Tor Functors

Definition

Tor functors are derived functors that measure the failure of a functor to be exact, specifically in the context of modules over a ring. They arise from taking the left derived functors of the tensor product, providing important information about the homological properties of modules, particularly in situations involving projective resolutions and exact sequences of chain complexes. Understanding Tor functors is crucial for analyzing relationships between modules and their structure through derived functors.

5 Must Know Facts For Your Next Test

1. Tor functors are denoted as $$\text{Tor}^n_R(M,N)$$, where $$R$$ is a ring, $$M$$ and $$N$$ are $$R$$-modules, and $$n$$ indicates the degree of the derived functor.
2. The first Tor functor, $$\text{Tor}^1_R(M,N)$$, measures the failure of the tensor product $$M \otimes_R N$$ to be exact.
3. Tor functors can be computed using a projective resolution of one of the modules involved, allowing for effective calculations in many scenarios.
4. For any fixed module $$N$$, the functor $$M \mapsto \text{Tor}^n_R(M,N)$$ is left exact but not necessarily right exact.
5. When working with free modules, Tor functors vanish, meaning that they yield zero for any free module involved in their computation.

Review Questions

• How do Tor functors relate to the concepts of exact sequences and derived functors?
  • Tor functors are derived functors specifically designed to measure the failure of the tensor product to preserve exactness when applied to modules. They emerge from examining exact sequences of chain complexes involving these modules. By applying a projective resolution to one module and using it in conjunction with another, we can analyze how well these modules interact through their tensor product.
• Explain how to compute Tor functors using a projective resolution and discuss its significance.
  • To compute Tor functors like $$\text{Tor}^n_R(M,N)$$, you start by taking a projective resolution of one module (say $$M$$) and then apply the tensor product with another module ($$N$$) at each stage. This results in an exact sequence that helps identify the kernel and cokernel at various levels. This process highlights how projective resolutions allow for better understanding of module interactions and their homological properties.
• Evaluate the implications of Tor functors vanishing for free modules in relation to module theory.
  • When Tor functors vanish for free modules, it indicates that free modules maintain a certain 'nice' structure that doesn't complicate relationships when involved in tensor products. This property significantly simplifies computations and allows us to focus on more complex modules by providing clear cases where derived functors yield trivial results. Understanding this aspect helps differentiate between various types of modules and their behaviors under homological operations.

Related terms

Derived Functors: Derived functors are constructions that extend the notion of classical functors to provide deeper insights into the properties of modules, capturing information that is lost when focusing solely on the original functor.

Exact Sequence: An exact sequence is a sequence of modules and homomorphisms between them where the image of one homomorphism equals the kernel of the next, reflecting a precise relationship between the structures involved.

Projective Resolution: A projective resolution of a module is an exact sequence of projective modules that approximates the module, allowing us to study its properties through simpler or more manageable objects.