When I first came across the concept of the localization of a ring, my immediate reaction upon reading the definition was one of "Well, why is it called the localization of a ring? Wouldn't the fractionalization of a ring be a far more appropriate term?"
Fortunately, I came across the definition in the context of classical algebraic geometry, and it didn't take me long after reading what followed in the texts I were consulting to understand that, ah yes, if $R$ is the coordinate ring of some algebraic set $V$, then a localization of $R$ to some multiplicatively closed subset corresponds to the restriction of $R$ to some open subset of $V$, and the whole "we are looking at the same ring as before, but in a local setting" became apparent to me.
My question today concerns the localization of modules. Specifically, what is the analogous way of thinking of localizations of modules so that it becomes apparent that we are looking at the same module as before, but in a local setting?
Look forward to your answers!
From the geometric point of view, modules $M$ over a ring $R$ correspond to quasicoherent sheaves on the spectrum $S$ of the ring, which are something like generalized vector bundles $E\to S$. Given such a bundle, one can recover the module as the global sections $\hom_S(S,E)$ of the bundle. Moreover, if $r\in R$ is an element such that the localization $R_r$ corresponds to an open subset $U\subset S$, then the localization $M_r$ is the $R_r$-module of global sections of the restricted bundle $E\vert_U\to U$ that is obtained by taking the inverse image of $U$ along the projection $E\to S$.