Projective and Injective Modules - Examples and Constructions

We explore concrete examples and methods for constructing projective and injective modules in various contexts.

ExampleProjective Modules over Local Rings

Over a local ring $(R, \mathfrak{m})$ , every finitely generated projective module is free. This follows from Nakayama's Lemma: if $P$ is finitely generated projective with minimal generating set $\{p_1, \ldots, p_n\}$ (minimal modulo $\mathfrak{m}$ ), then the natural map $R^n \to P$ is an isomorphism.

ExampleInjective Modules over Dedekind Domains

Over a Dedekind domain $R$ (e.g., $\mathbb{Z}$ or $\mathbb{C}[x]$ ), an $R$ -module $I$ is injective if and only if it is divisible. This characterization is particularly useful for explicit computations.

For example, $\mathbb{Q}$ is an injective $\mathbb{Z}$ -module, and $\mathbb{Q}/\mathbb{Z}$ is the injective hull of any finite cyclic group.

Definition3.11Flat Module

An $R$ -module $F$ is flat if the functor $F \otimes_R -$ is exact. Equivalently, for every injection $i: A \hookrightarrow B$ , the map $\text{id}_F \otimes i: F \otimes A \to F \otimes B$ is injective.

Theorem3.12Relationship Between Projective and Flat

Every projective module is flat. The converse holds over certain nice rings (e.g., perfect rings), but not in general.

Over a Noetherian ring, finitely generated flat modules are projective. Over a local ring, flat modules are free.

ExampleNon-Projective Flat Module

Over $R = \mathbb{Z}$ , the module $\mathbb{Q}$ is flat but not projective. It's flat because tensoring with $\mathbb{Q}$ is the same as localizing at $\mathbb{Z} \setminus \{0\}$ , which is exact. However, $\mathbb{Q}$ is not projective since it's not a direct summand of a free abelian group.

Definition3.13Projective Resolution

A projective resolution of an $R$ -module $M$ is an exact sequence: $\cdots \to P_2 \to P_1 \to P_0 \to M \to 0$

where each $P_i$ is projective. Resolutions are unique up to chain homotopy equivalence.

ExampleFree Resolution of $\mathbb{Z}/n\mathbb{Z}$

A minimal free resolution of $\mathbb{Z}/n\mathbb{Z}$ as a $\mathbb{Z}$ -module is: $0 \to \mathbb{Z} \xrightarrow{\times n} \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} \to 0$

This shows $\text{pd}_{\mathbb{Z}}(\mathbb{Z}/n\mathbb{Z}) = 1$ .

Definition3.14Injective Resolution

An injective resolution of an $R$ -module $M$ is an exact sequence: $0 \to M \to I^0 \to I^1 \to I^2 \to \cdots$

where each $I^i$ is injective. Injective resolutions always exist (by iteratively embedding into injective hulls) and are unique up to chain homotopy equivalence.

Remark

The existence of enough projectives and enough injectives guarantees that we can compute derived functors using either projective or injective resolutions, depending on which is more convenient for the problem at hand.