An injective resolution is a type of exact sequence of injective modules that allows one to represent a module as an extension by injective modules. This concept is crucial for understanding how injective modules can be used to study other modules and their homological properties. The construction of injective resolutions provides a way to compute derived functors, including Ext, and plays an important role in various homological contexts, such as sheaf cohomology and the determination of homological dimensions.
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An injective resolution of a module M is an exact sequence of the form 0 → M → I^0 → I^1 → ... where each I^n is an injective module.
Every module has an injective resolution, which may not be unique; however, all injective resolutions of a given module have the same length.
Injective resolutions are used to compute the Ext functors, which measure the extent to which modules fail to be projective.
The concept of injective resolution also relates closely to left derived functors, which can be computed using resolutions.
Injective resolutions play a significant role in sheaf cohomology by providing tools for understanding cohomological properties of sheaves on topological spaces.
Review Questions
How does an injective resolution help in computing the Ext functor?
An injective resolution provides a way to represent a module as an extension by injective modules, allowing for the application of Hom and tensor products. To compute Ext, one uses the long exact sequence obtained from applying Hom to the exact sequence in the injective resolution. This leads to exact sequences that reveal properties about extensions and give insight into how modules relate in terms of homomorphisms.
Discuss the significance of injective resolutions in the context of derived functors.
Injective resolutions are significant for derived functors as they provide a systematic method to obtain these functors by resolving modules. Specifically, derived functors like Ext can be computed by applying the functor to an injective resolution. This connection is essential because it allows one to analyze properties of modules and their relationships through derived functors, facilitating deeper insights into homological algebra.
Evaluate how injective resolutions influence the understanding of sheaf cohomology.
Injective resolutions significantly influence sheaf cohomology by providing techniques for calculating cohomological dimensions and understanding how sections over various open sets relate. When studying sheaves, one often constructs injective resolutions of sheaves over topological spaces, which facilitates computations in cohomology. This approach reveals important properties about global sections and how local data contribute to global behavior, linking algebraic concepts with geometric intuition.
An exact sequence is a sequence of module homomorphisms between modules where the image of one homomorphism equals the kernel of the next.
Derived Functor: Derived functors are functors that provide information about a functor's behavior through resolutions, particularly in relation to homological algebra.