Regular local ring

There are deep connections between algebraic (in fact, scheme-theoretic) notions of smoothness and regularity.

Definition

Let $A$ be a Noetherian local ring with maximal ideal Failed to parse (unknown function "\mathfrac"): {\displaystyle \mathfrac{m}} and residual field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=A/\mathfrac{m}} . The following conditions are equivalent:

1. The Krull dimension of $A$ is equal to the dimension of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrac{m}/\mathfrac{m}^2} as a $k$-vector space.