Tor functor

Definition

The tor functor is a fundamental tool in homological algebra that measures the failure of a sequence of modules to be exact. Specifically, it assigns to pairs of modules a sequence of derived functors that help track torsion elements in homology. It is closely connected to the study of abelian categories, where it plays a crucial role in understanding the relationships between modules and their derived categories, as well as in the applications of Kan extensions which facilitate the transfer of structures across different categories.

5 Must Know Facts For Your Next Test

1. The tor functor is denoted as $$\text{Tor}^i(M,N)$$, where $$M$$ and $$N$$ are modules and $$i$$ indicates the level in the derived series.
2. Tor functors can be computed using projective resolutions, making them essential in understanding relations between modules in an abelian category.
3. The first tor functor, $$\text{Tor}^1(M,N)$$, specifically captures torsion elements and is particularly useful when studying short exact sequences.
4. The tor functor behaves well with respect to base change; for example, if you change one of the modules in a pair, it can yield new insights into their relationship.
5. Tor functors also provide important information about flatness; if $$\text{Tor}^1(M,N) = 0$$, then one can conclude that $$N$$ is flat over the ring associated with $$M$$.

Review Questions

• How does the tor functor relate to the concepts of exact sequences and homological algebra?
  • The tor functor is deeply intertwined with exact sequences in homological algebra, as it helps identify when sequences fail to be exact by measuring torsion elements. In particular, it provides insights into how modules relate to each other through their derived sequences. Understanding tor functors allows us to analyze various properties of modules based on these exact sequences, which is foundational in studying their structure.
• Discuss how tor functors can be computed using projective resolutions and what this implies for studying module relationships.
  • To compute tor functors, one typically uses projective resolutions of the involved modules. This process involves resolving one or both modules in a given pair with projective modules and applying the tensor product followed by taking homology. The ability to compute tor using projective resolutions highlights its effectiveness as a tool for studying relationships between modules, as it reveals deeper structural properties that may not be immediately apparent through direct examination.
• Evaluate the implications of tor functors on flatness and how they influence our understanding of module properties within abelian categories.
  • Tor functors provide significant insights into flatness among modules. Specifically, if $$\text{Tor}^1(M,N) = 0$$ for two modules $$M$$ and $$N$$, this indicates that $$N$$ is flat over the ring associated with $$M$$. This result has profound implications for module theory and algebraic geometry because flatness ensures that certain properties hold under base change. Thus, studying tor functors not only informs us about torsion but also about broader categorical relationships that can simplify complex structures within abelian categories.

Related terms

Ext Functor: The Ext functor is another derived functor that measures the extent to which a module fails to be projective, providing insights into extension groups between modules.

Homological Dimension: Homological dimension refers to the length of the projective or injective resolutions of a module, which helps determine the complexity of its structure in terms of homological algebra.

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, indicating a precise relationship among the modules.