Computing Tor

Definition

Computing Tor is a process in homological algebra used to measure the derived functors of the tensor product, specifically quantifying the non-flatness of a module. It helps in understanding how modules interact under tensor operations, capturing information about their extensions and relationships. This is especially important in studying properties of modules over rings and allows for the investigation of resolutions and derived categories.

5 Must Know Facts For Your Next Test

1. Tor can be computed using projective or injective resolutions of the modules involved, which allows for finding the derived functors effectively.
2. The functor Tor is contravariant in both variables, meaning it reverses the direction of morphisms when computing.
3. For two modules M and N, $$\text{Tor}^i(M,N)$$ measures how many times you need to 'resolve' M to make it flat when tensored with N.
4. The first Tor group, $$\text{Tor}^1(M,N)$$, detects obstructions to lifting homomorphisms and provides insight into extensions between modules.
5. Computing higher Tor groups can provide information about higher extensions and derived functors in the context of spectral sequences.

Review Questions

• What role does projective resolution play in computing Tor, and why is it important?
  • Projective resolution is crucial for computing Tor because it provides a way to systematically resolve a module into projective modules, allowing one to capture the necessary algebraic information. By taking a projective resolution of one of the modules involved in the tensor product, we can compute the derived functors that give us the Tor groups. This process highlights how extensions and relationships between modules are structured and helps in understanding their behaviors under tensor operations.
• How do flat modules relate to the computation of Tor, particularly in understanding its properties?
  • Flat modules are significant when computing Tor because they ensure that certain exactness properties hold when performing tensor operations. If a module is flat, then tensoring it with another module does not introduce any torsion or change its exact sequence. Consequently, by analyzing how flatness affects computations of Tor, we gain insights into when certain obstructions arise and what this means for the underlying structure of the modules involved.
• Evaluate the implications of non-zero values of $$\text{Tor}^1(M,N)$$ on the structure and properties of the modules M and N.
  • Non-zero values of $$\text{Tor}^1(M,N)$$ indicate that there are obstructions to lifting homomorphisms between M and N, suggesting complexities in their interactions. This can reflect on the existence of extensions of M by N, meaning there might be non-trivial ways to 'fit' these modules together. Understanding these implications allows us to delve deeper into how modules relate to each other and how their structures inform us about possible decompositions or direct summands within category theory.

Related terms

Tensor Product: An operation that takes two modules and produces a new module, capturing bilinear mappings between them.

Flat Module: A module that preserves the exactness of sequences when tensored with another module, indicating that it behaves nicely under tensor products.

Projective Module: A module that satisfies a lifting property with respect to surjective homomorphisms, often used in constructing projective resolutions.