étale cohomology

Definition

Étale cohomology is a powerful tool in algebraic geometry that extends the notion of cohomology to schemes, allowing for the study of algebraic varieties over arbitrary fields. It provides a way to capture topological and algebraic information about these varieties, facilitating connections between geometry and number theory.

5 Must Know Facts For Your Next Test

1. Étale cohomology groups are particularly useful for studying the properties of schemes that have good reduction modulo prime ideals, especially in number theory.
2. It can be seen as a way to compute the Brauer group of a scheme, providing insights into the relationship between algebraic cycles and cohomological invariants.
3. The étale site allows for the definition of sheaves in a way that is compatible with finite étale morphisms, leading to important results like the Artin vanishing theorem.
4. The étale cohomology groups are known to have deep connections with other cohomology theories, such as singular cohomology and crystalline cohomology.
5. Étale cohomology is crucial for understanding the conjectures related to motives and their connections to various fields like arithmetic geometry.

Review Questions

• How does étale cohomology relate to the study of algebraic varieties over fields?
  • Étale cohomology provides a framework for understanding algebraic varieties by capturing essential geometric and topological properties through its cohomological groups. This approach allows mathematicians to connect algebraic geometry with number theory by examining how these varieties behave under various morphisms. The versatility of étale cohomology makes it an invaluable tool for exploring the structure of schemes over different fields.
• Discuss the significance of the étale site in defining sheaves and its implications for algebraic geometry.
  • The étale site is significant because it provides a setting where one can define sheaves that respect finite étale morphisms, allowing for a richer understanding of local and global properties of schemes. This leads to powerful results, such as the Artin vanishing theorem, which highlights how certain cohomological properties vanish under specific conditions. Such insights are crucial for further developments in motives and algebraic cycles.
• Evaluate how étale cohomology connects with other cohomology theories and its impact on broader mathematical concepts.
  • Étale cohomology serves as a bridge connecting various cohomological frameworks such as singular cohomology and crystalline cohomology. By establishing these connections, it influences our understanding of key conjectures in arithmetic geometry, particularly regarding motives and their properties. The interplay between these theories enriches our comprehension of both geometric structures and their associated algebraic invariants, demonstrating the profound impact étale cohomology has on contemporary mathematics.

Related terms

Cohomology:

A mathematical concept that studies the properties of shapes and spaces through algebraic structures, often used in topology to analyze the features of spaces.

Sheaf: A mathematical structure that encodes local data on a space, allowing for the systematic study of functions and sections over open subsets of a given space.

Schemes: Generalized spaces in algebraic geometry that extend the concept of algebraic varieties, providing a framework to work with solutions to polynomial equations in a more flexible setting.