In which sense modules generalize vector bundles?

Question

At https://ncatlab.org/nlab/show/module, we find the following statement:

The theory of monoids or rings and their modules, its “meaning” and usage, is naturally understood via the duality between algebra and geometry:

1. A ring R is to be thought of as the ring of functions on some space
2. An R-module is to be thought of as the space of sections of a vector bundle on that space.

I am very fascinated by the seeming depth of this fact and I would like to understand this better. I think I have the necessary "vocabulary" in the sense that I do know what a ring, a module, a vector bundle is, but sadly the entry is quite vague in the following.

How do we identify the space in 1? How do we identify the vector bundle in 2? It seems to me, these realization problems depend on the situation, especially from the chosen geometric context. Anyhow, the entry cites the fact that for algebraic varieties this can be made precise, and this assures the generality of the statement.

Any brief reference (such as a pamphlet or lecture notes) or help in understanding better would be grateful.

Answer 1

This statement isn’t literally true in any very intuitive sense. Rather, there is Swan’s theorem, which says that a vector bundle on a compact Hausdorff space is effectively the same thing as a finitely generated projective module over the ring of continuous real-valued functions on that space. Furthermore, the space can be recovered from the ring, by considering its set of maximal ideals with a suitable topology.

The situation in algebraic geometry generalizes by first simply defining the space associated to a ring $A$ to be its Zariski spectrum, which is a certain topologization of its set of prime ideals. Here there is no geometrically familiar notion of vector bundle, but one still thinks of a finitely generated projective $A$-module as being like a vector bundle over the spectrum of $A$. And more general correspondences are possible, though the definitions are more abstruse. See here for a definition valid in extreme generality: https://stacks.math.columbia.edu/tag/01M1