Tor

Definition

In the context of derived categories, Tor is a functor that measures the extent to which a given module fails to be projective. It is an important tool in homological algebra, as it captures information about the relationships between modules and their resolutions. The Tor functor is particularly relevant when dealing with derived categories because it helps in calculating derived functors, providing insights into how modules interact through exact sequences.

5 Must Know Facts For Your Next Test

1. Tor is denoted as Tor_n(A, B) where A and B are modules, and n indicates the level of the functor being calculated.
2. The computation of Tor can be done using projective resolutions of the modules involved, facilitating the extraction of deeper algebraic properties.
3. Tor measures not just the failure of exactness in sequences involving modules but also provides valuable information about extensions and syzygies.
4. For any two modules, if at least one is projective, then Tor between them vanishes for all levels, highlighting the significance of projective modules.
5. The Tor functor is right-exact but not left-exact, which means it preserves exact sequences only when they are exact at the end, reflecting its specific utility in homological studies.

Review Questions

• How does Tor relate to the concept of projective modules and their role in derived categories?
  • Tor is crucial for understanding the interactions between modules, particularly in how they fail to be projective. If one or both of the modules involved in computing Tor are projective, it indicates that certain exact sequences will preserve properties across them. This relationship emphasizes how projective modules simplify the understanding of derived categories by ensuring that Tor vanishes and thus provides a clear pathway for studying homological properties.
• Describe how one might compute Tor using projective resolutions and what this reveals about the involved modules.
  • To compute Tor, one typically starts by constructing a projective resolution of one of the modules involved. By taking the tensor product with another module along this resolution, one can derive sequences that reveal how much information is lost in terms of exactness. This process allows mathematicians to uncover deeper relationships between modules and highlights aspects like extensions or syzygies which may not be immediately visible without this technique.
• Evaluate the implications of Tor being right-exact but not left-exact within derived categories.
  • The fact that Tor is right-exact but not left-exact has significant implications in derived categories. It means that while it preserves some structures when moving to quotient objects or factoring through them, it fails to maintain all exactness when considering inclusions or injective scenarios. This characteristic affects how we interpret morphisms and extensions within derived categories, suggesting caution when applying Tor to infer properties related to initial inclusions or more complex categorical constructions.

Related terms

Derived Functor: A functor that arises from the construction of resolutions of objects in a category, allowing one to study properties of modules through homological methods.

Projective Module: A type of module that has the property that every short exact sequence of modules that ends with it splits, indicating that it is 'free' in a sense.

Homology: A mathematical concept that studies topological spaces through sequences of abelian groups or modules, revealing information about their structure and features.