Projective and Injective Modules - Main Theorem

Global dimension measures the homological complexity of a ring through the projective dimensions of its modules.

Definition3.15Global Dimension

The left global dimension of a ring $R$ is: $\text{gl.dim}(R) = \sup\{\text{pd}_R(M) : M \text{ is an } R\text{-module}\}$

Similarly, the right global dimension uses right modules. For commutative rings, these coincide.

Theorem3.16Global Dimension Theorem

For a ring $R$ , the following are equivalent:

$\text{gl.dim}(R) \leq n$
$\text{Ext}^{n+1}_R(M, N) = 0$ for all $R$ -modules $M, N$
Every $R$ -module has projective dimension at most $n$
Every $R$ -module has injective dimension at most $n$
$\text{pd}_R(R/I) \leq n$ for every left ideal $I$

Proof

(1) ⇒ (2): If $\text{gl.dim}(R) \leq n$ , then every module $M$ has a projective resolution of length at most $n$ : $0 \to P_n \to \cdots \to P_0 \to M \to 0$

Thus $\text{Ext}^{n+1}(M, N) = 0$ for all $N$ .

(2) ⇒ (5): For any left ideal $I$ , consider the quotient $R/I$ . The vanishing $\text{Ext}^{n+1}(R/I, N) = 0$ for all $N$ implies that $R/I$ has projective dimension at most $n$ .

(5) ⇒ (1): This follows from the fact that every module is a quotient of a free module, and projective dimension can be computed using resolutions.

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ExampleGlobal Dimensions of Common Rings

Fields: $\text{gl.dim}(k) = 0$ (every module is free, hence projective)
Principal Ideal Domains: $\text{gl.dim}(R) = 1$ (e.g., $\mathbb{Z}$ , $k[x]$ )
Regular Local Rings: $\text{gl.dim}(R)$ equals the Krull dimension
Polynomial Rings: $\text{gl.dim}(k[x_1, \ldots, x_n]) = n$

Theorem3.17Hilbert's Syzygy Theorem

If $R$ is a Noetherian ring with $\text{gl.dim}(R) = d < \infty$ , then: $\text{gl.dim}(R[x]) = d + 1$

In particular, for a field $k$ : $\text{gl.dim}(k[x_1, \ldots, x_n]) = n$

Remark

Hilbert's Syzygy Theorem is fundamental in algebraic geometry and commutative algebra. It shows that over polynomial rings, every finitely generated module has a finite free resolution, with length bounded by the number of variables.

Theorem3.18Auslander-Buchsbaum Formula

Let $R$ be a local Noetherian ring and $M$ a finitely generated $R$ -module with finite projective dimension. Then: $\text{pd}_R(M) + \text{depth}_R(M) = \text{depth}(R)$

where depth measures the length of maximal regular sequences.