Separation

Definition

In the context of étale morphisms, separation refers to a property of morphisms between schemes that ensures they behave nicely with respect to the underlying topological spaces. Specifically, a morphism is said to be separated if the diagonal morphism, which identifies points in the product space, is a closed immersion. This concept helps us understand how schemes can be distinguished from one another and ensures that various geometric properties are maintained in the transition between different schemes.

5 Must Know Facts For Your Next Test

1. Separation is crucial for ensuring that when working with schemes, certain pathological behaviors are avoided, which makes analysis easier.
2. In algebraic geometry, separated schemes correspond to varieties that can be nicely visualized without self-intersections.
3. For a morphism to be considered separated, it must satisfy the condition that its diagonal morphism is a closed immersion.
4. The notion of separation leads to a better understanding of fiber products and their properties in the context of étale morphisms.
5. Separatedness plays a role in defining various geometric concepts such as properness and completeness in the study of schemes.

Review Questions

• What does it mean for a morphism between schemes to be separated, and why is this property important?
  • A morphism between schemes is considered separated if its diagonal morphism is a closed immersion. This property is important because it ensures that points in the product space can be distinguished clearly, preventing unwanted overlaps. In practical terms, this helps maintain the integrity of geometric structures when performing operations such as taking fiber products or working with étale morphisms.
• How does separation relate to the concepts of closed immersion and diagonal morphism in the study of étale morphisms?
  • Separation directly connects to closed immersions through the requirement that the diagonal morphism must be a closed immersion for a morphism to be classified as separated. This relationship signifies that when two points are related through the diagonal, they do not intersect unnecessarily. By ensuring that these conditions hold, one can better analyze and understand how schemes interact under étale morphisms while preserving essential geometric features.
• Evaluate the impact of separation on the study of fiber products in algebraic geometry, particularly concerning étale morphisms.
  • The property of separation significantly impacts the study of fiber products in algebraic geometry by ensuring that these constructions behave predictably. When working with separated schemes, fiber products can be taken without introducing complications like excessive intersections or ambiguities in point identification. This predictability allows for cleaner and more effective analysis of properties such as flatness and unramifiedness within étale morphisms, which are crucial for advancing theories and applications in arithmetic geometry.

Related terms

Étale Morphism: A morphism of schemes that is flat and unramified, allowing for a local behavior resembling that of finite étale extensions of fields.

Closed Immersion: A type of morphism of schemes that identifies a subscheme as a closed subset, preserving the structure of the points and the topology.

Diagonal Morphism: A morphism from a scheme to its product with itself that identifies points, essentially capturing the notion of 'self-relation' within the scheme.