Total Derived Functor

Definition

The total derived functor is a construction in homological algebra that extends a functor to a derived functor, capturing the behavior of the original functor on complexes of modules. This concept is particularly important for understanding how functors like \text{Tor} and \text{Ext} behave when applied to chain complexes rather than just individual modules. By employing resolutions of modules, total derived functors facilitate the calculation of \text{Tor} and \text{Ext} groups in a systematic way.

5 Must Know Facts For Your Next Test

1. The total derived functor is used to compute \text{Tor} and \text{Ext} by taking resolutions of modules and applying the original functor to these resolutions.
2. Total derived functors can be defined for various types of functors, including additive and covariant functors, but their utility is most prominent in homological algebra.
3. The calculation of total derived functors often involves both projective and injective resolutions, reflecting the dual nature of many constructions in homological algebra.
4. When dealing with abelian categories, total derived functors have nice properties, such as being able to commute with filtered colimits under certain conditions.
5. Understanding total derived functors helps clarify relationships between different homological dimensions and can provide insight into deeper structural properties of modules.

Review Questions

• How does the concept of total derived functors relate to the computation techniques for \text{Tor} and \text{Ext}?
  • Total derived functors serve as a crucial tool for computing \text{Tor} and \text{Ext} by allowing us to apply these functors to chain complexes rather than just individual modules. This is done by taking resolutions of the involved modules, which transforms them into structures where the computations can be carried out systematically. Essentially, total derived functors help bridge the gap between module theory and homological methods, making it easier to extract meaningful results about relationships between modules.
• Discuss how resolutions are utilized in calculating total derived functors and why they are important.
  • Resolutions are used in calculating total derived functors by providing a way to replace complex modules with simpler, more manageable ones. Specifically, projective or injective resolutions allow us to express modules in terms of exact sequences, enabling the application of functors that may not be exact themselves. This technique is essential because it ensures that we can derive meaningful information about the behavior of our original functors in terms of more tractable components, thereby facilitating computations for \text{Tor} and \text{Ext}.
• Evaluate the significance of total derived functors in the broader context of homological algebra.
  • Total derived functors hold significant importance in homological algebra as they not only facilitate computations but also deepen our understanding of module relationships and properties. By connecting complex algebraic structures through resolutions and derived categories, they allow mathematicians to classify modules based on their homological characteristics. The insights gained from total derived functors extend into areas like category theory, algebraic geometry, and representation theory, underscoring their role as a foundational concept in modern mathematics.

Related terms

Derived Functor: A functor that measures how far a given functor deviates from being exact, typically by applying the functor to projective or injective resolutions.

Chain Complex: A sequence of abelian groups or modules connected by homomorphisms, where the composition of two consecutive homomorphisms is zero, used in the study of homological algebra.

Resolution: A sequence of morphisms between modules that approximates a module, typically used to compute derived functors by replacing modules with more manageable ones.