The Auslander-Buchsbaum Theorem

The Auslander-Buchsbaum theorem is a fundamental result connecting homological and algebraic properties of modules over local rings. It has profound consequences including the famous theorem that regular local rings are unique factorization domains.

The Auslander-Buchsbaum Formula

Theorem8.6Auslander-Buchsbaum Formula

Let $(R, \mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$ -module with finite projective dimension $\operatorname{pd}_R(M) < \infty$ . Then $\operatorname{pd}_R(M) + \operatorname{depth}(M) = \operatorname{depth}(R)$

This formula is remarkably powerful: it relates the homological invariant (projective dimension) to the algebraic invariant (depth) in a precise way.

ExampleDepth and projective dimension

Let $R = k[[x, y]]$ (regular, $\operatorname{depth} = 2$ ) and $M = R/(x)$ . Then $M$ has the free resolution $0 \to R \xrightarrow{x} R \to M \to 0$ , so $\operatorname{pd}(M) = 1$ . The formula gives $\operatorname{depth}(M) = 2 - 1 = 1$ , which is correct since $y$ is a regular element on $M = k[[y]]$ .

Consequences for Regular Local Rings

Theorem8.7Regular Local Rings are UFDs

Every regular local ring is a unique factorization domain (UFD).

Proof outline: By the Auslander-Buchsbaum-Serre theorem, a regular local ring has finite global dimension. Every height- $1$ prime $\mathfrak{p}$ satisfies $\operatorname{pd}(R/\mathfrak{p}) < \infty$ . By the Auslander-Buchsbaum formula, $\operatorname{pd}(R/\mathfrak{p}) = \operatorname{depth}(R) - \operatorname{depth}(R/\mathfrak{p}) = d - (d-1) = 1$ . Thus $R/\mathfrak{p}$ has a length- $1$ free resolution $0 \to R \to R \to R/\mathfrak{p} \to 0$ , showing $\mathfrak{p}$ is principal. By the Kaplansky criterion (a Noetherian domain is a UFD iff every height- $1$ prime is principal), $R$ is a UFD.

Theorem8.8Auslander-Buchsbaum-Serre Characterization

A Noetherian local ring $(R, \mathfrak{m})$ is regular if and only if it has finite global dimension: $\operatorname{gl.dim}(R) < \infty$ . In this case, $\operatorname{gl.dim}(R) = \dim R$ .

This homological characterization of regularity was a landmark achievement, showing that a purely ring-theoretic property (regularity) is equivalent to a homological property (finite global dimension).

RemarkLocalization of regularity

The Auslander-Buchsbaum-Serre theorem implies that regularity localizes: if $R$ is regular, then $R_\mathfrak{p}$ is regular for every prime $\mathfrak{p}$ . This was previously known only in special cases and is a deep consequence of the homological approach. The proof uses the fact that localization preserves projective dimension.