PRODUCTS IN HIGHER CHOW GROUPS AND MOTIVIC COHOMOLOGY
PRODUCTS IN HIGHER CHOW GROUPS AND MOTIVIC COHOMOLOGY
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高级 Chow 群中的积和 Motivic 上同调
DOI：
10.1090/pspum/067/1743246
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发表时间：
1998
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影响因子：
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通讯作者：
C. Weibel
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文献类型：
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作者：
C. Weibel
我们证明了一个相容性定理的各种产品已被定义在几个上同调理论：布洛赫的高周群CH（X，n）[B1];双变循环上同调Ar，n（S，X）[FV]; Voevodsky的motivic上同调H j M（X;Z（i））[V1];和étale上同调（有限系数）。最近，几个循环上同调群已被引入有限型方案在一个域K。其中包括Bloch [B1]的高阶Chow群CH（X，n），Friedlander和Voevodsky [FV]的双变循环上同调群Arn（X）= Arn（k，X），Voevodsky [V1]的动机上同调群HM（X，Z（i））.在合理的情况下，例如当X是光滑的并且域k允许奇点的分解时（例如，char（k）= 0），我们通过[FV]，[S2]和[V1]知道这些群是同构的：如果我们设置m + n = 2 i和r + i = dim（X），则（0.1）CH（X，n）n = Arn（X）n = H m M（X，Z（i））。本文的主要结果是这些同构分别标识了[B1]，[FV]和[V1]中定义的各种上同调积。主要定理0.2.设X在域k上是光滑的，且K上存在奇点分解。然后，（0.1）的同构确定了Bloch高阶Chow群上的乘积，双变循环上同调的乘积和动机上同调的乘积。也就是说，我们将识别布洛赫的产品[B1，5.7]（0.3）CH（X，m）= CH（X，n）→ CH（X，m + n），Friedlander-Voevodsky乘积[FV，8.6]（0.4）Arm（X）<$Asn（X）→ Ar+ s-d，m+n（X），d = dim（X），以及Voevodsky的动机上同调积[V1]（0.5）H M（X，Z（i））→ H m+n M（X，Z（i + j））。我们还将证明，从有限系数motivic上同调到étale上同调[SV]的自然映射也与乘积相容。
We prove a compatibility theorem for the various products which have been defined on several cohomology theories: Bloch’s higher Chow groups CH (X,n) [B1]; Bivariant cycle cohomology Ar,n(S,X) [FV]; Voevodsky’s motivic cohomology H j M (X;Z(i)) [V1]; and étale cohomology (for finite coefficients). Recently, several cycle cohomology groups have been introduced for schemes of finite type over a field k. These include the higher Chow groups CH (X, n) of Bloch [B1], the bivariant cycle cohomology groups Arn(X) = Arn(k, X) of Friedlander and Voevodsky [FV], and the motivic cohomology groups H M(X, Z(i)) of Voevodsky [V1]. In reasonable situations, such as when X is smooth and the field k admits resolution of singularities (e.g., char(k) = 0), we know by [FV], [S2] and [V1] that these groups are isomorphic: if we set m + n = 2i and r + i = dim(X) then (0.1) CH(X, n) ∼= Arn(X) ∼= H m M(X, Z(i)). The main result in this paper is that these isomorphisms identify the various cohomology products which have been defined in [B1], [FV] and [V1] respectively. Main Theorem 0.2. Let X be smooth over a field k which admits resolution of singularities. Then the isomorphism of (0.1) identifies the product on Bloch’s higher Chow groups, the product on bivariant cycle cohomology, and the product on motivic cohomology. That is, we shall identify Bloch’s product [B1, 5.7] (0.3) CH(X, m)⊗ CH(X, n)→ CH(X, m + n), the Friedlander-Voevodsky product [FV, 8.6] (0.4) Arm(X)⊗Asn(X)→ Ar+s−d,m+n(X), d = dim(X), and Voevodsky’s product in motivic cohomology [V1] (0.5) H M(X, Z(i))⊗H n M(X, Z(j))→ H m+n M (X, Z(i + j)). We will also show that the natural map from motivic cohomology with finite coefficients to étale cohomology [SV] is also compatible with products.