Derived functor

Definition

A derived functor is a concept in homological algebra that extends the idea of a functor by associating it with a sequence of objects (called derived objects) which capture the 'homological information' of the original functor. This allows one to study properties of functors that are not preserved under direct application, particularly in the context of resolving modules and understanding exact sequences.

5 Must Know Facts For Your Next Test

1. Derived functors can be computed using projective or injective resolutions, which provide a way to handle modules that are not necessarily free or injective.
2. The two most prominent derived functors are Tor and Ext, which arise from studying flat and projective resolutions respectively.
3. Derived functors play a crucial role in sheaf cohomology, allowing for the measurement of how local data glues together into global sections.
4. In derived categories, derived functors are represented as morphisms between complexes, allowing for a more abstract treatment of homological problems.
5. Understanding derived functors is essential for working with A-infinity algebras, as they provide a framework for defining higher homotopies and operations.

Review Questions

• How do derived functors relate to projective and injective resolutions, and why are these concepts important in homological algebra?
  • Derived functors utilize projective and injective resolutions to compute values associated with a functor that may not preserve exactness. Projective resolutions help in finding the Tor functor, while injective resolutions are essential for computing Ext. These resolutions allow us to resolve modules into simpler components where exact sequences can be analyzed, providing crucial insights into the structure and properties of modules.
• Discuss the significance of Tor and Ext as derived functors in the context of module theory and their applications in homological algebra.
  • Tor and Ext are fundamental derived functors that capture important aspects of module theory. Tor measures the non-flatness of a module relative to another, while Ext provides information about extensions between modules. They enable mathematicians to derive deeper properties about modules beyond just their direct relationships, making them vital tools for solving problems in homological algebra.
• Evaluate how derived functors influence our understanding of sheaf cohomology and its implications in modern mathematics.
  • Derived functors significantly enhance our understanding of sheaf cohomology by providing a framework to analyze local data's interaction with global sections. They help identify when local-to-global principles hold true, such as when sheaves represent cohomology classes. This understanding leads to powerful applications in algebraic geometry and topology, influencing theories related to vector bundles, schemes, and more complex structures in modern mathematics.

Related terms

Exact Sequence:

A sequence of algebraic objects and morphisms between them that reflects the structure of how these objects relate, specifically capturing when kernels and images match.

Projective Resolution:

A way to express a module as a quotient of a projective module, helping in computing derived functors like Tor and Ext.

Cohomology:

A mathematical tool that provides algebraic invariants for topological spaces, often related to derived functors in the context of sheaf theory and homological algebra.