Quasi-coherent sheaves and vector bundles

For an A-module M, the associated sheaf on Spec A has sections M_f on D(f) and stalk M_p at a prime p. A quasi-coherent sheaf is locally of this form. On an affine scheme, global sections and the associated-sheaf construction are inverse equivalences between modules and quasi-coherent sheaves. This is stronger than saying that a sheaf can be described by generators near one point: it controls the restriction maps on all affine localizations and the homomorphisms between such sheaves.

The affine equivalence is exact because localization is exact, and tensor products of associated sheaves correspond to tensor products of modules. On a locally Noetherian scheme, coherence is equivalent to being quasi-coherent and locally finitely generated. Outside that setting, coherence has extra relation-finiteness conditions and should not be replaced blindly by finite generation. Quasi-coherence itself does not require finite rank, finite presentation, or local freeness; all three are additional properties with different uses.

A finite locally free sheaf has an open cover on which it is isomorphic to a finite direct sum of the structure sheaf. Its rank is locally constant; different connected components may have different ranks. On Spec A, such sheaves correspond exactly to finite projective A-modules. Equivalently, the module is finitely presented and flat. A finite projective module need not be globally free, so a bundle may have bases locally without one preferred basis across the entire scheme.

Changing a local basis uses an invertible matrix with regular-function entries, and the transition matrices satisfy a cocycle on triple overlaps. This is the linear version of the gluing used for schemes. One can construct a geometric total space from the symmetric algebra, with a dual depending on the chosen convention; here the term vector bundle refers to its finite locally free sheaf of sections. A quotient module with torsion does not become a bundle simply because it has a finite presentation or one-dimensional fibers at some points.

The fiber at a point is a vector space over its residue field. Forming it sets every element of the local maximal ideal to zero, so it can hide nilpotent or torsion relations that remain visible in the stalk. A stalk can have a nonzero annihilator even when its residue-field fiber looks like an ordinary one-dimensional vector space. The correct local-freeness test asks for a neighborhood with a basis, or an equivalent finite-presentation and flatness criterion, rather than a visual inspection of one fiber.

For a morphism of schemes, pulling back a quasi-coherent sheaf is locally tensoring its module with the new coordinate ring. Tensor product is right exact, but it can destroy an injection. Flatness of the base-change algebra prevents that loss for all module sequences. This phenomenon also explains why the fiber of a subsheaf need not sit inside the fiber of the ambient sheaf: a nonzero local generator can map to zero after passing to the residue field.