Bloch's higher Chow group

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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

between motivic cohomology groups and higher Chow groups.

Motivation

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One of the motivations for higher Chow groups comes from homotopy theory. In particular, if   are algebraic cycles in   which are rationally equivalent via a cycle  , then   can be thought of as a path between   and  , and the higher Chow groups are meant to encode the information of higher homotopy coherence. For example,

 

can be thought of as the homotopy classes of cycles while

 

can be thought of as the homotopy classes of homotopies of cycles.

Definition

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

For each integer  , define

 

which is an algebraic analog of a standard q-simplex. For each sequence  , the closed subscheme  , which is isomorphic to  , is called a face of  .

For each i, there is the embedding

 

We write   for the group of algebraic i-cycles on X and   for the subgroup generated by closed subvarieties that intersect properly with   for each face F of  .

Since   is an effective Cartier divisor, there is the Gysin homomorphism:

 ,

that (by definition) maps a subvariety V to the intersection  

Define the boundary operator   which yields the chain complex

 

Finally, the q-th higher Chow group of X is defined as the q-th homology of the above complex:

 

(More simply, since   is naturally a simplicial abelian group, in view of the Dold–Kan correspondence, higher Chow groups can also be defined as homotopy groups  .)

For example, if  [2] is a closed subvariety such that the intersections   with the faces   are proper, then   and this means, by Proposition 1.6. in Fulton’s intersection theory, that the image of   is precisely the group of cycles rationally equivalent to zero; that is,

  the r-th Chow group of X.

Properties

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Functoriality

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Proper maps   are covariant between the higher chow groups while flat maps are contravariant. Also, whenever   is smooth, any map to   is contravariant.

Homotopy invariance

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If   is an algebraic vector bundle, then there is the homotopy equivalence

 

Localization

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Given a closed equidimensional subscheme   there is a localization long exact sequence

 

where  . In particular, this shows the higher chow groups naturally extend the exact sequence of chow groups.

Localization theorem

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(Bloch 1994) showed that, given an open subset  , for  ,

 

is a homotopy equivalence. In particular, if   has pure codimension, then it yields the long exact sequence for higher Chow groups (called the localization sequence).

References

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  1. ^ Lecture Notes on Motivic Cohomology (PDF). Clay Math Monographs. p. 159.
  2. ^ Here, we identify   with a subscheme of   and then, without loss of generality, assume one vertex is the origin 0 and the other is ∞.