Closed immersions

Definition

Closed immersions are morphisms in algebraic geometry that represent a way of embedding a closed subset of a scheme into another scheme while retaining the structure of the closed set. They can be thought of as the 'best' way to include one space inside another, ensuring that the image inherits a sheaf structure that is coherent. This concept is fundamental when discussing coherent sheaves, as closed immersions help define how sheaves behave on closed subsets of schemes.

5 Must Know Facts For Your Next Test

1. Closed immersions can be defined by a quotient of the ring of functions on the larger scheme, modding out by an ideal corresponding to the closed subset.
2. In terms of sheaves, if you have a closed immersion from a scheme X to a scheme Y, the pullback of a coherent sheaf on Y will yield a coherent sheaf on X.
3. The image of a closed immersion is not just a topological embedding but also has an induced structure that respects the algebraic properties of the original space.
4. Closed immersions are represented by the functoriality property, meaning they can be utilized in various categorical constructions and transformations.
5. Every closed immersion can be viewed locally as the vanishing locus of some set of functions, making them crucial for understanding local properties in algebraic geometry.

Review Questions

• How do closed immersions relate to the concept of coherent sheaves and their behavior on closed subsets?
  • Closed immersions play a vital role in understanding coherent sheaves as they allow for coherent sheaves defined on larger schemes to restrict naturally to closed subsets. When you have a coherent sheaf on a scheme Y and you perform a closed immersion into another scheme X, the structure allows you to pull back this sheaf to X, maintaining its coherence. This relationship illustrates how sheaves behave under embeddings, which is essential for studying local properties in algebraic geometry.
• Discuss the significance of closed immersions in defining algebraic structures within schemes.
  • Closed immersions are crucial in defining algebraic structures within schemes because they ensure that when one scheme is embedded into another, it retains its inherent properties. They allow for the transition between different schemes while preserving local algebraic data through their induced structures. Moreover, they form the basis for understanding how various algebraic objects behave under morphisms, enabling a deeper exploration into relationships between schemes and their associated sheaves.
• Evaluate how closed immersions can influence cohomological properties in algebraic geometry.
  • Closed immersions significantly influence cohomological properties in algebraic geometry by establishing how sheaves behave when transitioning between different spaces. Since closed immersions induce coherent sheaves through their pullbacks, they become essential in calculating cohomology groups. The understanding gained from examining these immersions can reveal key insights about the global sections of sheaves and their interactions with different topological spaces, leading to advancements in both theoretical and applied areas of geometry.

Related terms

Schemes: Mathematical structures that generalize algebraic varieties, consisting of a topological space along with a sheaf of rings, allowing for the study of solutions to polynomial equations.

Sheaf:

A mathematical tool that associates algebraic data to open sets of a topological space, allowing for local-to-global principles in topology and algebraic geometry.

Cohomology:

A branch of mathematics that studies the properties of topological spaces through algebraic invariants, often used to analyze sheaves and their associated cohomology groups.