étale cohomology

Definition

Étale cohomology is a powerful tool in algebraic geometry that studies the properties of schemes through the lens of sheaves over étale covers. It allows mathematicians to define and compute cohomology groups for schemes, providing a way to connect algebraic and topological concepts. This theory extends classical cohomology theories by utilizing étale morphisms, which help in understanding geometric properties by examining how sheaves behave under these morphisms.

5 Must Know Facts For Your Next Test

1. Étale cohomology provides a framework for studying the properties of schemes over arbitrary fields, including finite fields.
2. The étale cohomology groups can be computed using Čech cohomology methods, establishing a deep connection between these two cohomology theories.
3. This theory plays a critical role in the formulation of the Weil conjectures, linking algebraic geometry and number theory through its use of Galois representations.
4. Étale cohomology can be used to compute important invariants like the Picard group, which measures line bundles on a scheme.
5. It extends beyond classical topology by allowing for the analysis of non-closed and singular schemes in ways traditional cohomology cannot.

Review Questions

• How does étale cohomology relate to sheaf cohomology and why is it significant in algebraic geometry?
  • Étale cohomology builds on the foundation of sheaf cohomology by applying its concepts specifically to étale covers, which are important for understanding the structure of schemes. This relationship is significant because it allows algebraic geometers to extend classical results from topology into the realm of algebraic varieties, enabling deeper insights into their geometric properties and how they behave under different morphisms.
• In what ways can étale cohomology be connected to Čech cohomology, and why is this connection beneficial?
  • Étale cohomology can be computed using methods from Čech cohomology, as both frameworks deal with sheaves and their global sections. This connection is beneficial because it allows researchers to utilize existing tools and techniques from topological cohomology to study more complex algebraic structures. This duality also helps bridge gaps between different areas of mathematics, facilitating broader applications in number theory and arithmetic geometry.
• Evaluate the impact of étale cohomology on modern mathematical research, particularly in relation to the Weil conjectures.
  • Étale cohomology has significantly influenced modern mathematical research, especially with its role in the proof of the Weil conjectures, which propose profound relationships between algebraic geometry and number theory. By establishing connections through Galois representations, it has opened new avenues for understanding rational points on varieties and their zeta functions. This impact showcases how abstract concepts in algebraic geometry can lead to concrete results that transform our understanding of mathematical relationships across disciplines.

Related terms

Sheaf: A sheaf is a mathematical construct that systematically assigns data to open sets of a topological space, allowing for the localization of algebraic or analytic properties.

Cohomology: Cohomology is a branch of mathematics that studies the properties of spaces through algebraic invariants derived from the global sections of sheaves.

Étale Morphism: An étale morphism is a specific type of morphism between schemes that resembles a local isomorphism and has certain desirable lifting properties.