 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.