 Global-section localization for quasi-coherent sheaves

Choose a finite cover of an affine scheme by principal opens on which the quasi-coherent sheaf is associated with a module. The sheaf gluing axiom expresses global sections as the kernel of the difference map from the finite product of chart sections to the finite product of overlap sections. Localization commutes with kernels and finite products. Localizing at $a$ therefore gives the same kernel for the cover intersected with $D(a)$, proving the formula. The principal-open formulas identify the sheaf with that associated with its global-section module; no finite-generation hypothesis on the module is needed.