Derived Category

Definition

A derived category is a construction in homological algebra that provides a framework to systematically study complexes of objects and their morphisms, capturing the essential features of derived functors such as Tor and Ext. This concept allows for the manipulation of these complexes in a way that respects their homotopy properties, enabling us to derive useful information about the underlying categories. Derived categories also give rise to triangulated categories, which are essential for understanding relationships between different homological theories.

5 Must Know Facts For Your Next Test

1. Derived categories allow us to work with chain complexes directly, providing a more flexible and powerful tool for computations in homological algebra.
2. In derived categories, morphisms can be viewed up to homotopy, which enables a more refined approach to understanding complex relationships between objects.
3. The construction of derived categories involves taking an abelian category and formally adding 'homotopical' information, leading to the formation of quasi-isomorphic complexes.
4. Derived categories help facilitate the calculation of Ext and Tor groups by providing a setting in which these derived functors can be computed more easily.
5. The five lemma and nine lemma are key results that can be applied in derived categories to establish isomorphisms and exactness conditions between various sequences of morphisms.

Review Questions

• How do derived categories enhance our understanding of derived functors like Tor and Ext in homological algebra?
  • Derived categories enhance our understanding of derived functors like Tor and Ext by providing a systematic way to work with chain complexes and their morphisms. They allow for the manipulation of these complexes up to homotopy equivalence, meaning we can focus on their essential features without getting bogged down by technical details. This framework makes it easier to compute these functors as it captures their behavior across various objects in the category.
• Discuss the relationship between derived categories and triangulated categories, particularly how one leads to the other.
  • Derived categories and triangulated categories are closely linked; every derived category naturally has the structure of a triangulated category. The distinguished triangles in a triangulated category arise from the exact sequences associated with chain complexes in the derived category. This connection allows mathematicians to apply results from triangulated category theory to analyze derived categories, reinforcing our understanding of how homological properties interact with categorical structures.
• Evaluate how the five lemma and nine lemma play crucial roles in establishing results within derived categories.
  • The five lemma and nine lemma are fundamental tools in derived categories that help establish isomorphisms and verify exactness conditions. These lemmas can be applied to show that certain morphisms between complexes are isomorphisms if specific conditions are met, enabling us to draw conclusions about the relationships between various derived objects. By using these lemmas, mathematicians can transfer results across different contexts within homological algebra, making them indispensable for proving deeper theoretical results.

Related terms

Triangulated Category: A triangulated category is a type of category equipped with a distinguished class of triangles that satisfy certain axioms, allowing for the study of homological properties through its derived objects.

Exact Sequence:

An exact sequence is a sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next, providing a way to analyze relationships between various algebraic structures.

Derived Functor: A derived functor is a functor that arises from the study of chain complexes and provides an extension of classical functors like Hom and tensor products, capturing additional homological information.