# Local Rings of Global Dimension at Most Two

Posted by Jason Polak on 11. January 2017 · Write a comment · Categories: commutative-algebra

Wolmer Vasconcelos [1] gave the following classification theorem about commutative local rings of global dimension two:

Theorem. Let $A$ be a commutative local ring of global dimension two with maximal ideal $M$. If $M$ is principal or not finitely generated, then $A$ is a valuation domain. Otherwise $M$ is generated by a regular sequence of two elements, and $A$ will be Noetherian if and only if it is completely integrally closed.