Left derived functors

Definition

Left derived functors are a way to extend the notion of functors in homological algebra, allowing us to capture more information about the structure of a given category. They arise from the process of taking an abelian category and using projective resolutions to compute functors, especially when these functors do not behave well under direct limits or colimits. This concept is central to understanding how certain properties can be derived from an object in a category and has significant implications in the development of homological algebra.

5 Must Know Facts For Your Next Test

1. Left derived functors are denoted as $L_nF(A)$ for a functor $F$ and an object $A$ in an abelian category.
2. The calculation of left derived functors often requires taking a projective resolution of the object on which the functor is applied.
3. In general, left derived functors help to measure how far a functor is from being exact.
4. The first left derived functor is commonly known as the Ext group, which measures extensions of modules.
5. Applications of left derived functors include computing cohomology in algebraic topology and in the study of sheaves.

Review Questions

• How do left derived functors provide insight into the structure of a category when applied to an object?
  • Left derived functors give us a deeper understanding of the relationships between objects in a category by measuring how well a functor behaves with respect to exactness. By taking projective resolutions, we can derive information about extensions and how various objects can be built from simpler ones. This helps us to identify properties and potential issues within the category, thus enhancing our comprehension of its overall structure.
• Discuss how left derived functors differ from right derived functors in terms of their computation and application.
  • Left derived functors are computed using projective resolutions, while right derived functors use injective resolutions. This difference reflects their underlying principles: left derived functors often measure how far a functor is from being exact by exploring projective modules, while right derived functors evaluate properties related to injective modules. The choice between using left or right derived functors can significantly affect the conclusions drawn about objects within a category.
• Evaluate the importance of left derived functors in modern algebra and their influence on other areas such as topology or algebraic geometry.
  • Left derived functors play a critical role in modern algebra by providing tools for understanding complex structures through simpler components. Their applications extend beyond homological algebra; they are instrumental in fields like algebraic topology, where they aid in calculating cohomology groups, and in algebraic geometry for studying sheaf cohomology. By connecting various mathematical concepts, left derived functors illustrate the intricate web of relationships within mathematics, impacting research and applications across diverse domains.

Related terms

Abelian Category: A category where every morphism has a kernel and cokernel, allowing for the definition of exact sequences.

Projective Resolution: A sequence of projective modules that approximates a given module, used to compute derived functors.

Exact Sequence: A sequence of objects and morphisms such that the image of one morphism is equal to the kernel of the next, crucial for studying homological properties.