Blueprint for the Adjunction Formula: Proof of Auslander-Buchbaum-Serre

1.11 Proof of Auslander-Buchbaum-Serre

Lemma 1.11.1

Let \(R\) be a regular local ring. Then \(R\) has finite global dimension. That is, any finitely generated module \(R\) has finite projective dimension.

Proof ▼

Theorem 1.11.2 Ferrand-Vasconcelos, BH 2.2.8

Let \((R,\mathfrak {m},k)\) be a local noetherian ring. Let \(I\) be a nonzero ideal with finite projective dimension. If \(I / I^2\) is a free \(R\)-module, then \(I\) is generated by a regular sequence.

Proof ▼

Since \(I\) has finite projective dimension, it has a finite free resolution. Thus, by 1.4.6 it has must have an \(R\)-regular element \(x\). . . . finish the proof . . .

Theorem 1.11.3 Auslander-Buchsbaum-Serre Criterion, BH 2.2.7

Let \((R,\mathfrak {m},k\) be a noetherian local ring. The following are equivalent:

\(R\) is regular.
\(R\) has finite global dimension.
\(\text{proj dim}k {\lt} \infty \)

Proof ▼

(i) \(\implies \) (ii) is precisely Lemma 1.11.1

(ii) \(\implies \) (iii) follows by applying the definition of global dimension with \(M=k\)

(iii) \(\implies \) (i) is a special case of Theorem 1.11.2, using 1.5.10 to conclude regularity.

1.11.2.

Theorem 1.11.4 Regular Rings Localize, BH 2.2.9

Let \(R\) be a regular local ring, and let \(\mathfrak {p}\) be a prime ideal in \(R\). Then \(R_{\mathfrak {p}}\) is a regular local ring.

Proof ▼

By Auslander-Buchsbaum-Serre, it is enough to show that \(R_{\mathfrak {p}} / \mathfrak {p}R_{\mathfrak {p}}\) has finite projective dimension. Also by Auslander-Buchsbaum-Serre, we know that \(k = R / \mathfrak {m}\) has finite projective dimension. Then \(k\) has a minimal free resolution of finite length by Proposition ??. By the fact that the loclization of a resolution is a resolution, We get a finite resolution for \(R_{\mathfrak {p}} / \mathfrak {p}R_{\mathfrak {p}}\), thus any minimal resolution is also finite, giving us what we want.