Quasicoherence

Quasi-coherence, also written quasicoherence, is a structural property of sheaves of modules over a scheme. A sheaf F of O_X-modules on a scheme X is called quasi-coherent if for every open affine subset U = Spec A ⊆ X, the restriction F|_U is isomorphic to the sheaf associated to some A-module M. Equivalently, on Spec A one has F|_{Spec A} ≅ ˜M. This condition is local on X and stable under restriction to open subsets.

On an affine scheme X = Spec A, quasi-coherent sheaves correspond precisely to A-modules via the tilde correspondence. Consequently, the category of quasi-coherent sheaves on Spec A is equivalent to the category of A-modules. More generally, QCoh(X) is an abelian category: kernels, cokernels, and extensions of quasi-coherent sheaves are quasi-coherent, and operations such as tensor products preserve quasi-coherence under suitable hypotheses. The pullback of a quasi-coherent sheaf along any morphism of schemes is quasi-coherent, and if f: X → Y is quasi-compact and quasi-separated, the pushforward f_ of a quasi-coherent sheaf is quasi-coherent as well.

Quasi-coherent sheaves include many familiar objects, such as the structure sheaf O_X and any quasi-coherent sheaf associated to a coherent module on an open affine. The ideal sheaf of a closed subscheme is quasi-coherent, and locally free sheaves of finite rank (vector bundles) are quasi-coherent.

Coherence is a stronger condition: a coherent sheaf is quasi-coherent with additional finiteness properties. On Noetherian schemes, coherent sheaves are precisely the quasi-coherent sheaves that are locally of finite type. Not every sheaf is quasi-coherent; non-quasi-coherent examples arise on schemes as well.

The concept generalizes to algebraic spaces and stacks, where the category of quasi-coherent sheaves remains central to its structure and theory.