Derived Functors

Definition

Derived functors are a fundamental concept in category theory, representing a way to extend functors to obtain more information about the structure of objects in a category. They arise in contexts where regular functors may fail to capture essential properties, allowing mathematicians to explore derived categories and homological algebra. By studying derived functors, one can gain insights into resolutions and the relationships between different categories.

5 Must Know Facts For Your Next Test

1. Derived functors can be computed using projective or injective resolutions of objects in a category, helping to reveal hidden properties.
2. Common examples of derived functors include Ext and Tor, which provide information about extensions and torsion products in algebra.
3. Derived functors transform a covariant or contravariant functor into a sequence of functors that are easier to work with in terms of categorical properties.
4. In homological algebra, derived functors allow for the classification of modules and their relationships through derived categories.
5. The use of derived functors extends beyond pure mathematics and finds applications in fields like algebraic geometry and representation theory.

Review Questions

• How do derived functors enhance our understanding of regular functors and their limitations?
  • Derived functors enhance our understanding by providing a mechanism to extract deeper insights from regular functors that may not fully capture an object's structure. They are particularly useful when regular functors fail to preserve certain properties, enabling us to utilize resolutions to study the underlying relationships between objects. Through this extended lens, derived functors help identify cohomological dimensions and properties that would otherwise be obscured.
• Discuss the relationship between derived functors and resolutions in the context of computing these functors.
  • The relationship between derived functors and resolutions is pivotal for their computation. To derive these functors, one typically uses projective or injective resolutions of an object, which serve as tools for approximating the object and facilitating calculations. This method reveals how derived functors reflect the structure of the original object, capturing aspects like extensions and torsions that are crucial for understanding its categorization within homological algebra.
• Evaluate the implications of derived functors in broader mathematical contexts such as algebraic geometry or representation theory.
  • Derived functors have significant implications in broader mathematical contexts by providing powerful tools for understanding complex structures in areas like algebraic geometry and representation theory. For instance, they facilitate the study of sheaf cohomology, which is essential in geometric contexts, revealing properties of varieties. In representation theory, derived functors help classify representations by analyzing modules through homological techniques, contributing to a deeper understanding of the interconnections between various algebraic systems.

Related terms

Exact Functor: A functor that preserves the exactness of sequences, meaning it takes exact sequences to exact sequences.

Resolution: A sequence of morphisms that approximates an object in a way that allows for the computation of derived functors.

Homological Algebra: A branch of mathematics that studies homology in a general algebraic setting, often using derived functors to analyze relationships between algebraic structures.