Ext Functor

Definition

The Ext functor is a mathematical tool used in homological algebra that captures the extensions of modules. It measures how one module can be extended by another and plays a crucial role in understanding the structure of modules over a ring, particularly in classifying extensions and studying their properties.

5 Must Know Facts For Your Next Test

1. The Ext functor is denoted as \( \text{Ext}^n(M, N) \), where \( M \) and \( N \) are modules, and \( n \) indicates the level of extension being considered.
2. For \( n = 1 \), \( \text{Ext}^1(M, N) \) classifies the short exact sequences of modules, showing how \( N \) can be extended by \( M \).
3. The Ext functor is contravariant, meaning it reverses the direction of morphisms; if there is a homomorphism from \( M \) to \( M' \), it induces a map from \( \text{Ext}^n(M', N) \) to \( \text{Ext}^n(M, N) \).
4. When considering projective or injective modules, the Ext functor provides insight into whether certain extensions can exist or not, with injectives being related to vanishing Ext functors.
5. The relationship between Ext and Tor functors reveals deeper insights about the structure of modules, particularly through their respective sequences and derived functors.

Review Questions

• How does the Ext functor relate to the classification of short exact sequences in module theory?
  • The Ext functor, particularly \( \text{Ext}^1(M, N) \), is essential for classifying short exact sequences of modules. It identifies how one module can be extended by another and determines whether an extension exists. By examining these extensions, one can understand the interrelations between modules and gain insight into their structure.
• Discuss the implications of the contravariant nature of the Ext functor for morphisms between modules.
  • The contravariant property of the Ext functor means that when there is a homomorphism from module \( M \) to module \( M' \), it leads to an induced map from \( \text{Ext}^n(M', N) \) to \( \text{Ext}^n(M, N) \). This property highlights how relationships between modules can affect their extension groups, illustrating how changes in morphisms influence the possible extensions and derived structures of these modules.
• Analyze the relationship between Ext and Tor functors and how they contribute to our understanding of module extensions.
  • The Ext and Tor functors are deeply interconnected in homological algebra. While the Ext functor classifies extensions of modules, the Tor functor measures torsion in a module's structure. Their relationship is crucial in forming long exact sequences in homology, helping mathematicians understand how various algebraic properties interact. By analyzing both functors together, we gain comprehensive insight into module theory and its applications across different areas of mathematics.

Related terms

Modules: An abstract algebraic structure where a set of elements can be added together and multiplied by scalars from a ring, generalizing vector spaces.

Homological Algebra:

A branch of mathematics that studies homology in a general algebraic setting, focusing on the relationships between algebraic structures through their projective and injective properties.

Projective Modules: Modules that satisfy the lifting property with respect to epimorphisms, meaning they can 'lift' maps from quotient modules.