Ext functor

Definition

The Ext functor is a fundamental tool in homological algebra that measures the extent to which a module fails to be projective, providing a way to study extensions of modules. This concept connects deeply to various areas such as the historical development of homological algebra, category theory, and the computations involved in related functors like Tor.

5 Must Know Facts For Your Next Test

1. The Ext functor is typically denoted as $$\text{Ext}^n(A, B)$$, where it measures the n-th derived functor of the Hom functor.
2. In particular, $$\text{Ext}^1(A, B)$$ classifies short exact sequences of modules that fit into the context of extending module A by module B.
3. The Ext functor exhibits properties such as additivity and functoriality, making it a powerful tool for deriving information about module categories.
4. One can compute Ext groups using projective resolutions or injective resolutions of the modules involved, highlighting its computational aspects.
5. Ext functors find applications in various areas including algebraic topology, where they help in understanding cohomology groups and other topological features.

Review Questions

• How does the Ext functor provide insight into the structure of modules?
  • The Ext functor gives us valuable information about how modules can be extended and whether certain sequences can be split. Specifically, by measuring extensions, it allows us to understand when one module can be built from another through exact sequences. This reveals important structural properties, especially regarding projectivity and injectivity of modules.
• Discuss the significance of using projective resolutions in calculating Ext groups.
  • Using projective resolutions to compute Ext groups provides a systematic method for determining how modules can be expressed in terms of simpler components. When we take a projective resolution of a module, we can apply the Hom functor and obtain exact sequences that yield information about extensions. This technique underlines the importance of projective modules in understanding the broader category-theoretic framework of homological algebra.
• Evaluate the role of the Ext functor in connecting algebraic structures with topological concepts.
  • The Ext functor plays a crucial role in bridging algebra and topology by revealing cohomological properties associated with topological spaces through its applications in algebraic topology. By associating Ext groups with cohomology theories, it allows mathematicians to use algebraic tools to study topological features. This connection enhances our understanding of both fields, showing how homological methods can yield insights into complex topological behaviors.

Related terms

Hom functor: The Hom functor is a way to assign to each pair of modules the set of all homomorphisms between them, serving as a building block for defining other functors in homological algebra.

Projective module: A projective module is a type of module that satisfies a lifting property concerning homomorphisms, often simplifying calculations with Ext and Tor functors.

Injective module: An injective module is one that allows every homomorphism from a submodule to be extended to the entire module, playing a crucial role in the duality aspects of homological algebra.