Ext

Definition

Ext, short for 'ext functor', is a crucial concept in homological algebra that measures the extent to which a module fails to be projective. It captures the idea of extension groups, allowing mathematicians to study how modules can be 'extended' over one another through short exact sequences. In the context of derived categories, Ext plays a key role in understanding morphisms between complexes and helps identify relationships between objects in the derived category.

5 Must Know Facts For Your Next Test

1. Ext groups can be computed using projective resolutions, where Ext measures extensions of modules by looking at equivalence classes of exact sequences.
2. In derived categories, the objects are complexes of modules, and Ext helps to characterize morphisms between these complexes.
3. Ext can have significant implications in algebraic geometry and representation theory, where it relates to sheaf cohomology and representations of algebras.
4. For two modules A and B, Ext^1(A, B) classifies the short exact sequences that fit into an extension problem involving A and B.
5. The higher Ext groups, such as Ext^n(A, B), provide further insight into the complexity of extensions and can relate to higher derived functors.

Review Questions

• How does Ext help us understand relationships between modules in homological algebra?
  • Ext provides a framework for analyzing how one module can extend another through short exact sequences. By measuring the failure of a module to be projective, Ext allows us to classify extensions and understand how different modules interact with one another. This insight is essential for studying morphisms in derived categories and helps illustrate deeper connections between algebraic structures.
• Discuss how Ext relates to derived categories and its significance in this context.
  • In derived categories, Ext becomes a powerful tool for understanding morphisms between complexes of modules. It helps categorize these morphisms by providing essential information about their structure and relationships. The ability to derive Ext groups from complexes leads to a deeper comprehension of derived functors and influences the overall framework of homological algebra.
• Evaluate the role of Ext in cohomology theories and its broader implications in mathematics.
  • Ext plays a pivotal role in various cohomology theories by linking algebraic concepts with topological properties. Its utility in classifying extensions connects it with sheaf cohomology, making it instrumental in algebraic geometry. This connection highlights how Ext contributes not only to pure algebra but also enhances our understanding of geometry, thereby bridging different areas within mathematics.

Related terms

Derived Functor: A functor that arises from a given functor and captures information about the homological properties of modules or objects within a category.

Exact Sequence: A sequence of modules and homomorphisms between them such that the image of one homomorphism equals the kernel of the next, reflecting the notion of exactness in homological algebra.

Cohomology: A mathematical tool used to study topological spaces and algebraic structures, providing information about their shape and structure through algebraic invariants.