Tor

Definition

In the realm of homological algebra, 'tor' refers to a derived functor that arises from the tensor product of modules. Specifically, it measures the failure of a functor to be exact by quantifying how much torsion exists when relating two modules. This concept is crucial when analyzing the homological properties of rings and modules, providing insights into their structure through the lens of derived functors.

5 Must Know Facts For Your Next Test

1. The notation 'tor_n^R(A,B)' represents the nth derived functor of the tensor product of modules A and B over a ring R.
2. Tor is left exact in its first argument, which means it preserves finite limits but may not preserve colimits.
3. For free modules, tor vanishes, meaning that if either A or B is free, then tor will be trivial.
4. Computing tor often involves using projective resolutions to derive it from the tensor product, revealing insights about the structure of modules.
5. The values of tor can provide crucial information about relationships between modules, such as whether they are flat or have certain torsion elements.

Review Questions

• How does the tor functor relate to the concept of exact sequences in homological algebra?
  • The tor functor relates to exact sequences by providing a means to measure how far a given sequence is from being exact. When applying the tor functor to a sequence involving tensor products, it can help identify when exactness fails due to torsion. This is particularly important because understanding these failures allows mathematicians to gain deeper insights into module relationships and structure.
• Discuss how computing tor using projective resolutions can affect the understanding of module relationships.
  • Computing tor via projective resolutions highlights how torsion elements influence module interactions. By resolving modules into projective ones and taking the tensor product, we obtain insight into the derived functor behavior. This approach not only elucidates structural relationships between modules but also indicates whether they have properties such as flatness or torsion-free characteristics, which are essential in homological studies.
• Evaluate the implications of a module having non-zero values of tor when interacting with other modules in a given ring.
  • If a module exhibits non-zero values of tor when paired with another module in a ring, it signifies that there are torsion elements affecting their interaction. This situation indicates potential complications in understanding their homological properties and could suggest that one or both modules may not exhibit flatness or might have other restrictions in their structure. Such findings can lead to critical insights for further studies in algebraic topology, representation theory, or algebraic geometry, where these relationships often play a pivotal role.

Related terms

Tensor Product: An operation on two modules that produces another module, encapsulating bilinear relations and serving as a foundation for defining tor.

Ext: A derived functor related to the Hom functor that measures extensions of modules, complementary to the torsion quantified by tor.

Homological Dimension: The minimum length of projective resolutions for a module, indicating how 'complex' a module is from a homological perspective.