Smooth Schemes

Definition

Smooth schemes are algebraic varieties or spaces that have a well-behaved geometric structure, specifically meaning they are regular at every point, which indicates they do not have any singularities. In the context of arithmetic geometry, smooth schemes allow for the application of various cohomological techniques, facilitating the study of their properties through l-adic cohomology, which helps in understanding their topological and arithmetic aspects.

5 Must Know Facts For Your Next Test

1. Smooth schemes are characterized by having a smooth structure over their base field, which means they are locally isomorphic to affine spaces.
2. The notion of smoothness is crucial for the development of intersection theory and for understanding the behavior of morphisms between schemes.
3. A scheme is smooth if it is flat over a base and the fibers are geometrically regular, ensuring good behavior under base change.
4. In l-adic cohomology, smooth schemes provide the necessary conditions for applying various theorems, like the proper base change theorem, enabling deeper insights into their structure.
5. Many important results in number theory and arithmetic geometry depend on working with smooth schemes to ensure desired properties in computations.

Review Questions

• How do smooth schemes relate to singularities and why is this relationship important in arithmetic geometry?
  • Smooth schemes are essential because they lack singularities, which allows mathematicians to work with them without encountering pathological cases. The absence of singularities ensures that many algebraic operations behave nicely, facilitating the application of techniques from algebraic geometry and number theory. In arithmetic geometry, this relationship is crucial as it allows for more straightforward applications of cohomological methods, particularly when analyzing properties that depend on the absence of singular behavior.
• Discuss how the concept of smoothness in schemes contributes to our understanding of l-adic cohomology.
  • The concept of smoothness in schemes plays a significant role in l-adic cohomology because it ensures that the cohomological tools developed for smooth varieties can be effectively applied. Smooth schemes satisfy conditions that lead to desirable properties in l-adic cohomology, such as the proper base change theorem and consistency under morphisms. This connection allows mathematicians to derive important results regarding the relationship between algebraic structures and their topological features.
• Evaluate the implications of using smooth schemes for results in intersection theory and how this ties into broader mathematical contexts.
  • Using smooth schemes has profound implications for intersection theory because it guarantees that intersections behave well geometrically. Smoothness allows mathematicians to apply tools like intersection numbers and enumerative geometry without encountering complications from singular points. This leads to richer results that connect geometry with topology and number theory, ultimately contributing to our understanding of complex relationships within algebraic varieties and their applications in various fields such as moduli problems and arithmetic statistics.

Related terms

Regular Scheme: A regular scheme is a type of scheme where the local rings at every point are regular local rings, indicating that the scheme is nonsingular and behaves nicely geometrically.

Singularities: Points at which a mathematical object is not well-behaved in some way, such as points where derivatives do not exist or where the dimension changes.

Cohomology: A mathematical tool that studies the properties of topological spaces through algebraic structures, often used in conjunction with schemes to derive information about their structure and behavior.