Long exact sequence

Definition

A long exact sequence is a sequence of abelian groups or modules connected by homomorphisms such that the image of each map is equal to the kernel of the next, forming an exact chain. This structure plays a crucial role in connecting algebraic objects in a way that preserves information across different dimensions and can be especially useful in understanding properties like homology and cohomology.

5 Must Know Facts For Your Next Test

1. A long exact sequence can be derived from short exact sequences through a process of taking direct limits or colimits, preserving exactness over longer spans.
2. In algebraic topology, long exact sequences are often associated with pairs of spaces, allowing for the computation of homology or cohomology groups.
3. Long exact sequences can be constructed using functors, which helps in deriving new sequences from known ones while maintaining their exactness properties.
4. They provide a way to track how properties such as injectivity and surjectivity transfer through related algebraic structures in a systematic manner.
5. Long exact sequences often arise in various contexts, such as in the context of spectral sequences and derived functors, enriching the study of complex structures.

Review Questions

• How does a long exact sequence relate to the concept of exactness in shorter sequences?
  • A long exact sequence generalizes the idea of exactness found in short exact sequences by extending it across multiple groups or modules. In an exact sequence, the image of one map equals the kernel of the next, ensuring that information flows correctly from one object to another. The long exact sequence maintains this property over a broader framework, allowing for more complex relationships to be analyzed and understood.
• Discuss how long exact sequences can be utilized in homology theories to extract meaningful information about topological spaces.
  • Long exact sequences are powerful tools in homology theories as they connect homology groups associated with different spaces or pairs of spaces. By applying the long exact sequence to a pair of topological spaces, one can derive relationships between their respective homology groups. This enables mathematicians to gain insights into the connectivity and structure of these spaces, as well as how properties change when considering different dimensional aspects.
• Evaluate the significance of long exact sequences in advanced algebraic concepts such as derived functors and spectral sequences.
  • Long exact sequences play a critical role in advanced algebraic concepts like derived functors and spectral sequences by serving as foundational building blocks for understanding complex algebraic relationships. They allow for systematic connections between different layers of abstraction, enabling mathematicians to analyze how properties propagate through these constructs. This significance extends into areas such as cohomology theories where long exact sequences help unveil intricate relationships within algebraic topology and enhance our understanding of manifold structures.

Related terms

exact sequence: An exact sequence is a sequence of modules or groups where the image of one homomorphism equals the kernel of the next, ensuring that the structure remains intact throughout.

homology: Homology is a mathematical concept used to study topological spaces by associating sequences of algebraic structures, allowing for a systematic analysis of their shapes and connectivity.

cohomology: Cohomology is a dual concept to homology that assigns algebraic invariants to topological spaces, providing additional information about their structure and relationships.