I like your point of view here:

here R may be the structure sheaf of some ringed topos and accordingly the modules may be sheaves of modules.

which generalizes my earlier explicit listing of $\mathcal{O}$-modules. There are however two caveats: first of all we want that ringed topos is taken in the sense or topos with possibly *noncommutative* ring object, what was rarely a convention in the topos setup of $n$Lab (?), and more importantly, we would like in fact to mean a localization of the category of quasicoherent sheaves of modules, rather than a localization of the whole category of $\mathcal{O}$-modules. I do not know how to include this emphasis in noncommutative ringed topos language.

P.S. new stub Martindale quotient aka Martindale localization. Note also the related entry (to localization circle) in my personal $n$Lab: gluing categories from localizations (zoranskoda).

]]>Thanks! I have edited a bit further, adding some half-sentences with further remarks and pointers here and there. Also formatted a bit (of course :-)

]]>Major changes at localization of a module.

]]>Yes, later today or tomorrow. Now I have an urgent errant to do first.

]]>Okay, thanks. I guess I see. I never thought of it that way, to be frank. Maybe you have a sec to spell it out in the entry?

]]>But you wrote it in the flat case

$Q(M) = Q R\otimes_R M$is the localization functor for the left modules, the morphism part being obvious and $R$ the ground ring.

By localization one also means the localization map, what is the components of the unit of the corresponding adjunction between the localization and its right adjoint.,

]]>From that standard perspective I do not see why to separate into a separate entry.

There is no harm in having entries on special cases of other entries with more explicit details. I think that’s a good thing to have.

In this case: hm, what is the localization of the category of modules that you have in mind?

]]>If we have a localization functor $Q$ from some category $C$, then applying $Q$ on some object $m\in M$ is the localization of that object $m$< for any of the very many kind of categories and many kinds of localizations. If $M$ is a category of modules then it is a localization of a module. From that standard perspective I do not see why to separate into a separate entry.

Surely the motivating statements in the entry localization should explain intuition from covering classical special cases like the localization of categories of quasicoherent sheaves of modules (as well in future entries on spectra and modules over them) a la Pierre Gabriel’s thesis.

]]>You have “interpretation of modules as generalized vector bundles” pointing back to the same page.

Thanks, fixed.

]]>You have “interpretation of modules as generalized vector bundles” pointing back to the same page.

By the way, does localization cover all types of localization? I ask since the term cropped up in the context of the codensity monad, as idempotent completion.

]]>created *localization of a module*