Right derived functors

Definition

Right derived functors are a type of functor that arise in homological algebra, specifically used to measure the failure of exactness in a given functor. They are constructed using projective or injective resolutions of modules and provide insights into the structure of modules and their interactions with other algebraic objects. These functors help to generalize the concept of invariants associated with modules, such as Ext and Tor, which play a critical role in understanding module homomorphisms and their extensions.

5 Must Know Facts For Your Next Test

1. Right derived functors are denoted as $$R^nF$$ for a given functor $$F$$, where $$n$$ indicates the level of the derived functor.
2. The first right derived functor, $$R^1F$$, is often associated with measuring extensions and is closely related to the Ext functor.
3. Right derived functors are computed using projective or injective resolutions, depending on whether one is dealing with left or right derived functors.
4. The relationship between right derived functors and Tor can be understood through the use of the universal coefficient theorem in homological algebra.
5. In many contexts, right derived functors provide a means to classify modules up to certain homological dimensions, giving insight into their structures.

Review Questions

• How do right derived functors relate to exact sequences and what does this relationship imply about module interactions?
  • Right derived functors are closely tied to exact sequences because they help measure how far a given functor is from being exact. When an exact sequence fails to be exact, the derived functors quantify this failure by encapsulating information about extensions and resolutions. This implies that understanding these relationships can lead to deeper insights into how different modules interact with each other through their homological properties.
• Discuss the role of projective and injective resolutions in the computation of right derived functors and why this is important.
  • Projective and injective resolutions are crucial for computing right derived functors because they provide a way to approximate modules and their morphisms through exact sequences. By using these resolutions, one can derive information about the behavior of modules under certain functors, leading to the computation of invariants like Ext and Tor. This process is important as it helps uncover the underlying structure of modules, offering valuable tools for understanding their relationships in homological algebra.
• Evaluate how right derived functors contribute to the overall framework of homological algebra and their significance in module theory.
  • Right derived functors significantly enhance the framework of homological algebra by allowing mathematicians to explore the relationships between modules beyond just morphisms. They serve as tools for classifying modules up to homological dimensions and help elucidate structures like extensions and torsion. Their significance lies in their ability to provide deep insights into module theory, bridging connections between various algebraic constructs and offering avenues for further exploration in abstract algebra.

Related terms

Exact Sequence: A sequence of module homomorphisms between modules where the image of one homomorphism equals the kernel of the next, indicating that certain properties are preserved.

Projective Resolution: An exact sequence of projective modules that approximates a given module, helping in the computation of derived functors.

Injective Resolution: An exact sequence of injective modules used to study modules by providing a way to compute derived functors like Ext.