5.2 Definitions and properties of injective modules

Injective modules are crucial in homological algebra, offering a dual concept to projective modules. They're characterized by their ability to extend homomorphisms from submodules to larger modules, making them vital for constructing resolutions and studying cohomology.

Baer's criterion and essential extensions provide key ways to identify injective modules. The Matlis-Gabriel theorem reveals their structure over Noetherian rings, connecting them to prime ideals and opening up applications in local cohomology and Gorenstein ring theory.

Injective Modules and Divisible Modules

Definition and Properties of Injective Modules

• Injective module is a module $Q$ over a ring $R$ with the following property:
  • For any module homomorphism $f: M \to Q$ and any injective module homomorphism $i: M \to N$, there exists a module homomorphism $g: N \to Q$ such that $g \circ i = f$
  • Intuitively, this means that any homomorphism from a submodule of $N$ into $Q$ can be extended to a homomorphism from the entire module $N$ into $Q$
• Divisible module is an important example of an injective module
  • A module $M$ over a ring $R$ is divisible if for every $m \in M$ and every $r \in R$, there exists an element $x \in M$ such that $rx = m$
  • In the case of abelian groups (modules over $\mathbb{Z}$), a divisible group is one in which every element can be divided by any non-zero integer
  • Examples of divisible abelian groups include $\mathbb{Q}$ and $\mathbb{Q}/\mathbb{Z}$
• Injectivity is a categorical notion dual to the concept of projectivity
  • A module $Q$ is injective if and only if the functor $\text{Hom}_R(-, Q)$ is exact
  • This means that $\text{Hom}_R(-, Q)$ preserves short exact sequences

Extension Property and Applications

• Extension property is a key characteristic of injective modules
  • If $Q$ is an injective $R$-module, then for any $R$-module homomorphism $f: M \to Q$ and any injective $R$-module homomorphism $i: M \to N$, there exists an $R$-module homomorphism $g: N \to Q$ such that $g \circ i = f$
  • This property allows for the extension of homomorphisms from submodules to the entire module
• Injective modules have important applications in homological algebra
  • They are used in the construction of injective resolutions, which are crucial for computing derived functors
  • Injective modules also play a role in the study of cohomology theories, such as group cohomology and Ext functors

Characterizations of Injectivity

Baer's Criterion and Essential Extensions

• Baer's criterion is a necessary and sufficient condition for a module to be injective
  • An $R$-module $Q$ is injective if and only if for every ideal $I$ of $R$ and every $R$-module homomorphism $f: I \to Q$, there exists an $R$-module homomorphism $g: R \to Q$ such that $g|_I = f$
  • In other words, every homomorphism from an ideal of $R$ into $Q$ can be extended to a homomorphism from the entire ring $R$ into $Q$
• Essential extension is another characterization of injectivity
  • An $R$-module $M$ is an essential extension of a submodule $N$ if every non-zero submodule of $M$ has a non-zero intersection with $N$
  • A module $Q$ is injective if and only if it has no proper essential extensions

Injective Envelopes and Their Properties

• Injective envelope (also called injective hull) is a minimal injective module containing a given module
  • For any $R$-module $M$, there exists an injective $R$-module $E(M)$ containing $M$ as an essential submodule
  • The injective envelope $E(M)$ is unique up to isomorphism
• Properties of injective envelopes:
  • Every $R$-module can be embedded as an essential submodule of an injective $R$-module
  • If $M$ is a submodule of an injective module $Q$, then $M$ is essential in $Q$ if and only if $Q$ is an injective envelope of $M$
  • Injective envelopes are used in the construction of minimal injective resolutions

Structure Theorems

Matlis-Gabriel Theorem and Its Implications

• Matlis-Gabriel theorem describes the structure of injective modules over a Noetherian ring
  • Let $R$ be a commutative Noetherian ring. Then every injective $R$-module is a direct sum of indecomposable injective modules
  • Moreover, every indecomposable injective $R$-module is isomorphic to the injective envelope $E(R/\mathfrak{p})$ for some prime ideal $\mathfrak{p}$ of $R$
• Implications of the Matlis-Gabriel theorem:
  • It provides a classification of injective modules over Noetherian rings
  • The theorem establishes a correspondence between indecomposable injective modules and prime ideals of the ring
  • It allows for the study of injective modules through the lens of prime ideals and localizations
  • The theorem has applications in local cohomology and the theory of Gorenstein rings