Chow Groups

Definition

Chow groups are algebraic constructs that arise in the study of algebraic cycles on algebraic varieties. They provide a way to track and organize these cycles, which are formal sums of subvarieties, allowing for a rich interplay between geometry and cohomology. In the context of cohomology of sheaves, Chow groups help to understand the relationship between cycles and the corresponding cohomological invariants, enabling a deeper analysis of intersection theory and algebraic geometry.

5 Must Know Facts For Your Next Test

1. Chow groups are denoted by $CH^k(X)$ for a variety $X$, where $k$ indicates the codimension of the cycles being considered.
2. They form an abelian group structure, which allows for the addition of cycles and captures their algebraic properties.
3. The connection between Chow groups and cohomology is established through various results, such as the Chow ring, which encodes information about cycles in terms of cohomological classes.
4. Chow groups also play a crucial role in the study of rational equivalence among cycles, allowing mathematicians to classify cycles based on their relationships.
5. One important application of Chow groups is in the context of intersection numbers, which can be computed using their associated cohomology classes.

Review Questions

• How do Chow groups relate to the study of algebraic cycles on varieties?
  • Chow groups provide a framework for organizing algebraic cycles on varieties by defining them as formal sums of subvarieties. This organization helps mathematicians understand the properties and relationships between these cycles. In particular, Chow groups allow for the exploration of intersection theory by providing a way to add and compare cycles based on their geometrical and topological features.
• Discuss the significance of Chow rings in the context of Chow groups and cohomology.
  • Chow rings are significant because they establish a direct link between Chow groups and cohomology classes. By forming a ring structure from Chow groups, mathematicians can translate geometric problems into algebraic ones. This connection allows for powerful techniques from cohomology to be applied in studying algebraic cycles, particularly in calculating intersection numbers and understanding equivalences among cycles.
• Evaluate how Chow groups enhance our understanding of intersection theory within algebraic geometry.
  • Chow groups enhance our understanding of intersection theory by providing an organized way to consider the relationships between different algebraic cycles within varieties. They allow us to compute intersection numbers using their cohomological representations. By analyzing these intersections through Chow groups, we can derive deeper insights into properties like rational equivalence and derive important results that connect geometry with algebraic structures. This interplay not only enriches the study of intersection theory but also connects it with broader themes in algebraic geometry.

Related terms

Algebraic Cycle: An algebraic cycle is a formal sum of subvarieties of an algebraic variety, which can be used to study properties like intersection and equivalence.

Cohomology:

Cohomology is a mathematical tool used to study topological spaces through algebraic structures, often providing insights into the properties of sheaves.

Intersection Theory:

Intersection theory deals with the intersections of subvarieties within a given variety, capturing how these intersections behave geometrically and algebraically.