Tor Functor

Definition

The Tor functor is an important tool in homological algebra, used to measure the extent to which a module fails to be flat. Specifically, for two modules over a ring, the Tor functor computes the derived functors of the tensor product, providing crucial information about the structure of these modules. In the context of flatness, the Tor functor helps identify when a given module is flat by examining its behavior with respect to other modules.

5 Must Know Facts For Your Next Test

1. The Tor functor is denoted as \(\text{Tor}_i^R(M,N)\) and measures the failure of flatness between modules M and N over a ring R.
2. If \(\text{Tor}_1^R(M,N) = 0\), then M is flat over R with respect to N, which signifies that tensoring with M does not introduce any torsion.
3. The Tor functor can be computed using a projective resolution of one of the modules involved in the tensor product, facilitating computations in homological algebra.
4. For finitely generated modules over a Noetherian ring, the Tor functor is finitely presented, which means that it can be described in terms of generators and relations.
5. The vanishing of higher Tor groups, \(\text{Tor}_i^R(M,N) = 0\) for \(i > 0\), indicates that M has a particularly nice interaction with N, often leading to results about their dimensions or stability.

Review Questions

• How does the Tor functor relate to determining whether a module is flat?
  • The Tor functor provides insight into whether a module is flat by analyzing \(\text{Tor}_1^R(M,N)\). If this group vanishes, it indicates that M is flat over R concerning N. This connection allows mathematicians to use the properties of the Tor functor as a criterion for flatness when working with various modules.
• In what ways can you compute the Tor functor using projective resolutions?
  • To compute the Tor functor, one typically starts with a projective resolution of one of the modules involved. By taking a projective resolution of M, we can apply the tensor product with N and then take the homology of the resulting complex. This process reveals the structure and properties of the Tor groups and helps determine their values.
• Evaluate how the behavior of the Tor functor can influence our understanding of module interactions in commutative algebra.
  • The behavior of the Tor functor significantly impacts our understanding of module interactions because it encapsulates how various modules relate under tensor products. For instance, if higher Tor groups vanish, it can imply deep connections between their structures and dimensions. This understanding aids in classifying modules and exploring their properties within commutative algebra, revealing underlying patterns and relationships that may not be immediately apparent.

Related terms

Flat Module: A module is flat if the functor - ⊗_R M is exact for every injective homomorphism of R-modules, meaning it preserves the structure of short exact sequences.

Exact Sequence: An exact sequence is a sequence of module homomorphisms between modules such that the image of one homomorphism equals the kernel of the next, reflecting how algebraic structures interact.

Derived Functors: Derived functors are a way to extend a functor from a category to its derived category, capturing information about how far an object is from satisfying a given property, like being flat.