Affine module-sheaf equivalence ($A\text{-Mod}\simeq \mathrm{QCoh}(X)$)

On an affine variety or affine scheme with coordinate ring $A$, the global section functor and $M\mapsto \tilde{M}$ are inverse equivalences between modules and quasi-coherent sheaves. On every principal open subset, sections are the appropriate localization of a module. Localization detects exactness at maximal ideals, so these functors are exact even for modules which are not finitely generated.