Exact Sequence

Definition

An exact sequence is a sequence of algebraic objects and morphisms between them where the image of one morphism is exactly the kernel of the next. This concept is fundamental in many areas of mathematics, particularly in homology and cohomology theories, as it provides a way to understand relationships between different structures. In the context of cohomology groups, exact sequences help to capture important properties of spaces and their differential forms by relating them in a precise manner.

5 Must Know Facts For Your Next Test

1. Exact sequences can be used to derive long exact sequences from short exact sequences, allowing for deeper insights into the properties of topological spaces.
2. In the context of De Rham cohomology, exact sequences can show how different degrees of differential forms relate to each other through integration and differentiation.
3. A short exact sequence typically takes the form: $$0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$$ where the image of f is the kernel of g.
4. Exact sequences help identify isomorphisms between cohomology groups, indicating when two spaces have similar topological properties.
5. They are essential in proving important results like the Universal Coefficient Theorem, which connects homology and cohomology.

Review Questions

• How does an exact sequence relate the concepts of kernel and image in a sequence of morphisms?
  • An exact sequence explicitly captures the relationship between kernel and image across a series of morphisms. For an exact sequence $$0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$$, it states that the image of f (which is a subset of B) coincides exactly with the kernel of g (which is a subset of B). This condition ensures that any element in B that gets sent to zero in C must come from A, establishing a precise connection between these algebraic structures.
• Discuss how exact sequences can be applied to derive long exact sequences from short exact sequences.
  • Short exact sequences provide foundational relationships between three groups, allowing mathematicians to construct long exact sequences that reveal deeper connections among a larger collection of groups. For instance, when working with cohomology, one can start with a short exact sequence and use it to derive a long exact sequence that captures how cohomology groups change across dimensions. This process emphasizes how topological properties evolve and interact within larger structures and enables us to analyze more complex spaces systematically.
• Evaluate the role of exact sequences in proving the Universal Coefficient Theorem and their implications for homology and cohomology.
  • Exact sequences play a crucial role in establishing the Universal Coefficient Theorem, which links homology with cohomology by showing how both sets of groups relate through exactness. This theorem demonstrates that given a homology group, one can derive corresponding cohomology groups using an exact sequence. The implications are significant as they provide insights into how topological spaces can be understood through both lenses, offering powerful tools for classifying and comparing complex structures in mathematics.

Related terms

Cohomology: A mathematical tool used to study topological spaces through algebraic invariants that classify the shapes and structures within those spaces.

Kernel:

The set of elements that are mapped to zero by a given linear transformation, representing a form of symmetry or redundancy in the structure.

Image:

The set of outputs produced by applying a function or morphism to the elements of its domain, reflecting how structures are transformed.