Localization of commutative rings is a left exact left adjoint, so it behaves nicely with plenty of things. Local-to-global principles are also abundant in commutative algebra, and I thought some of them might actually be reflecting limits in disguise - that some types of limits in localizations must arise from limits in the original ring. Looking at localization as the tensor product of a symmetric monoidal category does not tell me much.


I did not give this much thought and I don't know much commutative algebra, so I hope it's not too stupid/basic to ask here. What inspired it was Nagata's lemma which says an element is prime iff it's either a unit or a prime in some localization. (Thought along the line of prime ideals as regular epis into integral domains.)


So, what kind of (co)limits does localization of commutative rings reflect?



What kinds of limits does localization of commutative rings reflect?
One of the fundamental results in commutative algebra is the following:



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> Let $M$ be an $A$-module. Then $M = 0$ if and only if $M\_\mathfrak{m} = 0$ for all maximal ideals $\mathfrak{m} \triangleleft A$.
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You can think of it as a local-to-global principle if you like. Regardless, it follows that:



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> The localisation functor $\mathbf{Mod} (A) \to \prod\_{\mathfrak{m} \in \operatorname{MaxSpec} A} \mathbf{Mod} (A\_\mathfrak{m})$ is conservative/faithful.
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In particular, $\mathbf{Mod} (A) \to \prod\_{\mathfrak{m} \in \operatorname{MaxSpec} A} \mathbf{Mod} (A\_\mathfrak{m})$ reflects finite limits and arbitrary colimits. This should be reminiscent of the fact in sheaf theory that the stalks functor $\mathbf{Sh} (X) \to \prod\_{x \in X} \mathbf{Set}$ reflects finite limits and arbitrary colimits.


