Smooth schemes

Definition

Smooth schemes are algebraic varieties that exhibit nice geometric properties, specifically characterized by having a well-behaved tangent space at every point. This property ensures that the scheme has no 'singularities' or abrupt changes in shape, allowing for a smooth structure that facilitates various mathematical applications, particularly in the realms of K-theory and arithmetic geometry. They play a crucial role in understanding the relationships between algebraic geometry, cohomology theories, and K-theory.

5 Must Know Facts For Your Next Test

1. Smooth schemes can be defined over any field, and their properties often extend to other algebraic structures, such as algebraic varieties.
2. The condition for a scheme to be smooth is locally equivalent to the vanishing of certain Jacobian determinants, which relate to how functions change in different directions.
3. Smoothness implies regularity in the context of the underlying ring of functions, meaning that they have no nilpotent elements in their structure.
4. In arithmetic geometry, smooth schemes are significant because they allow for the application of tools from both algebraic geometry and number theory without encountering obstructions from singular points.
5. The concept of smoothness is fundamental in defining important invariants in K-theory, which are essential for classifying vector bundles on schemes.

Review Questions

• How does the concept of smooth schemes relate to the properties of singular schemes in algebraic geometry?
  • Smooth schemes are distinguished from singular schemes primarily by their lack of singular points. While singular schemes can exhibit complex behavior due to their irregularities, smooth schemes maintain a consistent and well-defined geometric structure. This distinction is crucial for understanding how different types of schemes interact with tools from K-theory and cohomology, as smoothness provides a framework where these theories can be applied effectively without complications from singularities.
• Discuss the implications of smoothness on the flatness of morphisms between schemes and how this affects their algebraic properties.
  • Smoothness directly influences the flatness of morphisms between schemes, as smooth morphisms are always flat. This relationship is significant because flatness ensures that when transitioning from one scheme to another, properties such as dimension and regularity are preserved. Consequently, when working with smooth schemes, we can expect that algebraic properties will behave nicely across morphisms, enabling easier computations in both K-theory and cohomological contexts.
• Evaluate how the characteristics of smooth schemes contribute to the development of invariants in K-theory and their applications in arithmetic geometry.
  • The properties of smooth schemes play a pivotal role in formulating invariants within K-theory due to their regular structure and absence of singularities. These invariants can classify vector bundles and provide insights into more complex geometric structures. In arithmetic geometry, such invariants help bridge connections between number theory and algebraic geometry by enabling researchers to understand how various objects behave under morphisms and when viewed through different geometric lenses. The smoothness condition ensures that these evaluations remain consistent and reliable, ultimately enhancing our understanding of both fields.

Related terms

Singular schemes: Algebraic varieties that have points where the local behavior is not well-defined, often leading to singularities and complications in their geometric structure.

Flatness: A property of morphisms between schemes that ensures the fibers of the morphism vary continuously, essential for establishing good geometric behavior in families of schemes.

Cohomology: A mathematical tool used to study topological spaces and algebraic varieties through algebraic invariants, which helps in connecting various branches of mathematics.