Derived Functors

Definition

Derived functors are a powerful concept in homological algebra that arise from the study of functors, allowing for the measurement of the extent to which a functor fails to be exact. They provide insight into the relationships between different mathematical structures, particularly in relation to exact sequences and the properties of modules. These functors are especially useful in areas like topology and algebraic geometry, where they help analyze the behavior of sheaves and cohomology.

5 Must Know Facts For Your Next Test

1. Derived functors can be computed using projective or injective resolutions of modules, which helps understand their behavior in relation to various categories.
2. The first derived functor is often referred to as Ext, while the second is Tor, representing different ways in which derived functors measure exactness and non-exactness.
3. In the context of sheaf theory, derived functors can capture important topological information about spaces through cohomological methods.
4. Derived functors help characterize morphisms between objects by identifying when certain sequences are exact or when they fail to be.
5. They play a crucial role in modern algebraic geometry, linking geometric concepts with algebraic structures via cohomology theories.

Review Questions

• How do derived functors help in understanding exact sequences?
  • Derived functors assist in understanding exact sequences by measuring how much a given functor fails to be exact. When you have a sequence of modules connected by homomorphisms, derived functors can show whether this sequence holds true and provide information on the failure points. This helps mathematicians determine important properties about the structures involved, particularly when analyzing relationships between different modules.
• Discuss the relationship between derived functors and cohomology in topology.
  • Derived functors are deeply connected to cohomology because they allow for capturing essential topological features of spaces. Cohomology groups can be seen as derived functors that reflect how well-defined certain global sections are across different open sets in a space. This interplay enables mathematicians to analyze local properties and how they relate to global structures, making derived functors a key tool in modern topology.
• Evaluate the impact of derived functors on the development of modern algebraic geometry.
  • The introduction of derived functors has significantly advanced modern algebraic geometry by providing a framework for relating geometric objects with their algebraic properties. By using tools like sheaves and cohomology, derived functors help bridge gaps between these fields, allowing for more profound insights into how geometric structures behave under various transformations. This has led to new theories and applications in areas such as scheme theory and intersection theory, fundamentally shaping contemporary research.

Related terms

Exact Sequence: A sequence of mathematical objects and morphisms between them such that the image of one morphism equals the kernel of the next, providing a way to study the properties of modules.

Cohomology: A mathematical tool used to associate algebraic invariants to topological spaces, allowing for the analysis of their structure through derived functors.

Sheaf: A mathematical object that systematically captures local data associated with the open sets of a topological space, crucial in defining derived functors in the context of topology.