étale cohomology vs motivic cohomology

Definition

Étale cohomology and motivic cohomology are two different frameworks in algebraic geometry used to study the properties of algebraic varieties. Étale cohomology, developed by Grothendieck, focuses on the topology of varieties over a field, particularly using étale morphisms, while motivic cohomology extends this concept into the realm of motives, providing a link between geometry and algebraic cycles. Both cohomologies offer unique perspectives and tools for understanding the structure of varieties, particularly when examining their behavior under various geometric transformations.

5 Must Know Facts For Your Next Test

1. Étale cohomology provides a way to compute invariants of varieties that are more topological in nature, while motivic cohomology incorporates algebraic structures related to cycles and correspondences.
2. The key feature of étale cohomology is its ability to produce Galois representations, linking algebraic geometry with number theory through the action of the Galois group on the étale cohomology groups.
3. Motivic cohomology is often viewed as a bridge between algebraic topology and algebraic geometry, allowing for an integration of various cohomological theories.
4. Both types of cohomology can be used to define the Chow groups, but they approach the construction from different angles—étale focusing on topological properties and motivic concentrating on algebraic cycles.
5. Étale cohomology can be applied to any scheme over a field, while motivic cohomology requires a deeper understanding of motives and their relationship with algebraic varieties.

Review Questions

• How do étale cohomology and motivic cohomology differ in their approach to studying algebraic varieties?
  • Étale cohomology focuses on the topological aspects of algebraic varieties by employing étale morphisms and producing Galois representations that relate to number theory. In contrast, motivic cohomology explores the algebraic structures associated with cycles and correspondences, thereby linking various cohomological theories together. This fundamental difference means that while both methods can reveal important information about varieties, they do so from distinct perspectives—topological versus algebraic.
• Discuss the role of Galois representations in étale cohomology and how it connects to number theory.
  • Galois representations arise from étale cohomology as they describe how the Galois group acts on the étale cohomology groups of a variety. This action encodes significant arithmetic information about the variety, allowing mathematicians to draw connections between geometric properties and number-theoretic phenomena. The ability to associate these representations to specific cohomological invariants provides deep insights into the interplay between geometry and arithmetic in modern mathematics.
• Evaluate the significance of using both étale cohomology and motivic cohomology in understanding the properties of algebraic varieties.
  • Using both étale and motivic cohomology offers a comprehensive framework for analyzing algebraic varieties from multiple angles. While étale cohomology captures essential topological characteristics through Galois representations, motivic cohomology provides a more nuanced view that incorporates algebraic cycles and their relationships. This duality enriches our understanding of varieties by allowing us to connect seemingly disparate areas of mathematics, leading to broader applications in both geometry and number theory.

Related terms

Cohomology: A mathematical tool used to study topological spaces and algebraic structures by associating sequences of abelian groups or modules.

Motives: Abstract objects in algebraic geometry that serve as universal cohomological invariants, capturing essential features of algebraic varieties.

Étale Morphism: A type of morphism between schemes that is flat and unramified, allowing for a well-behaved notion of 'local' properties in the étale topology.