Short Exact Sequence

Definition

A short exact sequence is a sequence of algebraic objects and morphisms between them that captures essential relationships, where the image of one morphism equals the kernel of the next. It usually takes the form of a diagram with three groups and two homomorphisms, showing how these groups are connected. This concept is vital for understanding how chain complexes relate to homology and how to work with exact sequences in various mathematical contexts.

5 Must Know Facts For Your Next Test

1. A short exact sequence is denoted as $$0 \to A \to B \to C \to 0$$, indicating that A injects into B, B surjects onto C, and both A and C are exact.
2. The concept allows for the transfer of properties between objects; for instance, if A is projective, then B will also have properties related to projectivity.
3. Short exact sequences can be used to derive long exact sequences in homology, which play a critical role in computing homology groups.
4. In the context of chain complexes, short exact sequences help to analyze and understand how different complexes are related through their homology.
5. Short exact sequences can be applied to both finite and infinite dimensional cases in algebraic topology, making them widely applicable in various areas.

Review Questions

• How does a short exact sequence illustrate the relationship between chain complexes and their homology?
  • A short exact sequence provides a structured way to connect different chain complexes and allows for the transfer of information about their homology. By analyzing a short exact sequence like $$0 \to A \to B \to C \to 0$$, one can determine how the homology of A relates to that of B and C. This relationship is crucial in deriving long exact sequences in homology, which gives deeper insights into the properties of spaces represented by these complexes.
• Discuss the significance of exactness in a short exact sequence and its implications for understanding algebraic structures.
  • Exactness in a short exact sequence means that the image of one morphism is equal to the kernel of the next. This property ensures that no information is lost as you move through the sequence. For example, if you know properties about A, you can conclude certain properties about B and C. Understanding this helps mathematicians manipulate algebraic structures effectively, particularly when working with modules and groups in different contexts.
• Evaluate how short exact sequences are utilized in deriving long exact sequences in homology theory, including an example.
  • Short exact sequences are foundational in deriving long exact sequences in homology theory, allowing for a systematic approach to calculating homology groups across various spaces. For instance, if you have a pair of spaces where a short exact sequence arises from an inclusion map, you can use this to create a long exact sequence that connects the homology of both spaces. This process not only illustrates relationships between different homological dimensions but also helps in understanding how certain topological features persist or change under continuous transformations.

Related terms

Chain Complex: A sequence of abelian groups or modules connected by homomorphisms where the composition of two consecutive homomorphisms is zero.

Homology:

A mathematical tool used to associate a sequence of abelian groups or modules with a topological space, providing information about its shape and structure.

Exactness:

A property of a sequence of morphisms where the image of one morphism equals the kernel of the next, indicating that the sequences do not lose information at each step.