# Tag Archives: local ring

## 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 […]

## A Case of No Positive Finite Projective Dimension

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 \$ […]