Bloch's higher Chow group

In algebraic geometry, Bloch's higher Chow groups, a generalization of Chow group, is a precursor and a basic example of motivic cohomology (for smooth varieties). It was introduced by Spencer Bloch Template:Harv and the basic theory has been developed by Bloch and Marc Levine.

In more precise terms, a theorem of Voevodsky implies: for a smooth scheme X over a field and integers p, q, there is a natural isomorphism

${\mathrm{H}}^p(X;\mathbb{Z}(q))\simeq {\mathrm{CH}}^q(X,2q-p)$

between motivic cohomology groups and higher Chow groups.

Motivation

One of the motivations for higher Chow groups comes from homotopy theory. In particular, if

$\alpha ,\beta \in Z_{*}(X)$

are algebraic cycles in

$X$

which are rationally equivalent via a cycle

$\gamma \in Z_{*}(X\times {\mathrm{\Delta }}^1)$

, then

$\gamma$

can be thought of as a path between

$\alpha$

and

$\beta$

, and the higher Chow groups are meant to encode the information of higher homotopy coherence. For example,

${\text{CH}}^{*}(X,0)$

can be thought of as the homotopy classes of cycles while

${\text{CH}}^{*}(X,1)$

can be thought of as the homotopy classes of homotopies of cycles.

Definition

Let X be a quasi-projective algebraic scheme over a field (“algebraic” means separated and of finite type).

For each integer $q\ge 0$, define

${\mathrm{\Delta }}^q=\mathrm{Spec}(\mathbb{Z}[t_0,\dots ,t_q]/(t_0+\dots +t_q-1)),$

which is an algebraic analog of a standard q-simplex. For each sequence $0\le i_1<i_2<\cdots <i_r\le q$, the closed subscheme $t_{i_1}=t_{i_2}=\cdots =t_{i_r}=0$, which is isomorphic to ${\mathrm{\Delta }}^{q-r}$, is called a face of ${\mathrm{\Delta }}^q$.

For each i, there is the embedding

${\partial }_{q,i}:{\mathrm{\Delta }}^{q-1}\stackrel{\sim }{\to }\{t_i=0\}\subset {\mathrm{\Delta }}^q.$

We write $Z_i(X)$ for the group of algebraic i-cycles on X and $z_r(X,q)\subset Z_{r+q}(X\times {\mathrm{\Delta }}^q)$ for the subgroup generated by closed subvarieties that intersect properly with $X\times F$ for each face F of ${\mathrm{\Delta }}^q$.

Since ${\partial }_{X,q,i}={\mathrm{id}}_X\times {\partial }_{q,i}:X\times {\mathrm{\Delta }}^{q-1}\hookrightarrow X\times {\mathrm{\Delta }}^q$ is an effective Cartier divisor, there is the Gysin homomorphism:

$\underset{X,q,i}{\overset{*}{\partial }}:z_r(X,q)\to z_r(X,q-1)$,

that (by definition) maps a subvariety V to the intersection $(X\times \{t_i=0\})\cap V.$

Define the boundary operator $d_q=\sum _{i=0}^q(-1{)}^i\underset{X,q,i}{\overset{*}{\partial }}$ which yields the chain complex

$\cdots \to z_r(X,q)\stackrel{d_q}{\to }{z}_r(X,q-1)\stackrel{d_{q-1}}{\to }\cdots \stackrel{d_1}{\to }{z}_r(X,0).$

Finally, the q-th higher Chow group of X is defined as the q-th homology of the above complex:

${\mathrm{CH}}_r(X,q):={\mathrm{H}}_q(z_r(X,\cdot )).$

(More simply, since $z_r(X,\cdot )$ is naturally a simplicial abelian group, in view of the Dold–Kan correspondence, higher Chow groups can also be defined as homotopy groups ${\mathrm{CH}}_r(X,q):={\pi }_q{z}_r(X,\cdot )$.)

For example, if $V\subset X\times {\mathrm{\Delta }}^1$ is a closed subvariety such that the intersections $V(0),V(\mathrm{\infty })$ with the faces $0,\mathrm{\infty }$ are proper, then $d_1(V)=V(0)-V(\mathrm{\infty })$ and this means, by Proposition 1.6. in Fulton’s intersection theory, that the image of $d_1$ is precisely the group of cycles rationally equivalent to zero; that is,

${\mathrm{CH}}_r(X,0)=$ the r-th Chow group of X.

Properties

Functoriality

Proper maps $f:X\to Y$ are covariant between the higher chow groups while flat maps are contravariant. Also, whenever $Y$ is smooth, any map to $Y$ is contravariant.

Homotopy invariance

If

$E\to X$

is an algebraic vector bundle, then there is the homotopy equivalence

${\text{CH}}^{*}(X,n)\cong {\text{CH}}^{*}(E,n)$

Localization

Given a closed equidimensional subscheme

$Y\subset X$

there is a localization long exact sequence

$\begin{array}{c}\cdots \\ {\text{CH}}^{*-d}(Y,2)\to {\text{CH}}^{*}(X,2)\to {\text{CH}}^{*}(U,2)\to & \\ {\text{CH}}^{*-d}(Y,1)\to {\text{CH}}^{*}(X,1)\to {\text{CH}}^{*}(U,1)\to & \\ {\text{CH}}^{*-d}(Y,0)\to {\text{CH}}^{*}(X,0)\to {\text{CH}}^{*}(U,0)\to & 0\end{array}$

where

$U=X-Y$

. In particular, this shows the higher chow groups naturally extend the exact sequence of chow groups.

Localization theorem

Template:Harv showed that, given an open subset $U\subset X$, for $Y=X-U$,

$z(X,\cdot )/z(Y,\cdot )\to z(U,\cdot )$

is a homotopy equivalence. In particular, if $Y$ has pure codimension, then it yields the long exact sequence for higher Chow groups (called the localization sequence).

References

• Template:Cite journal
• Template:Cite journal
• Peter Haine, An Overview of Motivic Cohomology
• Vladmir Voevodsky, “Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic,” International Mathematics Research Notices 7 (2002), 351–355.