Properness

Definition

Properness is a property of a morphism of schemes that generalizes the idea of compactness in topology. It ensures that the morphism behaves well with respect to certain limit processes and guarantees that the pre-image of a compact set under the morphism is compact. This concept is crucial when working with Hilbert schemes and quotient schemes, as it allows for control over the behavior of families of schemes over a base scheme.

5 Must Know Facts For Your Next Test

1. A morphism is proper if it satisfies the valuative criterion for properness, which relates to the behavior of maps from discrete valuation rings.
2. Properness implies that the image of a compact subset under the morphism is also compact, making it useful in various constructions in algebraic geometry.
3. In the context of Hilbert schemes, properness ensures that families of subschemes behave well and are parameterized nicely.
4. For quotient schemes, properness helps ensure that the quotient behaves well under group actions and that limits exist for families.
5. A key result related to properness is that any proper morphism is universally closed, meaning it remains closed when composed with any base change.

Review Questions

• How does the concept of properness relate to the behavior of morphisms in algebraic geometry?
  • Properness ensures that morphisms exhibit good behavior regarding limit processes and compactness. Specifically, if a morphism is proper, it guarantees that the pre-image of compact sets remains compact and that various geometric structures are preserved under base changes. This makes proper morphisms essential when working with families of schemes, such as those encountered in Hilbert schemes and quotient schemes.
• What role does properness play in understanding families of schemes within Hilbert schemes?
  • In Hilbert schemes, properness plays a crucial role by ensuring that families of subschemes parameterized by the scheme are well-behaved. This means that not only do these families form coherent structures, but they also allow for controlled limits when considering moduli problems. Proper morphisms guarantee that we can effectively study how subschemes vary in a family, ensuring stability and consistency across parameters.
• Evaluate the implications of properness on quotient schemes and their applications in algebraic geometry.
  • Properness in the context of quotient schemes ensures that when we form quotients by group actions, we retain nice geometric properties. This means that we can expect limits to exist and behave consistently under varying group actions. Evaluating these implications shows how proper quotient schemes lead to moduli spaces that are well-structured and manageable. The ability to control the geometry through properness enhances our understanding of geometric properties tied to symmetry and invariant theory within algebraic geometry.

Related terms

Flatness:

A property of morphisms that indicates how fibers behave uniformly, allowing for good control over their structure in deformation problems.

Quasi-compactness: A property related to schemes where every open cover has a finite subcover, serving as a weaker condition than properness.

Separatedness: A property indicating that the diagonal morphism is a closed immersion, ensuring distinctness of points in certain contexts, closely related to properness.