étale cohomology

Definition

Étale cohomology is a type of cohomology theory in algebraic geometry that studies the properties of schemes using étale morphisms, which are a generalization of covering maps. This approach allows for the computation of topological invariants and provides a bridge between algebraic and topological methods, making it particularly useful in the context of schemes over fields and in the study of their geometric properties.

5 Must Know Facts For Your Next Test

1. Étale cohomology provides a way to compute sheaf cohomology in the context of schemes, making it an important tool for modern algebraic geometry.
2. The fundamental group of an algebraic variety can be understood via étale cohomology, leading to insights into its geometric properties and underlying structure.
3. The étale site is crucial for defining étale cohomology, as it allows one to work with étale morphisms that capture essential features of schemes.
4. Étale cohomology has connections to number theory, particularly through its use in the proof of the Weil conjectures, which relate to the zeta functions of algebraic varieties.
5. This type of cohomology also facilitates comparison between various types of cohomological theories, such as singular cohomology and sheaf cohomology, thereby enriching the understanding of their relationships.

Review Questions

• How does étale cohomology relate to the study of schemes in algebraic geometry?
  • Étale cohomology is fundamentally tied to the study of schemes because it allows mathematicians to analyze their geometric properties through the lens of étale morphisms. These morphisms act like covering spaces in topology, enabling a way to derive topological invariants from algebraic data. This connection not only enhances our understanding of schemes but also provides tools for computations and comparisons within algebraic geometry.
• Discuss the significance of the étale site in defining étale cohomology and how it differs from other sites used in cohomological studies.
  • The étale site is essential for defining étale cohomology as it encapsulates all étale morphisms between schemes, allowing for a systematic approach to studying their properties. Unlike other sites used in sheaf theory, such as the Zariski site or the Zariski topology, the étale site offers a finer perspective that captures more geometric information about schemes. This difference plays a crucial role in understanding both local and global properties in algebraic geometry.
• Evaluate how étale cohomology impacts the relationship between algebraic geometry and number theory, particularly regarding the Weil conjectures.
  • Étale cohomology significantly impacts the relationship between algebraic geometry and number theory by providing tools that link geometric concepts to arithmetic properties. The Weil conjectures, which concern zeta functions of algebraic varieties over finite fields, were proven using results from étale cohomology. This connection illustrates how geometric techniques can yield deep insights into number-theoretic questions, reinforcing the idea that these two fields are profoundly intertwined.

Related terms

Cohomology:

A mathematical tool that studies the global properties of topological spaces or algebraic varieties through the use of sequences of abelian groups or modules.

Sheaf:

A mathematical structure that encodes local data associated with open subsets of a topological space, allowing for the study of global sections and cohomological properties.

Topos Theory: A branch of mathematics that generalizes set theory and category theory, providing a framework for doing geometry in a more abstract setting, particularly through the concept of a topos.