Long exact sequences

Definition

Long exact sequences are sequences of abelian groups or modules linked by homomorphisms, where the kernel of one map is equal to the image of the previous map, forming a chain that continues indefinitely. They are essential in algebraic topology and algebraic geometry, particularly in the study of cohomology of sheaves, providing a systematic way to relate different cohomology groups through exactness conditions.

5 Must Know Facts For Your Next Test

1. Long exact sequences arise naturally when dealing with short exact sequences and can be derived from them by applying functors such as Hom.
2. They play a crucial role in connecting cohomology groups, particularly in establishing long exact sequences of cohomology associated with a short exact sequence of sheaves.
3. The long exact sequence in cohomology often allows for computations involving sheaf cohomology by providing information about the relationship between different cohomology groups.
4. In many cases, long exact sequences reveal the vanishing or non-vanishing of certain cohomology groups, giving insights into the geometric properties of the underlying space.
5. These sequences facilitate the application of various key results in algebraic topology and algebraic geometry, such as Mayer-Vietoris sequences and spectral sequences.

Review Questions

• How do long exact sequences connect different cohomology groups within the context of sheaves?
  • Long exact sequences connect different cohomology groups by establishing a continuous relationship between them through exactness. When we have a short exact sequence of sheaves, applying the functor that computes cohomology yields a long exact sequence. This sequence shows how the vanishing or existence of one cohomology group can influence others, thereby allowing us to derive important information about the underlying topological structure.
• Discuss the significance of long exact sequences in deriving results about sheaf cohomology.
  • Long exact sequences are significant in deriving results about sheaf cohomology because they allow mathematicians to track how properties propagate through related sheaves. For instance, if we have a short exact sequence of sheaves, the corresponding long exact sequence in cohomology can provide insights into when certain groups are zero or non-zero. This connection helps to simplify complex calculations and offers a way to approach problems systematically by breaking them down into manageable parts.
• Evaluate how long exact sequences influence the computations in cohomology theories across different mathematical contexts.
  • Long exact sequences influence computations in cohomology theories by providing powerful tools for linking disparate parts of algebraic topology and geometry. For example, they are instrumental in Mayer-Vietoris arguments where one computes the cohomology of a space by breaking it down into simpler pieces. This connection across various contexts enables deeper understanding and techniques that can be applied universally, impacting both theoretical aspects and practical applications in algebraic geometry and beyond.

Related terms

Exact Sequence: A sequence of abelian groups or modules where the image of one homomorphism is equal to the kernel of the next, ensuring that certain algebraic properties hold.

Cohomology: A mathematical tool used to study topological spaces via algebraic invariants, often implemented through sheaf theory in algebraic geometry.

Sheaf: A mathematical structure that assigns data to open sets of a topological space, allowing for local-to-global principles in algebraic geometry and topology.