Closed embedding

Definition

A closed embedding is a specific type of morphism between two algebraic varieties that identifies the image as a closed subset of the target variety. This means that not only does the morphism send points from one variety to another, but it also respects the structure of these varieties by making the image 'closed' in a topological sense. A key characteristic of closed embeddings is that they can be thought of as a way to view one variety as sitting inside another, preserving certain geometric properties.

5 Must Know Facts For Your Next Test

1. Closed embeddings arise naturally in algebraic geometry when one variety is contained within another in a well-defined manner.
2. The image of a closed embedding is always equipped with the Zariski topology, which allows for effective manipulation and analysis of algebraic sets.
3. Every closed embedding is also a regular immersion, meaning it can be locally described by regular functions.
4. In the context of schemes, closed embeddings correspond to the inclusion of schemes defined by ideals, giving them a concrete algebraic interpretation.
5. Closed embeddings help define schemes that are 'subschemes' of other schemes, playing a crucial role in the study of their geometric properties.

Review Questions

• How does a closed embedding differ from an open embedding in terms of topological properties and geometric representation?
  • A closed embedding contrasts with an open embedding primarily in terms of how the image relates to the target variety. In a closed embedding, the image forms a closed subset, which means that it includes its limit points and adheres to the closure properties in topology. On the other hand, an open embedding results in an image that is an open subset, excluding its boundary points. This distinction plays a crucial role in understanding how varieties interact with each other geometrically.
• Discuss the significance of closed embeddings in relation to schemes and ideals in algebraic geometry.
  • Closed embeddings are significant in algebraic geometry because they correspond directly to the inclusion of schemes defined by ideals. When one scheme is defined by an ideal within another, it provides a structured way to see how subvarieties or subschemes are positioned relative to their ambient varieties. This relationship facilitates analysis and manipulation of properties like dimension and intersection theory within algebraic geometry.
• Evaluate how closed embeddings contribute to understanding more complex structures in algebraic geometry, particularly in higher dimensions.
  • Closed embeddings play an essential role in understanding complex structures in algebraic geometry, especially as one moves into higher dimensions. They allow mathematicians to study subvarieties through their relationships with larger varieties, preserving geometric and algebraic information. By analyzing these embeddings, one can deduce properties such as singularity behavior and deformation theory, which are vital for grasping intricate geometrical configurations and their transformations.

Related terms

isomorphism: An isomorphism is a morphism between two varieties that has an inverse, meaning it establishes a one-to-one correspondence between the points of the two varieties and preserves their structures.

open embedding: An open embedding is a morphism where the image of the source variety is an open subset of the target variety, contrasting with closed embeddings where the image is closed.

morphism: A morphism is a general term for a structure-preserving map between two algebraic varieties, which can be thought of as a function that respects their algebraic structures.