Long exact sequence

Definition

A long exact sequence is a sequence of abelian groups or modules connected by homomorphisms, which satisfies exactness at every point, indicating that the image of each homomorphism equals the kernel of the next. This concept is crucial in understanding the behavior of homology and cohomology theories, allowing one to relate different algebraic structures through their exact sequences and facilitating computations in various contexts.

5 Must Know Facts For Your Next Test

1. Long exact sequences can be derived from short exact sequences by taking appropriate functors, showcasing a deeper connection between different algebraic structures.
2. In homological algebra, long exact sequences often arise when working with derived functors, particularly Ext and Tor, linking these concepts to algebraic topology.
3. Long exact sequences can also be applied in the context of sheaf cohomology, where they help describe relationships among cohomology groups associated with open coverings.
4. These sequences are essential for understanding how various algebraic constructs interact, particularly in the study of extension problems and derived categories.
5. Diagram chasing is a powerful technique used with long exact sequences to deduce properties or compute unknown elements within the sequence.

Review Questions

• How do long exact sequences relate to short exact sequences in terms of homological properties?
  • Long exact sequences can be derived from short exact sequences by applying functors, demonstrating that they preserve certain properties of the underlying objects. For example, when you have a short exact sequence of chain complexes, applying the homology functor leads to a long exact sequence in homology. This connection allows for greater flexibility in studying relationships between different algebraic structures and their associated invariants.
• Discuss how long exact sequences are utilized in sheaf cohomology to relate cohomology groups of different open sets.
  • In sheaf cohomology, long exact sequences provide a way to connect the cohomology groups associated with various open sets within a topological space. Specifically, when considering a good cover or a Mayer-Vietoris sequence, the long exact sequence helps us understand how the cohomology of intersections relates to that of individual open sets. This is crucial for computing global sections and understanding sheaf-theoretic properties across different topological contexts.
• Evaluate the significance of long exact sequences in the context of derived functors and their universal properties.
  • Long exact sequences play a vital role in understanding derived functors, such as Ext and Tor, by illustrating how these functors interact with different algebraic structures. They highlight universal properties by showing how certain constructions lead to canonical maps between groups or modules that reveal information about extensions and torsion. The presence of a long exact sequence can often simplify complex relationships into manageable forms, providing deep insights into both algebraic and topological phenomena.

Related terms

Exactness: The property of a sequence of modules or groups where the image of one homomorphism equals the kernel of the next, ensuring no information is lost between mappings.

Chain Complex: A sequence of abelian groups or modules connected by boundary maps, where the composition of two consecutive boundary maps is zero, allowing for the definition of homology.

Cohomology: A mathematical tool that assigns algebraic invariants to topological spaces, often using long exact sequences to study the relationship between homology groups and derived functors.