When does it suffice to show statements about rings only for the local ring after localizing at a prime?

Question

I'm learning commutative Algebra with the book from Eisenbud. But I'm having trouble understanding some of his Proofs. Often we have a Statement about a ring R or an R-module M, which we'd like to proof. Then he assumes wlog that R is a local ring and continues proving the Statement. Now I know the fact, that if $p\subset R$ is a prime, then $R_p$, its localization at p, is a local ring. But I don't know why it suffices to show the Statement for $R_p$ instead of R. I could give you examples, but this 'trick' is used in very different settings, so I'd rather have a general answer about when it is reasonable to make this reduction and when it is not.

Answer 1

First of all, I love this question.

Secondly, as Max says in the comments, what needs to be true in order to use this trick is that the statement "Property A holds for ring $R$ (or module $M$)" needs to follow from the statement "Property A holds for all localizations $R_p$ (or $M_p$), as $p$ ranges over all primes of $R$." Properties of which this is true are said to be local properties. Some properties are local by definition, i.e. the definition is first given for local rings and then defined to hold for general rings whenever it holds for every localization (e.g. the notion of a regular ring), but other properties' localness requires a theorem.

For example, injectivity is a local property of a map: If $\phi: M\rightarrow N$ is a module homomorphism over a ring $R$, and for every prime $p\triangleleft R$, we have $\phi_p:M_p \rightarrow N_p$ is injective, then $\phi$ is injective. This statement requires proof (it is Proposition 3.9 in Atiyah-MacDonald's classic Introduction to Commutative Algebra and I'm sure it's also proven in Eisenbud), but once we have proved it, it means we can afterward assess injectivity just by considering localizations.