Serre's theorem on regular local rings

Remarks

If $R$ is regular, then $\mathfrak{m}$ is generated by a regular sequence of length $\dim R$, and its Koszul complex is a free resolution of $k$ of that length. For a Noetherian local ring, the global dimension equals $\operatorname{pd}_R k$, which gives the implication from 1 to 2. The converse is the deep part: a finite resolution of $k$ forces the minimal number of generators of $\mathfrak{m}$ to equal $\dim R$. Example: $R = K[x]/(x^2)$ is not regular, and indeed $\cdots \to R \xrightarrow{x} R \xrightarrow{x} R \to K \to 0$ is a minimal resolution of $K$ that never stops. A major consequence is that every localization of a regular local ring at a prime ideal is again regular.

Going deeper

For a student

The theorem was proved by Serre and, independently, by Auslander and Buchsbaum. The homological criterion gives a short proof that a localization of a regular local ring is regular, since finite global dimension passes to localizations. It also gives Hilbert's syzygy theorem in the local setting, and, combined with the Auslander–Buchsbaum formula, it controls the resolutions of all finitely generated modules over a regular local ring.