Embedding dimension

Summary

The embedding dimension of a Noetherian local ring $(R, \mathfrak m, k)$ is the dimension of its cotangent space $\mathfrak m / \mathfrak m^2$ over $k$. By Nakayama's lemma it equals the minimal number of generators of $\mathfrak m$. It is always at least the Krull dimension of $R$, and equality defines a regular local ring.