Derived functors

Definition

Derived functors are a concept in category theory that provide a way to measure the failure of a functor to be exact. They arise from the need to extend the notion of homological algebra, where they help in computing objects like Ext and Tor, which reveal deeper properties of modules over rings. Derived functors play a crucial role in connecting various areas of mathematics, allowing one to understand the relationships between different algebraic structures.

5 Must Know Facts For Your Next Test

1. Derived functors are computed using projective or injective resolutions of objects, allowing one to derive new information about these objects.
2. The most commonly studied derived functors include Ext and Tor, which help in understanding extensions and torsion products of modules.
3. Derived functors help in the study of how certain functors behave when applied to exact sequences, giving insights into their exactness.
4. They are essential in the field of homological algebra, providing tools for analyzing complex relationships in various algebraic contexts.
5. In categorical terms, derived functors can be viewed as a way to encode obstructions to lifting morphisms across certain functorial settings.

Review Questions

• How do derived functors relate to the concept of exactness in category theory?
  • Derived functors measure how much a functor fails to be exact by analyzing its behavior on exact sequences. When applying a functor to an exact sequence, if the resulting sequence is not exact, derived functors help capture the 'deficiency' or obstruction present. This relationship emphasizes the importance of derived functors in understanding when and why certain algebraic structures maintain their properties under transformations.
• Discuss the significance of projective and injective resolutions in the computation of derived functors.
  • Projective and injective resolutions serve as foundational tools for computing derived functors. By constructing these resolutions for an object, one can systematically derive new insights about its structure through the lens of homological algebra. Specifically, these resolutions provide a means to define Ext and Tor functors, allowing mathematicians to explore module properties and interrelations by focusing on how these resolutions approximate the original object.
• Evaluate the impact of derived functors on modern mathematical research, particularly in areas like algebraic topology and representation theory.
  • Derived functors have had a profound impact on modern mathematical research by providing essential tools for studying complex structures in algebraic topology and representation theory. In algebraic topology, they facilitate the exploration of homology theories, while in representation theory, they help analyze how groups act on vector spaces through representations. This dual utility showcases how derived functors not only deepen our understanding of individual subjects but also foster connections between various fields, enriching mathematical knowledge as a whole.

Related terms

Functor:

A functor is a mapping between categories that preserves the structure of those categories, sending objects and morphisms from one category to another.

Exact Sequence: An exact sequence is a sequence of objects and morphisms in which the image of one morphism equals the kernel of the next, reflecting a relationship among these objects.

Homology: Homology refers to a mathematical concept that studies topological spaces by associating sequences of algebraic objects, helping to identify and classify their properties.