Derived categories

Definition

Derived categories are a fundamental concept in modern algebraic geometry and homological algebra, representing a way to systematically organize and manipulate complexes of objects. They allow mathematicians to study derived functors and provide a framework for understanding the relationships between different algebraic structures through the lens of triangulated categories.

5 Must Know Facts For Your Next Test

1. Derived categories allow for the manipulation of chain complexes, making it easier to work with cohomological dimensions and derived functors.
2. In the context of l-adic cohomology, derived categories help to understand how different l-adic sheaves behave under various morphisms and pullbacks.
3. The homotopy category can be viewed as a specific example of a derived category where morphisms are defined up to homotopy equivalence.
4. The notion of quasi-isomorphism plays a crucial role in derived categories, allowing one to relate complexes that may not be isomorphic but have the same cohomology.
5. Derived categories facilitate the construction of spectral sequences, which are powerful tools in both algebraic geometry and number theory for calculating invariants.

Review Questions

• How do derived categories improve our understanding of cohomology theories?
  • Derived categories enhance our understanding of cohomology theories by providing a structured way to handle chain complexes and derived functors. They allow mathematicians to focus on the relationships between objects rather than just their individual properties. This approach leads to better insights into how different cohomological dimensions interact and how they can be calculated more efficiently through spectral sequences.
• Discuss the role of triangulated categories in the framework of derived categories and their implications in algebraic geometry.
  • Triangulated categories serve as the foundational structure within which derived categories operate. They provide a way to categorize morphisms and complexes that maintain certain exactness properties. In algebraic geometry, this means one can study the relationships between sheaves, schemes, and their derived counterparts while preserving essential geometric information. The interplay between triangulated categories and derived categories allows for deeper insights into the behavior of morphisms in various algebraic contexts.
• Evaluate the significance of quasi-isomorphisms in derived categories and their impact on computations in l-adic cohomology.
  • Quasi-isomorphisms are crucial in derived categories as they establish an equivalence relation among chain complexes based on their cohomological data rather than their specific forms. This abstraction enables mathematicians to focus on the essential features of objects without being bogged down by extraneous details. In l-adic cohomology, this means one can simplify calculations by working with representative complexes that share the same cohomological invariants, facilitating easier computations and enhancing understanding of how different l-adic sheaves relate to each other.

Related terms

triangulated category: A triangulated category is a type of category equipped with an auto-equivalence that helps to generalize the concept of exact sequences in homological algebra.

cohomology: Cohomology is a mathematical tool used to study the properties of spaces through algebraic invariants, often represented as functors from topological spaces to graded abelian groups.

sheaf: A sheaf is a mathematical construct that encodes local data attached to the open sets of a topological space, allowing for the study of global properties through local observations.