Norm residue homomorphism

Definition

The norm residue homomorphism is a crucial tool in algebraic K-theory that connects Milnor K-theory with Galois cohomology. It takes elements from the Milnor K-group of a field, providing a way to translate questions about the field's residues into the context of algebraic cycles and higher K-groups, highlighting important relationships between different areas of number theory.

5 Must Know Facts For Your Next Test

1. The norm residue homomorphism plays a vital role in establishing a bridge between Milnor K-theory and the étale cohomology of fields, allowing for a deeper understanding of their structure.
2. It has implications for the Bloch-Kato conjecture, suggesting that it can help determine whether certain elements in Galois cohomology can be represented by cycles.
3. This homomorphism is particularly significant when considering fields with non-trivial Galois groups, as it captures the behavior of norms and residues under field extensions.
4. One application of the norm residue homomorphism is in the study of local-global principles, which involve understanding how properties defined locally can extend globally across fields.
5. Understanding the norm residue homomorphism is essential for proving results related to the Merkurjev-Suslin theorem, particularly regarding the linking of rational points and K-groups.

Review Questions

• How does the norm residue homomorphism relate elements of Milnor K-theory to Galois cohomology?
  • The norm residue homomorphism serves as a critical link between Milnor K-theory and Galois cohomology by translating elements from Milnor K-groups into Galois cohomological terms. This mapping allows researchers to interpret questions about residues in a field within the framework of algebraic cycles and higher K-groups. As a result, it provides insights into how these two areas interact, paving the way for deeper understanding in number theory.
• Discuss how the norm residue homomorphism impacts our understanding of the Bloch-Kato conjecture.
  • The norm residue homomorphism is central to the Bloch-Kato conjecture as it proposes a profound connection between Milnor K-theory and Galois cohomology. By establishing this homomorphism, it becomes possible to investigate whether elements in Galois cohomology can be represented by algebraic cycles. Thus, it not only supports the conjecture but also opens up new avenues for proving various results within this theoretical framework.
• Evaluate the significance of the norm residue homomorphism in proving aspects of the Merkurjev-Suslin theorem.
  • The norm residue homomorphism is crucial for demonstrating key aspects of the Merkurjev-Suslin theorem as it connects rational points with Milnor K-theory. By leveraging this homomorphism, mathematicians can show how rational points on projective spaces correspond to elements in higher K-groups. This linkage is vital for proving results related to fields with specific properties and highlights how deeply intertwined these concepts are within algebraic K-theory.

Related terms

Milnor K-theory: A branch of algebraic K-theory that studies fields through their Milnor K-groups, which are formed by taking the values of functions on the fields and examining their properties under Galois actions.

Galois cohomology: A cohomological tool that provides information about field extensions by analyzing how Galois groups act on various algebraic structures, essential for understanding symmetries in number theory.

Bloch-Kato conjecture: A conjecture proposing a deep connection between Milnor K-theory, Galois cohomology, and the structure of algebraic cycles, aiming to unify these areas within the framework of algebraic K-theory.