Tor

Definition

Tor is a functor that measures the failure of flatness between two modules in a category, particularly in the context of homological algebra. It arises as a left derived functor of the Hom functor, which means it captures important information about how well a module behaves with respect to another module when we take exact sequences into account. This concept is essential for understanding derived functors and their universal properties in various algebraic structures.

5 Must Know Facts For Your Next Test

1. Tor is typically denoted as $$\text{Tor}_n(A, B)$$, where $$A$$ and $$B$$ are two modules and $$n$$ indicates the degree of the derived functor.
2. The computation of Tor can be done using projective resolutions, allowing for an explicit way to analyze the relationships between modules.
3. Tor has a significant role in defining the notion of derived categories and their applications in algebraic geometry and representation theory.
4. For flat modules, $$\text{Tor}_n(A, B) = 0$$ for all $$n > 0$$, which highlights how flatness relates to the vanishing of Tor.
5. Tor satisfies various properties such as being a contravariant functor in its first argument, which affects how it interacts with morphisms in the category.

Review Questions

• How does Tor relate to the concepts of exact sequences and flat modules?
  • Tor measures how much a module fails to be flat when we consider exact sequences. If a module is flat, then Tor will vanish for all positive degrees, meaning that taking the tensor product with a flat module preserves the exactness of sequences. This connection emphasizes the role of Tor in understanding module behavior and properties in homological algebra.
• Explain how Tor can be computed using projective resolutions and why this is important.
  • To compute Tor, we can use projective resolutions of the modules involved. By constructing a projective resolution for one of the modules and applying the tensor product with another module along with taking homology, we can derive values for Tor. This method is significant because it provides a systematic way to extract information about module interactions and helps us understand the underlying structure in homological algebra.
• Critically analyze the implications of Tor's properties on derived functors and their applications across different areas of mathematics.
  • The properties of Tor as a derived functor not only illustrate its role in measuring failures of exactness but also have profound implications across various mathematical disciplines. For instance, its vanishing conditions relate closely to flatness and projectivity, which are pivotal in algebraic geometry and commutative algebra. Additionally, Tor’s behavior under change of base ring or morphisms reflects broader themes in deformation theory and representation theory, showcasing its utility beyond mere calculations and into conceptual frameworks that influence modern mathematical research.

Related terms

Hom: The Hom functor assigns to each pair of modules the set of all homomorphisms between them, serving as a foundation for many constructions in homological algebra.

Flat Module: A module is flat if taking the tensor product with it preserves exact sequences, which is crucial for defining Tor and understanding its applications.

Derived Functors: Derived functors extend the idea of functors to measure how much certain properties fail in categories, with Tor being one of the primary examples used to study these failures.