What are some local properties?

Question

What I mean by the question is for example the following statement:

Let $R$ be a ring and suppose that for each prime ideal $\mathfrak{p}$ the local ring $R_{\mathfrak{p}}$ has no nilpotent elements. Then $R$ itself also has no nilpotent elements.

So we call having nilpotent elements a local property. What are other local properties or what are non-local properties? For example being an integral domain is not a local property.

Answer 1

$\newcommand{\p}{\mathfrak p} \newcommand{\m}{\mathfrak m}$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.

Answer 2

• 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.