Injective Module

Definition

An injective module is a type of module that has the property that any homomorphism from a submodule into it can be extended to a homomorphism from the entire module. This concept is closely related to various properties of rings and modules, particularly in the context of depth, regular sequences, and Cohen-Macaulay rings, as it influences their structure and classification. Injective modules play a significant role in understanding projective resolutions and the overall homological dimensions of algebraic structures.

5 Must Know Facts For Your Next Test

1. Injective modules are characterized by their ability to allow extension of homomorphisms, which makes them crucial for building resolutions in homological algebra.
2. Every injective module is also a flat module, but not all flat modules are injective; this distinction can affect how we analyze modules in different contexts.
3. In the category of modules over a Noetherian ring, injective modules can be represented as direct summands of the injective hulls of certain modules.
4. Injective modules correspond to specific functors in homological algebra, notably related to Ext functors, impacting their applications in depth and regular sequences.
5. Injective dimension is an important invariant associated with injective modules, reflecting how far one must go in extending modules to achieve injectivity.

Review Questions

• How does the property of an injective module relate to extensions of homomorphisms and what implications does this have for regular sequences?
  • The property of an injective module allows any homomorphism from a submodule to be extended to the entire module. This characteristic plays a significant role when working with regular sequences because it ensures that when we have regular elements in our ring, we can extend maps without losing structure. This capability is crucial for studying depth and Cohen-Macaulay rings since these concepts often involve analyzing how elements relate through extensions.
• Discuss how injective modules can influence the classification of Cohen-Macaulay rings and their associated depth.
  • Injective modules significantly influence the classification of Cohen-Macaulay rings as they provide insight into their depth properties. In Cohen-Macaulay rings, having injective modules allows for better handling of homological dimensions, which helps determine if the ring possesses the desired depth conditions. The interplay between injectivity and depth gives rise to richer structural insights about how these rings behave under various operations.
• Evaluate the relationship between injective dimension and the concept of depth within Noetherian rings and explain its significance.
  • The relationship between injective dimension and depth within Noetherian rings is pivotal for understanding their homological properties. Specifically, the injective dimension provides bounds on how far we need to extend a module before achieving an injective resolution. This is significant because it allows us to classify rings based on their depth; low injective dimensions indicate a rich structure related to regular sequences and Cohen-Macaulay properties, whereas high dimensions may suggest complications or deficiencies in modular behavior.

Related terms

Cohen-Macaulay Ring: A Cohen-Macaulay ring is a ring that satisfies a certain depth condition, where the depth equals the Krull dimension, indicating balanced dimensions between ideals and their generating sets.

Regular Sequence: A regular sequence is a sequence of elements in a ring that satisfies specific conditions related to non-vanishing and independence, playing an essential role in defining depth and Cohen-Macaulay properties.

Depth: Depth is a measure of the 'height' of a module or ring, determined by the length of the longest regular sequence contained within it, which can be crucial for understanding its structural properties.