Cotangent space of a local ring

Summary

The cotangent space of a local ring $(R, \mathfrak m, k)$ is the vector space $\mathfrak m / \mathfrak m^2$ over the residue field $k$. Its dual is the Zariski tangent space at the point that $R$ describes. Its dimension is the embedding dimension of $R$, which is used to define regular local rings, that is, smooth points.