Ext^n

Definition

The functor $$\text{Ext}^n$$ is a key concept in homological algebra, representing the derived functors of the Hom functor. It captures the information about extensions of modules, particularly concerning how modules can be built from others and how they can fit into exact sequences. The notation $$\text{Ext}^n(A, B)$$ describes the set of equivalence classes of extensions of the module $$A$$ by the module $$B$$, where $$n$$ indicates the degree of the extension.

5 Must Know Facts For Your Next Test

1. $$\text{Ext}^0(A, B)$$ is isomorphic to $$\text{Hom}(A, B)$$, representing the set of all homomorphisms between modules $$A$$ and $$B$$.
2. For projective modules, the group $$\text{Ext}^n(A, B)$$ vanishes for all $$n > 0$$, simplifying many calculations involving extensions.
3. The functor $$\text{Ext}^1(A, B)$$ specifically classifies extensions of the module $$B$$ by $$A$$, indicating how one module can be built from another.
4. $$\text{Ext}^n(A, B)$$ is a contravariant functor in its first argument, meaning that reversing the order of inputs changes the behavior and properties.
5. In many algebraic contexts, understanding $$\text{Ext}^n$$ allows for insights into the structure of modules and their relationships within categories.

Review Questions

• How does the Ext functor relate to exact sequences and what implications does this relationship have for understanding module extensions?
  • The Ext functor provides a way to study module extensions through exact sequences. Specifically, if you have an exact sequence involving two modules, applying the Hom functor can lead to insights about extensions via the Ext functor. This relationship allows mathematicians to understand when modules can fit into certain sequences and how they can be decomposed or constructed from others.
• Discuss the importance of projective modules in relation to Ext groups and how they simplify computations involving module extensions.
  • Projective modules play a crucial role in homological algebra because they guarantee that Ext groups vanish for higher degrees. This means that if one of the modules involved in an extension is projective, then all higher Ext groups will be zero. This greatly simplifies calculations since one does not need to consider complex extensions beyond the first level when working with projective modules.
• Evaluate how understanding Ext groups impacts broader concepts in algebraic topology and category theory.
  • Understanding Ext groups enriches various areas such as algebraic topology and category theory by providing deeper insights into how structures interact. In algebraic topology, for instance, Ext groups can relate to homology theories by revealing connections between different spaces through their respective algebraic invariants. Moreover, in category theory, Ext groups highlight important relationships between objects and morphisms, facilitating a richer understanding of derived categories and functors in general.

Related terms

Hom Functor: The Hom functor assigns to each pair of modules a new module whose elements are the homomorphisms from one module to another.

Exact Sequence: An exact sequence is a sequence of module homomorphisms such that the image of one homomorphism equals the kernel of the next.

Projective Module: A projective module is a module that satisfies a lifting property, allowing it to be 'free' in some sense and often simplifying the computation of extensions.