Right derived functors

Definition

Right derived functors are a type of functor that arises in homological algebra, specifically associated with a left exact functor. They measure the failure of a left exact functor to be exact, extending its action to obtain information about the cohomological properties of objects in a category. By utilizing projective resolutions of modules, right derived functors allow us to study properties like homology and cohomology in various mathematical contexts, providing deep insights into the structure of algebraic objects.

5 Must Know Facts For Your Next Test

1. Right derived functors are typically denoted as $R^nF$, where $F$ is the left exact functor and $n$ is a non-negative integer representing the degree of derivation.
2. To compute right derived functors, one often uses projective resolutions of modules to extend the action of a left exact functor.
3. Common examples of right derived functors include Ext and Tor, which provide important tools for studying module categories.
4. Right derived functors can capture significant topological and algebraic information about sheaves and other mathematical structures.
5. The relationship between left and right derived functors is crucial for understanding the overall framework of homological algebra.

Review Questions

• How do right derived functors relate to left exact functors in terms of measuring properties within categories?
  • Right derived functors arise from left exact functors and serve as a tool to measure how far these left exact functors are from being exact. Specifically, they help in understanding the cohomological aspects by extending the analysis beyond exact sequences. While left exact functors preserve finite limits, right derived functors utilize projective resolutions to analyze the failure of these properties, revealing deeper insights into the algebraic structures involved.
• Discuss the process of calculating right derived functors using projective resolutions and how this impacts their application.
  • Calculating right derived functors involves first finding a projective resolution of the module in question. This means we take our module and express it as an increasing sequence of projective modules that approximates it. We then apply the left exact functor to this resolution, allowing us to obtain information about the homological properties of the original module. This process not only enables us to compute values like Ext and Tor but also influences our understanding of module behavior across different categories.
• Evaluate the significance of right derived functors in the broader context of homological algebra and their implications for mathematical theory.
  • Right derived functors play a pivotal role in homological algebra by linking abstract algebraic concepts with topological and geometric ideas. Their ability to extend analysis beyond basic module theory allows mathematicians to derive results about sheaves, cohomology groups, and spectral sequences. This evaluation reveals how right derived functors help build bridges between different mathematical fields, leading to richer theories and applications, such as in algebraic geometry or representation theory. Their implications resonate throughout many areas of mathematics, making them foundational tools for both theoretical exploration and practical problem-solving.

Related terms

Left derived functors: These are derived functors associated with right exact functors, measuring how far a functor is from being left exact.

Exact sequences: Sequences of objects and morphisms that provide information on how to build complex structures in a category, particularly in relation to kernels and cokernels.

Projective resolutions: A way to resolve a module using projective modules, which play a key role in defining derived functors.