Left derived functors

Definition

Left derived functors are a way to extend a functor from a category of modules to a more general category by capturing information about the derived limits. They are constructed using projective resolutions, which allows one to compute how much the functor deviates from being exact. Left derived functors play a significant role in homological algebra, particularly in the context of defining Ext and Tor functors, where they help understand the relationships between different modules and their morphisms.

5 Must Know Facts For Your Next Test

1. Left derived functors are denoted as L_nF for a functor F, where n indicates the degree of derivation.
2. The first left derived functor, L_1F, corresponds to the Ext functor and measures extensions between modules.
3. Higher left derived functors can capture more complex relationships between modules, going beyond just extensions.
4. The process of calculating left derived functors involves taking projective resolutions of modules and applying the original functor to these resolutions.
5. Left derived functors are crucial in establishing connections between different algebraic structures, especially in studying homological dimensions.

Review Questions

• How do left derived functors relate to projective resolutions in their computation?
  • Left derived functors rely on projective resolutions to extend their application beyond exact categories. By taking a projective resolution of a module, one can apply the original functor to this resolution to compute the left derived functors. This process captures how far the functor is from being exact and provides a way to derive important invariants associated with modules.
• Discuss the significance of left derived functors in understanding Ext and Tor functors.
  • Left derived functors are fundamentally linked to Ext and Tor functors, which are essential tools in homological algebra. The first left derived functor is essentially the Ext functor, which helps classify extensions between modules. Tor, on the other hand, arises as a higher left derived functor that captures torsion information. Together, these derived functors help in analyzing the relationships between modules and contribute to our understanding of their structure.
• Evaluate how left derived functors impact the study of homological dimensions and their applications.
  • Left derived functors significantly impact the study of homological dimensions by providing a systematic way to explore module properties across different contexts. Their ability to measure extensions and torsion allows mathematicians to classify modules based on their complexity and relationships with other modules. This classification aids in resolving problems in various areas such as algebraic geometry and representation theory, highlighting their essential role in modern mathematics.

Related terms

Projective Resolution: A sequence of projective modules that approximates a given module, allowing for the computation of derived functors.

Exact Sequence: A sequence of modules and homomorphisms where the image of one homomorphism equals the kernel of the next, indicating no 'gaps' in the sequence.

Tor Functor: A derived functor that measures the failure of flatness of modules, providing insights into the torsion phenomena in homological algebra.