Algebraic cycles

Definition

Algebraic cycles are formal sums of subvarieties of a given algebraic variety, which play a significant role in the study of both motivic cohomology and algebraic K-Theory. These cycles help in understanding the properties of varieties through their intersection theory and provide a way to relate geometry and topology in a coherent framework. They are essential for defining classes in cohomology theories that correspond to various algebraic invariants.

5 Must Know Facts For Your Next Test

1. Algebraic cycles can be classified into different types, such as 0-cycles, 1-cycles, and higher-dimensional cycles, each with distinct properties and implications.
2. The Chow ring is generated by algebraic cycles and captures important information about the geometry of varieties, including their intersection numbers.
3. One key aspect of algebraic cycles is their role in defining the Grothendieck group of varieties, which relates to the structure of algebraic K-Theory.
4. Algebraic cycles can be used to formulate conjectures like the Hodge conjecture, which connects algebraic geometry with topology by relating cycles to cohomology classes.
5. The relationship between algebraic cycles and motives provides a bridge between different cohomology theories, unifying concepts from algebraic geometry and number theory.

Review Questions

• How do algebraic cycles contribute to our understanding of the structure of varieties through intersection theory?
  • Algebraic cycles provide a framework for studying how subvarieties intersect within a larger variety. By analyzing these intersections, we gain insights into the geometric properties and relationships between different components of the variety. This understanding allows us to develop tools like intersection numbers and refine our classifications of varieties based on these interactions.
• In what ways do algebraic cycles relate to cohomology theories and their applications in modern mathematics?
  • Algebraic cycles serve as a crucial link between algebraic geometry and cohomology theories by providing representatives for classes in cohomology. This connection allows us to apply tools from topology to study the properties of varieties, leading to significant results like the formulation of conjectures that involve both geometric and topological concepts. By connecting these areas, we deepen our understanding of the underlying structures present in algebraic varieties.
• Evaluate the implications of algebraic cycles on conjectures such as the Hodge conjecture and their role in shaping contemporary mathematics.
  • Algebraic cycles have far-reaching implications for conjectures like the Hodge conjecture, which posits a deep relationship between algebraic geometry and topology. By investigating how these cycles correspond to cohomology classes, mathematicians aim to uncover fundamental truths about the nature of varieties. The ongoing research into these relationships not only advances our knowledge of algebraic structures but also drives developments in related fields like number theory and arithmetic geometry, showcasing the interconnectedness of modern mathematical ideas.

Related terms

Chow group: The Chow group is a mathematical structure that classifies algebraic cycles up to rational equivalence, allowing for the study of their properties in a systematic way.

Cohomology:

Cohomology is a mathematical tool used to study topological spaces through algebraic invariants, which can be applied to analyze the properties of algebraic varieties.

Intersection theory: Intersection theory is the study of how subvarieties intersect within an algebraic variety, providing insights into their geometric properties and leading to results about algebraic cycles.