Long exact sequences

Definition

Long exact sequences are sequences of algebraic objects, often associated with homology or cohomology theories, that connect different K-groups in a way that reveals deep structural properties. These sequences are essential in algebraic K-theory as they help in understanding the relationships between various K-groups and facilitate the application of techniques like the Q-construction and the plus construction, as well as the Bott periodicity theorem.

5 Must Know Facts For Your Next Test

1. Long exact sequences arise from short exact sequences through a process called 'longening', allowing for the study of more complex relationships among K-groups.
2. In algebraic K-theory, long exact sequences provide crucial information when studying the behavior of K-theory under various constructions, such as the Q-construction.
3. The long exact sequence in K-theory is often derived from an initial short exact sequence of ring homomorphisms, leading to important insights into the structure of K-groups.
4. These sequences facilitate calculations by providing a way to relate different groups, particularly in the context of connecting K-theoretic invariants across different spaces or constructions.
5. The Bott periodicity theorem is often demonstrated using long exact sequences, showing how K-theory exhibits periodicity over certain categories.

Review Questions

• How do long exact sequences help to reveal relationships between different K-groups in algebraic K-theory?
  • Long exact sequences illustrate how different K-groups are interconnected through morphisms. They allow mathematicians to track changes and shifts among these groups resulting from various algebraic constructions or inclusions. By following these sequences, one can gain insight into how one group's properties may influence or be influenced by another, thus forming a cohesive understanding of their collective behavior.
• Discuss how long exact sequences can be derived from short exact sequences and their implications for calculating K-groups.
  • Long exact sequences can be constructed from short exact sequences by extending them through a process known as 'longening'. This process involves considering additional homomorphisms that connect the original groups involved in the short exact sequence. The implications for calculating K-groups are significant because these long sequences provide a systematic way to relate various groups, making it easier to derive useful information about their structure and properties.
• Evaluate the role of long exact sequences in establishing the Bott periodicity theorem for algebraic K-theory.
  • Long exact sequences play a pivotal role in demonstrating the Bott periodicity theorem by showing how certain K-theoretic invariants exhibit periodic behavior. Specifically, they illustrate how mappings between different K-groups cycle through patterns, providing a structured understanding of K-theory's periodic nature. This connection not only supports theoretical claims but also aids in practical calculations involving these groups, reinforcing the importance of long exact sequences in this area of mathematics.

Related terms

Homology: A mathematical concept that studies topological spaces through algebraic invariants, helping to understand their structure and classification.

Exact Sequence: A sequence of algebraic objects and morphisms between them where the image of one morphism equals the kernel of the next, highlighting relationships within a mathematical structure.

K-Theory: A branch of algebra that studies vector bundles and projective modules through the use of K-groups, which serve as invariants in various mathematical contexts.