Algebraic cycles

Definition

Algebraic cycles are formal sums of subvarieties of algebraic varieties, which play a significant role in algebraic geometry and K-theory. They help in understanding the relationships between algebraic varieties, their cohomological properties, and serve as a bridge between geometric and topological concepts. The study of algebraic cycles also leads to important conjectures and theorems that connect various areas of mathematics.

5 Must Know Facts For Your Next Test

1. Algebraic cycles can be understood in terms of their dimension; for example, a 0-cycle corresponds to a formal sum of points, while higher-dimensional cycles correspond to subvarieties of higher dimensions.
2. The Chow groups are used to define a systematic approach to studying algebraic cycles, allowing mathematicians to analyze their properties and relationships.
3. The Bloch-Beilinson conjectures relate algebraic cycles to the structure of motives, which aim to unify various branches of mathematics by categorizing these cycles.
4. Algebraic cycles are closely linked with the theory of motives, suggesting deep connections between algebraic geometry and other areas like number theory.
5. Applications of algebraic cycles extend to computational aspects in arithmetic geometry, where they can help solve problems related to rational points on varieties.

Review Questions

• How do algebraic cycles contribute to the understanding of geometric properties in algebraic geometry?
  • Algebraic cycles provide a framework for relating subvarieties within an algebraic variety, helping to understand their geometric properties through tools like Chow groups. By studying how these cycles interact via intersection theory, mathematicians can derive insights into the structure and relationships between various varieties. This connection allows for deeper exploration into the cohomological aspects of these geometries.
• Discuss the role of algebraic cycles in relation to the Bloch-Kato conjecture and its implications for K-theory.
  • The Bloch-Kato conjecture posits a relationship between algebraic cycles and Galois representations in number theory. It implies that certain properties of algebraic cycles can have direct consequences on K-theory, particularly in how they relate to the motives associated with these cycles. Understanding this relationship could lead to profound insights into both K-theory and the arithmetic properties of varieties.
• Evaluate the impact of algebraic cycles on modern mathematics, especially in bridging different fields like geometry and number theory.
  • Algebraic cycles significantly impact modern mathematics by serving as a link between geometry and number theory. Their study leads to advanced topics like motives and intersection theory, which unify various mathematical concepts under one umbrella. As researchers investigate these connections further, they uncover new relationships that enrich both fields, fostering collaboration and innovation across different areas of study.

Related terms

Chow group: A group that consists of algebraic cycles modulo rational equivalence, providing a way to classify cycles and study their properties.

Intersection theory: A branch of algebraic geometry that studies the intersection of subvarieties within a given variety, which is crucial for understanding the structure of algebraic cycles.

K-theory: A field in mathematics that studies vector bundles and their generalizations, particularly through algebraic cycles, linking them to topological and cohomological properties.