Short exact sequence

Definition

A short exact sequence is a sequence of module homomorphisms between three modules that provides a way to study the relationships among them. It is typically expressed in the form 0 → A → B → C → 0, where the image of each homomorphism equals the kernel of the next, indicating that A embeds into B, and B surjects onto C. This concept is foundational in understanding properties of modules and rings, especially in characterizing exactness and deriving information about the modules involved.

5 Must Know Facts For Your Next Test

1. In a short exact sequence, the first and last terms are always zero, denoting that the sequence is exact at those points.
2. The kernel of the surjective map from B to C is isomorphic to the image of the injective map from A to B, establishing an important relationship between the modules.
3. Short exact sequences can be used to define and understand derived functors, which have deep implications in homological algebra.
4. They serve as a building block for longer exact sequences, allowing for analysis of complex relationships between more than three modules.
5. In the context of Koszul complexes, short exact sequences help illustrate how resolutions can be constructed and how they relate to certain algebraic structures.

Review Questions

• How do short exact sequences illustrate the concept of exactness in module theory?
  • Short exact sequences directly embody the concept of exactness by showing how one module embeds into another while maintaining relationships through kernels and images. The condition that the image of one homomorphism equals the kernel of the next highlights how modules interact within the sequence. This not only emphasizes structural relationships but also sets up frameworks for understanding more complex algebraic structures.
• Discuss how short exact sequences are related to derived functors and their significance in module theory.
  • Short exact sequences play a critical role in defining derived functors because they allow mathematicians to create resolutions of modules. By taking an exact sequence and applying functors to it, one can derive new information about modules that might not be visible directly. This process leads to significant tools like Ext and Tor, which help in understanding extensions and torsion in module categories.
• Evaluate the importance of short exact sequences in understanding Koszul complexes and their applications in algebra.
  • Short exact sequences are vital in analyzing Koszul complexes because they allow for constructing resolutions that can simplify complex algebraic problems. By embedding modules into larger ones via short exact sequences, one can dissect problems into manageable parts. This method also connects deeply with homological techniques, which find applications across various areas in mathematics such as algebraic geometry and representation theory.

Related terms

Exactness: A property of a sequence of modules where the image of one homomorphism equals the kernel of the next, ensuring that certain algebraic properties hold across the sequence.

Module Homomorphism: A structure-preserving map between two modules that respects addition and scalar multiplication, crucial for defining relationships in exact sequences.

Kernel:

The set of elements in a module that are mapped to zero by a homomorphism, representing a measure of the 'failure' of the homomorphism to be injective.