Short Exact Sequence

Definition

A short exact sequence is a sequence of algebraic structures and homomorphisms where the image of one morphism equals the kernel of the next, typically represented as 0 → A → B → C → 0. This definition serves as a foundation for understanding various concepts in algebra, particularly in relation to how structures relate to one another through homomorphisms and their exactness properties.

5 Must Know Facts For Your Next Test

1. Short exact sequences provide a way to study relationships between modules, allowing one to derive properties about them through their connections.
2. In a short exact sequence, if A injects into B and B surjects onto C, then one can conclude important facts about the structure of C based on A and B.
3. Short exact sequences are fundamental in defining derived functors like Ext, which helps in understanding extensions of modules.
4. The concept of a short exact sequence is essential for proving existence theorems that guarantee certain types of module structures can be formed.
5. In terms of diagram chasing, short exact sequences are often used as a tool to extract information about morphisms and relationships within larger diagrams.

Review Questions

• How do short exact sequences illustrate the relationship between different algebraic structures?
  • Short exact sequences highlight the way different algebraic structures are connected through homomorphisms. The property that the image of one morphism equals the kernel of another indicates a precise relationship between modules. For example, if we have 0 → A → B → C → 0, the injective map from A to B and the surjective map from B to C demonstrate how elements transition from one structure to another while preserving essential information about their interactions.
• In what ways do short exact sequences relate to chain complexes and their exactness?
  • Short exact sequences serve as foundational elements in the study of chain complexes by providing clear examples of exactness. When dealing with chain complexes, each component must satisfy the condition that the image of one map equals the kernel of the next. This principle is crucial when analyzing homology theories and helps illustrate how complex structures can be broken down into simpler components while retaining important relationships among them.
• Evaluate how understanding short exact sequences can enhance our knowledge of Ext functors and their applications.
  • Understanding short exact sequences is key to grasping how Ext functors operate and their significance in module theory. The Ext functor arises naturally from considering extensions defined by short exact sequences. By evaluating these sequences, we can derive critical information about module extensions and classify them according to their properties. This foundational knowledge allows us to apply Ext in various contexts, including cohomology theories and obstruction problems, which further enrich our understanding of algebraic structures.

Related terms

Exactness: A property of a sequence of modules and homomorphisms where the image of one map is equal to the kernel of the next, indicating a perfect 'fit' in the sequence.

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

Kernel: The set of elements that map to zero under a given homomorphism, providing insight into the structure and behavior of algebraic objects.