Finite morphism

Definition

A finite morphism is a type of morphism between schemes that is affine and satisfies the condition of being finitely presented, meaning it corresponds to a finite type of ring homomorphism. In simpler terms, this means that the preimage of any affine open set is a scheme that is covered by finitely many affine opens. Finite morphisms are crucial in algebraic geometry as they allow for the analysis of algebraic structures while maintaining a level of compactness and control.

5 Must Know Facts For Your Next Test

1. Finite morphisms are automatically quasi-finite, meaning they have finitely many points in each fiber over a point.
2. If a finite morphism is also of finite presentation, it allows for the study of its underlying algebraic structure using tools from commutative algebra.
3. Finite morphisms can be used to study the behavior of varieties and their singularities, particularly when blowing up singular points.
4. In characteristic zero, finite morphisms correspond to finite extensions of fields, establishing a deep connection between geometry and algebra.
5. Finite morphisms preserve the property of being integral, which means they relate closely to schemes that have no embedded components.

Review Questions

• How does the concept of finite morphism enhance our understanding of the structure of schemes?
  • Finite morphisms provide a framework for analyzing schemes in a controlled way by ensuring that any affine open set remains manageable with finitely many generators and relations. This is especially important when studying the global properties of schemes, as it allows one to apply algebraic techniques to geometrical problems. Moreover, finite morphisms maintain key features like integrality and quasi-finiteness, which help us draw connections between algebraic structures and geometric insights.
• Discuss the relationship between finite morphisms and blowing up processes in algebraic geometry.
  • The process of blowing up introduces new structures to manage singularities in varieties, and finite morphisms play a critical role in this context. When performing a blow-up, we often end up with a space that has new geometric properties, and finite morphisms help control these new features. They ensure that the resultant scheme from a blow-up remains compactly represented while also maintaining connections back to the original variety's structure. This relationship facilitates further exploration of singularities and resolutions within algebraic geometry.
• Evaluate how finite morphisms impact the study of integral schemes and their properties within algebraic geometry.
  • Finite morphisms are fundamental in understanding integral schemes because they maintain integrality across their mapping. An integral scheme has no embedded points, and through a finite morphism, we can ensure that this property is preserved under various transformations. This preservation allows mathematicians to explore more complex geometrical structures while ensuring that basic properties like irreducibility are intact. Additionally, studying how finite morphisms interact with integral schemes can lead to significant insights into their geometric behavior and underlying algebraic characteristics.

Related terms

Affine Morphism: A morphism between schemes where the preimage of any affine open set is again affine.

Finitely Presented Module: A module over a ring that can be generated by a finite number of elements and has a finite number of relations among those generators.

Galois Morphism: A morphism that exhibits properties similar to those of Galois extensions in field theory, characterized by a certain symmetry and finiteness conditions.