Left derived functors

Definition

Left derived functors are a sequence of functors that arise from applying a left exact functor to a projective resolution of an object in a category. They help extend the concept of functors to measure the failure of certain exactness properties, particularly in homological algebra. These derived functors provide crucial insights into the structure and properties of algebraic objects by capturing how far they deviate from exactness when applied to complexes.

5 Must Know Facts For Your Next Test

1. Left derived functors are denoted as L_nF for a left exact functor F, where n indicates the level of derivation.
2. They are computed using projective resolutions, which help ensure that the results capture information about the module's structure.
3. The first left derived functor, L_1F, often corresponds to the kernel or cokernel related to the original functor application.
4. Left derived functors provide valuable tools in algebraic topology and algebraic geometry by measuring cohomological properties.
5. The study of left derived functors allows for deeper exploration of invariants and transformations in various algebraic contexts.

Review Questions

• How do left derived functors differ from right derived functors in terms of their construction and purpose?
  • Left derived functors are constructed using left exact functors applied to projective resolutions, focusing on the failure of exactness at the left side. In contrast, right derived functors arise from right exact functors and target the failure of exactness at the right side. This distinction is crucial as it shapes their respective applications in capturing different aspects of module behavior in homological algebra.
• Discuss the significance of projective resolutions in calculating left derived functors and how they influence the results obtained.
  • Projective resolutions play an essential role in calculating left derived functors because they provide an exact sequence that approximates the module being studied. By applying a left exact functor to this resolution, one can effectively track how well the original object retains its structure under transformation. The choice of projective modules within these resolutions directly influences the derived functors' values, impacting their interpretation in various algebraic settings.
• Evaluate the role of left derived functors in understanding cohomological dimensions and their implications for algebraic structures.
  • Left derived functors are instrumental in analyzing cohomological dimensions, as they reveal how well different algebraic structures behave under various transformations. They help identify crucial invariants within these structures, influencing our understanding of their complexity and classification. By assessing these invariants through left derived functors, mathematicians can derive significant conclusions about dimensionality and connectivity within algebraic frameworks, leading to deeper insights into their interrelations.

Related terms

Right derived functors:

Right derived functors are constructed from applying a right exact functor to a projective resolution, capturing information about the cohomology of the complex.

Projective resolution: A projective resolution is an exact sequence of projective modules that approximates another module, serving as a tool to compute derived functors.

Exact sequence: An exact sequence is a sequence of objects and morphisms such that the image of one morphism equals the kernel of the next, reflecting important properties in homological algebra.