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?

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    $\begingroup$ You can't really expect a single localisation functor to reflect anything. For instance, inverting $0$ is an example of localisation. $\endgroup$ – Zhen Lin Dec 9 '15 at 14:30
  • $\begingroup$ I don't understand what functor you're naming with the phrase "localization of commutative rings." Do you mean the localization functor on modules or what? $\endgroup$ – Qiaochu Yuan Dec 9 '15 at 18:11

One of the fundamental results in commutative algebra is the following:

Let $M$ be an $A$-module. Then $M = 0$ if and only if $M_\mathfrak{m} = 0$ for all maximal ideals $\mathfrak{m} \triangleleft A$.

You can think of it as a local-to-global principle if you like. Regardless, it follows that:

The localisation functor $\mathbf{Mod} (A) \to \prod_{\mathfrak{m} \in \operatorname{MaxSpec} A} \mathbf{Mod} (A_\mathfrak{m})$ is conservative/faithful.

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.


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