Exact Sequences

Definition

Exact sequences are sequences of algebraic objects and morphisms where the image of one morphism equals the kernel of the next. This concept is crucial in understanding how different spaces or structures interact with one another, highlighting relationships such as cohomology and homology. In various contexts, exact sequences can provide powerful tools for studying properties like K-theory and Gysin homomorphisms, as well as their connections to algebraic structures.

5 Must Know Facts For Your Next Test

1. Exact sequences can be classified into short exact sequences, long exact sequences, and split exact sequences, each serving different purposes in algebraic topology and K-theory.
2. In the context of Gysin homomorphisms, exact sequences help relate the K-theory of a space to that of its subspaces, establishing significant relationships in algebraic geometry.
3. Quillen's higher algebraic K-theory heavily relies on the notion of exact sequences to understand how various objects can be related within an overarching framework.
4. Exact sequences can lead to long exact sequences in cohomology, which are vital for computing the cohomology groups of complex structures.
5. In algebraic K-theory of schemes and varieties, exact sequences help reveal how morphisms between schemes affect their K-groups and provide a deeper understanding of their geometric properties.

Review Questions

• How do exact sequences facilitate the relationship between different spaces in K-theory?
  • Exact sequences play a vital role in K-theory by illustrating how the image of one morphism corresponds to the kernel of another. This connection helps researchers study the relationships between various spaces and their associated K-groups. For instance, when considering Gysin homomorphisms, exact sequences allow for a clear understanding of how subspaces influence the overall structure of the space's K-theory.
• Discuss how long exact sequences are constructed from short exact sequences and their significance in cohomology.
  • Long exact sequences are constructed by taking a series of short exact sequences, where each short sequence contributes additional information about the relationships between different homology or cohomology groups. The significance lies in their ability to connect multiple levels of algebraic invariants derived from topological spaces. This allows mathematicians to compute complex cohomology groups effectively, revealing deeper insights into the underlying geometric properties.
• Evaluate the impact of exact sequences on our understanding of K-theory for schemes and varieties.
  • Exact sequences significantly enhance our understanding of K-theory for schemes and varieties by establishing critical connections between their morphisms and K-groups. By analyzing how these exact sequences behave under various operations, mathematicians can derive important results regarding the invariance properties of K-theory. This leads to a richer comprehension of both algebraic and geometric structures within this realm, illustrating how foundational concepts tie together multiple aspects of modern algebraic geometry.

Related terms

Kernel: The kernel of a morphism is the set of elements that map to zero, providing insights into the structure of the algebraic object.

Cohomology:

Cohomology is a mathematical tool used to study topological spaces through algebraic invariants, often linked with exact sequences to derive useful information about these spaces.

Homology: Homology provides a way to associate algebraic objects to a topological space, offering insights into its shape and structure, often examined through exact sequences.