Algebraic cycle

Definition

An algebraic cycle is a formal sum of subvarieties of an algebraic variety, where the subvarieties can have different dimensions and are assigned integer coefficients. This concept is crucial in understanding the structure of algebraic varieties and plays a significant role in both Milnor's K-theory and Quillen's K-theory, as it helps to study the relationships between algebraic cycles and various invariants of varieties.

5 Must Know Facts For Your Next Test

1. Algebraic cycles can be used to define important invariants like intersection numbers, which measure how cycles intersect within a given variety.
2. In Milnor's K-theory, algebraic cycles play a role in relating algebraic varieties to topological spaces through the study of their K-groups.
3. Quillen's K-theory further explores algebraic cycles by relating them to stable homotopy types, offering insights into the algebraic structures involved.
4. The degree of an algebraic cycle is an important concept, determining how 'many' times a subvariety appears in the cycle's formal sum.
5. Algebraic cycles are foundational for understanding various conjectures and theorems in algebraic geometry, such as the conjecture linking cycles with cohomological invariants.

Review Questions

• How do algebraic cycles relate to Chow groups and their role in algebraic geometry?
  • Algebraic cycles are central to the construction of Chow groups, which classify these cycles under equivalence relations. Chow groups help organize algebraic cycles into manageable structures, allowing mathematicians to study their properties and relationships. Understanding how these cycles fit within Chow groups provides insights into the geometric and topological properties of varieties.
• Discuss how intersection theory utilizes algebraic cycles and its importance in both Milnor's K-theory and Quillen's K-theory.
  • Intersection theory leverages algebraic cycles to analyze how different subvarieties intersect within a larger variety. This interplay is crucial for both Milnor's and Quillen's K-theories, as they use intersection numbers derived from these cycles to connect geometric properties with topological invariants. The results from intersection theory contribute significantly to understanding the relationship between algebraic geometry and homotopy theory.
• Evaluate the impact of algebraic cycles on modern developments in motivic cohomology and their implications for future research.
  • Algebraic cycles are fundamental to motivic cohomology as they provide a bridge between classical cohomological techniques and modern algebraic geometry. Their use in defining motivic invariants allows researchers to formulate deep connections between different areas of mathematics, such as number theory and topology. As research progresses, the study of algebraic cycles will likely yield further insights into unresolved questions in algebraic geometry and enhance our understanding of its underlying structures.

Related terms

Chow group: A Chow group is an algebraic structure that captures the equivalence classes of algebraic cycles on a variety, providing a way to study their relationships and properties.

Intersection theory: Intersection theory studies how subvarieties intersect within a larger variety, helping to understand their combinatorial and geometric properties.

Motivic cohomology: Motivic cohomology is a type of cohomology theory that extends classical notions of cohomology to algebraic varieties, particularly using the framework of algebraic cycles.