Exact Sequence

Definition

An exact sequence is a sequence of mathematical objects and morphisms between them, such that the image of each morphism matches the kernel of the next. This concept is crucial in algebraic topology and homological algebra, as it helps in understanding relationships between different mathematical structures. In particular, exact sequences can reveal important properties of sheaves, cohomology theories, and morphisms, making them essential tools in various areas of mathematics.

5 Must Know Facts For Your Next Test

1. Exact sequences can be classified into types such as short exact sequences, long exact sequences, and exact triangles, each serving different purposes in homological algebra.
2. In an exact sequence, if you have a sequence like $A \xrightarrow{f} B \xrightarrow{g} C$, it is exact at $B$ if $\text{Im}(f) = \text{Ker}(g)$.
3. Exact sequences play a key role in establishing isomorphisms between cohomology groups, as seen in Čech cohomology.
4. In sheaf theory, exact sequences help describe the behavior of sheaves and provide conditions for sheaf cohomology to be computable.
5. Exactness can also be studied through injective resolutions, where exact sequences are used to construct resolutions that provide insights into the homological properties of sheaves.

Review Questions

• How do exact sequences relate to injective resolutions and what purpose do they serve in this context?
  • Exact sequences are fundamental in injective resolutions because they allow us to construct a series of morphisms that reveal the structure of modules or sheaves. In this context, an exact sequence ensures that every module can be approximated by injective modules. This is crucial for computing derived functors, which further helps in understanding various properties of sheaves and their extensions.
• What is the significance of exact sequences in Čech cohomology and how do they help in computing cohomology groups?
  • In Čech cohomology, exact sequences are used to relate different cohomology groups. They establish connections between the local data provided by sheaves over open covers and global properties of the space. By forming long exact sequences from short ones, we can derive important relationships that simplify the computation of cohomology groups, such as using Mayer-Vietoris sequences for spaces with specific decompositions.
• Evaluate the impact of exact sequences on understanding morphisms of ringed spaces and their associated sheaves.
  • Exact sequences play a critical role in analyzing morphisms of ringed spaces because they enable us to understand how sheaves interact under continuous maps. By studying exactness at different points, we can determine conditions under which sheaf properties are preserved or reflected through these morphisms. This evaluation sheds light on coherence conditions, ensuring that certain algebraic structures maintain their integrity when transitioning between different topological contexts.

Related terms

Kernel: The kernel of a morphism is the set of elements that are mapped to zero, capturing information about the structure of the morphism.

Cohomology:

A mathematical framework that provides a way to assign algebraic invariants to topological spaces, helping to study their properties and relationships.

Functor:

A functor is a mapping between categories that preserves the structure of categories, often used in defining relationships between different mathematical contexts.