Ext functor

Definition

An ext functor is a functor that encodes the relationships between morphisms in a category, specifically in the context of abelian categories and homological algebra. It generalizes the notion of derived functors, capturing how extensions of objects relate to each other. This concept plays a crucial role in studying the structure of modules and homological dimensions, making it essential for understanding both extensions in abelian categories and applications through Kan extensions.

5 Must Know Facts For Your Next Test

1. Ext functors are denoted as $$\text{Ext}^n(A, B)$$, where $$A$$ and $$B$$ are objects in an abelian category and $$n$$ indicates the degree of extension.
2. They classify extensions of a module or object by describing all possible ways to fit one object into another.
3. The first ext functor, $$\text{Ext}^1(A, B)$$, captures equivalence classes of short exact sequences that split, providing critical insight into how modules can be extended.
4. In homological algebra, the higher ext groups $$\text{Ext}^n(A, B)$$ provide information about derived categories and can reflect properties like projectivity and injectivity.
5. Ext functors have a dual counterpart called $$\text{Tor}$$, which measures the failure of flatness of modules and is crucial for exploring homological dimensions.

Review Questions

• How do ext functors relate to short exact sequences in abelian categories?
  • Ext functors are closely related to short exact sequences by classifying extensions between objects. Specifically, $$\text{Ext}^1(A, B)$$ represents equivalence classes of short exact sequences that connect objects $$A$$ and $$B$$. This connection allows for understanding how one object can be viewed as an extension of another, providing important insights into their structural relationships.
• Discuss the importance of ext functors in the context of homological algebra and their impact on module theory.
  • Ext functors play a pivotal role in homological algebra as they help classify extensions and provide crucial information about module theory. They reveal how modules can be extended, thus influencing concepts like projectivity and injectivity. Understanding ext groups allows mathematicians to delve into deeper properties of modules, enabling a better grasp of their homological dimensions and relationships.
• Evaluate how ext functors and Kan extensions interact within categorical frameworks, particularly in understanding derived categories.
  • Ext functors and Kan extensions interact significantly in categorical frameworks by facilitating deeper insights into derived categories. While ext functors classify extensions between objects in abelian categories, Kan extensions provide a way to extend these concepts further along different morphisms. This interplay enhances our understanding of how functors can be constructed or modified in relation to existing structures, ultimately enriching our perspective on homological algebra and categorical topology.

Related terms

Derived Functor: A functor that extends a given functor to a derived category, providing insight into the homological properties of the objects involved.

Abelian Category: A category in which morphisms can be added together and where kernels and cokernels exist, allowing for a well-defined notion of exact sequences.

Kan Extension: A generalization of functorial constructions that allows for extending functors along other functors, crucial for the study of universal properties.