Long exact sequence

Definition

A long exact sequence is a sequence of abelian groups or modules and homomorphisms that captures the relationships between different cohomology groups and is fundamental in algebraic topology and sheaf theory. It provides a way to connect the cohomology of a space or a sheaf with the cohomology of its subspaces or subsheaves, allowing for a deeper understanding of their structure and properties.

5 Must Know Facts For Your Next Test

1. A long exact sequence typically arises from a short exact sequence of sheaves, where the long sequence tracks how cohomology groups relate to one another.
2. The long exact sequence is especially useful in studying derived functors, such as sheaf cohomology, which links local data to global properties.
3. One common example of a long exact sequence is the one associated with a pair of spaces, where it relates the cohomology groups of the space and its subspace.
4. Long exact sequences help in proving various important results in algebraic geometry, such as the vanishing of certain cohomology groups under specific conditions.
5. In practice, long exact sequences provide crucial tools for computing cohomology groups, often simplifying complex calculations by breaking them down into more manageable parts.

Review Questions

• How does a long exact sequence relate different cohomology groups, and what significance does this have in understanding topological spaces?
  • A long exact sequence connects different cohomology groups by providing a continuous link through its homomorphisms. This relationship helps reveal how local properties of a space influence its global topology, allowing mathematicians to derive important insights about the structure and behavior of spaces. Essentially, it acts as a bridge between smaller and larger spaces or sheaves, giving us valuable information about their interdependencies.
• Discuss how long exact sequences can be used to compute the cohomology of complex spaces or sheaves effectively.
  • Long exact sequences are instrumental in simplifying complex computations in cohomology by breaking them into smaller, more manageable parts. When dealing with a short exact sequence of sheaves, one can derive a long exact sequence in cohomology that relates the cohomology groups of the original sheaves. This allows for effective computation by leveraging known properties of simpler sheaves or spaces to draw conclusions about more complicated structures.
• Evaluate the role of long exact sequences in proving key results within algebraic geometry, particularly regarding derived functors.
  • Long exact sequences play a critical role in algebraic geometry by establishing connections between derived functors like sheaf cohomology and allowing for proofs of significant results. They provide a framework for understanding how certain conditions lead to vanishing cohomology groups, which can reveal important properties about schemes and varieties. This relationship highlights how long exact sequences serve not only as computational tools but also as foundational elements that underpin various theoretical advancements within the field.

Related terms

Cohomology: A mathematical tool that assigns algebraic invariants to topological spaces, helping to classify them based on their shape and structure.

Sheaf: A mathematical construct that allows for the systematic tracking of local data attached to the open sets of a topological space.

Exact Sequence: A sequence of mathematical objects and morphisms such that the image of one morphism equals the kernel of the next, ensuring continuity of relationships in the sequence.