Closed embedding

Definition

A closed embedding is a type of morphism in algebraic geometry that represents an inclusion of one algebraic variety into another, such that the image of the included variety is closed in the ambient variety. This concept is crucial for understanding how varieties can be related and how push-forward maps operate in the context of Gysin homomorphisms, where one often needs to consider the properties of varieties under these embeddings.

5 Must Know Facts For Your Next Test

1. Closed embeddings are characterized by their ability to allow the inclusion of a variety while preserving certain topological properties, particularly compactness.
2. In algebraic geometry, a closed embedding is often denoted by a specific kind of ring homomorphism, typically represented by the ideal of the embedded variety.
3. The image of a closed embedding is always a closed subset in the Zariski topology, which is fundamental for applying various geometric and topological tools.
4. Closed embeddings are essential in defining the Gysin homomorphism because they help transfer information between different cohomology groups related to different varieties.
5. When dealing with push-forward maps, closed embeddings ensure that any class pushed forward retains important geometric and topological properties, allowing for meaningful calculations in cohomology.

Review Questions

• How does a closed embedding influence the relationship between two algebraic varieties?
  • A closed embedding creates a clear relationship between two algebraic varieties by ensuring that one is included in another in a way that the structure and properties are preserved. Specifically, it allows for the image of the included variety to be considered a closed subset of the larger variety. This relationship is crucial when utilizing tools like push-forward maps and Gysin homomorphisms since they rely on understanding how classes and properties behave under this inclusion.
• Discuss how closed embeddings facilitate the use of Gysin homomorphisms in algebraic geometry.
  • Closed embeddings serve as the foundation for Gysin homomorphisms by providing a framework through which cohomological data can be transferred between varieties. When one variety is closed embedded in another, it allows mathematicians to create Gysin homomorphisms that effectively push forward cohomology classes from the embedded variety to the ambient space. This operation is essential because it connects various geometric properties and allows for deeper insights into the relationships between different varieties.
• Evaluate the role of closed embeddings in determining properties of push-forward maps and their implications for cohomology calculations.
  • Closed embeddings play a significant role in determining properties of push-forward maps by ensuring that classes pushed forward from a subvariety retain their structural integrity and relevant information. This preservation is critical for accurate cohomology calculations since it allows researchers to draw meaningful conclusions about how different varieties interact. The behavior of push-forward maps in relation to closed embeddings can reveal intricate details about singularities, intersections, and other geometric features, highlighting their importance in advanced algebraic geometry.

Related terms

Morphism: A continuous function between two algebraic varieties that preserves the structure of the varieties.

Gysin homomorphism: A homomorphism associated with a closed embedding that allows one to transfer cohomological information from the included variety to the ambient variety.

Push-forward map: A map that takes classes from the cohomology of a subvariety and pushes them forward to the cohomology of the ambient variety, often associated with closed embeddings.