Tor Functor

Definition

The Tor functor is a construction in homological algebra that measures the extent to which a module fails to be projective. Specifically, it computes the derived functors of the tensor product, revealing important information about the relationships between modules over a ring. The Tor functor is useful for understanding various properties of modules, including their extensions and the interactions with other algebraic structures.

5 Must Know Facts For Your Next Test

1. The Tor functor is denoted as $$\text{Tor}_n^R(M,N)$$ where R is a ring and M, N are R-modules, and it can be computed using projective resolutions.
2. For a flat module, $$\text{Tor}_1^R(M,N) = 0$$, indicating that flat modules do not introduce torsion when taking the tensor product.
3. The functor is left exact, meaning it preserves finite limits but not colimits, highlighting its role in understanding exact sequences.
4. The values of the Tor functor provide insight into the structure of modules; specifically, $$\text{Tor}_1^R(M,N)$$ can be seen as capturing obstructions to lifting maps in exact sequences.
5. In practical terms, Tor functors can be computed using homological methods such as spectral sequences or via derived categories in more advanced contexts.

Review Questions

• How does the Tor functor relate to projective modules and what does it reveal about non-projective modules?
  • The Tor functor reveals how much a module deviates from being projective by measuring the failure of exactness in certain sequences involving tensor products. For instance, if a module M is projective, then $$\text{Tor}_1^R(M,N) = 0$$ for any R-module N. This means that projective modules are nice in that they preserve exactness under tensoring, while non-projective modules might introduce torsion and complicate the structure of the resulting sequences.
• Discuss how one can compute the Tor functor using projective resolutions and its implications for understanding module relationships.
  • To compute the Tor functor $$\text{Tor}_n^R(M,N)$$, one first constructs a projective resolution of one of the modules (typically M) and then applies the tensor product with the other module (N) followed by taking homology. This process reveals important information about how M and N interact under extension and provides insight into their derived behavior. The resulting groups give a measure of how torsion-like behavior arises when moving between these modules.
• Evaluate the significance of the Tor functor in homological algebra and its applications beyond mere computation of derived functors.
  • The Tor functor plays a crucial role in homological algebra by providing deeper insights into module theory and categorical relationships. Beyond computation, it helps identify obstructions to splitting exact sequences and sheds light on the structure of Ext groups as well. Its applications extend into various areas such as algebraic topology and algebraic geometry, where understanding how objects relate through derived functors leads to important results in cohomology theories and sheaf cohomology.

Related terms

Homological Algebra: A branch of mathematics that studies homology in a general algebraic setting, focusing on structures like chain complexes and derived functors.

Tensor Product: An operation that takes two modules and produces a new module, which captures the linear combinations of the elements from both original modules.

Projective Module: A type of module that has properties similar to free modules, specifically that every surjective homomorphism onto it splits, allowing for nice lifting properties.