Suppose $R$ is a Noetherian local ring with unique maximal ideal $m\subset R$. We say that $R$ is **regular** if the dimension of $R$ is equal to the dimension of $m/m^2$ as an $R/m$-vector space. Regular local rings arise as the local rings of varieties over a field corresponding to smooth points, and this gives an abundant supply of them: for a field $k$, the rings $k[x,y]_{(x,y)}$ and $k[x,y]_{(x-a,y-a^2)}/(y – x^2)$ for instance are regular local rings.

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A commutative Noetherian local ring $ R$ with maximal ideal $ M$ is called a regular local ring if the Krull dimension of $ R$ is the same as the dimension of $ M/M^2$ as a $ R/M$-vector space.

In studying regular local rings one often uses the following lemma in inductive arguments: if $ R$ is an arbitrary commutative Noetherian local ring with maximal ideal $ M$, and $ M$ consists entirely of zero divisors, then the projective dimension $ \mathrm{pd}_R(A)$ is either zero or infinity for any finitely-generated $ R$-module $ A$. In other words, the only $ R$-modules of finite projective dimension are the projective (hence free) modules.

This is a neat little result that has a fun More »