Bloch's higher Chow group explained

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^[1] implies: for a smooth scheme X over a field and integers p, q, there is a natural isomorphism

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

CH ^{*(X,0)}

can be thought of as the homotopy classes of cycles while

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}( Z [t _{0,} ... , t _{q]/(t} _{0} + ... + t _{q} - 1 )), which is an algebraic analog of a standard q-simplex. For each sequence 0 \le i _{1} < i _{2} < … < i _{r} \le q , the closed subscheme t _{i _{1}} = t _{i _{2}} = … = 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,} : \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 x \Delta ^{q)} for the subgroup generated by closed subvarieties that intersect properly with X x F for each face F of \Delta ^{q} .

Since

\partial _{X,} = \operatorname{id} _{X} x \partial _{q,} : X x \Delta ^{q-1} \hookrightarrow X x \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 x \{ t _{i} = 0 \}) \cap V.

Define the boundary operator

d _{q} = \sum ^{q} _{i =0} ( -1 ) ^{i} \partial ^{*} _{X, q, i} which yields the chain complex… \to z _{r(X,} q) \overset{d _{q}\to} z _{r(X,} q -1 ) \overset{d _{q-1} }\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{CH} _{r(X,} q) : = \operatorname{H} _{q(z} _{r(X,} ⋅ )).

(More simply, since

z _{r(X,} ⋅ ) 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,} ⋅ ) .)

For example, if

V \subset X x \Delta ^{1} ^[2] is a closed subvariety such that the intersections V( 0 ), V( inf ty) with the faces 0 , inf ty are proper, thend _{1(V)} = V( 0 ) - V( inf ty) 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

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

Localization

Given a closed equidimensional subscheme

Y \subset X there is a localization long exact sequence

\begin{align} … \\ CH ^{*-d} (Y, 2 ) \to CH ^{*} (X, 2 ) \to CH ^{*} (U, 2 ) \to & \\ CH ^{*-d} (Y, 1 ) \to CH ^{*} (X, 1 ) \to CH ^{*} (U, 1 ) \to & \\ CH ^{*-d} (Y, 0 ) \to CH ^{*} (X, 0 ) \to CH ^{*} (U, 0 ) \to & 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, ⋅ )/z(Y, ⋅ ) \to z(U, ⋅ ) 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

• Bloch . Spencer . Algebraic cycles and higher K-theory . . 61 . September 1986 . 267–304 . 10.1016/0001-8708(86)90081-2 . free.
• Bloch . Spencer . The moving lemma for higher Chow groups . Journal of Algebraic Geometry . 3 . 537–568 . 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.

Notes and References

1. Book: Lecture Notes on Motivic Cohomology. Clay Math Monographs. 159.
2. Here, we identify \Delta ^{1} with a subscheme of P ^{1} and then, without loss of generality, assume one vertex is the origin 0 and the other is ∞.