Derived Functor

Definition

A derived functor is a construction in homological algebra that measures the extent to which a functor fails to be exact. It provides a systematic way of obtaining new functors that capture information about the original functor and its failure to preserve certain structures, like exact sequences. Derived functors arise in various contexts, including the calculation of K-theory groups, where they help analyze properties of algebraic objects and their relationships.

5 Must Know Facts For Your Next Test

1. Derived functors are typically denoted as $ ext{L}F$ for left derived functors and $ ext{R}F$ for right derived functors, where $F$ is the original functor being studied.
2. The first derived functor is commonly referred to as the Ext functor, which measures extensions of modules and can provide insights into their relationships.
3. In K-theory, derived functors can be used to compute Grothendieck groups, connecting abstract algebraic properties to topological and geometric interpretations.
4. Derived functors are calculated using projective or injective resolutions, allowing one to systematically address non-exactness in sequences.
5. The concept of derived functors can be generalized to more complex settings, such as derived categories, which provide a framework for understanding homological properties in a broader context.

Review Questions

• How do derived functors relate to exact sequences, and why is this relationship significant?
  • Derived functors provide a way to understand the failure of a given functor to preserve exact sequences. When a functor is not exact, derived functors help measure how far it deviates from preserving exactness. This relationship is significant because it allows mathematicians to analyze properties of algebraic structures by studying how derived functors interact with these sequences, leading to valuable insights into their behavior.
• Discuss the process of calculating derived functors and its importance in algebraic K-theory.
  • To calculate derived functors, one typically uses projective or injective resolutions of objects involved. This process systematically addresses non-exactness in sequences by allowing for the application of homological techniques. In algebraic K-theory, these calculations are crucial as they facilitate the determination of Grothendieck groups, which serve as important invariants capturing essential properties of algebraic objects and their interrelations.
• Evaluate how derived functors enhance our understanding of both homology and cohomology theories within algebraic contexts.
  • Derived functors enrich our understanding of homology and cohomology theories by providing tools that measure deviations from exactness in these contexts. They reveal deeper connections between different algebraic structures by linking them through exact sequences and resolutions. This enhanced understanding allows mathematicians to explore complex relationships within algebraic systems, thereby facilitating advancements in both theoretical research and practical applications within various branches of mathematics.

Related terms

Exact Sequence: An exact sequence is a sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. Exactness reflects the preservation of certain algebraic structures through the sequence.

Homology: Homology is a mathematical concept that associates a sequence of abelian groups or modules with a topological space or algebraic structure, capturing information about its shape or structure through algebraic invariants.

Cohomology: Cohomology is similar to homology but focuses on the dual perspective, associating groups or modules with spaces based on functions defined on them. It often provides deeper insights into the structure of spaces than homology.