Gysin homomorphisms

Definition

Gysin homomorphisms are algebraic constructions in K-theory that arise from the consideration of a proper morphism of schemes. They provide a way to understand the behavior of K-theory under certain geometric operations, especially in relation to the localization sequence, which connects the K-theory of spaces with the K-theory of their subspaces and their complements.

5 Must Know Facts For Your Next Test

1. Gysin homomorphisms are used to construct a map between K-groups that reflects the inclusion of a submanifold into a manifold.
2. They can be understood as a generalization of push-forward maps in cohomology, allowing for the transfer of K-theoretic information from one space to another.
3. In the context of the localization sequence, Gysin homomorphisms play a crucial role in relating the K-theory of a space to that of its complements.
4. The Gysin homomorphism associated with a proper morphism can often be represented using the Euler class of a vector bundle, linking geometric and topological properties.
5. They are essential for proving important results in algebraic K-theory, including the Riemann-Roch theorem for varieties.

Review Questions

• How do Gysin homomorphisms relate to the localization sequence in K-theory?
  • Gysin homomorphisms are fundamental in understanding the localization sequence because they help establish connections between the K-theory of a space and its open subspaces. When considering a proper morphism, Gysin homomorphisms allow us to map K-groups in a way that reflects how subspaces relate to their complements, thus facilitating the construction of long exact sequences in K-theory. This interplay is key in extracting valuable algebraic information from geometric structures.
• Discuss how Gysin homomorphisms generalize push-forward maps in cohomology and their implications for algebraic K-theory.
  • Gysin homomorphisms can be viewed as a generalization of push-forward maps found in cohomology because they provide a method for transferring information about K-theoretic classes from one space to another under proper morphisms. This is significant for algebraic K-theory as it allows mathematicians to apply techniques from cohomology while studying vector bundles and projective modules. The implications extend to various results, including how these maps can be used to prove correspondences between different types of spaces.
• Evaluate the importance of Gysin homomorphisms in proving results like the Riemann-Roch theorem for varieties.
  • Gysin homomorphisms play a critical role in proving results such as the Riemann-Roch theorem for varieties by facilitating connections between algebraic geometry and topological invariants. They enable researchers to utilize properties of vector bundles over varieties and link them with cohomological data, thus providing insights into the geometric structure of these varieties. The interplay between Gysin homomorphisms and other algebraic tools ultimately leads to deeper understandings and significant advances in the field.

Related terms

Localization sequence: A long exact sequence in K-theory that connects the K-theory of a space with that of its open subspaces and their complements.

Proper morphism: A type of morphism in algebraic geometry that satisfies certain compactness conditions, allowing for the application of Gysin homomorphisms.

K-theory: A branch of mathematics that studies vector bundles and projective modules through the lens of stable isomorphism classes, providing tools for understanding various topological and algebraic properties.