Derived category

Definition

A derived category is a construction in homological algebra that allows for the study of complexes of objects within an abelian category, focusing on their homotopical properties. It enables mathematicians to work with quasi-isomorphisms and captures the idea of 'derived functors', which can provide a more refined understanding of morphisms between objects, especially in contexts where exact sequences are crucial.

5 Must Know Facts For Your Next Test

1. Derived categories allow for the manipulation of chain complexes as single objects, making it easier to perform homological algebra without worrying about specific representatives.
2. In derived categories, morphisms correspond to quasi-isomorphisms between chain complexes, providing a way to treat homotopically equivalent complexes as equivalent.
3. The process of forming derived categories can be seen as a way to 'localize' the homotopy category with respect to quasi-isomorphisms.
4. Derived functors such as Ext and Tor are naturally associated with derived categories, allowing for the extraction of significant invariants from complexes.
5. The derived category of an abelian category has a rich structure that often supports additional operations such as triangulated structures and various equivalences.

Review Questions

• How does the concept of derived categories enhance our understanding of morphisms in abelian categories?
  • Derived categories enhance our understanding by treating complexes of objects as single entities, allowing us to focus on their homotopical properties rather than specific elements. This perspective helps in studying morphisms via quasi-isomorphisms, leading to insights about the relationships between complex structures. As a result, mathematicians can better analyze situations where exact sequences play a key role, improving their ability to derive important properties and invariants from these morphisms.
• What role do derived functors play in the framework of derived categories, and why are they significant?
  • Derived functors play a critical role in the framework of derived categories as they allow us to systematically study invariants associated with complexes. For instance, functors like Ext and Tor emerge naturally within this context, providing valuable tools for extracting homological information from chain complexes. Their significance lies in their ability to generalize classical notions while still respecting the relationships between objects in abelian categories, thereby deepening our understanding of algebraic structures.
• Critically analyze how derived categories relate to triangulated categories and their importance in modern algebraic geometry.
  • Derived categories relate closely to triangulated categories since they both provide frameworks for studying homological properties through triangles. The derived category's triangulated structure facilitates the manipulation of exact sequences and provides a way to express complex relationships between geometric objects. In modern algebraic geometry, this connection is crucial as it allows for advanced techniques such as derived functoriality and cohomology theories, ultimately contributing to richer geometric insights and more robust theoretical developments.

Related terms

abelian category: An abelian category is a category where morphisms can be added, has kernels and cokernels for every morphism, and satisfies certain axioms that ensure the existence of exact sequences.

complex: In the context of homological algebra, a complex is a sequence of abelian groups or modules connected by morphisms that satisfy the property that the composition of any two consecutive morphisms is zero.

triangulated category: A triangulated category is a type of category equipped with an additional structure that allows for the formal manipulation of triangles, which is essential for discussing derived categories.