Derived Category

Definition

A derived category is a construction in homological algebra that allows for the manipulation of chain complexes and the study of their homological properties by formally adding 'homotopies' between them. It serves as an essential tool for understanding the relationships between objects in an abelian category, particularly in relation to exact sequences. By moving to the derived category, one can simplify many complex problems involving exact sequences, leading to more manageable computations and deeper insights into the structure of the original category.

5 Must Know Facts For Your Next Test

1. Derived categories are constructed from an abelian category by taking its chain complexes and localizing with respect to quasi-isomorphisms, allowing one to study derived functors effectively.
2. In a derived category, morphisms can represent not just ordinary maps but also homotopies between maps, which provides a richer framework for analysis.
3. The derived category has a natural triangulated structure, which facilitates the definition of important concepts such as distinguished triangles that capture the behavior of certain exact sequences.
4. The relationship between derived categories and abelian categories helps in defining and computing derived functors like Ext and Tor, which are pivotal in homological algebra.
5. Derived categories are used extensively in algebraic geometry and representation theory, enabling deep insights into cohomology theories and D-module theory.

Review Questions

• How does the concept of derived categories enhance our understanding of exact sequences within abelian categories?
  • Derived categories enhance our understanding of exact sequences by providing a framework that allows us to study chain complexes related to these sequences. In this context, one can examine morphisms between complexes and their homotopy types rather than just focusing on individual objects. This approach reveals deeper relationships and properties that might be obscured when only considering abelian categories directly.
• Discuss how derived functors like Ext and Tor can be computed using derived categories and their significance in homological algebra.
  • Derived functors like Ext and Tor are computed within derived categories by representing them as functors from the derived category to abelian groups or modules. The derived category allows for the application of spectral sequences and other techniques that simplify these computations significantly. This capability is crucial because it enables mathematicians to derive meaningful invariants and understand the underlying structure of modules or sheaves through their homological properties.
• Evaluate how the triangulated structure of derived categories impacts their applications in fields like algebraic geometry or representation theory.
  • The triangulated structure of derived categories provides a powerful tool for analyzing complex relationships among objects through distinguished triangles. In algebraic geometry, this structure helps in understanding cohomological properties and the behavior of sheaves under various operations. In representation theory, it aids in exploring relationships between different representations, facilitating advancements in the understanding of module categories and enhancing techniques for studying their extensions and restrictions. This interplay greatly enriches both fields by providing a cohesive language for dealing with various mathematical constructs.

Related terms

Chain Complex: A sequence of abelian groups or modules connected by homomorphisms where the composition of consecutive maps is zero, forming a foundational concept in homological algebra.

Abelian Category: A category in which morphisms can be added, and every pair of morphisms with a common codomain has a kernel and cokernel, allowing for a well-defined notion of exact sequences.

Exact Sequence: A sequence of objects and morphisms in an abelian category such that the image of one morphism equals the kernel of the next, capturing the idea of 'exactness' in terms of preserving structure.