Projective Dimension

Definition

Projective dimension is a homological invariant that measures the complexity of a module by determining the length of the shortest projective resolution of that module. It connects deeply with various concepts in homological algebra, such as resolutions, derived functors, and cohomology theories, showcasing how modules relate to projective modules and their role in understanding other homological properties.

5 Must Know Facts For Your Next Test

1. The projective dimension of a module is zero if and only if the module is projective.
2. Every finitely generated module over a Noetherian ring has finite projective dimension if it has finite length.
3. The projective dimension is an additive invariant; that is, for a direct sum of modules, the projective dimension equals the maximum of their individual projective dimensions.
4. There exist modules with infinite projective dimension, particularly over rings that are not Noetherian.
5. Projective dimensions are closely related to derived functors like Tor and Ext, as they can help determine the behavior of these functors when computed on modules.

Review Questions

• How does the concept of projective dimension relate to resolutions and what role do these resolutions play in understanding module properties?
  • Projective dimension is fundamentally connected to resolutions because it quantifies how many steps it takes to resolve a module using projective modules. A projective resolution provides insight into the structure of the module by enabling computations of derived functors such as Ext and Tor. Understanding these resolutions helps reveal important information about the properties and behaviors of modules in relation to projectivity.
• Discuss how the projective dimension can be used to differentiate between modules over different types of rings, such as Noetherian versus non-Noetherian rings.
  • The projective dimension serves as an important tool in distinguishing modules over Noetherian rings from those over non-Noetherian rings. In Noetherian settings, finitely generated modules have finite projective dimensions, providing a structured way to classify them. However, in non-Noetherian rings, one may encounter modules with infinite projective dimensions, indicating more complex structures and behaviors. This differentiation helps in understanding how the ring's properties influence its modules.
• Evaluate the implications of having an infinite projective dimension for a module and how this affects its derived functors.
  • Having an infinite projective dimension for a module suggests that it cannot be resolved using a finite number of projective modules, indicating significant complexity in its structure. This complexity impacts derived functors like Tor and Ext since they may not behave well or yield straightforward results when computed on such modules. Understanding these implications can lead to deeper insights into module theory and its applications in various mathematical contexts.

Related terms

Projective Module: A module is projective if it satisfies the lifting property with respect to surjective homomorphisms, meaning that any surjective map onto a quotient can be lifted to a map onto the module.

Injective Dimension: The injective dimension of a module is defined similarly to projective dimension but uses injective resolutions instead, measuring how far a module is from being injective.

Homological Dimension: Homological dimension refers to the maximum of the projective and injective dimensions of a module, providing a unified way to measure its complexity in terms of both projective and injective resolutions.