étale-locally criterion for separability

Definition

The étale-locally criterion for separability is a concept in algebraic geometry that provides a condition to determine whether a morphism between schemes is separable. Specifically, it states that a morphism is separable if, after passing to an étale cover, the corresponding ring homomorphism becomes a separable extension. This connection between étale morphisms and separability is crucial for understanding how algebraic structures behave over various fields and how they interact with each other.

5 Must Know Facts For Your Next Test

1. The étale-locally criterion for separability allows one to check separability by working with local étale neighborhoods instead of the global structure.
2. In practice, this criterion helps simplify problems in algebraic geometry by reducing them to easier computations in local rings.
3. The criterion plays a significant role in the theory of function fields and their separable extensions, connecting geometric properties with algebraic ones.
4. Using this criterion, one can deduce properties of varieties over algebraically closed fields from their behavior over more general fields.
5. The étale-locally criterion is essential in defining and understanding concepts like normality and smoothness in schemes.

Review Questions

• How does the étale-locally criterion for separability simplify the process of determining whether a morphism is separable?
  • The étale-locally criterion for separability simplifies the process by allowing you to check separability through local étale neighborhoods rather than dealing with the entire morphism at once. When you pass to an étale cover, you can focus on smaller, more manageable pieces where the corresponding ring homomorphism reflects separability. This localized approach makes it easier to analyze complex structures without getting overwhelmed by global considerations.
• Discuss the significance of separating morphisms in the context of algebraic geometry and how they relate to étale morphisms.
  • Separating morphisms are crucial in algebraic geometry as they ensure that certain geometric properties are preserved when moving between varieties. The relationship with étale morphisms lies in the fact that étale covers provide a framework where local behavior can be analyzed effectively. This connection allows mathematicians to derive important results about how geometric objects interact, particularly when working over different fields or when analyzing fiber products.
• Evaluate the impact of the étale-locally criterion for separability on our understanding of algebraic structures in fields of varying characteristics.
  • The impact of the étale-locally criterion for separability on our understanding of algebraic structures is profound, particularly when dealing with fields of different characteristics. This criterion helps establish connections between geometric properties and algebraic behavior, enabling mathematicians to understand how varieties behave under various extensions. By confirming separability through local conditions, it becomes possible to tackle broader questions regarding singularities, smoothness, and other significant attributes of schemes, thereby enriching the field's depth and application.

Related terms

Separable Morphism: A morphism of schemes is separable if its corresponding field extension is a separable field extension, meaning it does not have repeated roots in its minimal polynomial.

Étale Morphism: An étale morphism is a type of morphism between schemes that behaves like a locally finite disjoint union of copies of the base scheme, preserving many properties of local rings and being flat.

Flat Morphism: A flat morphism between schemes is one where the fibers behave nicely with respect to base change, ensuring that the local rings of the fibers vary continuously with respect to the base scheme.