Ext and tor functors

Definition

Ext and Tor functors are fundamental tools in homological algebra used to study the properties of modules over a ring. Ext measures the extent to which a module fails to be projective, while Tor quantifies how far a module is from being flat. Together, they provide critical insights into the relationships between modules, their resolutions, and various homological dimensions.

5 Must Know Facts For Your Next Test

1. The Ext functor, denoted as \(\text{Ext}^n(A,B)\), calculates the n-th derived functors of the Hom functor, providing information on extensions of modules.
2. The Tor functor, denoted as \(\text{Tor}_n(A,B)\), represents the n-th derived functors of the tensor product, capturing information about torsion in modules.
3. Ext and Tor can be used to determine projective dimensions and flat dimensions of modules, which are crucial for understanding their homological properties.
4. Both functors satisfy various naturality properties, making them useful in constructing long exact sequences and relating different homological dimensions.
5. The relationship between Ext and Tor is encapsulated in several important results, such as the universal coefficient theorem for homology, linking algebraic topology with homological algebra.

Review Questions

• How do Ext and Tor functors relate to the concepts of projective and flat modules?
  • Ext functors help identify whether a module is projective by examining extensions between modules. If \(\text{Ext}^1(A,B) = 0\), it indicates that A is a projective module in relation to B. In contrast, Tor functors evaluate the flatness of modules by determining if tensoring with A preserves exact sequences. If \(\text{Tor}_1(A,B) = 0\), then A is a flat module with respect to B.
• Discuss how Ext and Tor functors can be utilized to compute homological dimensions of modules.
  • Homological dimensions are determined by analyzing Ext and Tor functors' values. For instance, the projective dimension of a module can be computed using \(\text{Ext}^n(A, R) = 0\) for all \(n > pd(A)\), indicating that A has finite projective dimension. Similarly, flat dimension is determined using \(\text{Tor}_n(A,B) = 0\) for all \(n > fd(A)\), highlighting the depth of flatness. These dimensions provide critical insights into module behavior and their relationships.
• Evaluate the implications of Ext and Tor functors on the structure theory of modules over a ring.
  • The analysis of Ext and Tor functors significantly impacts our understanding of module structure by revealing underlying relationships between different types of modules. These functors help classify modules based on their projectivity and flatness, leading to essential results like the existence of resolutions. Additionally, they assist in establishing long exact sequences that connect different homological invariants, allowing for a deeper understanding of module behavior over rings and providing tools for addressing complex algebraic problems.

Related terms

Projective Module:

A module that satisfies the property that every surjective module homomorphism onto it can be lifted to a homomorphism from the domain.

Injective Module:

A module that satisfies the property that every injective module homomorphism from it can be extended to a homomorphism to any module containing it.

Flat Module:

A module that preserves the exactness of sequences when tensored with other modules, meaning it does not introduce new torsion elements.