Projective and Injective Modules - Core Definitions

Projective and injective modules are dual concepts that play central roles in homological algebra, serving as building blocks for resolutions.

Definition3.1Projective Module

An $R$ -module $P$ is projective if it satisfies the following equivalent conditions:

Lifting property: For every surjection $g: B \twoheadrightarrow C$ and homomorphism $f: P \to C$ , there exists a lift $\tilde{f}: P \to B$ with $g \circ \tilde{f} = f$
Splitting property: Every short exact sequence $0 \to A \to B \to P \to 0$ splits
Direct summand: $P$ is a direct summand of a free module
Exactness: The functor $\text{Hom}_R(P, -)$ is exact

ExampleFree Modules are Projective

Every free $R$ -module $F = \bigoplus_{i \in I} R$ is projective. Given a surjection $g: B \twoheadrightarrow C$ and map $f: F \to C$ , we can lift each basis element $e_i$ independently to $b_i \in B$ with $g(b_i) = f(e_i)$ , then extend by linearity.

Definition3.2Injective Module

An $R$ -module $I$ is injective if it satisfies the following equivalent conditions:

Extension property: For every injection $f: A \hookrightarrow B$ and homomorphism $g: A \to I$ , there exists an extension $\tilde{g}: B \to I$ with $\tilde{g} \circ f = g$
Splitting property: Every short exact sequence $0 \to I \to B \to C \to 0$ splits
Exactness: The functor $\text{Hom}_R(-, I)$ is exact

ExampleDivisible Abelian Groups

An abelian group $D$ is divisible if for every $d \in D$ and non-zero $n \in \mathbb{Z}$ , there exists $x \in D$ with $nx = d$ . Divisible groups are precisely the injective $\mathbb{Z}$ -modules.

Examples include $\mathbb{Q}$ , $\mathbb{Q}/\mathbb{Z}$ , and $\mathbb{R}$ .

Definition3.3Projective Dimension

The projective dimension of a module $M$ , denoted $\text{pd}_R(M)$ , is the minimum length $n$ such that there exists a projective resolution: $0 \to P_n \to \cdots \to P_1 \to P_0 \to M \to 0$

If no such finite resolution exists, we say $\text{pd}_R(M) = \infty$ .

Remark

Projective dimension measures "how far" a module is from being projective. Modules with $\text{pd}_R(M) = 0$ are precisely the projective modules.

Definition3.4Injective Dimension

The injective dimension of a module $M$ , denoted $\text{id}_R(M)$ , is the minimum length $n$ such that there exists an injective resolution: $0 \to M \to I^0 \to I^1 \to \cdots \to I^n \to 0$

Modules with $\text{id}_R(M) = 0$ are precisely the injective modules.

ExampleFinitely Generated Modules over PIDs

Over a principal ideal domain $R$ , every finitely generated module has projective dimension at most 1. This follows from the structure theorem: every such module is a direct sum of cyclic modules $R/(a_i)$ and copies of $R$ .

Remark

The dual notions of projective and injective dimensions are related by various duality theorems. In nice situations (e.g., over Noetherian rings with finite global dimension), these dimensions encode deep information about the ring structure.