Derived Functor

Definition

A derived functor is a tool in homological algebra that extends the notion of a functor to measure the failure of a functor to be exact. Derived functors provide a way to systematically study how algebraic structures, like modules or sheaves, behave under certain transformations. They allow mathematicians to extract deeper information from complexes, especially in cohomological contexts such as Čech cohomology.

5 Must Know Facts For Your Next Test

1. Derived functors are often calculated using projective or injective resolutions of modules, which help to define and compute them explicitly.
2. The most common derived functors are Ext and Tor, which provide information about extensions and tensor products of modules, respectively.
3. Derived functors can be viewed as cohomology theories when applied to sheaves on topological spaces, linking algebra with topology.
4. In the context of Čech cohomology, derived functors help to compute cohomology groups using covers of the space and show how local data can reflect global properties.
5. The derived functor framework is crucial for understanding deeper properties of sheaves and their cohomology in the study of algebraic varieties.

Review Questions

• How do derived functors relate to the concept of exactness in the study of algebraic structures?
  • Derived functors are fundamentally linked to the concept of exactness because they measure how much a given functor fails to be exact. When we apply a functor to an exact sequence and it fails to preserve that exactness, derived functors help capture that failure by providing additional algebraic invariants. This connection is key in analyzing sequences and their implications for module behavior under transformations.
• What role do projective and injective resolutions play in the computation of derived functors, particularly in the context of modules?
  • Projective and injective resolutions are essential for computing derived functors because they provide a means to transform a module into a more manageable form. By resolving a module into either projective or injective components, one can define derived functors such as Ext and Tor explicitly. This method facilitates the examination of how these modules behave under various operations, ultimately leading to deeper insights into their structure.
• Evaluate the significance of derived functors in connecting local data with global properties within the realm of Čech cohomology.
  • Derived functors hold significant importance in connecting local data with global properties in Čech cohomology by allowing for the computation of cohomology groups through local covers. They enable us to derive global information about sheaves by examining their behavior over small open sets. This bridge between local behavior and global characteristics is vital for understanding complex spaces, such as algebraic varieties, highlighting the interplay between local sections and global sections.

Related terms

Cohomology:

A mathematical concept that studies the properties of spaces through algebraic invariants derived from chain complexes.

Exact Sequence: A sequence of algebraic objects and morphisms between them that encodes information about the relationships and structure of those objects.

Homology: A mathematical construct used to associate a sequence of abelian groups or modules with a topological space, reflecting its structure and properties.