Algebraic cycles

Definition

Algebraic cycles are formal combinations of subvarieties of an algebraic variety, used to study the intersection theory and properties of varieties in algebraic geometry. They are critical in understanding motives and play a significant role in the Brauer-Manin obstruction, connecting geometric properties to arithmetic characteristics.

5 Must Know Facts For Your Next Test

1. Algebraic cycles can be thought of as generalized 'subvarieties' that help encapsulate information about the geometry of varieties.
2. The study of algebraic cycles is essential for defining and working with Chow groups, which provide insight into the equivalence classes of these cycles.
3. The intersection product allows for the operation on algebraic cycles, leading to a deeper understanding of how different cycles interact geometrically.
4. Algebraic cycles are instrumental in understanding the relationship between geometry and arithmetic through motives, which serve as a bridge between these two fields.
5. In the context of the Brauer-Manin obstruction, algebraic cycles help identify potential obstructions to finding rational points on varieties.

Review Questions

• How do algebraic cycles relate to Chow groups and what role do they play in understanding geometric properties?
  • Algebraic cycles serve as elements within Chow groups, which classify them up to rational equivalence. This classification allows mathematicians to explore geometric properties and their implications on the structure of algebraic varieties. By studying these cycles and their interactions, one gains insights into how different geometrical aspects influence one another.
• Discuss the significance of algebraic cycles in the context of motives and how they contribute to our understanding of cohomology theories.
  • Algebraic cycles are fundamental in defining motives, which act as a unifying concept across various cohomology theories in algebraic geometry. They allow for a systematic study of the relationships between different cohomological constructs and facilitate comparisons among diverse geometrical objects. The presence of algebraic cycles in this framework enables deeper investigations into how geometry informs arithmetic and vice versa.
• Evaluate the implications of algebraic cycles on the Brauer-Manin obstruction and their impact on rational points in algebraic varieties.
  • Algebraic cycles significantly influence the Brauer-Manin obstruction by providing a framework for identifying possible obstructions to the existence of rational points on algebraic varieties. This interplay highlights how geometric data encapsulated in these cycles can affect arithmetic outcomes. Understanding this relationship is crucial for formulating effective strategies for determining rationality properties within algebraic geometry.

Related terms

Homology: A mathematical tool used to study topological spaces by associating a sequence of abelian groups or modules, allowing for an understanding of the shapes and structures within algebraic geometry.

Chow Group: A group that classifies algebraic cycles modulo rational equivalence, providing a way to capture the geometric properties of varieties.

Motives: Abstract objects in algebraic geometry that aim to unify various cohomology theories, connecting them through a common framework that includes algebraic cycles.