AN ADDITIVE VERSION OF HIGHER CHOW

GROUPS

UNE VERSION ADDITIVE DES GROUPES DE CHOW SUPÉRIEURS

SPENCER BLOCH AND HÉLÈNE ESNAULT

Abstract. The cosimplicial scheme

Δ^{•} = Δ^{0} \overrightarrow{→} Δ^{1} \begin{matrix}→ \\ → \\ → \end{matrix} …; Δ^{n} := Spec (k[t_{0}, …, t_{n}]/(∑ t_{i} − t))

was used in[3] to define higher Chow groups. In this note, we let t tend to 0 and replace Δ^{•} by a degenerate version

Q^{•} = Q^{0} \overrightarrow{→} Q^{1} \begin{matrix}→ \\ → \\ → \end{matrix} …; Q^{n} := Spec (k[t_{0}, …, t_{n}]/(∑ t_{i}))

to define an additive version of the higher Chow groups. For a field k, we show the Chow group of 0-cycles on Q^{n} in this theory is isomorphic to the group of absolute (n − 1)-Kähler forms Ω_{k}^{n−1}.

An analogous degeneration on the level of de Rham cohomology associated to “constant modulus” degenerations of varieties in various contexts is discussed. Résumé en français: Le schéma cosimplicial

Δ^{•} = Δ^{0} \overrightarrow{→} Δ^{1} \begin{matrix}→ \\ → \\ → \end{matrix} …; Δ^{n} := Spec (k[t_{0}, …, t_{n}]/(∑ t_{i} − t))

a été utilisé dans [3] afin de définir des groupes de Chow supérieurs. Dans cette note, nous définissons une version additive des groupes de Chow supérieurs en faisant tendre t vers 0 et en remplaçant Δ^{•} par une version dégénérée

Q^{•} = Q^{0} \overrightarrow{→} Q^{1} \begin{matrix}→ \\ → \\ → \end{matrix} …; Q^{n} := Spec (k[t_{0}, …, t_{n}]/(∑ t_{i})).

Nous montrons que sur un corps k, le groupe de Chow des 0-cycles dans cette théorie est isomorphe aux formes de Kähler absolues de degré (n − 1).

Nous discutons une dégénerescence analogue en cohomologie de de Rham apparaissant dans diverses situations pour des familles à module constant.

Date: July 21, 2002. 1991 Mathematics Subject Classification. Primary 14C15, 14C35.

1. Introduction

The purpose of this note is to study a common sort of limiting phenomenon which occurs in the study of motives. Here is a simple example. Let k be a field. Let S = 𝔸_{t}^{1} = Spec (k[t]) and let T = Spec (k[x, t]/(x(x − t)) ↪ 𝔸_{t,x}^{2}. Over S[1/t] the Picard scheme Pic(𝔸_{x,t}^{2}, T)/S is represented by 𝔾_{m,S[1/t]}. On the other hand, when t = 0 one gets Pic(𝔸_{x}^{1}, {x^{2} = 0}) ≅ 𝔾_{a,k}. In some sense, 𝔾_{m} has “jumped” to 𝔾_{a}.

In higher dimension, for t ≠ 0, the homology of the group of algebraic cycles associated to the cosimplicial scheme

(1.1) Δ^{•} = Δ^{0} \overrightarrow{→} Δ^{1} \begin{matrix}→ \\ → \\ → \end{matrix} …; Δ^{n} := Spec (k[t_{0}, …, t_{n}]/(∑ t_{i} − t))

is known to give motivic cohomology ([9]). What can one say about the algebraic cycle groups (1.3) of the degenerate cosimplicial complex:

(1.2) Q^{•} = Q^{0} \overrightarrow{→} Q^{1} \begin{matrix}→ \\ → \\ → \end{matrix} …; Q^{n} := Spec (k[t_{0}, …, t_{n}]/(∑ t_{i}))?

Our main result is a calculation of the Chow groups of 0-cycles on Q^{•}. Let Z^{n}(Q^{r}) be the free abelian group on codimension n algebraic cycles on Q^{r} satisfying a suitable general position condition with respect to the face maps. Let SH^{n}(k, r) be the cohomology groups of the complex

(1.3)  … Z^{n}(Q^{r+1}) → Z^{n}(Q^{r}) → Z^{n}(Q^{r−1}) → …

where the boundary maps are alternating sums of pullbacks along face maps. Write Ω_{k}^{•} for the absolute Kähler differentials.

Theorem 1.1. SH^{n}(k, n) ≅ Ω_{k}^{n−1}.

Note for n = 1, this is the above 𝔾_{a}.

We explain another example of this limiting phenomenon. We consider ℙ^{n+1} with homogeneous coordinates U_{0}, …, U_{n+1} over a field k. Let

(1.4)  X : f(U_{0}, …, U_{n+1}) = 0

be a hypersurface X defined by a homogeneous polynomial f of degree n + 2 (e.g. an elliptic curve in ℙ^{2}).

Write u_{i} = U_{i}/U_{0}, and define a log (n + 1)-form on ℙ^{n+1} \ X

(1.5)  ω_{f} := \frac{du_{1} ∧ … ∧ du_{n+1}}{f(1, u_{1}, …, u_{n+1})}.

As well known, this form generates ω_{ℙ^{n+1}}(X) ≅ O_{ℙ^{n+1}}.

Let r_{1}, …, r_{n+1} ≥ 0 be integers. We consider the action of 𝔾_{m} given by the substitutions u_{i} = t^{−r_{i}}v_{i}. Let N be minimal such that t^{N}f(1, t^{−r_{1}}u_{1}, …, t^{−r_{n+1}}u_{n+1}) is integral in t. Assume s := N − ∑ r_{i} > 0 and s is invertible in k. One checks easily that in the coordinates v_{i} one has

(1.6)  ω_{f} = t^{s}ν_{f}(t, v_{1}, …, v_{n+1}) + st^{s−1}dt ∧ γ_{f}(t, v_{1}, …, v_{n+1})

with

ν_{f}(t, v_{1}, …, v_{n+1}) = \frac{dv_{1} ∧ … ∧ dv_{n+1}}{t^{N}f(1, t^{−r_{1}}v_{1}, …, t^{−r_{n+1}}v_{n+1})}

γ_{f}(t, v_{1}, …, v_{n+1}) = \frac{1}{s} ∑_{i}(−1)^{i}r_{i}v_{i} \frac{dv_{1} ∧ … ∧ d̂v_{i} ∧ … ∧ dv_{n+1}}{t^{N}f(1, t^{−r_{1}}v_{1}, …, t^{−r_{n+1}}v_{n+1})}.

Thus ν_{f} and γ_{f} are forms in dv_{i} of degrees n + 1 and n respectively which are integral in t. As ω_{f} is an (n + 1)-form on ℙ^{n+1} in the ucoordinates, it is closed, so

st^{s−1}dt ∧ (ν_{f}(t, v_{1}, …, v_{n+1}) − dγ_{f}(t, v_{1}, …, v_{n+1}))

+st^{s−1} · \frac{t}{s}dν_{f}(t, v_{1}, …, v_{n+1}) = 0.

Dividing by t^{s−1} and restricting to t = 0 yields

(1.7)  ν_{f}|_{t=0} = dγ_{f}|_{t=0}.

In other words, under the action of the 1-parameter subgroup, ν_{f} degenerates to the exact form dγ_{f}.

To see the relationship with the 0-cycles, take

Δ_{n+1} : f(1, u_{1}, …, u_{n+1}) = u_{1}u_{2} ⋯ u_{n+1}(1 − u_{1} − … − u_{n+1}) = 0.

Substitute v_{i} = tu_{i}, 1 ≤ i ≤ n + 1. Then s = 1, and a calculation yields

γ_{Δ_{n+1}}(:= γ_{f}) = \frac{∑_{i=1}^{n+1}(−1)^{i}d log(v_{1}) ∧ … ∧ { \atop d log(v_{i})} ∧ … ∧ d log(v_{n+1})}{t − v_{1} − … − v_{n+1}}; (1.8)

ν_{Δ_{n+1}}(:= ν_{f}) = \frac{dv_{1} ∧ … ∧ dv_{n+1}}{v_{1}v_{2} ⋯ v_{n+1}(t − v_{1} − … − v_{n+1})}.

Note the limiting configuration as t → 0 is (compare (1.2))

v_{1}v_{2} ⋯ v_{n+1}(v_{1} + … + v_{n+1}) = 0.

For convenience we write

(1.9)  γ_{n} := γ_{Δ_{n+1}}|_{t=0}

We view ν_{Δ_{n+1}}|_{t=1} (resp. γ_{n}) as a map

Z_{0}(𝔸_{k}^{n+1} \ {(1 − u_{1} + ⋯ + u_{n+1})u_{1} ⋯ u_{n+1} = 0}) → Ω_{k}^{n+1} (resp.

Z_{0}(𝔸_{k}^{n+1} \ {v_{1}v_{2} ⋯ v_{n+1}(v_{1} + ⋯ + v_{n+1}) = 0}) → Ω_{k}^{n}).

Here Z_{0} denotes the free abelian group on closed points (0-cycles) and the maps are respectively

(1.10)  x ↦ Tr_{k(x)/k}ν|_{{x}}; x ↦ Tr_{k(x)/k}γ|_{{x}}.

In the first case, the Nesterenko-Suslin-Totaro theorem ([7], [8]) identifies the zero cycles modulo relations coming from curves in 𝔸^{n+2} with the Milnor K-group K_{n+1}^{M}(k). The evaluation map (1.10) passes to the quotient, and the resulting map K_{n+1}^{M}(k) → Ω_{k}^{n+1} is given on symbols by the d log-map

(1.11)  {x_{1}, …, x_{n+1}} ↦ d log(x_{1}) ∧ ⋯ ∧ d log(x_{n+1}).

In the second case, factoring out by the relations coming from curves on 𝔸^{n+2} as in (1.3) yields the Chow group of 0-cycles SH^{n+1}(k, n + 1), and our main result is that evaluation on γ_{n} gives an isomorphism

(1.12)  SH^{n+1}(k, n + 1) ≅ Ω_{k}^{n}.

Sections 2-5 contain the proof of theorem 1.1. Section 6 contains some brief remarks on specialization of forms as it relates to Aomoto’s theory of configurations, to 0-cycles on hypersurfaces, and to Goncharov’s theory of hyperbolic motives. Finally, section 7 computes SH^{1}(k, n), responding to a question of S. Lichtenbaum.

The authors wish to thank A. Goncharov and S. Lichtenbaum for several very inspiring conversations, also the referee for very careful reading and helpful remarks.

2. The additive Chow groups

In this section, we consider a field k, and a k-scheme X of finite type. We will throughout use the following notations.

Notations 2.1. We set Q^{n} = Spec k[t_{0}, …, t_{n}]/(∑_{i=0}^{n} t_{i}), together with the faces

∂_{j} : Q^{n−1} → Q^{n}; ∂_{j}^{∗}(t_{i}) = \begin{cases}t_{i} & i < j \\ 0 & i = j \\ t_{i−1} & i > j \end{cases} .

One also has degeneracies

π_{j} : Q^{n} → Q^{n−1}; π_{j}^{∗}(t_{i}) = \begin{cases}t_{i} & i < j \\ t_{i} + t_{i+1} & i = j \\ t_{i+1} & i > j \end{cases} .

We denote by {0} ∈ Q^{n} the vertex defined by t_{i} = 0. We write Q_{X}^{n} = Q^{n} ×_{Spec (k)} X. The above face and degeneracy maps make Q_{X}^{•} a cosimplicial scheme.

Definition 2.2. Let SZ_{q}(X, n) be the free abelian group on irreducible, dimension q subvarieties in Q_{X}^{n} with the property:

(i) They don’t meet {0} × X. (ii) They meet all the faces properly, that is in dimension ≤ q.

Thus the face maps induce restriction maps

∂_{i} : SZ_{q}(X, n) → SZ_{q−1}(X, n − 1); i = 0, … n; ∂ = ∑_{i=0}^{n}(−1)^{i}∂_{i};

yielding complexes SZ_{q−•}(X, •):

… →^{∂} SZ_{q+1}(X, n + 1) →^{∂} SZ_{q}(X, n) →^{∂} SZ_{q−1}(X, n − 1) →^{∂} …

Definition 2.3. The additive higher Chow groups are given for n ≥ 1 by

SH_{q}(X, n) = H_{n}(SZ_{q−•}(X, •))

and (for X equidimensional)

SH^{p}(X, n) = SH_{dimX−p}(X, n).

The groups are not defined for n = 0.

Remarks 2.4. (i) The above should be compared with the higher Chow groups CH^{p}(X, n) defined as above with Q^{•} replaced by Δ^{•}, where Δ^{n} := Spec (k[t_{0}, …, t_{n}]/(∑ t_{i} − 1)).

(ii) The cosimplicial scheme Q^{•} admits an action of 𝔾_{m,k}, which we define by

x ⋆ (t_{0}, …, t_{n}) := (t_{0}/x, …, t_{n}/x).

(The reason for the inverse will be clear below.) By functoriality, we obtain a k^{×}-action on the SH^{p}(X, n).

(iii) Let f : X′ → X be a proper map with n = dim X′ − dim X. Then one has a push-forward map

f_{∗} : SZ^{p}(X′, •) → SZ^{p−n}(X, •).

On homology this yields SH^{p}(X′, n) → SH^{p−n}(X, n). We will be particularly interested in the case X = Spec (k), X′ = Spec (k′) with [k′ : k] < ∞. We write tr_{k′/k} : SH^{p}(k′, n) → SH^{p}(k, n) for the resulting map. This trace map is compatible with the action of k^{×} from (ii) in the sense that for x ∈ k^{×} and a′ ∈ SH^{p}(k′, n) we have x ⋆ tr_{k′/k}(a′) = tr_{k′/k}(x ⋆ a′).

Lemma 2.5. The ⋆ action of k^{×} on SH^{n}(k, n) extends to an action of the multiplicative monoid k by setting 0 ⋆ x = 0. This action comes from a k-vector space structure on SH^{n}(k, n).

Proof. We have to show that for a closed point x = (u_{0}, …, u_{n}) ∈ Q^{n} \ ∪_{i=0}^{n}∂_{i}(Q^{n−1}), and a, b ∈ k, one has (a + b) ⋆ x = a ⋆ x + b ⋆ x. For either a or b = 0, this is trivial. Thus we assume ab ≠ 0. Let k′ = k(x). Then the class in SH^{n}(k, n) of x is the trace from k′ to k of a k′-rational point x′ ∈ SH^{n}(k′, n). Using the compatibility of ⋆ and trace from Remarks 2.4(iii) above, we reduce to the case x k-rational.

Write (t_{0}−u_{0}, …, t_{n}−u_{n}) for the ideal of x. Define ℓ(t) = −\frac{ab}{u_{0}} t+a+b. Consider the curve W ⊂ Q^{n+1} defined parametrically by

W = \{(t, −t + \frac{u_{0}}{ℓ(t)}, \frac{u_{1}}{ℓ(t)}, …, \frac{u_{n}}{ℓ(t)})\}.

To check that this parametrized locus is Zariski-closed, we consider the ideal:

I_{W} = ((t_{1} + t_{0})ℓ(t_{0}) − u_{0}, t_{2}ℓ(t_{0}) − u_{1}, …, t_{n}ℓ(t_{0}) − u_{n}).

If y = (y_{0}, …, y_{n+1}) is a geometric point in the zero locus of I_{W}, then since the u_{i} ≠ 0 we see that ℓ(y_{0}) ≠ 0. Substituting t = y_{0}, we see that y lies on the parametrized locus W.

The equation −t + \frac{u_{0}}{ℓ(t)} = 0 leads to a quadratic equation in t with solutions t = \frac{u_{0}}{a} , t = \frac{u_{0}}{b}. If a + b ≠ 0 we have

∂_{0}(W) = (\frac{u_{0}}{a + b}, …, \frac{u_{n}}{a + b}) = (a + b) ⋆ (u_{0}, …, u_{n});

∂_{1}(W) = (\frac{u_{0}}{a} , …, \frac{u_{n}}{a}) + (\frac{u_{0}}{b} , …, \frac{u_{n}}{b})

= a ⋆ (u_{0}, …, u_{n}) + b ⋆ (u_{0}, …, u_{n})

∂_{i}(W) = 0; i ≥ 2,

so the lemma follows in this case. If a + b = 0, then ∂_{0}W = 0 as well, and again the assertion is clear. □

3. Additive Chow groups and Milnor K-theory

We consider the map (compatible with faces) ι : Δ^{•} → Q^{•+1} defined on k-rational points by (u_{0}, …, u_{n}) ↦ (−1, u_{0}, …, u_{n}). It induces a map of complexes Z^{p}(k, •) → SZ^{p+1}(k, • + 1), which in turn induces a map

ι : CH^{p}(k, n) → SH^{p+1}(k, n + 1). (3.1)

By [7] and [8], one has an isomorphism

K_{n}^{M}(k) ≅ CH^{n}(k, n) (3.2)

of the higher Chow groups of 0-cycles with Milnor K-theory. It is defined by:

(u_{0}, …, u_{n}) ↦ \{−\frac{u_{0}}{u_{n}} , …, −\frac{u_{n−1}}{u_{n}} \} (3.3)

\{b_{1}, …, b_{n}\} ↦ (\frac{b_{1}}{c} , …, \frac{b_{n}}{c} , −\frac{1}{c}); c = −1 + ∑_{i=1}^{n} b_{i}. (3.4)

Note that if ∑_{i=1}^{n} b_{i} = 1, then the symbol β := {b_{1}, …, b_{n}} is trivial in Milnor K-theory, and one maps β to 0.

In this way, one obtains a map

(3.5) K_{n−1}^{M}(k) → SH^{n}(k, n); {x_{1}, …, x_{n−1}} ↦

( − 1, \frac{x_{1}}{−1 + ∑_{i=1}^{n−1} x_{i}} , …, \frac{x_{n−1}}{−1 + ∑_{i=1}^{n−1} x_{i}} , \frac{−1}{−1 + ∑_{i=1}^{n−1} x_{i}} ).

4. Differential forms

In this section we construct a k-linear map Ω_{k}^{n−1} → SH^{n}(k, n). (Here Ω_{k}^{i} are the absolute Kähler differential i-forms.)

The following lemma is closely related to calculations in [5].

Lemma 4.1. As a k-vector space, the differential forms Ω_{k}^{n−1} are isomorphic to (k ⊗_{ℤ} ∧^{n−1}k^{×})/R. The k-structure on k ⊗_{ℤ} ∧^{n−1}k^{×} is via multiplication on the first argument. The relations R, for n ≥ 2, are the k-subspace spanned by

a ⊗ (a ∧ b_{1} ∧ … ∧ b_{n−2}) + (1 − a) ⊗ ((1 − a) ∧ b_{1} ∧ … ∧ b_{n−2}),

for b_{i} ∈ k^{×}, a ∈ k. The map (k ⊗_{ℤ} ∧^{n−1}k^{×})/R → Ω_{k}^{n−1} is then defined by (a, b_{1}, …, b_{n−1}) ↦ ad log b_{1} ∧ … ∧ d log b_{n−1}.

Proof. Write ′Ω^{∗} := (k ⊗_{ℤ} ∧^{∗}k^{×})/R. This is a quotient of the graded k-algebra k ⊗_{ℤ} ∧^{∗}k^{×} by the graded ideal R and hence has a graded kalgebra structure, generated in degree 1. There is an evident surjection of graded algebras ′Ω^{∗} ↠ Ω^{∗}, so, by the universal mapping property of the exterior algebra Ω^{∗}, it suffices to check ′Ω^{1} ≅ Ω^{1}.

Define D : k → (k ⊗_{ℤ} k^{×})/R by D(a) = a ⊗ a for a ∈ k^{×}, else D(0) = 0. To define the required inverse, it suffices to show D is a derivation. Clearly, D(ab) = aD(b) + bD(a). Also 1 ⊗ −1 is trivial in k ⊗ k^{×}, so D(−a) = −D(a). Given a, b ∈ k^{×}, write b = −ac. We have

(4.1) D(a + b) = D(a − ac) = aD(1 − c) + (1 − c)D(a) = − aD(c) + D(a) − cD(a) = D(a) + D(−ac) = D(a) + D(b).

Hence D is a derivation so the inverse map Ω^{1} →′ Ω^{1} is defined. □

Remark 4.2. We will frequently use the relations in the equivalent form a ⊗ a ∧ (⋯) − (1 + a) ⊗ (1 + a) ∧ (⋯) ∼ 0.

Proposition 4.3. One has a well-defined k-linear map

ϕ : Ω_{k}^{n−1} → SH^{n}(k, n)

α := ad log b_{1} ∧ … ∧ d log b_{n−1} ↦ a ⋆ (−1, \frac{b_{1}}{γ} , …, \frac{b_{n−1}}{γ} , −\frac{1}{γ})

where γ = −1 + ∑_{i=1}^{n−1} b_{i}. (Define α = 0 when γ = 0.) The diagram

K_{n−1}^{M}(k) ⟶^{d log} Ω_{k}^{n−1}
ι↓^{(3.5)}  ^{ϕ}↓
SH^{n}(k, n)  SH^{n}(k, n)

is commutative.

Proof. We write c = −1 + a + ∑_{i=2}^{n−1} b_{i}. By Lemma 4.1, we have to show

0 = ρ :=

a ⋆ (−1, \frac{a}{c}, …, \frac{b_{n−1}}{c} , −\frac{1}{c}) − (a + 1) ⋆ (−1, \frac{a + 1}{c + 1}, …, \frac{b_{n−1}}{c + 1}, −\frac{1}{c + 1}).

If a = 0, then one has

ρ = −(−1, \frac{1}{c}, …, \frac{b_{n−1}}{c} , −\frac{1}{c}) = −ι\{1, b_{2}, …, b_{n−1}\} = 0.

Similarly, ρ = 0 if a = −1. Assume now a ≠ 0, −1. Set b = −a ∈ k \ {0, 1}. One defines, for n ≥ 3 and (u_{1}, …, u_{n−1}) ∈ (Δ^{n−2} \ ∪_{i=1}^{n−1}Δ^{n−3})(k), the parametrized curve

Γ(b, u) := \{(\frac{−1}{b} + t, \frac{1}{b − 1}, −t, \frac{−u_{1}}{b(b − 1)}, …, \frac{−u_{n−1}}{b(b − 1)})\} ⊂ Q^{n+1}

and for n = 2

Γ(b) := \{(\frac{−1}{b} + t, \frac{1}{b − 1}, −t, \frac{−1}{b(b − 1)})\} ⊂ Q^{3}. (4.2)

(See [8] for the origin of this definition). This curve is indeed in good position, so it lies in SZ^{n}(Q^{n+1}). One computes

∂Γ(b, u) = (1 − b) ⋆ (−1, 1 − \frac{1}{b}, \frac{u_{1}}{b} , …, \frac{u_{n−1}}{b}) (4.3)

+b ⋆ (−1, \frac{b}{b − 1}, \frac{−u_{1}}{b − 1}, …, \frac{−u_{n−1}}{b − 1} ).

(Resp. in the case n = 2

∂Γ(b) = (1 − b) ⋆ (−1, 1 − \frac{1}{b}, \frac{1}{b}) + b ⋆ (−1, \frac{b}{b − 1}, \frac{−1}{b − 1}).)

Now one has

(−1, 1 − \frac{1}{b}, \frac{u_{1}}{b} , …, \frac{u_{n−1}}{b}) = ι\{\frac{1 − b}{u_{n−1}}, −\frac{u_{1}}{u_{n−1}} , …, −\frac{u_{n−2}}{u_{n−1}} \}

= ι[\{1 − b, −\frac{u_{1}}{u_{n−1}} , …, −\frac{u_{n−2}}{u_{n−1}} \} − \{u_{n−1}, u_{1}, …, u_{n−2}\}],

as the rest of the multilinear expansion contains only symbols of the shape {…, u_{n−1}, …, −u_{n−1}, …}. On the other hand, since ∑_{i=1}^{n−1} u_{i} = 1, one has {u_{n−1}, u_{1}, …, u_{n−2}} = 0. Similarly, one has

(−1, \frac{b}{b − 1}, \frac{−u_{1}}{b − 1}, …, \frac{−u_{n−1}}{b − 1} ) = ι\{\frac{b}{u_{n−1}} , −\frac{u_{1}}{u_{n−1}} , …, −\frac{u_{n−2}}{u_{n−1}} \}.

The same argument yields that this is

ι\{b, −\frac{u_{1}}{u_{n−1}} , …, −\frac{u_{n−2}}{u_{n−1}} \}.

It follows now from (4.3) that for n ≥ 3 we have the relation in SH^{n}(k, n)

(4.4) (1 − b) ⋆ ι\{1 − b, −\frac{u_{1}}{u_{n−1}} , …, −\frac{u_{n−2}}{u_{n−1}} \}+

b ⋆ ι\{b, −\frac{u_{1}}{u_{n−1}} , …, −\frac{u_{n−2}}{u_{n−1}} \} = 0

(The analogous relation for n = 2 is similar.) □

Proposition 4.4. With notation as above, the map

ϕ : Ω_{k}^{n−1} → SH^{n}(k, n)

is surjective. In particular, SH^{n}(k, n) is generated by the classes of k-rational points in Q^{n}.

Proof. It is easy to check that the image of ϕ coincides with the subgroup of SH^{n}(k, n) generated by k-points. Clearly, SH^{n}(k, n) is generated by closed points, and any closed point is the trace of a k′-rational point for some finite extension k′/k. We first reduce to the case k′/k separable. If x ∈ Q_{k}^{n} is a closed point in good position (i.e. not lying on any face) such that k(x)/k is not separable, then a simple Bertini argument shows there exists a curve C in good position on Q^{n+1} such that ∂C = x + y where y is a zero cycle supported on points with separable residue field extensions over k. Indeed, let W ⊂ Q^{n+1} be the union of the faces. View x ∈ W. Since x is in good position, it is a smooth point of W. Bertini will say that a non-empty open set in the parameter space of n-fold intersections of hypersurfaces of large degree containing x will meet W in x plus a smooth residual scheme. Since k is necessarily infinite, there will be such an n-fold intersection defined over k. Since the residual scheme is smooth, it cannot contain inseparable points. Then x ≡ −y which is supported on separable points.

We assume now k′/k finite separable, and we must show that the trace of a k′-point is equivalent to a zero cycle supported on k-points. Since the image of ϕ is precisely the subgroup generated by k-rational points, it suffices to check that the diagram

(4.5)  \begin{matrix}Ω_{k′}^{n−1} & ⟶^{ϕ} & SH^{n}(k′, n) \\ ↓^{Tr}_{k′/k} & ↓^{Tr}_{k′/k} \\ Ω_{k}^{n−1} & ⟶^{ϕ} & SH^{n}(k, n) \end{matrix}

commutes. Because k′/k is separable, one has Ω_{k}^{n−1} ↪ Ω_{k′}^{n−1}, and Ω_{k′}^{n−1} = k′ · Ω_{k}^{n−1}. One reduces to showing, for α = (−1, α_{1}, …, α_{n}) ∈ Q^{n}(k) and t ∈ k′, that Tr(t ⋆ α) = (Tr(t)) ⋆ α in SH^{n}(k, n).

Let P (V) = V^{N} + a_{N−1}V^{N−1} + … + a_{1}V + a_{0} ∈ k[V] be the minimal polynomial of −\frac{1}{t}. We set b_{N} = \frac{−1}{α_{n}}, b_{i} = \frac{−a_{i}}{α_{n}} , i = N − 1, …, 2 and b_{i} = a_{i}, i = 1, 0. We define the polynomial Q(V, u) = b_{N}V^{N−1}u + … b_{2}V u + b_{1}V +b_{0} ∈ k[V, u], which by definition fulfills Q(V, −α_{n}V) = P (V). We define the ideal

I = (Q(V_{0}, u), V_{1} + α_{1}V_{0}, …, V_{n−1} + α_{n−1}V_{0})
⊂ k[V_{0}, …, V_{n−1}, u].

It defines a curve W ⊂ 𝔸^{n+1}. We think of 𝔸^{n+1} as being Q^{n+1} with the faces V_{0} = 0, …, V_{n−1} = 0, u = 0, u + ∑_{i=0}^{n−1} V_{i} = 0. Then this curve is in general position and defines a cycle in SZ^{1}(k, n + 1).

Since b_{0} ≠ 0, and α_{i} ≠ 0, one has

∂_{i}W = 0, i = 0, 1, …, n − 1. (4.6)

One has

∂_{u}W defined by (a_{1}V_{0} + a_{0}, V_{1} + α_{1}V_{0}, …, V_{n−1} + α_{n−1}V_{0}).

To compute the last face, we observe that the ideal

(u + ∑_{i=0}^{n−1} V_{i}, V_{1} + α_{1}V_{0}, …, V_{n−1} + α_{n−1}V_{0}),

contains u + α_{n}V_{0}. Consequently ∂_{u+∑_{i=0}^{n−1} V_{i}}W is defined by

(Q(V_{0}, −α_{n}V_{0}), V_{1} + α_{1}V_{0}, …, V_{n−1} + α_{n−1}V_{0}),

with Q(V_{0}, −α_{n}V_{0}) = P (V_{0}). Thus one obtains

0 ≡ (−1)^{n}∂W = \frac{a_{1}}{a_{0}} ⋆ (−1, α) − t ⋆ (−1, α).

Since P is the minimal polynomial of −\frac{1}{t}, \frac{a_{1}}{a_{0}} is the trace of t. □

5. The main theorem

Recall (1.8) we have a logarithmic (n − 1)-form γ_{n−1} on Q^{n} = Spec (k[v_{0}, …, v_{n}]/(∑_{i=0}^{n} v_{i}))

(5.1) γ_{n−1} = \frac{1}{v_{0}} ∑_{1}^{n}(−1)^{i}d log(v_{1}) ∧ … ∧ { \atop d log(v_{i})} ∧ … ∧ d log(v_{n})

dγ_{n−1} = ν_{n} = \frac{dv_{1} ∧ … ∧ dv_{n}}{v_{0}v_{1} ⋯ v_{n}}

Writing v_{i} = V_{i}/V_{n+1}, we can view γ_{n−1} as a meromorphic form on ℙ^{n} = Proj(k[V_{0}, …, V_{n+1}]/(∑_{0}^{n} V_{i})). Let A : V_{0} ⋯ V_{n} = 0; ∞ : V_{n+1} = 0. The fact that dγ_{n−1} has log poles on the divisors V_{i} = 0, 0 ≤ i ≤ n implies that

(5.2)  γ_{n−1} ∈ Γ(ℙ^{n}, Ω_{ℙ^{n}}^{n−1}(log(A + ∞))(−∞))

In particular, γ_{n−1} has log poles, so we can take the residue along components of A. (The configuration A does not have normal crossings. The sheaf Ω_{ℙ^{n}}^{n−1}(log(A + ∞)) is defined to be the subsheaf of j_{∗}Ω_{ℙ^{n}−A−∞}^{n−1}, where j : ℙ^{n} \ (A ∪ ∞) → ℙ^{n} is the open embedding, generated by forms without poles and the evident log forms with residue 1 along one hyperplane and (−1) along another one. According to [1], the global sections of this naturally defined log sheaf compute de Rham cohomology. )

In the following, we adopt the sign convention Res_{t=0} \frac{dt}{t} ∧ ω = ω. This yields Res_{t=0}d(\frac{dt}{t} ∧ ω) = −d(Res_{t=0} \frac{dt}{t} ∧ ω).

Lemma 5.1. We have the following residue formulae

Res_{v_{i}=0}γ_{n} = (−1)^{i}γ_{n−1}; 0 ≤ i ≤ n + 1.

Proof. One can either compute directly or argue indirectly as follows:

(5.3) − dRes_{v_{i}=0}γ_{n} = Res_{v_{i}=0}dγ_{n} = Res_{v_{i}=0}ν_{n+1} =

(−1)^{i+1}ν_{n} = −d((−1)^{i}γ_{n−1}) ≠ 0.

To conclude now, it suffices to show that the space of global sections (5.2) has dimension 1. To verify the dimension 1 property, let A′ ⊂ A be defined by V_{1} ⋯ V_{n} = 0. Then A′+∞ consists of n+1 hyperplanes in general position in ℙ^{n}, so Ω_{ℙ^{n}}^{n−1}(log(A′+∞)) = ∧^{n−1}Ω_{ℙ^{n}}^{1}(log(A′+∞)) ≅ O_{ℙ^{n}}^{⊕n}. One looks at the evident residue

Ω_{ℙ^{n}}^{n−1}(log(A + ∞))(−∞) → Ω_{ℙ^{n−1}}^{n−2}(log(A + ∞))(−∞).

along V_{0} = 0.) □

Remark 5.2. The computation of the lemma shows that γ_{n−1} is the unique (n − 1)-differential form in Γ(ℙ^{n}, Ω_{ℙ^{n}}^{n−1}(log(A + ∞))(−∞)) with dγ_{n−1} = ν_{n}. The configuration A being given, fixing ∞ fixes γ_{n−1}.

Theorem 5.3. The assignment x ↦ Tr_{k(x)/k}(γ(x)) gives an isomorphism for n ≥ 1

SH^{n}(k, n) ≅ Ω_{k}^{n−1}.

Proof. Let X ⊂ Q_{k}^{n+1} be a curve in good position. For a zero-cycle c in good position on Q^{n} we write Trγ_{n−1}(c) ∈ Ω_{k}^{n−1} (absolute differentials) for the evident linear combination of traces from residue fields of closed points. We must show Trγ_{n−1}(∂X) = 0. Let \overline{X} denote the closure of X in ℙ^{n}. We consider γ_{n−1}|\overline{X} ∈ Γ(\overline{X}, Ω_{\overline{X}/ℤ}^{n−1}(∗D)), where D is the pole set of γ_{n−1}|\overline{X}. The form γ_{n−1} dies when restricted to ∞. Thus D = ∪_{j=0}^{n}D_{j} ⊂ X, with D_{j} := ∂_{j}(Q^{n})∩\overline{X}. We define the residue along D_{j} as follows. One has an exact sequence

(5.4) 0 → Ω_{k}^{n} ⊗ O\overline{X} → Ωn ; \overline{X}(log D) → Ω_{k}^{n−1} ⊗ ω\overline{X}(log D) → 0.

The residue map followed by the trace

(5.5) ω\overline{X}(log D) \underrightarrow{Tr∘∑ Res_{D_{j}}} k

yields, by the reciprocity formula, a vanishing residue map on global n-forms

(5.6) H^{0}(\overline{X}, Ω_{\overline{X}/ℤ}^{n}(log D)) → Ω_{k}^{n−1} ⊗_{k} H^{0}(\overline{X}, ω\overline{X}(log D)) ⟶^{0} Ω_{k}^{n−1}.

On the other hand, the residue decomposes as

(5.7) H^{0}(\overline{X}, Ω_{\overline{X}/ℤ}^{n}(log D)) \underrightarrow{∑_{j} Res_{D_{j}}} ⊕_{j=0}^{n}Ω_{D_{j}}^{n−1} \underrightarrow{∑_{j} Tr} Ω_{k}^{n−1}.

By Lemma 5.1, γ_{n−1}(D_{j}) = (Res_{v_{j}=0}γ_{n})(D_{j}). The desired vanishing follows.

We now have

Ω_{k}^{n−1} ↠^{ϕ} SH^{n}(k, n) ⟶^{γ_{n−1}} Ω_{k}^{n−1}.

It suffices to check the composition is multiplication by (−1)^{n+1}. Given b_{1}, …, b_{n−1} ∈ k with c := ∑ b_{i} − 1 ≠ 0, the composition is computed to be (use (5.1) and Proposition 4.3)

a · d log(b_{1}) ∧ … ∧ d log(b_{n−1}) ↦

− a ∑(−1)^{i}d log(\frac{b_{1}}{ac}) ∧ … ∧ { \atop d log(\frac{b_{i}}{ac})} ∧ … ∧ d log(\frac{−1}{ac})

Expanding the term on the right yields

− a ∑(−1)^{i}d log(\frac{b_{1}}{c}) ∧ … ∧ { \atop d log(\frac{b_{i}}{c})} ∧ … ∧ d log(\frac{−1}{c})+

− a · d log(a) ∧ ( …),

and it is easy to check that the terms involving d log(a) cancel. In this way, one reduces to the case a = 1. Here

(5.8) − ∑(−1)^{i}d log(\frac{b_{1}}{c}) ∧ … ∧ { \atop d log(\frac{b_{i}}{c})} ∧ … ∧ d log(\frac{−1}{c}) =

(−1)^{n+1} \frac{db_{1}}{b_{1}} ∧ … ∧ \frac{db_{n−1}}{b_{n−1}} + \frac{dc}{c} ∧ ( …).

Again the terms involving d log(c) cancel formally, completing the proof. □

Challenge 5.4. Finally, as a challenge we remark that the Kähler differentials have operations (exterior derivative, wedge product,...) which are not evident on the cycles SZ. For example, one can show that the map

∇(x_{0}, …, x_{n}) = (x_{0}, −\frac{x_{1}x_{0}}{1 − x_{0}} , …, −\frac{x_{n}x_{0}}{1 − x_{0}} , −\frac{x_{0}}{1 − x_{0}} ) (5.9)

satisfies

γ_{n}(∇(x)) = (−1)^{n}dγ_{n−1}(x) (5.10)

and hence induces the exterior derivative on the 0-cycles. The map is not uniquely determined by this property, and this particular map does not preserve good position for cycles of dimension > 0. Can one find a geometric correspondence on the complex SZ^{•} which induces d on the 0-cycles? What about the pairings (a, b) ↦ a ∧ b or (a, b) ↦ a ∧ db?

6. Specialization of Forms

In this section we consider again the specialization of differential forms as in §1. Recall f(U_{0}, …, U_{n+1}) is homogeneous of degree n + 2, u_{i} = U_{i}/U_{0}, and v_{i} = t^{r_{i}}u_{i}. N is minimal such that t^{N}f(1, t^{−r_{1}}v_{1}, …, t^{−r_{n+1}}v_{n+1}) is integral in t, and we assume s = N − ∑ r_{i} > 0. We have forms

(6.1) ω_{f} := \frac{d(t^{−r_{1}}v_{1}) ∧ … ∧ d(t^{−r_{n+1}}v_{n+1})}{f(1, t^{−r_{1}}v_{1}, …, t^{−r_{n+1}}v_{n+1})} = t^{s}ν_{f} + st^{s−1}dt ∧ γ_{f}

and ν_{f}|_{t=0} = dγ_{f}|_{t=0}.

We have already mentioned the case

f = Δ_{n+1} = (t^{−1}v_{1}) ⋯ (t^{−1}v_{n+1})(1 − t^{−1}v_{1} − … − t^{−1}v_{n+1}).

The forms ν_{Δ_{n+1}} and γ_{Δ_{n+1}} are given in (1.8). As before, we write ν_{n+1} := ν_{Δ_{n+1}}|_{t=0}; γ_{n} := γ_{Δ_{n+1}}|_{t=0}. The differential form ν_{n+1} plays an interesting rôle in the computation of de Rham cohomology of the complement of hyperplane configurations. Let A_{t} be the configuration in ℙ^{n} of (n+1) hyperplanes in general position with with affine equation u_{0}u_{1} ⋯ u_{n} = 0, u_{0} + u_{1} + … + u_{n} = t ≠ 0. Then H^{n}(ℙ^{n} \ A_{t}) = H^{0}(ℙ^{n}, Ω_{ℙ^{n}}^{n}(log A_{t})) is a pure Tate structure generated by ν_{Δn+1}. Now make t tend to 0 and consider the degenerate configuration A_{0} with affine equation u_{0}u_{1} ⋯ u_{n} = 0, u_{0} + u_{1} + … + u_{n} = 0. Exactness of ν_{n+1} = dγ_{n} in the case (1.7) follows from Aomoto’s theory ([1], [4]). Due to the special shape of the configurations A_{t} considered, however, more is true. We have fixed ∞, and the differential form γ_{n} lies in H^{0}(ℙ^{n+1}, Ω^{n}(log(A_{0} + ∞)(−∞)). That is, it has log poles along all components of A_{0}, and it vanishes as a differential form at ∞ (see Lemma 5.1). This k-vector space of differential forms is 1dimensional. In other words, the differential form γ_{n} is uniquely defined by the vanishing condition at ∞ and the secondary data dγ_{n} = ν_{n+1} (see remark 5.2). Do such primitives for more general degenerating configurations admit an interpretation identifying 0-cycles with spaces of differential forms?

Here is another example of specialization and 0-cycles. Define

(6.2)  f(1, u_{1}, u_{2}) = u_{1}^{2} − u_{2}^{3} − au_{2} − b; v_{1} = t^{3}u_{1}, v_{2} = t^{2}u_{2}

One computes

(6.3)  ν = \frac{dv_{1} ∧ dv_{2}}{v_{1}^{2} − v_{2}^{3} − at^{4}v_{2} − bt^{6}} ; γ = \frac{2v_{2}dv_{1} − 3v_{1}dv_{2}}{v_{1}^{2} − v_{2}^{3} − at^{4}v_{2} − bt^{6}} .

One checks that Res(γ|_{t=0}) = v_{2}/v_{1} and

(6.4)  d(v_{2}/v_{1}) = −dv_{2}/2 = dRes(ν|_{t=0}) on v_{1}^{2} − v_{2}^{3} = 0

The assignment x ↦ Tr_{k(x)/k}(v_{2}/v_{1})(x) identifies the jacobian of the special fibre v_{1}^{2} − v_{2}^{3} = 0 with 𝔾_{a}(k) = k.

A final example of specialization, which we understand less well, though it was an inspiration for this article, concerns the hyperbolic motives of Goncharov [6]. The matrix coefficients of his theory (in the sense of [2]) are the objects H^{2n−1}(ℙ^{2n−1} \ Q, M \ Q ∩ M). Here Q ⊂ ℙ^{2n−1} is a smooth quadric and M is a simplex (union of 2n hyperplanes in general position). The subschemes Q and M are taken in general position with respect to each other. The notation is intended to suggest a sort of abstract relative cohomology group. In de Rham cohomology, the non-trivial class in H_{DR}^{2n−1}(ℙ^{2n−1} \ Q) is represented by

(6.5)  ω = \frac{du_{1} ∧ … ∧ du_{2n−1}}{(u_{1}^{2} + … + u_{2n−1}^{2} − 1)^{n}}

Substituting u_{i} = v_{i}t, we get

(6.6) ω =

\frac{t^{2n−1}dv_{1} ∧ … ∧ dv_{2n−1} + t^{2n−2}dt ∧ ∑(−1)^{i}v_{i}dv_{1} ∧ … d̂v_{i} … ∧ dv_{2n−1}}{(t^{2}(v_{1}^{2} + … + v_{2n−1}^{2}) − 1)^{n}}

We get ν|_{t=0} = ±dv_{1} ∧ … ∧ dv_{2n−1} and γ|_{t=0} = \frac{1}{2n−1} ∑(−1)^{i}v_{i}dv_{1} ∧ … d̂v_{i} … ∧ dv_{2n−1}. Let Δ_{M} ∈ H_{2n−1}(ℙ^{2n−1}, M; ℤ) be a generator. The hyperbolic volume

(6.7)  ∫_{Δ_{M}} ω

is the real period ([6] section 4.1) of the Hodge structure associated to H^{2n−1}(ℙ^{2n−1} \ Q). Goncharov remarks (op. cit., Question 6.4 and Theorem 6.5) that this volume degenerates to the euclidean volume as t → 0. (More precisely, from (6.7), we see that as a relative form, dv_{1} ∧ … ∧ dv_{2n−1} = lim_{t→0} t^{1−2n}ω.) He asks for an interpretation of the degenerated volume in terms of some sort of motive over k[t]/(t^{2}).

In the Goncharov picture we can view Q as fixed and degenerate M. Suppose M : L_{0}L_{1} ⋯ L_{2n−1} = 0 where

L_{i} = L_{i}(v_{1}, …, v_{2n−1}) = L_{i}(u_{1}/t, …, u_{2n−1}/t)

Assuming the L_{i} are general, clearing denominators and passing to the limit t → 0 yields a degenerate simplex M_{0} consisting of 2n hyperplanes meeting at the point v = 0. This limiting configuration leads to the Chow groups SH^{∗}(k, 2n − 1), but we do not see how to relate γ|_{t=0} to cycles.

7. Cycle Groups of Divisors

We compute SH^{1}(k, n) for n ≥ 1.

Let O_{Q^{n},0} be the local ring at (0, …, 0) ∈ Q^{n}. Let

(7.1)  I^{n} ⊂ J^{n} ⊂ O_{Q^{n},0}

be the ideals of all (resp. all but the last) face.

Lemma 7.1. SH^{1}(k, n) ≅ (1 + I^{n})^{×}/∂_{n+1}(1 + J^{n+1})^{×}.

Proof. Let D ⊂ Q^{n} be an effective divisor. Let

f ∈ k_{n} := k[t_{0}, …, t_{n}]/(∑ t_{i})

be a defining equation for D. Then D meets faces properly in our sense if and only if f ∈ O_{Q^{n},0}^{×}. Conversely, any f ∈ O_{Q^{n},0}^{×} defines a (not necessarily effective) divisor on Q^{n} meeting faces properly. Further

(7.2)  (f) · Q_{i}^{n−1} = 0 ⇔ f ∈ k^{×} · (1 + t_{i}O_{Q^{n},0}).

By general simplicial considerations, the SH^{1}(k, n) is given by divisors meeting all faces trivially, modulo the restriction to the last face of divisors on Q^{n+1} meeting all but the last face trivially. Scaling the f above to remove the factor k^{×}, this is precisely the assertion of the lemma. □

Proposition 7.2. We have SH^{1}(k, 1) = k, and SH^{1}(k, n) = (0) for n ≥ 2.

Proof. The first assertion is part of theorem 5.3. The exact sequence

(7.3)  0 → I^{n+1} → J^{n+1} ⟶^{∂_{n+1}} I^{n} → 0

yields the surjectivity 1+J^{n+1} ↠ 1+I^{n}. We observe that when n = 1, we have I^{1} = (t_{0}) but ∂_{2}(J^{2}) = (t_{0}^{2}). □

References

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[2] Beilinson, A., Goncharov, A., Schechtman, V., Varchenko, A.: Aomoto dilogarithms, mixed Hodge structures, and motivic cohomology of a pair of triangles in the plane, The Grothendieck Festschrift, Birkhäuser (1990), 131-172.

[3] Bloch, S.: Algebraic cycles and higher K-theory, Adv. in Math. 61 (1986), no 3, 267-304.

[4] Brieskorn, E.: Sur les groupes de tresses. In Sém. Bourbaki 401, Lect. Notes in Math. 317, 21-44, Berlin Heideleberg New York, Springer (1973).

[5] Cathelineau, J-L.: Remarques sur les différentielles des polylogarithmes uniformes, Ann. Inst. Fourier, Grenoble 46 (1996) no. 5, 1327-1347.

[6] Goncharov, A.: Volumes of hyperbolic manifolds and mixed Tate motives, JAMS 12 (1999), no 2, 569-618.

[7] Nesterenko, Yu., Suslin, A.: Homology of the general linear group over a local ring, and Milnor’s K-theory, Izv. Akad. Nauk SSSR Ser. Math. 53 (1989), no 1, 121-146, translation in Math. USSR-Izv. 34 (1990), no 1, 121-145.

[8] Totaro, B.: Milnor K-theory is the simplest part of algebraic K-theory, K-Theory 6 (1992), no 2, 177-189.

[9] Voevodsky, V.: Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. Math. Res. Not. 7 ( 2002), 7, 351-355.

Dept. of Mathematics, University of Chicago, Chicago, IL 60637, USA

E-mail address: bloch@math.uchicago.edu

Mathematik, Universität Essen, FB6, Mathematik, 45117 Essen, Germany

E-mail address: esnault@uni-essen.de