Projective and Injective Modules - Applications

Projective and injective modules have profound applications throughout algebra and beyond.

Theorem3.19Bass's Theorem on Injective Dimensions

A ring $R$ is Noetherian if and only if direct sums of injective $R$ -modules are injective.

Remark

Bass's Theorem provides a homological characterization of Noetherian rings. It shows that the preservation of injectivity under direct sums is a strong finiteness condition.

Theorem3.20Matlis Duality

Over a complete Noetherian local ring $(R, \mathfrak{m}, k)$ with residue field $k$ , there is a duality between finitely generated $R$ -modules and Artinian $R$ -modules given by: $D(M) = \text{Hom}_R(M, E(k))$

where $E(k)$ is the injective hull of the residue field.

ExampleApplication to Cohen-Macaulay Rings

For a Cohen-Macaulay local ring $R$ of dimension $d$ , the canonical module $\omega_R$ (when it exists) satisfies: $\text{Ext}^i_R(k, R) = \begin{cases} 0 & i \neq d \\ \omega_R & i = d \end{cases}$

This characterization is central to duality theory in commutative algebra.

Theorem3.21Grothendieck's Vanishing Theorem

Let $X$ be a Noetherian scheme of finite type over a field $k$ , and let $\mathcal{F}$ be a coherent sheaf on $X$ . Then for any affine open $U = \text{Spec}(R)$ and sufficiently large $n$ : $H^i(U, \mathcal{F}(n)) = 0 \quad \text{for all } i > 0$

This uses the injective resolution of sheaves and their derived functors.

ExampleSerre's Criterion for Normality

A Noetherian integral domain $R$ is normal if and only if:

$R$ is regular in codimension 1 (i.e., $R_{\mathfrak{p}}$ is a DVR for all height 1 primes $\mathfrak{p}$ )
Serre's condition $(S_2)$ holds

This can be reformulated using projective and injective dimensions of certain modules.

Theorem3.22Quillen-Suslin Theorem

Every finitely generated projective module over a polynomial ring $k[x_1, \ldots, x_n]$ (where $k$ is a field) is free.

Remark

The Quillen-Suslin Theorem resolved Serre's conjecture from algebraic geometry. It shows that vector bundles over affine space are trivial, which has important consequences for algebraic K-theory.

ExampleApplication to Module Categories

The category of projective $R$ -modules forms an additive category that is fundamental to algebraic K-theory. The Grothendieck group $K_0(R)$ is defined using the category of finitely generated projective $R$ -modules, measuring the failure of the category to be closed under extensions.