Properness

Definition

Properness refers to a property of a morphism between schemes, indicating that the morphism is universally closed and has finite fibers. This concept connects closely to other important features in algebraic geometry, ensuring that certain topological and algebraic conditions are met, which facilitate various constructions, such as Néron models and arithmetic surfaces. Properness often implies nice behavior regarding compactness and the extension of functions, making it a vital concept in understanding the structure of algebraic varieties.

5 Must Know Facts For Your Next Test

1. Properness is a crucial condition for the existence of good compactifications in algebraic geometry.
2. In the context of Néron models, properness ensures that the model behaves well at all places of the base field, particularly at the infinite places.
3. For arithmetic surfaces, properness can guarantee that certain morphisms do not 'escape' to infinity, maintaining control over their geometric properties.
4. A proper morphism can be seen as a generalization of a proper map in topology, where it extends the notion of compactness to algebraic varieties.
5. In algebraic geometry, properness often implies the existence of sufficient global sections for line bundles, which is critical for many constructions.

Review Questions

• How does properness relate to Néron models and what implications does it have for their behavior?
  • Properness ensures that Néron models behave well across all places of a Dedekind domain. Specifically, it guarantees that the model remains well-defined and compact at all primes and at infinity. This allows for a smooth transition between different completions of the field, making it possible to study the reduction of abelian varieties without losing essential properties or information about their structure.
• Discuss the significance of properness when analyzing arithmetic surfaces and how it influences their geometric properties.
  • Properness is significant for arithmetic surfaces as it controls their behavior at infinity and prevents them from becoming 'too large' or 'escaping.' This condition enables one to effectively manage the interactions between different fibers and ensures that various morphisms are well-behaved. Properness thus plays a key role in establishing fundamental properties such as finiteness of fibers and closure of images, which are crucial for understanding the global geometry of these surfaces.
• Evaluate how properness impacts the extension of functions on schemes and why this is essential in algebraic geometry.
  • Properness profoundly impacts the extension of functions on schemes by ensuring that morphisms allow for coherent lifting of regular functions across fibers. This property is essential in algebraic geometry because it leads to several important results, such as the ability to extend meromorphic functions from a dense open subset to the entire space. Without properness, many results regarding divisors, sheaves, and other constructions would not hold, severely limiting our ability to work with algebraic varieties effectively.

Related terms

Universally Closed: A morphism is universally closed if its image under any base change is a closed set in the target scheme.

Finite Fibers: A morphism has finite fibers if for every point in the target, the preimage consists of finitely many points.

Néron Model: A Néron model is a special type of proper model of an abelian variety over a Dedekind domain, designed to facilitate the study of its reduction modulo prime ideals.