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 (Bloch 1986) 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;\mathbb {Z} (q))\simeq \operatorname {CH} ^{q}(X,2q-p)

between motivic cohomology groups and higher Chow groups.

Definition

Let X be an algebraic scheme over a field (“algebraic” means separated and of finite type).

For each integer $q\geq 0$ , 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 $0\leq i_{1}<i_{2}<\cdots <i_{r}\leq q$ , 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 $V\subset X\times \triangle ^{1}$ ^[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, 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.

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

References

↑ Voevodsky 2002
↑ Here, we identify $\triangle ^{1}$ with a subscheme of $\mathbb {P} ^{1}$ and then, without loss of generality, assume one vertex is the origin 0 and the other is ∞.

S. Bloch, “Algebraic cycles and higher K-theory,” Adv. Math. 61 (1986), 267–304.
S. Bloch, “The moving lemma for higher Chow groups,” J. Algebraic Geom. 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.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Voevodsky 2002

[2] Here, we identify $\triangle ^{1}$ with a subscheme of $\mathbb {P} ^{1}$ and then, without loss of generality, assume one vertex is the origin 0 and the other is ∞.