higher Chow groups

higher Chow groups
Name	Higher Chow groups
Field	Algebraic geometry
Introduced	1980s
Introduced by	Spencer Bloch

higher Chow groups are a family of homological invariants in algebraic geometry that extend classical Chow groups to a graded theory parameterized by an integer called the weight. They were introduced to provide cycle-theoretic models for motivic cohomology and to relate algebraic cycles to algebraic K-theory and regulators. The theory connects work of Spencer Bloch, constructions in Quillen's algebraic K-theory, and conjectures of Beilinson and Bloch–Kato.

Definition and construction

Bloch defined the groups using a complex of algebraic cycles on a scheme X, now called Bloch's higher Chow complex, built from algebraic simplices Δ^n with faces and degeneracies analogous to the singular chain complex on a singular homology model. For a quasi-projective scheme X over a field k, one considers the free abelian group generated by integral closed subschemes of X×Δ^n meeting faces properly; boundary maps come from alternating sums of pullbacks along face inclusions, producing a chain complex Z^p(X,·). The higher Chow group CH^p(X,q) is the qth homology of Z^p(X,·), recovering the classical Chow group CH^p(X) when q=0. Bloch's construction uses techniques related to moving lemmas and comparisons with intersection theory on smooth projective varieties such as those studied by Fulton.

Basic properties and functoriality

Higher Chow groups satisfy homotopy invariance for regular schemes: for a smooth scheme X over a field, the pullback along the projection X×A^1→X induces isomorphisms CH^p(X,q) ≅ CH^p(X×A^1,q). They admit contravariant functoriality for flat pullback and covariant functoriality for proper pushforward, compatible with Gysin maps for regular embeddings defined using excess intersection formalism. There are localization long exact sequences associated to closed immersions and open complements reminiscent of those in Quillen's Q-construction for K-theory and of the localization sequence in étale cohomology. Cycle class maps relate CH^p(X,q) to cohomology theories such as Deligne cohomology, Betti cohomology, and crystalline cohomology via regulator maps constructed by Beilinson, Bloch, and Soulé.

Relationship to motivic cohomology and K-theory

Higher Chow groups provide a concrete model for motivic cohomology groups H^{2p-q}_M(X, Z(p)), fitting into the framework developed by Vladimir Voevodsky and collaborators. For smooth schemes over a field, there are isomorphisms CH^p(X,q) ≅ H^{2p-q}_M(X,Z(p)), and these groups sit in spectral sequences relating motivic cohomology to algebraic K-theory, notably the Bloch–Lichtenbaum spectral sequence and the motivic spectral sequence of Voevodsky and Levine. Furthermore, regulators from K-theory to analytic or Hodge theoretic invariants factor through higher Chow groups, linking work of Borel, Beilinson, and Deligne on special values of L-functions and on the regulator maps in the Beilinson conjectures.

Examples and computations

For X = Spec k a field, CH^p(k,q) vanishes for p>0 when q=0, while for p=0 the groups recover Z for q=0 and often connect to Milnor K-theory for q>0 via isomorphisms CH^p(k,p) ≅ K^M_p(k) in the presence of certain vanishing results of Nesterenko–Suslin and Suslin's rigidity. For smooth quasi-projective curves C over a field, CH^1(C,1) relates to the Jacobian and to K_1 of the function field; explicit calculations appear in work of Bloch and Levine. For surfaces, computations link to regulator maps studied by Beilinson and to examples by Murre and Collino. Over finite fields, results of Geisser and Levine compute motivic cohomology and show finiteness properties reminiscent of Weil conjectures phenomena.

Bloch's higher Chow complex and homotopy invariance

Bloch's complex Z^p(X,·) is constructed from cycle groups Z^p(X×Δ^n) with boundary maps induced by the simplicial structure of Δ^·. Homotopy invariance for smooth schemes follows from moving lemmas and from comparison with the cycle complex for X×A^1, using techniques analogous to those in Gabber's refinement of purity and to Thomason's work on descent. The complex is contravariantly functorial for flat maps and admits cap and cup product structures compatible with external products arising from Künneth formulas and intersections as developed by Fulton and Bloch–Ogus.

Applications and conjectures

Higher Chow groups play a central role in formulations of the Beilinson conjectures on special values of L-functions for motives, in Bloch–Kato conjectures relating motivic cohomology to Galois cohomology, and in the study of mixed motives envisioned by Deligne and formalized by Voevodsky and Ayoub. They appear in explicit descriptions of regulators used in computations of periods for motives associated to modular forms studied by Shimura and Manin, and in recent advances on rationality questions and decompositions of diagonal linked to work by Voisin and Colliot-Thélène. Conjectural relationships connect vanishing of certain CH^p(X,q) to statements about algebraic cycles and to the existence of motivic t-structures posited by Beilinson and Jannsen.

Variants and generalizations

Variants include higher Chow groups with coefficients (e.g., torsion coefficients or Q-coefficients), cycles with modulus reflecting relative situations studied by Kerz and Saito, and equivariant higher Chow groups for schemes with group actions modeled on constructions by Thomason and Atiyah–Segal. Derived and triangulated enhancements appear in the setting of triangulated categories of motives of Voevodsky, Bondarko, and Ayoub, while syntomic and p-adic analogues connect to work of Besser and Nekovář. There are also noncommutative and DG-enhanced versions motivated by Kontsevich's homological noncommutative geometry and by applications to Hochschild homology and cyclic homology in the sense of Connes.

Category:Algebraic geometry