Cycle class map

Definition

The cycle class map is a homomorphism that connects algebraic cycles on a variety to cohomology classes, typically in the context of Chow groups. This map plays a crucial role in bridging the world of algebraic geometry with topology, as it allows one to translate geometric properties into algebraic invariants. By linking cycles to their corresponding classes in motivic cohomology, the cycle class map also facilitates deeper investigations into the relationship between cycles and other mathematical structures.

5 Must Know Facts For Your Next Test

1. The cycle class map is usually denoted by a specific symbol, often denoting its connection to Chow groups and cohomology theories.
2. It provides a way to define intersection numbers in algebraic geometry, allowing for the computation of important invariants associated with varieties.
3. The cycle class map can be defined in various contexts, such as for smooth projective varieties or more general schemes.
4. In many cases, the image of the cycle class map captures significant information about the geometry of the underlying variety.
5. The study of cycle class maps has led to important results in arithmetic geometry, particularly in relation to the conjectures of Grothendieck and Beilinson.

Review Questions

• How does the cycle class map connect algebraic cycles to cohomology classes, and why is this relationship significant?
  • The cycle class map establishes a homomorphism from the Chow groups, which represent algebraic cycles, to cohomology groups that reflect topological features. This connection is significant because it allows mathematicians to use topological methods to study algebraic varieties, bridging two seemingly different areas of mathematics. By translating geometric properties into cohomological invariants, it opens up pathways for understanding deep results in both fields.
• Discuss how the cycle class map is applied in intersection theory and its implications for algebraic geometry.
  • In intersection theory, the cycle class map enables mathematicians to compute intersection numbers of subvarieties within a given variety. This computation is crucial for understanding how different geometric objects interact with each other. By providing a systematic way to assign numerical invariants to these intersections, the cycle class map helps derive essential properties about the structure of varieties and their classifications within algebraic geometry.
• Evaluate the impact of the cycle class map on modern arithmetic geometry and its relation to Grothendieck's conjectures.
  • The impact of the cycle class map on modern arithmetic geometry is profound, particularly in its role in proving or supporting Grothendieck's conjectures. By linking cycles to motivic cohomology through the cycle class map, mathematicians have been able to explore connections between algebraic cycles and various arithmetic properties. This has led to significant advancements in understanding how these structures behave under rational equivalence and has enriched the field by revealing deep relationships among cohomological theories.

Related terms

Chow groups: Groups that classify algebraic cycles on a variety modulo rational equivalence, serving as important tools in algebraic geometry.

motivic cohomology: A homology theory that extends classical cohomology theories to capture both topological and algebraic properties of varieties.

algebraic cycle: A formal sum of subvarieties of a given variety, used to study the intersection theory and cohomological properties of varieties.