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 {{harv|Bloch|1986}} and the basic theory has been developed by Bloch and Marc Levine.

In more precise terms, a theorem of Voevodsky{{Cite book|url=http://www.claymath.org/library/monographs/cmim02.pdf|title=Lecture Notes on Motivic Cohomology|publisher=Clay Math Monographs|pages=159}} implies: for a smooth scheme X over a field and integers p, q, there is a natural isomorphism

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

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 $\gamma \in Z_*(X\times \Delta^1)$ , 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

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

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 $\Delta^{q-r}$ , is called a face of $\Delta^q$ .

For each i, there is the embedding

: $\partial_{q, i}: \Delta^{q-1} \overset{\sim}\to \{ t_i = 0 \} \subset \Delta^q.$

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

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

: $\partial_{X, q, i}^*: z_r(X, q) \to z_r(X, q-1)$ ,

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

Define the boundary operator $d_q = \sum_{i=0}^q (-1)^i \partial_{X, q, i}^*$ which yields the chain complex

: $\cdots \to z_r(X, q) \overset{d_q}\to z_r(X, q-1) \overset{d_{q-1}}\to \cdots \overset{d_1}\to z_r(X, 0).$

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

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

(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 $\operatorname{CH}_r(X, q) := \pi_q z_r(X, \cdot)$ .)

For example, if $V \subset X \times \Delta^1$ Here, we identify $\Delta^1$ with a subscheme of $\mathbb{P}^1$ and then, without loss of generality, assume one vertex is the origin 0 and the other is ∞. is a closed subvariety such that the intersections $V(0), V(\infty)$ with the faces $0, \infty$ are proper, then

$d_1(V) = V(0) - V(\infty)$ 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,

: $\operatorname{CH}_r(X, 0) =$ 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

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

where

U = X-Y

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

Localization theorem

{{harv|Bloch|1994}} 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

{{cite journal | last1=Bloch | first1=Spencer | title=Algebraic cycles and higher K-theory | journal=Advances in Mathematics | volume=61 | date=September 1986 | pages=267–304 | doi=10.1016/0001-8708(86)90081-2 | doi-access=free}}
{{cite journal | last1=Bloch | first1=Spencer | title=The moving lemma for higher Chow groups | journal=Journal of Algebraic Geometry | volume=3 | pages=537–568 | date=1994}}
Peter Haine, [http://math.mit.edu/~phaine/files/Motivic_Overview.pdf 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.

Category:Algebraic geometry