étale cohomology

Definition

Étale cohomology is a powerful tool in algebraic geometry that allows for the study of algebraic varieties over fields, especially in relation to their rational points and the geometric properties they possess. It provides a way to compute cohomological invariants of schemes that are not necessarily smooth or projective, enabling a deeper understanding of their structure through a more flexible approach compared to classical cohomology theories.

5 Must Know Facts For Your Next Test

1. Étale cohomology can be used to derive results about rational points on varieties, particularly through applications like the Brauer-Manin obstruction.
2. It is designed to work well with p-adic fields and schemes, making it vital in understanding the properties of p-adic manifolds.
3. The comparison theorems show a connection between étale cohomology and other types of cohomology, such as singular or de Rham cohomology, highlighting its versatility.
4. The Weil conjectures relate étale cohomology to zeta functions of varieties over finite fields, providing profound insights into number theory and algebraic geometry.
5. Étale cohomology is particularly useful in studying arithmetic threefolds by allowing mathematicians to analyze their properties and understand their rational points.

Review Questions

• How does étale cohomology contribute to our understanding of rational points on algebraic varieties?
  • Étale cohomology plays a crucial role in determining rational points on algebraic varieties by providing tools that help analyze their structure and properties. Specifically, it enables the use of the Brauer-Manin obstruction, which relates to how certain cohomological classes can obstruct the existence of rational points. This connection deepens our comprehension of when and why rational solutions may exist or fail to exist.
• Discuss the relationship between étale cohomology and p-adic manifolds, particularly in terms of their applications.
  • Étale cohomology is intrinsically linked to p-adic manifolds as it provides a framework for studying these objects over p-adic fields. The methods developed within étale cohomology allow for an understanding of how these manifolds behave under various morphisms and transformations. This is particularly significant in arithmetic geometry, where p-adic techniques offer insights into the properties of schemes that are not accessible through classical methods.
• Evaluate the significance of the Weil conjectures in connecting étale cohomology with number theory and algebraic geometry.
  • The Weil conjectures have far-reaching implications as they establish deep connections between étale cohomology, zeta functions, and number theory. By relating the topological properties of algebraic varieties over finite fields to their arithmetic characteristics, these conjectures allow mathematicians to draw conclusions about counting rational points. This interplay not only enriches our understanding of algebraic geometry but also leads to breakthroughs in number theory, revealing profound relationships between seemingly disparate areas.

Related terms

Cohomology groups: Algebraic structures that represent the number of solutions to equations defined on topological spaces or algebraic varieties, used in various cohomology theories to extract topological and algebraic information.

Sheaf theory: A mathematical framework that allows for the systematic study of local data attached to the open sets of a topological space or a scheme, serving as a foundation for defining cohomology.

Galois cohomology: A branch of cohomology theory that studies the actions of Galois groups on the field extensions and their implications for algebraic structures, closely related to étale cohomology through the study of field extensions.