= Open affine representability criterion
A <Zariski sheaf> $F$ is a <functor represented by a scheme> if it has a set of <open subfunctors> represented by <affine schemes> whose pullbacks cover every test <scheme>. The overlaps are <open subschemes> in both charts, and their identifications satisfy the compatibility on triple overlaps. <Gluing of schemes along open subschemes> produces a representing <scheme>; the <Zariski sheaf> condition identifies its <functor of points> with $F$. Conversely, every <affine open cover> of a <scheme> supplies such a presentation. A family of local charts covers in this sense without requiring every global element to belong to a single chart.
Back to article page
