Derived Functors

Definition

Derived functors are a fundamental concept in homological algebra, arising from the need to study the behavior of functors when applied to modules or objects that may not be projective or injective. They are constructed using projective or injective resolutions, allowing us to measure the failure of a functor to be exact, thereby giving insight into the cohomological properties of the objects involved. This concept connects deeply with various structures like long exact sequences in cohomology, sheaf cohomology, and the interplay between sheaves and algebraic topology.

5 Must Know Facts For Your Next Test

1. Derived functors are typically denoted as $\mathrm{L}F$ for left-derived functors and $\mathrm{R}F$ for right-derived functors, where $F$ is the functor in question.
2. They help to compute invariants like Ext and Tor, which measure the extent to which certain exact sequences fail to split.
3. The construction of derived functors involves taking a resolution of the input object and applying the functor to this resolution.
4. In sheaf theory, derived functors provide crucial insights into the cohomology of sheaves, allowing us to analyze their global sections and related properties.
5. The long exact sequence in cohomology arises naturally from derived functors, establishing connections between different degrees of cohomological dimensions.

Review Questions

• How do derived functors relate to the concept of injective resolutions in measuring the failure of a functor?
  • Derived functors are closely tied to injective resolutions because they utilize these resolutions to analyze how a functor behaves when applied to modules that are not injective. By constructing an injective resolution of an object and applying a left exact functor, we can derive information about how far the functor is from being exact. This relationship helps quantify the 'failure' of the functor by examining the cohomological characteristics captured by the derived functors.
• What is the significance of long exact sequences in cohomology in relation to derived functors?
  • Long exact sequences in cohomology play a significant role in understanding derived functors as they emerge from taking derived functors of short exact sequences. When applying derived functors to such sequences, we get a long exact sequence that connects various cohomological groups. This connection reveals how properties of one group influence others, showing the intricate relationships between them and demonstrating how derived functors provide a comprehensive view of cohomological dimensions.
• Evaluate how derived functors enhance our understanding of sheaf cohomology and its applications in algebraic topology.
  • Derived functors enrich our understanding of sheaf cohomology by enabling us to study global sections of sheaves over complex topological spaces through homological techniques. They allow us to compute cohomology groups that reflect essential topological features and help us understand how sheaves behave under various operations. In algebraic topology, this insight proves crucial for connecting local data captured by sheaves with global topological invariants, facilitating deeper exploration into spaces' structural properties and relationships.

Related terms

Projective Resolution: A projective resolution is an exact sequence of projective modules that approximates a given module, allowing us to study its properties through homological techniques.

Cohomology:

Cohomology is a mathematical tool that assigns algebraic invariants to topological spaces or algebraic structures, capturing their global properties and relationships.

Exact Sequence:

An exact sequence is a sequence of algebraic objects and morphisms between them such that the image of one morphism equals the kernel of the next, providing a framework for understanding relationships between different structures.