Existence of injective resolutions

Definition

The existence of injective resolutions refers to the property that every module can be embedded into an injective module, which can then be extended to an injective resolution. This concept is crucial in homological algebra as it facilitates the study of modules through their resolutions, especially in relation to derived functors and Ext groups.

5 Must Know Facts For Your Next Test

1. Every module over a Noetherian ring has an injective resolution, highlighting a crucial property in homological algebra.
2. Injective resolutions are used to compute derived functors, specifically the Ext groups, which classify extensions of modules.
3. An injective resolution can be constructed using Zorn's Lemma, emphasizing the relationship between set theory and homological algebra.
4. The length of an injective resolution can vary depending on the chosen injective modules used in the construction.
5. Injective resolutions play a vital role in the theory of derived categories, providing a framework for understanding morphisms between complexes.

Review Questions

• How does the existence of injective resolutions facilitate the computation of Ext groups?
  • The existence of injective resolutions allows for the classification of extensions of modules through the computation of Ext groups. By embedding a module into an injective module and extending it through an injective resolution, one can systematically study morphisms and derive information about possible extensions. This connection helps connect module theory with derived functors, making it easier to analyze complex relationships between modules.
• In what way do injective resolutions demonstrate the importance of Zorn's Lemma in homological algebra?
  • Injective resolutions showcase the importance of Zorn's Lemma because constructing these resolutions relies on selecting injective modules from a partially ordered set. Zorn's Lemma guarantees that every non-empty chain has an upper bound, which ensures that we can find injective modules that extend homomorphisms consistently. This foundational aspect links set theory with module theory, revealing deeper structural insights in homological algebra.
• Evaluate how the properties of injective resolutions impact the overall structure of derived categories in homological algebra.
  • Injective resolutions significantly impact derived categories by providing a systematic way to handle morphisms between complexes. The existence of such resolutions leads to the definition of derived functors and allows for a rich interplay between homology and cohomology theories. This relationship enhances our understanding of how different mathematical structures interact, paving the way for advanced concepts such as triangulated categories and stable homotopy theory.

Related terms

Injective module: A module is called injective if it satisfies the property that every homomorphism from a submodule can be extended to the entire module.

Projective resolution: A projective resolution is an exact sequence of projective modules that approximates a given module, used to compute derived functors like Tor.

Ext functor: The Ext functor measures the extent to which modules fail to be projective and is used to classify extensions of modules.