# Bloch's higher Chow group

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In [algebraic geometry](/source/Algebraic_geometry), **Bloch's higher Chow groups**, a generalization of [Chow group](/source/Chow_group), is a precursor and a basic example of [motivic cohomology](/source/Motivic_cohomology) (for smooth varieties). It was introduced by [Spencer Bloch](/source/Spencer_Bloch) (Bloch 1986) and the basic theory has been developed by Bloch and [Marc Levine](/source/Marc_Levine_(mathematician)).

In more precise terms, a theorem of Voevodsky[1] implies: for a [smooth scheme](/source/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](/source/Algebraic_cycle) 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](/source/Proper_intersection) 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](/source/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](/source/Scheme-theoretic_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](/source/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](/source/Abelian_group), in view of the [Dold–Kan correspondence](/source/Dold%E2%80%93Kan_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[2] 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](/source/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](/source/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](/source/Exact_sequence) of chow groups.

## Localization theorem

(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](/source/Codimension), then it yields the long exact sequence for higher Chow groups (called the localization sequence).

## Rational coefficients

With rational coefficients, higher Chow groups identify with a piece of [algebraic K-theory](/source/Algebraic_K-theory)[3]

- \operatorname{H}^p(X; \mathbb{Q}(q)) \simeq \operatorname{CH}^q(X, 2q - p)\otimes\mathbb Q \simeq K_{2p-q}(X)^{(q)},

namely the eigenspace of the [Adams operation](/source/Adams_operation) \Psi^l associated to the eigenvalue l^q (which does not depend on choice of l).

## References

1. [*Lecture Notes on Motivic Cohomology*](http://www.claymath.org/library/monographs/cmim02.pdf). Clay Math Monographs. p. 159.

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 ∞.

1. Levine, Marc (1992). ["Bloch's higher Chow groups revisited"](https://www.numdam.org/item/?id=AST_1994__226__235_0). *[Astérisque](/source/Ast%C3%A9risque)*. '***(226): 235-320. – via Numdam.***

- Bloch, Spencer (September 1986). "Algebraic cycles and higher K-theory". *[Advances in Mathematics](/source/Advances_in_Mathematics)*. **61**: 267–304. [doi:10.1016/0001-8708(86)90081-2](https://doi.org/10.1016/0001-8708(86)90081-2)
- Bloch, Spencer (1994). "The moving lemma for higher Chow groups". *Journal of Algebraic Geometry*. **3**: 537–568.
- Peter Haine, [An Overview of Motivic Cohomology](http://math.mit.edu/~phaine/files/Motivic_Overview.pdf)
- Vladmir Voevodsky, “Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic,” International Mathematics Research Notices 7 (2002), 351–355.

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