What are some local properties?

• Being Cohen-Macaulay is a local property, also by definition.
• So is being Gorenstein.
• Being universally catenary is local.
• Being Noetherian is not quite local.
• Being finitely generated is local.

$\newcommand{\p}{\mathfrak p} \newcommand{\m}{\mathfrak m}$I took these properties from Atiyah-Macdonald. The first key local property is : if $M$ is a $A$-module, then there is an equivalence between :

• $M=0$
• For all prime $\p$, $M_{\p} = 0$
• For all maximal ideals $\m$, $M_{\m} = 0$

We have similar equivalence for the following properties :

• Injectivity of a morphism of $A$-module $f : M \to N$.
• Flatness of a $A$-module $M$.
• $M$ is torsion-free.
• $A$ is integrally closed.
• $M$ is an invertible fractional ideal.

After, some properties are by definition local, for example being smooth or normal. Sometimes, properties can be checked locally but you have to ask different conditions. For example, $A$ is absolutely flat if and only if for all maximal ideals $\m$, $A_{\m}$ is a field.