## Example: Cohen-Macaulay Ring that is Not Regular

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 […]