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

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 $$V \subset X \times \Delta^1$$ 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.

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
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).