Tor

Definition

In algebra, Tor is a functor that measures the non-free part of a module over a ring and is denoted as \text{Tor}^i_R(A, B), where A and B are R-modules. It captures the notion of how two modules interact with each other through their tensor product, providing insights into their derived functors and homological properties.

5 Must Know Facts For Your Next Test

1. Tor can be computed using projective resolutions of the modules involved, which allows one to find the 'depth' of the interaction between the two modules.
2. Tor is left exact in its first argument but not necessarily right exact, meaning that while it preserves limits, it may not preserve colimits.
3. The value of \text{Tor}^0_R(A, B) is equal to the tensor product A \otimes_R B, providing a bridge between homological algebra and tensor operations.
4. Higher Tor groups, such as \text{Tor}^1_R(A, B) and beyond, are important for understanding the obstructions to splitting exact sequences.
5. In the context of Koszul complexes, Tor can be calculated using the resolution provided by these complexes, allowing for computation in various algebraic settings.

Review Questions

• How does Tor relate to the concepts of projective resolutions and homological algebra?
  • Tor is intrinsically connected to projective resolutions as it is computed using these resolutions. When you take a projective resolution of one of the R-modules involved in Tor, you can derive higher Tor groups by looking at how the modules fail to be flat. This makes Tor a central player in homological algebra because it provides insights into the depth and structure of module interactions.
• Discuss the significance of higher Tor groups in understanding module properties and their relationships.
  • Higher Tor groups are crucial because they help identify obstructions to exactness in sequences involving modules. For instance, \text{Tor}^1_R(A, B) gives information about whether a certain short exact sequence splits. In essence, these higher groups allow mathematicians to analyze complex relationships between modules and understand how their structures interweave through extensions.
• Evaluate the role of Koszul complexes in calculating Tor and how they enhance our understanding of module interactions.
  • Koszul complexes play a pivotal role in calculating Tor by providing explicit resolutions that facilitate computation. They allow for systematic ways to find higher Tor groups, particularly when dealing with modules that arise from algebraic varieties or rings with specific properties. This connection not only simplifies calculations but also enriches our understanding of how modules interact by linking geometric ideas with algebraic structures through homological techniques.

Related terms

Ext: Ext is another derived functor that measures extensions of modules and is denoted as \text{Ext}^i_R(A, B), providing information about how modules can be embedded into larger modules.

Hom: Hom is the functor that assigns to each pair of modules A and B the set of R-linear homomorphisms from A to B, written as \text{Hom}_R(A, B).

Derived Functors: Derived functors are functors that arise from the process of applying the homological algebra tools to measure the 'failure' of certain functors to be exact.