Bloch's Higher Chow Groups

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 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
:$\operatorname^p(X; \mathbb(q)) \simeq \operatorname^q(X, 2q - p)$
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,

$\text^*(X,0)$

can be thought of as the homotopy classes of cycles while

$\text^*(X,1)$

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(\mathbb _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 $t_ = t_ = \cdots = t_ = 0$, which is isomorphic to $\Delta^$, is called a face of $\Delta^q$.

For each ''i'', there is the embedding
:$\partial_: \Delta^ \overset\to \ \subset \Delta^q.$

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

Since $\partial_ = \operatorname_X \times \partial_: X \times \Delta^ \hookrightarrow X \times \Delta^q$ is an effective Cartier divisor, there is the Gysin homomorphism:
:$\partial_^*: z_r(X, q) \to z_r(X, q-1)$,
that (by definition) maps a subvariety ''V'' to the intersection $(X \times \) \cap V.$

Define the boundary operator $d_q = \sum_^q (-1)^i \partial_^*$ which yields the chain complex
:$\cdots \to z_r(X, q) \overset\to z_r(X, q-1) \overset\to \cdots \overset\to z_r(X, 0).$

Finally, the ''q''-th higher Chow group of ''X'' is defined as the ''q''-th homology of the above complex:
:$\operatorname_r(X, q) := \operatorname_q(z_r(X, \cdot)).$

(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 $\operatorname_r(X, q) := \pi_q z_r(X, \cdot)$.)

For example, if $V \subset X \times \Delta^1$Here, we identify $\Delta^1$ with a subscheme of $\mathbb^1$ 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,
:$\operatorname_r(X, 0) =$ 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 $X$ is smooth, any map from $X$ is covariant.

Homotopy invariance

If $E \to X$ is an algebraic vector bundle, then there is the homotopy equivalence

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

Localization

Given a closed equidimensional subscheme $Y \subset X$ there is a localization long exact sequence

$\begin \cdots \\ \text^(Y,2) \to \text^(X,2) \to \text^(U,2) \to & \\ \text^(Y,1) \to \text^(X,1) \to \text^(U,1) \to & \\ \text^(Y,0) \to \text^(X,0) \to \text^(U,0) \to & \text0 \end$

where $U = X-Y$. In particular, this shows the higher chow groups naturally extend the exact sequence of chow groups.

Localization theorem

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=The moving lemma for higher Chow groups , journal=Journal of Algebraic Geometry , volume=3 , pages=537–568 , date=1994
*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.

Algebraic geometry