∗(F 1n+1 − (m+1)Fn+1,0)] ≥ 0, which is the modulus condition for V .

In case n = 1, we have δ̃∗1g∗(F2,0) = f

∗δ∗1(F2,0) = f

∗(F1,0) = 0 so that [δ̃∗1g∗(F 12 − (m+ 1)F2,0)] =

[δ̃∗1g∗(F 12 )] ≥ 0. Hence again by Lemma 2.4.1 we have g∗[F 12 − (m+ 1)F2,0] ≥ 0, which is the modulus condition

for V .

Definition 4.1.3. For any irreducible admissible cycle Z ∈ Tzq(X,n;m), we write δn(Z) for δn(Z×), or furtherδ(Z), if the reference to n is apparent or unnecessary. We extend it Z-linearly.

15

Proposition 4.1.4. If Z ∈ Tzq(X,n;m), then δ(Z) ∈ Tzq+1(X,n+ 1;m). Furthermore, δ and ∂ satisfy therelation ∂δ + δ∂ = 0.

Proof . We may assume that Z is irreducible. We first prove the following.Claim : (i) ∂�n+1,1 ◦ δn = 0 for � = 0,∞. (ii) ∂�n+1,i ◦ δn = δn−1 ◦ ∂�n,i−1 for i ≥ 2 and � = 0,∞.

Here ∂�n+1,i is the i-th face ∂�i on Bn+1. For (i), note from the definition of δn that δn(Z) does not intersect

{y1 = 0} in X ×Bn+1 for otherwise it must intersect the divisor {t =∞} nontrivially in X ×Bn+1, but thisdivisor is empty. If the cycle δn(Z) intersects {y1 =∞}, then it must intersect the divisor {t = 0} in X ×Bn+1.But, by Lemma 4.1.2, δn(Z) satisfies the modulus condition so that the cycle possibly intersects the divisor{t = 0} only in X ×A1 × (P1)n, not in X ×Gm ×�n = X ×Bn+1. Hence, ∂�n+1,1δn(Z) = 0.

For (ii), observe from the definition of δn that for i ≥ 2 and � = 0,∞, the diagram

G×m ×�n−2ι�i−1//

δn−1

��

G×m ×�n−1

δn

��

Gm ×�n−1ι�i // Gm ×�n

(4.1.3)

is Cartesian. Hence δn−1((ι�i−1)

∗(Z)) = (ι�i)∗(δn(Z)). This proves the Claim.

Using this and the proper intersection property of Z, we deduce that δ(Z) intersects properly with the faces.We also saw in Lemma 4.1.2 that δ(Z) satisfies the modulus condition. Since δ does not change the dimensionof cycles, we now have δ(Z) ∈ Tzq+1(X,n+ 1;m).

Finally, we have ∂ ◦ δn(Z) =∑n

i=1 (−1)i[∂∞n+1,i ◦ δn(Z)− ∂0n+1,i ◦ δn(Z)]

=†n∑i=2

(−1)i[δn−1 ◦ ∂∞n,i−1(Z)− δn−1 ◦ ∂0n,i−1(Z)]

= −n−1∑i=1

(−1)i[δn−1 ◦ ∂∞n,i(Z)− δn−1 ◦ ∂0n,i(Z)]

= −δn−1

(n−1∑i=1

(−1)i[∂∞n,i(Z)− ∂0n,i(Z)]

)= −δn−1 ◦ ∂(Z),

where the equality † follows from the Claim. This proves the proposition.

Corollary 4.1.5. For every q ≥ 1, the above δ defines a chain map δ : Tzq(X, •;m)→ Tzq+1(X, •+ 1;m).

Proof . It is clear from (4.1.1) that δ preserves the degenerate cycles. The corollary follows thus fromProposition 4.1.4.

Lemma 4.1.6. When X = Spec (k), the operator δ restricted on the zero-cycles coincides with the differentialoperator D of [Rülling, 2007, (3.9.3)] in TCHn(k, n;m).

Proof . The definition of D is equivalent to the following: consider a rational map δ′ : Gm ×�n−1 → Gm ×�ngiven by (t, y1, · · · , yn−1) 7→ (t, t, y1, · · · , yn−1). Then, for a closed point p on G×m ×�n−1, define D(p) := −δ′(p).So, it is enough to show that δn(p) + δ′(p) is the boundary of an admissible 1-cycle.

We first consider the case where p is a k-rational point, say p = (s, z1, · · · , zn−1) with s 6= 0, zi 6= 1 ∈ k. Then,we consider the parameterized admissible 1-cycle γ : (y 7→ (s, y, (sy − 1)/(y − 1), z1, · · · , zn−1) ∈ Gm ×�n+1,whose boundaries are ∂

∞1 (γ) = (s, s, z1, · · · , zn−1) = δ′(p), ∂01(γ) = 0,∂∞2 (γ) = 0, ∂

02(γ) = (s, 1/s, z1, · · · , zn−1) = δn(p),

∂�i (γ) = 0, for i ≥ 3, � ∈ {0,∞}.(4.1.4)

Hence, ∂(−γ) = δ′(p) + δn(p), as desired.Now suppose p is not a k-rational point. In this case, p may not be of the form (s, z1, · · · , zn−1) over k.

Nevertheless, by the good position assumption, the point p lies in none of the codimension 1 face {yi = �}, where� = 0,∞, so that we have a closed immersion ι : Spec (κ(p))→ G×m × (P1\{0, 1,∞})n−1 = G×m × (G×m)n−1. Thisgives n values s, z1, · · · zn−1 ∈ κ(p)\{0, 1} with κ(p) = k(s, z1, · · · , zn−1) such that there is a (not necessarilyunique) factorization of ι

Spec (κ(p))ιp→ G×m,κ(p) ×κ(p) (G

×m,κ(p))

n−1 π→ G×m × (G×m)n−1,


where π is induced by the map π : Spec (κ(p))→ Spec (k), and ιp sends the singleton to the κ(p)-rational point(s, z1, · · · , zn−1) (call it p′) of G×m,κ(p) ×κ(p) (G

×m,κ(p))

n−1. Then, by the first case, with the base field k replaced

by κ(p), we have −δ′(p′) ≡ δn(p′) in TCHn(κ(p), n;m). Since π∗(p′) = p as 0-cycles and π∗ obviously commuteswith δn and δ

′, by the existence of the trace push-forward π∗ : TCHn(κ(p), n;m)→ TCHn(k, n;m) of [Rülling,

2007, Lemma 3.18], we have −δ′(p) ≡ δn(p) in TCHn(k, n;m).

4.2 Computation of δ2

Now suppose that X is a smooth quasi-projective variety. Note that δ2(Z) = δ(δ(Z)) is a nonzero cycle. Ournext goal is to show that δ2 is zero modulo a boundary. We consider certain cycles in z2(Gm, 3) that are twodimensional variants of some 1-cycles in [Totaro, 1992]. For t ∈ Gm, we let u := t−1.

For 1 ≤ j ≤ 5, and l = 1, 2, let Γlj ⊂ Gm ×�3 be the 2-cycles defined by the rational maps ψlj : Gm ×�→Gm ×�3 given as follows: for t ∈ Gm\{1,−1} and x ∈ �\{0}

ψl1(t, x) =(t, ul, x, (1−u

l)x−(1−ul)/(1−tl)x−(1−ul)/(1−tl)

),

ψl2(t, x) =(t, ul, x, (1−t

l)x−1x−1

), ψl3(t, x) =

(t, x, u

lx−1x−1 , 1− t

),

ψl4(t, x) =(t, x, 1− x, u

l−x1−x

), ψl5(t, x) =

(t, x, 1− x, t

l−x1−x

).

(4.2.1)

For an admissible irreducible 1-cycle α ⊂ Gm ×�2 defined by a rational map

ϕ : Gm → Gm ×�2, ϕ(t) = (ϕ(t)(0), ϕ(t)(1), ϕ(t)(2)), (4.2.2)

we often regard α as the parameterization (ϕ(t)(0), ϕ(t)(1), ϕ(t)(2)) to simplify the notations.It is easy to check from the definitions that all Γlj are closed in Gm ×�3 and they define admissible cycles

in z2(Gm, 3). One can also check in a straightforward way that they have the following boundaries:

∂Γl1 =(t, ul, 1− ul

)−(t, ul, (1− ul)/(1− tl)

)+(t, ul, 1/(1− tl)

),

∂Γl2 =(t, ul, 1− tl

)+(t, ul, 1/(1− tl)

), ∂Γl3 = −

(t, ul, 1− tl

)−(t, tl, 1− tl

),

∂Γl4 =(t, ul, 1− ul

), ∂Γl5 =

(t, tl, 1− tl

).

(4.2.3)

Since u = t−1, note that 1−ul

1−tl =1−t−l1−tl =

tl−11−tl · t

−l = −t−l = −ul. Hence, we have(t, ul,−ul

)=(

t, ul, (1− ul)/(1− tl)). Hence, using (4.2.3) one checks that for l = 1, 2,

(t, ul,−ul) = ∂Γl, Γl := −Γl1 + Γl2 + Γl3 + Γl4 + Γl5. (4.2.4)

As in (4.2.1), we consider two additional 2-cycles Γ26 and Γ27 ⊂ Gm ×�3 associated to the rational maps ψ26 and

ψ27 : Gm ×�→ Gm ×�3 given by

ψ26(t, x) =(t, x, ux−u

2

x−u2 ,−u2), ψ27(t, x) =

(t, u, x, ux+u

2

x+u2

), (4.2.5)

and their boundaries are

∂Γ26 = −2(t, u,−u2) + (t, u2 − u2), ∂Γ27 = (t, u, u)− (t, u,−u2) + (t, u,−u). (4.2.6)

Combining (4.2.4) and (4.2.6), we obtain in z2(Gm, 2)

2(t, u, u) = ∂Γ, Γ := 2Γ1 + Γ2 − Γ26 + 2Γ27. (4.2.7)

Given an admissible cycle Z ∈ Tzq(X,n;m), we can regard it as a higher Chow cycle in zq(X ×Gm, n− 1).For any Γlj ∈ z2(Gm, 3), we get the exterior product Γlj × Z ∈ zq+2(X ×Gm ×Gm, n+ 2). Let

∆Gm : X ×Gm ×�n+1 → X ×Gm ×Gm ×�n+1 (4.2.8)

∆Gm(x, t, x1, x2, y1, · · · , yn−1) = (x, t, t, x1, x2, y1, · · · , yn−1)

be the diagonal map.From the definitions of the cycles, Γlj × Z intersects X ×Gm ×�n+1 properly under the closed embedding

∆Gm . Since X is smooth, we can pull-back the cycles to define Γlj ? Z := ∆

∗Gm(Γ

lj × Z) in zq+2(X ×Gm, n+ 2).

17

Lemma 4.2.1. The cycle Γlj ? Z lies in Tzq+2(X,n+ 3;m) under the natural inclusion Tzq+2(X,n+ 3;m) ↪→

zq+2(X ×Gm, n+ 2).

Proof . It is easy to check that V ′ := |Γlj ? Z| is the closure of the image of Z ×� under the rational map

Ψlj : X ×Gm ×�n → X ×Gm×,�n+2 (4.2.9)

Ψlj(x, t, y, y1, · · · , yn−1) = (x, ψlj(0), ψlj(1), ψlj(2), ψlj(3), y1, · · · , yn−1),

in the notation of (4.2.2).We only need to show that V ′ satisfies the modulus condition. We follow the proof of Lemma 4.1.2 to prove

the modulus condition for V ′. Consider the following commutative diagram

ZN × P1

Ψ̃lj��

f ′// X × P1 × P1 × (P1)n−1

Ψlj

��

V′N g′

// X × P1 × (P1)n+2.

(4.2.10)

Here f ′ = f × Id, where f : ZN → X × P1 × (P1)n−1 is the normalization map for Z as in (4.1.2). Note thatthe map Ψ

l

j is defined since the rational maps ψlj naturally extend to morphisms ψ

lj : P1 × P1 → P1 × (P1)

3.

Since Z satisfies the modulus condition, we have [f∗(F 1n)− (m+ 1)f∗{t = 0}] ≥ 0 on ZN

. As f ′ is

identity on P1, this implies that [∑n+1

i=3 f′∗{yi = 1} − (m+ 1)f ′∗{t = 0}] ≥ 0 on Z

N. Since Ψ

l

j is identity on

the last (n− 1)-copies of P1, we conclude that f ′∗ ◦Ψlj∗[∑n+1

i=3 F1n+3,i − (m+ 1)Fn+3,0] ≥ 0, which in turn

yields (Ψ̃lj)∗ ◦ g′∗[

∑n+1i=3 F

1n+3,i − (m+ 1)Fn+3,0] ≥ 0 on Z

N × P1. Now we deduce from Lemma 2.4.1 thatg′∗[∑n+1

i=3 F1n+3,i − (m+ 1)Fn+3,0] ≥ 0. In particular, g′

∗[F 1n+3 − (m+ 1)Fn+3,0] ≥ 0. Hence, V ′ (thus Γlj ? Z)

satisfies the modulus condition.

Our main interest for δ2 is the following.

Proposition 4.2.2. Let X be a smooth quasi-projective variety over a perfect field of char(k) 6= 2 and letα ∈ Tzq(X,n;m) be such that ∂(α) = 0. Then δ2(α) = 0 as a class in TCHq+2(X,n+ 2;m). In particular,δ descends to a map of additive higher Chow groups δ : TCHq(X,n;m)→ TCHq+1(X,n+ 1;m) such thatδ2 = 0.

Proof . The last part of the proposition follows from Corollary 4.1.5 once we prove the first part. We begin withthe following.Claim : For any α ∈ Tzq(X,n;m) and Γ as in (4.2.7), one has ∂(Γ× α) = ∂Γ× α− Γ× ∂α in zq+2(X ×Gm ×Gm, n+ 2).

This is easy; since Γ is a Z-linear combination of Γlj ’s, it suffices to prove the claim for each Γlj . Notethat for � ∈ {0,∞}, ∂�i (Γlj × α) = ∂�i (Γlj)× α if 1 ≤ i ≤ 3, and Γlj × ∂�i−3(α) if 4 ≤ i ≤ n+ 2. From this, westraightforwardly deduce ∂(Γlj × α) = ∂(Γlj)× α− Γlj × ∂α. This proves the claim.

Now, from the definition of δn in (4.1.1), for any irreducible admissible cycle Z ∈ Tzq(X,n;m), δ2(Z) isthe image of Z under the rational map

X ×Gm ×�n−1 → X ×Gm ×�n+1 (4.2.11)

(x, t, y1, · · · , yn−1) 7→ (x, t, t−1, t−1, y1, · · · , yn−1) = (x, t, u, u, y1, · · · , yn−1).

It follows from (4.2.11) and (4.2.8) that δ2(Z) is an irreducible admissible cycle which is the scheme-theoretic inverse image of the irreducible cycle (t, u, u)× Z ⊂ X ×G2m ×�n+1 (see (4.2.7)) under the closedimmersion ∆Gm .

Since the diagram

Gm ×�n+1ι�i //

∆Gm��

Gm ×�n+2

∆Gm��

Gm ×Gm ×�n+1ι�i // Gm ×Gm ×�n+2


is Cartesian, for any α ∈ Tzq(X,n;m) with ∂(α) = 0, we get in zq+2(X ×Gm, n+ 1):

2δ2(α) = 2(∆∗Gm((t, u, u)× α)) = ∆∗Gm(2(t, u, u)× α)) =

† ∆∗Gm(∂Γ× α)= ∆∗Gm (∂Γ× α− Γ× ∂α) =

‡ ∆∗Gm (∂(Γ× α)) = ∂(∆∗Gm(Γ× α)

)= ∂ (Γ ? α) ,

where † holds by (4.2.7), and ‡ holds by the Claim. Since Γ ? α ∈ Tzq+2(X,n+ 3;m) by Lemma 4.2.1, we concludethat 2δ2(α) = 0 as a class in TCHq+2(X,n+ 2,m). Since char(k) 6= 2, we conclude from Corollary 3.5.4 thatthe class of δ2(α) in TCHq+2(X,n+ 2,m) is zero.

4.3 Leibniz rule

We now show that the differential δ is a graded derivation for the internal product ∧ on the additive higherChow cycle classes.

Let X be a smooth quasi-projective k-variety, let (x, t1, t2, y1, · · · , yn−1, y, λ) be the coordinates ofX ×B2,n+3. Let TX be the closed subscheme of X ×B2,n+3 defined by the equation t1y − 1 = λ(t1t2y − 1),and let TX be its Zariski closure in X ×B2,n+3.

Definition 4.3.1. Let Z ⊂ X ×B2,n+1 be an irreducible closed subvariety. Define CZ := TX · (Z ×�2) onX ×B2,n+3. Extend it to all cycles Z-linearly.

Lemma 4.3.2. Let m1,m2 ≥ 1 be integers. Let Z be a cycle on X ×B2,n+1 satisfying the modulus (m1,m2)condition. Then, CZ satisfies the modulus (m1,m2) condition on X ×B2,n+3.

Proof . We may assume that Z is irreducible. Let Z and CZ be the Zariski closures of Z and CZ in X ×B2,n+1and X ×B2,n+3, respectively. Then, we have the commutative diagram

CN

Z

prN

��

νC // CZ� ιC //

pr

��

X ×B2,n+3

pr

��

ZN νZ // Z

� ιZ // X ×B2,n+1

where pr : X ×B2,n+3 → X ×B2,n+1 is the projection that ignores the last two coordinates y, λ, the maps νZ , νCare the normalizations, and the dominant maps pr, prN are induced by pr. Since Z satisfies the modulus (m1,m2)

condition, we have∑2

j=1(mj + 1)[ν∗Zι∗Z{tj = 0}] ≤

∑n−1i=1 [ν

∗Zι∗Z{yi = 1}]. Then, by pulling back via prN and

using the commutativity of the diagram, we have∑2

j=1(mj + 1)[ν∗Cι∗Cpr

∗{tj = 0}] ≤∑n−1

i=1 [ν∗Cι∗Cpr

∗{yi = 1}].But pr∗{tj = 0} = {tj = 0} for j = 1, 2 and pr∗{yi = 1} = {yi = 1} for 1 ≤ i ≤ n− 1. Hence,

2∑j=1

(mj + 1)[ν∗Cι∗C{tj = 0}] ≤

n−1∑i=1

[ν∗Cι∗C{yi = 1}] ≤

n+1∑i=1

[ν∗Cι∗C{yi = 1}],

which is the modulus (m1,m2) condition for CZ .

Corollary 4.3.3. Let m1,m2 ≥ 1. Let Z be a cycle on X ×B2,n+1 satisfying the modulus (m1,m2) condition.Then, µ∗(CZ) as in Definition 3.1.3 on X ×Bn+2 is well-defined, and it satisfies the modulus min{m1,m2}condition.

Proof . By Lemma 4.3.2, CZ satisfies the modulus (m1,m2) condition so that µ∗(CZ) satisfies the modulusmin{m1,m2} condition by Proposition 3.1.6.

Definition 4.3.4. n = n1 + n2 − 1. Let Zi ∈ Tzqi(X,ni;m) for i = 1, 2. Let Z1 ×µ′ Z2 be the cycle σ ·µ∗(CZ1×Z2) on X ×X ×Bn+2, where σ = (n+ 1, n, · · · , 1)2 is the square of the cyclic permutation on{1, 2, · · · , n+ 1}.

Remark 4.3.5. One can describe Z1 ×µ′ Z2 as follows: consider the rational map

µ′ : X ×X ×Gm ×Gm ×�n−1 ×�→ X ×X ×Gm ×�n+1 (4.3.1)

µ′ (x1, x2, t1, t2, y1, · · · , yn−1, y) = (x1, x2, t1t2, y,t1y − 1t1t2y − 1

, y1, · · · , yn−1).

Then, set-theoretically Z1 ×µ′ Z2 is equal to the Zariski closure of µ′(Z1 × Z2 ×�).

19

Remark 4.3.6. The above product ×µ′ is motivated by the computation in [Park, 2007, Lemma 2.5].For t1, t2 ∈ Gm, let Ct1,t2 := {t1}×µ′{t2} ⊂ Gm ×�2, which is the parameterized curve (y ∈ k) 7→(t1t2, y, (t1y − 1)/(t1t2y − 1)) ∈ Gm ×�2. This is an admissible 1-cycle and its boundary is given (loc.cit.) by∂Ct1,t2 = −(t1t2, 1/t1)− (t1t2, 1/t2) + (t1t2, 1/t1t2).

Let n = n1 + n2 − 1 and q = q1 + q2 − 1.

Proposition 4.3.7. Z1×µ′Z2 is an admissible cycle in Tzq+1(X ×X,n+ 2;m).

Proof . By Corollary 4.3.3, the cycle Z1 ×µ′ Z2 satisfies the modulus m condition.We now prove that Z ′ := Z1 ×µ′ Z2 intersects properly with all faces. It is easy to see from Definition 4.3.4

and (4.3.1) that when σn1 is the permutation (n1, n1 − 1, · · · , 2, 1), we have∂∞1 (Z

′) = σn1 · (Z1 ×µ δ(Z2)) , ∂01(Z ′) = 0,∂∞2 (Z

′) = δ (Z1 ×µ Z2) , ∂02(Z ′) = δ(Z1)×µ Z2,

∂�i (Z′) =

{∂�i−2(Z1)×µ′Z2, if 3 ≤ i ≤ n1 + 1,Z1×µ′∂�i−n1−1(Z1), if n1 + 2 ≤ i ≤ n+ 1.

(4.3.2)

Since Zi’s are admissible, this shows the proper intersection property of Z′, by Corollary 3.1.7, Proposition 4.1.4,

and induction on the codimension of the faces.

By Proposition 4.3.7, we have the Z-bilinear map

×µ′ : Tzq1(X,n1;m)⊗ Tzq2(X,n2;m)→ Tzq+1(X ×X,n+ 2;m). (4.3.3)

Proposition 4.3.8 (Leibniz rule). Let X be a smooth quasi-projective variety, and let ξ ∈ Tzq1(X,n1;m), η ∈Tzq2(X,n2;m). Let n = n1 + n2 − 1, q = q1 + q2 − 1. Then, in the group Tzq+1(X ×X,n+ 1;m), we have

δ(ξ ×µ η)− δξ ×µ η − (−1)n1−1ξ ×µ δη = ∂(ξ ×µ′ η)− ∂ξ ×µ′ η − (−1)n1−1ξ ×µ′ ∂η + ∂γ (4.3.4)

where γ is an admissible cycle as in Lemma 3.3.1, up to sign.

Proof . It follows from (4.3.2) that

∂ (ξ ×µ′ η) =n+1∑i=1

(−1)i(∂∞i − ∂0i ) (ξ ×µ′ η)

= δ (ξ ×µ η)− {σn1 · (ξ ×µ δη) + δξ ×µ η}+n1+1∑i=3

(−1)i{(∂∞i−2 − ∂0i−2)(ξ)}×µ′η

+n+1∑

i=n1+2

(−1)iξ×µ′{(∂∞i−n1−1 − ∂0i−n1−1)(η)}

= δ (ξ ×µ η)− {σn1 · (ξ ×µ δη) + δξ ×µ η}+ (∂ξ)×µ′ η + (−1)n1−1ξ×µ′(∂η).

This is equivalent to the identity

δ (ξ ×µ η)− δξ ×µ η − σn1 · (ξ ×µ δη) = ∂ (ξ ×µ′ η)− ∂ξ×µ′η − (−1)n1−1ξ×µ′∂η. (4.3.5)

But, by Lemma 3.3.1, we have σn1 · (ξ ×µ δη) = (−1)n1−1ξ ×µ δη + ∂γ for an admissible cycle γ. Hence, wededuce (4.3.4) as desired.

We put together all pieces so far to conclude another main result of this paper.

Theorem 4.3.9. Let X be a smooth projective variety over a perfect field k of char 6= 2. Then, the additivehigher Chow groups (TCH(X;m),∧, δ) form a graded-commutative differential graded algebra, where δ is thegraded derivation for the internal product ∧. The derivation commutes with the pull-back and push-forwardmaps of additive higher Chow groups.

Proof . This follows from Theorem 3.5.1 and Proposition 4.3.8.

Remark 4.3.10. (cf. Remark 3.0.3) If the moving lemma [Krishna and Park, 2012a, Theorem 4.1] holds alsofor smooth quasi-projective varieties, then Theorem 4.3.9 extends to all smooth quasi-projective varieties aswell.


5 Witt-complex structure

So far, we considered mostly additive higher Chow groups with a fixed modulus m ≥ 1. In this section, weconsider the projective system of additive higher Chow groups over all modulus m ≥ 1, and we show that forsmooth projective varieties, they form a restricted Witt-complex over k. Recall:

Definition 5.0.11 ([Rülling, 2007, Definition 1.14], [Hesselholt and Madsen, 2001, Definition 1.1.1]). Let Abe a commutative ring with unity. A restricted Witt-complex over A is defined to be a projective system ofdifferential graded Z-algebras

((Em)m∈N, R : Em+1 → Em),

together with families of homomorphisms of graded rings

(Fr : Erm+r−1 → Em)m,r∈N,

and homomorphisms of graded groups

(Vr : Em → Erm+r−1)m,r∈N,

satisfying the following relations for all n, r ∈ N:

(i) RFr = FrRr, RrVr = VrR, F1 = V1 = Id, FrFs = Frs, VrVs = Vrs.

(ii) FrVr = r. When (r, s) = 1, then FrVs = VsFr on Erm+r−1.(iii) Vr(Fr(x)y) = xVr(y) for all x ∈ Erm+r−1 and y ∈ Em.(iv) FrdVr = d (where d denotes the differential of the DGAs).

Furthermore, there is a homomorphism of projective systems of rings (λ : Wm(A)→ E0m)m∈N that commuteswith Fr and Vr, and satisfies

(v) Frdλ([a]) = λ([a]r−1)dλ([a]) for all a ∈ A and r ∈ N.

Here, Wm(A) is the Witt ring of A associated to the truncation set {1, 2, · · · ,m}, and [a] is the Teichmüller liftof a.

The category of restricted Witt-complexes over A has an initial object given by the projective system(WmΩ•A) of de Rham-Witt complexes of [Hesselholt and Madsen, 2001], and when A is any field k of characteristic6= 2, it is proven in [Rülling, 2007] that this projective system is realized by the collection of additive higherChow groups of zero cycles for X = Spec (k). Our goal is to study its higher dimensional analogue, Theorem5.3.1: when X is smooth projective, the system of DGAs {TCH(X;m)}m≥1 has a structure of a restrictedWitt-complex over k.

5.1 Frobenius operators

For r ≥ 1, consider the r-fold product φr : Gm → Gm, a 7→ ar. This naturally extends to φr : A1 → A1. In bothcases, the maps φr are finite and flat. For each equidimensional k-scheme X, we have the induced finite flatmorphisms X ×Gm → X ×Gm and X ×Bn → X ×Bn, also denoted by φr.

Lemma 5.1.1. Let Z ∈ Tzq(X,n; rm+ r − 1) be an irreducible admissible cycle. Then, the push-forwardφr∗(Z) as an abstract algebraic cycle is admissible, and lies in Tz

q(X,n;m).

Proof . The morphism φr : X ×Gm → X ×Gm is finite. Hence, the cycle φr∗(Z) is in zq(X ×Gm, n− 1), i.e.,it is in good position with all faces.

To show that φr∗(Z) satisfies the modulus m condition, it is enough to show that the image W := φr(Z)satisfies the modulus condition. (N.B. It is not a corollary of [Krishna and Levine, 2008, §3.3].) Let Z, W bethe Zariski closures of Z and W in X ×Bn, and let νZ and νW be the normalizations of the closures. Then, wehave a commutative diagram

ZN νZ //

φNr��

Z � ιZ //

φr��

X ×Bn

φr

��

WN νW // W �

ιW // X ×Bn.

Here, let Dm :=∑n−1

i=1 {yi = 1} − (m+ 1){t = 0}. Then, φ∗rD =∑n−1

i=1 {yi = 1} − r(m+ 1){t = 0} =Drm+r−1. By the given assumption, we have [ν

∗Zι∗ZDrm+r−1] ≥ 0, which is equivalent to [(φNr )∗ν∗W ι∗WDm] ≥ 0. By

Lemma 2.4.1, we have [ν∗W ι∗WDm] ≥ 0, i.e., W satisfies the modulus m condition. This completes the proof.

21

Definition 5.1.2. For r ≥ 1, let Fr = φr∗. We call it the r-th Frobenius on additive higher Chow cycles. Onenotes immediately that Fr commutes with the operations of intersecting with the faces so that Fr gives amorphism of complexes Fr : Tz

q(X, •; rm+ r − 1)→ Tzq(X, •;m) and a homomorphism Fr : TCHq(X,n; rm+r − 1)→ TCHq(X,n;m).

Lemma 5.1.3. Let f : X → Y be a morphism of quasi-projective varieties over k, where Y is smooth andprojective. Then, the pull-backs f∗ : TCHq(Y, n;m)→ TCHq(X,n;m) constructed in [Krishna and Park, 2012a,Theorem 7.1] are compatible with the Frobenius map Fr for each r ≥ 1.

Proof . From the proof of [Krishna and Park, 2012a, Theorem 7.1], for some finite collectionW of locally closed

subsets of Y , we have chain maps TzqW(Y, •;m)pr∗2→ Tzq{Γf}(X × Y, •;m)

gr∗f→ Tzq(X, •;m), whose compositiongives f∗. Here, TzqW(Y, n;m) is defined in [Krishna and Park, 2012a, Definition 4.2]. Since all of Fr, pr

∗2 , gr

∗f

are equivariant with respect to taking faces, we need to check that the following diagram commutes, wheremr := rm+ r − 1:

TzqW(Y, n;mr)pr∗2 //

Fr

��

Tzq{Γf}(X × Y, n;mr)

Fr

��

gr∗f// Tzq(X,n;mr)

Fr

��

TzqW(Y, n;m)pr∗2 // Tzq{Γf}(X × Y, n;m)

gr∗f// Tzq(X,n;m).

But the equality Fr ◦ pr∗2 = pr∗2 ◦ Fr just holds by definition, as long as the flat-pull-back pr∗2 and finite push-forward Fr are well-defined, which is the case. (One can also deduce it from [Fulton, 1998, Proposition1.7] regarding the cycles as Chow cycles.) Similarly, we have the equality Fr ◦ gr∗f = gr∗f ◦ Fr since gr∗f isthe well-defined Gysin pull-back, by [Krishna and Park, 2012a, Corollary 7.2]. Taking the homology groupsand using the moving lemma [Krishna and Park, 2012a, Theorem 4.1], we obtain the desired compatibilityFr ◦ f∗ = f∗ ◦ Fr.

Lemma 5.1.4. Let X and Y be equidimensional k-schemes. Then, the external product ×µ is compatible withthe Frobenius Fr for each r ≥ 1. More precisely, the following diagram commutes, where mr := rm+ r − 1:

TCH(X;mr)⊗ TCH(Y ;mr)×µ−−−−→ TCH(X × Y ;mr)yFr⊗Fr yFr

TCH(X;m)⊗ TCH(Y ;m) ×µ−−−−→ TCH(X × Y ;m).

Proof . This immediately follows from Theorem 3.5.1 together with the fact that (ab)r = ar · br for a, b ∈Gm.

Corollary 5.1.5. Let X be a smooth projective variety over k. Then, the internal product ∧ is compatiblewith the Frobenius map Fr for each r ≥ 1. In particular, Fr is a graded ring homomorphism with respect to theinternal product ∧.

Proof . The statement is equivalent to the commutativity of the outer rectangle of the following diagram, wheremr := rm+ r − 1:

TCH(X;mr)⊗ TCH(X;mr)×µ

//

Fr⊗Fr��

TCH(X ×X;mr)

Fr

��

∆∗ // TCH(X;mr)

Fr

��

TCH(X;m)⊗ TCH(X;m)×µ

// TCH(X ×X;m) ∆∗// TCH(X;m).

But, by Lemma 5.1.3 the right square is commutative and by Lemma 5.1.4 the left square is commutative.Hence, the outer rectangle is commutative, as desired.


5.2 Verschiebung operators

We now define the Verschiebung operators. Let X be an equidimensional k-scheme, and let r ≥ 1 and φr be asin Section 5.1.

Lemma 5.2.1. Let Z ∈ Tzq(X,n;m) be an irreducible admissible cycle. Then, the pull-back φ∗r(Z) as anabstract algebraic cycle is admissible, thus lies in Tzq(X,n; rm+ r − 1).

Proof . The morphism φr : X ×Gm → X ×Gm is flat. Hence, the cycle φ∗r(Z) is in zq(X ×Gm, n− 1), i.e., itis in good position with all faces.

It remains to show that φ∗r(Z) satisfies the modulus (rm+ r − 1) condition. (N.B. It is not a corollaryof [Krishna and Levine, 2008, §3.4].) Let W be an irreducible component of φ∗r(Z). It is enough to prove themodulus condition for W . Let W and Z be the Zariski closures of W and Z in X ×Bn, and let νW and νZ bethe normalizations of the closures. Then, we have a commutative diagram

WN

φNr��

νW // W � ιW //

φr��

X ×Bn

φr

��

ZN νZ // Z �

ιZ // X ×Bn.

Here, let Dm :=∑n−1

i=1 {yi = 1} − (m+ 1){t = 0}. By the modulus m condition satisfied by Z, we have[ν∗Zι

∗ZDm] ≥ 0. Hence, [(φNr )∗ν∗Zι∗ZDm] ≥ 0. This is equivalent to [ν∗W ι∗Wφ∗rDm] ≥ 0. But, φ∗rDm = Drm+r−1

as seen in the proof of Lemma 5.1.1, so that W satisfies the modulus (rm+ r − 1) condition, as desired.

Definition 5.2.2. For r ≥ 1, let Vr = φ∗r . We call it the r-th Verschiebung on additive higher Chow cycles. Onenotes immediately that Vr commutes with intersections with the faces so that Vr gives a morphism of complexesVr : Tz

q(X, •,m)→ Tzq(X, •, rm+ r − 1) and a homomorphism Vr : TCHq(X,n;m)→ TCHq(X,n; rm+ r −1).

Lemma 5.2.3. Let f : X → Y be a morphism of quasi-projective varieties over k, where Y is smooth projective.Then, the pull-backs f∗ : TCHq(Y, n;m)→ TCHq(X,n;m) constructed in [Krishna and Park, 2012a, Theorem7.1] are compatible with the Verschiebung map Vr for each r ≥ 1.

Proof . As in the proof of Lemma 5.1.3, by [Krishna and Park, 2012a, Theorem 7.1], for some finite collectionW

of locally closed subsets of Y , we have chain maps TzqW(Y, •;m)pr∗2→ Tzq{Γf}(X × Y, •;m)

gr∗f→ Tzq(X, •;m) whosecomposition gives f∗. Since all of Vr, pr

∗2 , gr

∗f are equivariant with respect to taking faces, it is enough to check

that the following diagram commutes, where mr := rm+ r − 1:

TzqW(Y, n;m)pr∗2 //

Vr

��

Tzq{Γf}(X × Y, n;m)

Vr

��

gr∗f// Tzq(X,n;m)

Vr

��

TzqW(Y, n;mr)pr∗2 // Tzq{Γf}(X × Y, n;mr)

gr∗f// Tzq(X,n;mr).

But, the equalities Vr ◦ pr∗2 = pr∗2 ◦ Vr and Vr ◦ gr∗f = gr∗f ◦ Vr just hold by definition as long as all pull-backsin the expressions make sense, which is the case, because pr2 is flat and gr

∗f is the Gysin pull-back of [Krishna

and Park, 2012a, Corollary 7.2]. Taking the homology groups and using the moving lemma [Krishna and Park,2012a, Theorem 4.1], we obtain the desired compatibility Vr ◦ f∗ = f∗ ◦ Vr.

The following observations about the Frobenius and the Verschiebung maps in the positive characteristicwill be useful in proving some of the axioms of the Witt-complex structure on the additive higher Chow groups.

Lemma 5.2.4. Assume that the ground field k is of characteristic p > 0 and let Z ( X ×Gm ×�n−1 be anintegral closed subvariety representing an element [Z] ∈ Tzq(X,n;m), where X is any smooth quasi-projectivevariety over k. Let Z ′ denote the scheme theoretic inverse image of Z under the map φp. Then, Z

′ is irreducibleand one of the following holds:

(i) Z ′ is reduced. In this case, [κ(Z ′) : κ(Z)] = [κ(δ(Z ′)) : κ(φp ◦ δ(Z ′))] = p and φ∗p([Z]) = [Z ′].(ii) Z ′ is non-reduced. In this case, [κ(Z ′red) : κ(Z)] = [κ(δ(Z

′red)) : κ(φp ◦ δ(Z ′red))] = 1 and φ∗p([Z]) = p[Z ′red].

23

In both cases, one has [κ(φp ◦ δ(Z ′red)) : κ(χn+1,p ◦ φp ◦ δ(Z ′red))] = 1.

Proof . Since k is of characteristic p > 0, it follows easily from the definition of φp that Z′ is irreducible,

and it is either reduced, or defined by the p-th power of the ideal defining Z ′red in X ×Gm ×�n−1. One hasφ∗p([Z]) = [Z

′] in the first case and φ∗p([Z]) = p[Z′red] in the second. Since Fp ◦ Vp = p (see Claim (ii) of the proof

of Theorem 5.3.1, Part 1),we must have [κ(Z ′) : κ(Z)] = p in the first case and [κ(Z ′red) : κ(Z)] = 1 in the second.To check the degrees of the other function field extensions, we use the commutative squares of finite maps

(cf. (5.5.1))

Z ′redδ //

φp

��

δ(Z ′red)

χn+1,p◦φp��

δ(Z ′red)pr//

φp

��

Z ′red

φp

��

Zδ // δ(Z) φp ◦ δ(Z ′red)

pr// Z.

(5.2.1)

Here, the horizontal maps in the right square are induced by the projection map X ×Gm ×�n → X ×Gm ×�n−1 which ignores the first box. The assertions about the degrees of the other field extensions follow easilyfrom the commutativity of the above squares and the fact that the horizontal maps in the two squares areisomorphisms.

Given a finite map f : X → Y of smooth quasi-projective varieties over a field k and a cycle γ =d∑i=1

miVi ∈

Tzq(X,n;m), let f(γ) ∈ Tzq(X,n;m) be the cycle f(γ) :=d∑i=1

mif(Vi).

Lemma 5.2.5. Let X be a smooth quasi-projective variety over a field k of characteristic p > 0. Let

γ =d∑i=1

miVi ∈ Tzq(X,n;m) be a cycle such that χn,p(γ) = 0. Then γ = 0.

Proof . This is an immediate consequence of the fact that χn,p : Vi → χn,p(Vi) is bijective for each i and thatTzq(X,n;m) is a free abelian group on integral admissible cycles on X ×Gm ×�n−1.

Lemma 5.2.6. Let X be a smooth quasi-projective variety over a field k of characteristic p > 0. Given

a cycle γ =d∑i=1

miVi ∈ Tzq(X,n;m), let ξ =d∑i=1

miφp(Vi). Assume that ∂(ξ) = 0. Then, ∂0,∞n+1,1(φpδ(γ)) = 0 =

∂(φpδ(γ)).

Proof . We have shown in the proof of Proposition 4.1.4 that (δ(Vi)) ∩ {y1 = 0,∞} = ∅. Since the restrictionto a face of �n commutes with φp, we see that (φpδ(Vi)) ∩ {y1 = 0,∞} = ∅. Since χ−1n+1,p(�) = � for � =0,∞, we also get (χn+1,pφpδ(Vi)) ∩ {y1 = 0,∞} = ∅. We conclude for each i ≥ 1 that ∂0,∞n+1,1(φpδ(Vi)) = 0 =∂0,∞n+1,1(χn+1,pφpδ(Vi)). In particular, we get

∂0,∞n+1,1(φpδ(γ)) = 0 = ∂0,∞n+1,1(χn+1,pφpδ(γ)). (5.2.2)

On the other hand, we know that

0 = −δ∂(ξ) = ∂(δ(ξ)) = ∂(δφp(γ)) = ∂(χn+1,pφpδ(γ)),

where the second and the fourth equalities follow from Proposition 4.1.4 and the diagram (5.5.1) below,respectively. We conclude from (5.2.2) that

∑ni=2(−1)i(∂∞i − ∂0i )(χn+1,pφpδ(γ)) = 0. As ∂�iχn+1,p = χn+1,p∂�i

for 2 ≤ i ≤ n and � = 0,∞, we get χn+1,p(∑n

i=2(−1)i(∂∞i − ∂0i )(φpδ(γ))) = 0. Applying Lemma 5.2.5, this yields∑ni=2(−1)i(∂∞i − ∂0i )(φpδ(γ)) = 0. Combining this with (5.2.2), we conclude that ∂(φpδ(γ)) = 0.

5.3 Witt-complex structure

From [Rülling, 2007], recall that for each m ≥ 1, we have an isomor

Documents

KAIST 수리과학과 | - DGA-structure on additive higher Chow ...mathsci.kaist.ac.kr/~jinhyun/papers/CDGA.pdfMumbai, India and 2Department of Mathematical Sciences, KAIST, 291 Daehak-ro,