Extension Groups

Definition

Extension groups are a concept in homological algebra that measures the ways in which a module can be extended by another module. They help to classify and understand modules by capturing information about the extensions of one module by another, revealing relationships between modules through short exact sequences. This concept is particularly useful in the study of projective and injective resolutions, where these extensions provide insight into the structure of modules.

5 Must Know Facts For Your Next Test

1. Extension groups can be denoted as Ext(A, B), where A is the module being extended and B is the module providing the extension.
2. The first extension group, Ext^1(A, B), captures all extensions of A by B, and is crucial for understanding how modules can fit together.
3. If a short exact sequence exists, it indicates that A can be viewed as a quotient of an extension of B by some other module.
4. For projective modules, the extension groups vanish; that is, Ext^1(A, P) = 0 for any projective module P.
5. Injective resolutions help compute extension groups, as they allow one to resolve modules in a way that reveals their extensions with injective modules.

Review Questions

• How do extension groups help us understand the relationships between different modules?
  • Extension groups reveal how one module can be extended by another, essentially showing different ways two modules can interact or combine. By examining these groups, we can see whether there are extensions and how they might form short exact sequences. This understanding aids in classifying modules and determining their structure relative to each other.
• In what way does the existence of an exact sequence relate to extension groups?
  • An exact sequence provides information about how one module can be expressed in terms of others, particularly through extensions. If an exact sequence exists, it signifies that there are relationships among the modules involved that allow for a clean transition from one to another. This directly connects to extension groups, as they characterize the nature of these transitions and indicate how modules can be constructed from one another.
• Critically analyze how projective and injective resolutions contribute to calculating extension groups and understanding their implications.
  • Projective and injective resolutions serve as tools for resolving modules into simpler components that reveal their structure. Projective resolutions are essential for establishing when extension groups vanish, indicating that a module is projective, while injective resolutions provide a means to compute extension groups effectively. By using these resolutions, we can draw conclusions about module behavior under extensions, thus deepening our understanding of both individual modules and their interrelations within algebraic structures.

Related terms

Exact Sequence: A sequence of modules and module homomorphisms that is exact at each module, meaning the image of one homomorphism equals the kernel of the next.

Projective Module: A module that satisfies the property that every surjective homomorphism onto it can be lifted to a homomorphism from the preimage.

Injective Module: A module that satisfies the property that every injective homomorphism into it can be extended to a homomorphism from any containing module.