Arithmetic homology and an integral version of Katos conjecture

TL;DR

This paper introduces an integral homology theory over finite fields, called arithmetic homology, and an integral version of Kato homology, proposing their properties and relationships with higher Chow groups.

Contribution

It defines new integral homology theories over finite fields and explores their expected properties and connections to existing algebraic structures.

Findings

Proposes an integral Borel-Moore homology theory over finite fields.

Introduces an integral version of Kato homology.

Suggests these groups are finitely generated and relate to higher Chow groups.

Abstract

We define an integral Borel-Moore homology theory over finite fields, called arithmetic homology, and an integral version of Kato homology. Both types of groups are expected to be finitely generated, and sit in a long exact sequence with higher Chow groups of zero-cycles.

Equations

$$0\to C{H}_0(\bar{X},i+1{)}_G\to C{H}_0(X,i)\to C{H}_0(\bar{X},i{)}^G\to 0.$$

$$\cdots \to {\oplus }_{x\in X_{(s)}}{K}_s^M(k(x){\otimes }_{{\mathbb{F}}_q}{\bar{\mathbb{F}}}_q{)}_G\to {\oplus }_{x\in X_{(1)}}(k(x){\otimes }_{{\mathbb{F}}_q}{\bar{\mathbb{F}}}_q{)}_G^{\times }\to \cdots \to {\oplus }_{x\in X_{(0)}}\mathbb{Z},$$

$$\cdots \to H_i^K(X,\mathbb{Z})\stackrel{\times m}{⟶}{H}_i^K(X,\mathbb{Z})\to H_i^K(X,\mathbb{Z}/m)\to \cdots ,$$

$$\cdots \to C{H}_0(X,i)\to H_{i+1}^c(X_{\text{ar}},\mathbb{Z})\to H_{i+1}^K(X,\mathbb{Z})\to C{H}_0(X,i-1)\to \cdots .$$

$$0\to (k(C){\otimes }_{{\mathbb{F}}_q}{\bar{\mathbb{F}}}_q{)}_G^{\times }\to \underset{x\in C_{(0)}}{⨁}\mathbb{Z}\to \mathbb{Z}\to 0.$$

$$0\to (k(C){\otimes }_{{\mathbb{F}}_q}{\bar{\mathbb{F}}}_q{)}_G^{\times }\to k(C{)}_{\mathbb{Q}}^{\times }\to \mathrm{Br}k(C)\to 0,$$

$$0\to C{H}_0(C)\to H_1^c(C_{\text{ar}},\mathbb{Z})\to H_1^K(C,\mathbb{Z})\to 0,$$

$$C{H}_0(X)\stackrel{{\theta }_X}{\to }{H}_1^c(X_{\text{ar}},\mathbb{Z})\to {\pi }_1(X{)}^{ab}.$$

$$C{H}_n(X,i-2n)=H_i(X_{\text{Zar}},{\mathbb{Z}}^c(n))=:H_i^c(X_{\text{Zar}},\mathbb{Z}(n))$$

$$E_{s,t}^1={\oplus }_{x\in X_{(s)}}{H}^{s-t}(k(x),\mathbb{Z}(s-n))\Rightarrow H_{s+t}^c(X_{\text{Zar}},\mathbb{Z}(n)).$$

$$H_i^c(X_{\text{et}},A(n))=H^{-i}R\Gamma (X_{\text{et}},A\otimes {\mathbb{Z}}^c(n)).$$

$$R{f}^{!}\mathbb{Z}/m\cong {\mathbb{Z}}^c/m(0).$$

$$C{H}_n(X,i-2n)=H_i^c(X_{\text{Zar}},\mathbb{Z}(n))\to H_i^c(X_{\text{et}},\mathbb{Z}(n))$$

$$E_{s,t}^1={\oplus }_{x\in X_{(s)}}{H}^{s-t}(k(x{)}_{\text{et}},\mathbb{Z}(s-n))\Rightarrow H_{s+t}^c(X_{\text{et}},\mathbb{Z}(n)).$$

$$H_i^c(X_{\text{ar}},A(n))=H^{1-i}R{\Gamma }_{G}R\Gamma ({\bar{X}}_{\text{et}},A\otimes {\mathbb{Z}}^c(n)).$$

$$0\to H_i^c({\bar{X}}_{\text{et}},\mathbb{Z}(n){)}_G\to H_i^c(X_{\text{ar}},\mathbb{Z}(n))\to H_{i-1}^c({\bar{X}}_{\text{et}},\mathbb{Z}(n){)}^G\to 0.$$

$$H_i^c(X_{\text{ar}},\mathbb{Z}(n))\cong H^{2d+1-i}(X_W,\mathbb{Z}(d-n)),$$

$$\cdots \to H^i(X_{\text{et}},\mathbb{Z}(p))\to H^i(X_W,\mathbb{Z}(p))\to H^{i-1}(X_{\text{et}},\mathbb{Q}(p))\to \cdots .$$

$$\begin{gathered} \cdots \to H_{i-1}^c(X_{\text{et}},\mathbb{Z}(n))\to H_i^c(X_{\text{ar}},\mathbb{Z}(n)) \\ \to C{H}_n(X,i-2n{)}_{\mathbb{Q}}\to H_{i-2}^c(X_{\text{et}},\mathbb{Z}(n))\to \cdots , \end{gathered}$$

$$H_i^c(X_{\text{et}},\mathbb{Z}/m(n))\cong H_{i+1}^c(X_{\text{ar}},\mathbb{Z}/m(n)).$$

$$H_i^c(X_{\text{ar}},\mathbb{Q}(n))\cong C{H}_n(X,i-2n{)}_{\mathbb{Q}}\oplus C{H}_n(X,i-2n-1{)}_{\mathbb{Q}}.$$

$$\cdots \to H_i^c(Z_{\text{ar}},\mathbb{Z}(n))\to H_i^c(X_{\text{ar}},\mathbb{Z}(n))\to H_i^c(U_{\text{ar}},\mathbb{Z}(n))\to \cdots$$

$$H_i^c({\mathbb{P}}^r\times X_{\text{ar}},\mathbb{Z}(n))\cong {\oplus }_{j=0}^r{H}_{i-2j}^c(X_{\text{ar}},\mathbb{Z}(n-j)).$$

$$H_i^c({\mathbb{A}}^r\times X_{\text{ar}},\mathbb{Z}(n))\cong H_{i-2r}^c(X_{\text{ar}},\mathbb{Z}(n-r)).$$

$${\mathbb{Z}}^c(n)(\bar{X})\otimes A[G]\stackrel{1-\phi }{\to }{\mathbb{Z}}^c(n)(\bar{X})\otimes A[G],$$

$$0\to C{H}_n(\bar{X},i-2n{)}_G\to H_i^c(X_{\text{ar}},\mathbb{Z}(n))\to C{H}_n(\bar{X},i-1-2n{)}^G\to 0.$$

Full text

1 Introduction

For a separated scheme $X$ of finite type over a field $k$, let ${{\mathbb{Z}}}^{c}(0)(X)$ be Bloch’s complex of relative zero-cycles, generated in degree $-i$ by cycles of dimension $i$ on $X\times\Delta^{i}$ in good position. The higher Chow groups $CH_{0}(X,i)$ are defined as the homology $H_{i}({{\mathbb{Z}}}^{c}(0)(X))$ of ${{\mathbb{Z}}}^{c}(0)(X)$. Varying $X$, we obtain a complex of etale sheaves ${{\mathbb{Z}}}^{c}(0)$ and can consider its etale hypercohomology $H_{i}(X_{\text{\rm et}},{{\mathbb{Z}}}^{c}(0))$. If $k$ is algebraically closed, then it is a consequence of the Beilinson-Lichtenbaum conjecture that $CH_{0}(X,i)\cong H_{i}(X_{\text{\rm et}},{{\mathbb{Z}}}^{c}(0))$ (ichdual, Thm.3.1), but in general, these groups are not isomorphic. Over a finite field, Kato kato defined for each $m$ a complex with homology $H_{i}^{K}(X,{{\mathbb{Z}}}/m)$ and conjectured that if $X$ is smooth, proper and connected, then $H_{i}^{K}(X,{{\mathbb{Z}}}/m)$ vanishes for $i>0$, and $H_{0}^{K}(X,{{\mathbb{Z}}}/m)\cong{{\mathbb{Z}}}/m$. Jannsen and Saito jannsensaitoalt observed that Kato homology measures the difference between the finite coefficient versions $CH_{0}(X,i,{{\mathbb{Z}}}/m)$ and $H_{i}(X_{\text{\rm et}},{{\mathbb{Z}}}^{c}/m(0))$, and proved Kato’s conjecture assuming resolution of singularities jannsensaito.

In this paper, we construct Borel-Moore homology groups $H_{i}^{c}(X_{\text{\rm ar}},{{\mathbb{Z}}})$, which are a substitute for the (pathological) etale higher Chow groups $H_{i}(X_{\text{\rm et}},{{\mathbb{Z}}}^{c}(0))$, and define an integral version of Kato’s complex whose homology groups $H_{i}^{K}(X,{{\mathbb{Z}}})$ measure the difference between $CH_{0}(X,i)$ and $H_{i}^{c}(X_{\text{\rm ar}},{{\mathbb{Z}}})$. The analog of Kato’s conjecture is that if $X$ is smooth, proper and connected, then $H_{i}^{K}(X,{{\mathbb{Z}}})$ vanishes for $i>0$, and $H_{i}^{K}(X,{{\mathbb{Z}}})={{\mathbb{Z}}}$.

The Borel-Moore homology theory, which we call arithmetic homology, is constructed by applying the Weil-etale formalism of Lichtenbaum to ${{\mathbb{Z}}}^{c}(0)$: Given a scheme $X$ over the finite field ${{\mathbb{F}}}_{q}$, $H_{i}^{c}(X_{\text{\rm ar}},{{\mathbb{Z}}})$ is the $-i$th cohomology group of the complex $R\Gamma_{G}R\Gamma(\bar{X}_{\text{\rm et}},{{\mathbb{Z}}}^{c}(0))[1]$, where $\bar{X}=X\times_{{{\mathbb{F}}}_{q}}\bar{{\mathbb{F}}}_{q}$, and $G$ is the Weil group of ${{\mathbb{F}}}_{q}$ acting on $\bar{{\mathbb{F}}}_{q}$. These groups are expected to be finitely generated. More precisely, we have

Theorem 1.1

Let $X$ be smooth and proper. Assume resolution of singularities up to the dimension of $X$ and the Beilinson-Lichtenbaum conjecture (see below). Then the following statements are equivalent:

1. a)

$CH_{0}(X,i)_{{\mathbb{Q}}}=0$* for all $i>0$.* 2. b)

There is an isomorphism $CH_{0}(X,i)\cong H_{i+1}^{c}(X_{\text{\rm ar}},{{\mathbb{Z}}})$ of finitely generated abelian groups for all $i\geq 0$. 3. c)

There are short exact sequences for all $i$,

[TABLE]

The vanishing of the Chow groups is a special case of Parshin’s conjecture $K_{i}(X)_{{\mathbb{Q}}}=0$ for $i>0$, and in particular it follows from Tate’s conjecture together with Beilinson’s conjecture that rational and homological equivalence agree up to torsion ichtate over finite fields, or from finite dimensionality of smooth and projective schemes over ${{\mathbb{F}}}_{q}$ in the sense of Kimura-O’Sullivan ichparshin.

The integral analog of Kato homology comes into play when comparing higher Chow groups with arithmetic homology for not necessarily smooth or proper schemes over ${{\mathbb{F}}}_{q}$: Consider the complex

[TABLE]

where $X_{(s)}$ denote the points of $X$ of dimension $s$, and $K_{s}^{M}(k(x)\otimes_{{{\mathbb{F}}}_{q}}\bar{{\mathbb{F}}}_{q})_{G}$ is the group of Frobenius coinvariants of the Milnor $K$-group of the finite product of fields $k(x)\otimes_{{{\mathbb{F}}}_{q}}\bar{{\mathbb{F}}}_{q}$. The maps in the complex are induced by boundary maps in localization sequences of higher Chow groups via the identification $K_{s}^{M}(k(x)\otimes_{{{\mathbb{F}}}_{q}}\bar{{\mathbb{F}}}_{q})\cong H^{s}(k(x)\otimes_{{{\mathbb{F}}}_{q}}\bar{{\mathbb{F}}}_{q},{{\mathbb{Z}}}(s))$. The group $H_{i}^{K}(X,{{\mathbb{Z}}})$ is the $i$th homology of this complex. Assuming the Beilinson-Lichtenbaum conjecture, we have a long exact sequence

[TABLE]

which justifies calling the groups $H_{i}^{K}(X,{{\mathbb{Z}}})$ an integral version of Kato homology. As an analog of Kato’s conjecture, we propose the following

Conjecture 1.2

If $X$ is smooth, proper and connected, then $H_{i}^{K}(X,{{\mathbb{Z}}})=0$ for $i>0$ and $H_{0}^{K}(X,{{\mathbb{Z}}})\cong{{\mathbb{Z}}}$.

It is easy to use the known results on the (torsion) Kato conjecture to show that the conjecture is true in degree [math], and that $H_{i}^{K}(X,{{\mathbb{Z}}})\cong CH_{0}(X,i)_{{\mathbb{Q}}}$ for smooth and proper $X$ and $i=1,2$. Regarding the conjecture and the relationship between higher Chow groups and arithmetic homology, we show

Theorem 1.3

Assuming resolution of singularities and the Beilinson-Lichtenbaum conjecture, the following statements are equivalent:

a) For every smooth and proper $X$ over ${{\mathbb{F}}}_{q}$, $CH_{0}(X,i)_{{\mathbb{Q}}}=0$ for $i>0$.

b) Conjecture 1.2 holds, and for every separated scheme of finite type $X$ over ${{\mathbb{F}}}_{q}$, there is a long exact sequence of finitely generated groups

[TABLE]

The exact sequence of the Theorem exists unconditionally in degrees $i\leq 1$. We show that Conjecture 1.2 holds for curves, i.e. there is an exact sequence

[TABLE]

for smooth and proper curves $C$. Since the sequence comparing Weil-etale to etale cohomology ichweil gives a short exact sequence

[TABLE]

we obtain $(k(C)\otimes_{{{\mathbb{F}}}_{q}}\bar{{\mathbb{F}}}_{q})^{\times}_{G}\otimes{{\mathbb{Q}}}/{{\mathbb{Z}}}\cong\operatorname{Br}k(C)$, and (1) is an integral version of the classical short exact sequence for the Brauer group of $k(C)$. For an arbitrary curve $C$ over ${{\mathbb{F}}}_{q}$, we obtain $H_{0}^{c}(C_{\text{\rm ar}},{{\mathbb{Z}}})\cong H_{0}^{K}(C,{{\mathbb{Z}}})$, there is a short exact sequence

[TABLE]

and $CH_{0}(C,i)\cong H_{i+1}^{c}(C_{\text{\rm ar}},{{\mathbb{Z}}})$ for $i\geq 1$. The latter groups are finitely generated for $i=1$ and zero for $i>1$. For curves, Kato homology is easy to calculate recursively; for example if $C$ is proper with dual graph $\Gamma_{C}$, then $H_{i}^{K}(C,{{\mathbb{Z}}})=H_{i}(\Gamma_{C},{{\mathbb{Z}}})$.

Arithmetic homology can be applied to study abelian class field theory of proper schemes. The main observation is that the group $H_{1}^{c}(X_{\text{\rm ar}},{{\mathbb{Z}}})$ (which is conjecturally finitely generated) becomes isomorphic to the abelianized fundamental group $\pi_{1}(X)^{ab}$ after profinite completion, and the reciprocity map factors as

[TABLE]

Notation: For an abelian group $A$, $A^{\wedge}=\operatornamewithlimits{lim}A/n$ is the profinite completion, and $A^{*}=\operatorname{Hom}(A,{{\mathbb{Q}}}/{{\mathbb{Z}}})$ the Pontrjagin dual. All schemes over a field are separated and of finite type; from section 3 we fix a finite field as the base field.

Acknowledgements: Parts of this paper were written while the author visited the University of Tokyo, and we thank T.Saito and the University for the inspiring atmosphere they provided. We are indebted to S.Saito for inspiring discussions.

2 Higher Chow groups

For a scheme $X$ over a field $k$, Bloch’s higher Chow complex $z_{n}(X,*)$ is defined as follows bloch. In degree $-i$, it is be the free abelian group $z_{n}(X,i)$ generated by cycles of dimension $n+i$ on $X\times_{k}\Delta^{i}$ which meet all faces properly. The differentials are given by taking the alternating sum of intersection with face maps. Higher Chow groups $CH_{n}(X,i)$ are defined as the homology of this complex. We let ${{\mathbb{Z}}^{c}}(n)_{X}=z_{n}(-,*)[2n]$ be the complex of etale sheaves on $X$ with $z_{n}(-,-2n-i)$ in degree $i$. For a proper map $f:X\to Y$ we have a push-forward $f_{*}{{\mathbb{Z}}^{c}}(n)_{X}\to{{\mathbb{Z}}^{c}}(n)_{Y}$, and for a flat, equidimensional map $f:X\to Y$ of relative dimension $d$, we have a pull-back $f^{*}{{\mathbb{Z}}^{c}}(n)_{Y}\to{{\mathbb{Z}}^{c}}(n+d)_{X}[-2d]$. If $X$ is smooth of dimension $d$, then there is a quasi-isomorphism of complexes of Zariski sheaves ${{\mathbb{Z}}^{c}}(n)\cong{{\mathbb{Z}}}(d-n)[2d]$, where the right hand side is the motivic complex of Voevodsky voevodsky. For a finitely generated field $F$ over $k$, we define $CH_{n}(F,i)=\operatornamewithlimits{colim}_{U}CH_{n}(U,i)$, where the colimit runs through $U$ of finite type over $k$ with field of functions $F$. If $F$ has transcendence degree $d$ over $k$, then $CH_{n}(F,i)\cong H^{2d-2n-i}(F,{{\mathbb{Z}}}(d-n))$, where the right hand side is motivic cohomology, and vanishes for $i<d-n$, and agrees with $K_{d-n}^{M}(F)$ for $i=d-n$. As a formal consequence of localization for higher Chow groups, one obtains an isomorphism

[TABLE]

and spectral sequences

[TABLE]

In particular, $H_{i}^{c}(X_{\text{\rm Zar}},{{\mathbb{Z}}}(n))=0$ for $i<n$. For an abelian group $A$, we define

[TABLE]

If $A={{\mathbb{Q}}}$, then $H_{i}^{c}(X_{\text{\rm et}},{{\mathbb{Q}}}(n))=H_{i}^{c}(X_{\text{\rm Zar}},{{\mathbb{Q}}}(n))=CH_{n}(X,i-2n)_{{\mathbb{Q}}}$. In ichdual, we proved that for every integer $m$ and every scheme $f:X\to k$ over a perfect field $k$, there is a quasi-isomorphism

[TABLE]

Here $Rf^{!}$ is the extraordinary inverse image of SGA 4 XVIII. Hence the etale homology groups $H_{i}^{c}(X_{\text{\rm et}},{{\mathbb{Z}}}/m(0))$ with coefficients ${{\mathbb{Z}}}^{c}/m(0)$ agree with usual etale homology groups $H_{i}(X_{\text{\rm et}},{{\mathbb{Z}}}/m):=H^{-i}(X_{\text{\rm et}},Rf^{!}{{\mathbb{Z}}}/m)$ of Laumon laumon. For a finitely generated field $F$ over $k$, we define $H_{i}^{c}(F_{\text{\rm et}},A(n))=\operatornamewithlimits{colim}_{U}H_{i}^{c}(U_{\text{\rm et}},A(n))$.

The Beilinson-Lichtenbaum conjecture over $k$ in homological weight $n$ says that if $X$ is a smooth scheme of dimension $d$ over $k$, then the canonical map

[TABLE]

is an isomorphism for $i\geq n+d-1$. By considering cohomological dimension, it then must be an isomorphism for all $i$ if $k$ is algebraically closed and $n\leq 0$. As a special case, the conjecture implies that $H^{d-n+1}(F_{\text{\rm et}},{{\mathbb{Z}}}(d-n))=0$ for fields $F$ of transcendence degree $d$ over $k$; this statement is often called ”Hilbert’s Theorem 90”. Over an algebraically closed field, Suslin suslinetale shows that the groups $CH_{0}(X,i,{{\mathbb{Z}}}/m)$ and $H_{i}^{c}(X_{\text{\rm et}},{{\mathbb{Z}}}/m(0))$ are isomorphic for $\operatorname{char}k\not|m$, but it is not clear that the isomorphism is induced by the canonical change of topology map. In marcI, it is shown that the canonical map is an isomorphism if $m$ is a power of the characteristic. In ichdual, we use Suslin’s theorem to show the following result.

Proposition 2.1

Let $k$ be a perfect field and $n\leq 0$.

a) (Localization) For a closed embedding $i:Z\to X$, the canonical map ${\mathbb{Z}}^{c}_{Z}(n)\to Ri^{!}{\mathbb{Z}}^{c}_{X}(n)$ is a quasi-isomorphism.

b) (Homotopy formula) If $p:X\times{\mathbb{A}}^{m}\to X$ is the projection, then we have a quasi-isomorphism of complexes of etale sheaves $Rp_{*}{\mathbb{Z}}^{c}_{X\times{\mathbb{A}}^{m}}(n)\cong{\mathbb{Z}}^{c}_{X}(n-m)$.

c) (Niveau spectral sequence) There is a spectral sequence

[TABLE]

In particular, $H_{i}^{c}(X_{\text{\rm et}},{{\mathbb{Z}}}(n))=0$ for $i<n$.

The last statement follows because $s+t<n$ implies that $s<n$ or $t<0$, both of which imply that $E_{s,t}^{1}$ vanishes. Note that by the homotopy formula, the Beilinson-Lichtenbaum conjecture in homological weight $n\leq 0$ implies the Beilinson-Lichtenbaum conjecture in all weights less than $n$.

3 Arithmetic homology with compact support

We fix a finite field ${{\mathbb{F}}}_{q}$ with Galois group $\hat{G}=\operatorname{Gal}(\bar{{\mathbb{F}}}_{q}/{{\mathbb{F}}}_{q})$ and let $G\subset\hat{G}$ be the Weil group of ${{\mathbb{F}}}_{q}$, i.e. the subgroup of $\hat{G}$ generated by the Frobenius endomorphism $\varphi$. Given a separated scheme of finite type $X$ over ${{\mathbb{F}}}_{q}$, let $\bar{X}=X\times_{{{\mathbb{F}}}_{q}}\bar{{\mathbb{F}}}_{q}$. For an abelian group $A$, we define arithmetic homology groups with compact support and coefficients in $A$ as the homology groups of the complex $R\Gamma_{G}R\Gamma(\bar{X}_{\text{\rm et}},A\otimes{{\mathbb{Z}}^{c}}(n))[1]$, where $G$ acts on $\bar{{\mathbb{F}}}_{q}$, so that

[TABLE]

The Leray spectral sequence for composition of functors degenerates into short exact sequences

[TABLE]

In particular, $H_{i}^{c}(X_{\text{\rm ar}},{{\mathbb{Z}}}(n))=0$ for $i<n\leq 0$ by Proposition 2.1. If $X$ is smooth of dimension $d$, then the quasi-isomorphism ${{\mathbb{Z}}}^{c}(n)\cong{{\mathbb{Z}}}(d-n)[2d]$ implies

[TABLE]

where the right hand side are the Weil-etale cohomology groups of licht; ichweil. (For arbitrary $X$ of finite type over ${{\mathbb{F}}}_{q}$, one has to replace etale cohomology by eh-cohomology and Weil-etale cohomology by arithmetic cohomology of ichweilII to obtain well-behaved cohomology groups). Recall from (ichweil, Thm.7.1) that for every smooth $X$ over ${{\mathbb{F}}}_{q}$, there is a long exact sequence

[TABLE]

The same proof shows

Theorem 3.1

There are long exact sequences

[TABLE]

hence for every integer $m\geq 1$ an isomorphism

[TABLE]

With rational coefficients, the long exact sequence splits, and

[TABLE]

$\Box$

From now on, we assume that $n\leq 0$. Then for a closed subscheme $Z$ of $X$ with open complement $U$ we have by Proposition 2.1 localization sequences

[TABLE]

Corollary 3.2

Let $n\leq 0$.

a) (Projective bundle formula) Let ${\mathbb{P}}^{r}_{X}\to X$ be a projective bundle of relative dimension $r$. Then

[TABLE]

b) (Homotopy formula) For every $r\geq 0$,

[TABLE]

Proof

The homotopy formula follows from Prop. 2.1 via (7), and the projective bundle formula can be derived from this using localization and induction. $\Box$

The Beilinson-Lichtenbaum conjecture over $\bar{{\mathbb{F}}}_{q}$ implies that, for $n\leq 0$, $R\Gamma(X_{\text{\rm ar}},A\otimes{{\mathbb{Z}}^{c}}(n))$ has the explicit representative

[TABLE]

with $\varphi$ acting diagonally, and that the sequence (7) takes the form

[TABLE]

For example, $H_{0}^{c}(X_{\text{\rm ar}},{{\mathbb{Z}}}(0))=CH_{0}(\bar{X})_{G}$.

In the following, we restrict ourselves to the case $n=0$ and write $H_{i}^{c}(X_{\text{\rm ar}},A)$ for $H_{i}^{c}(X_{\text{\rm ar}},A(0))$. This is no loss of generality in view of the homotopy formula.