A spectral site is an algebraic site C with an affine basis B which is an affine spectral site. We show that ASch and GASch are affine spectral sites. Thus Spec A is sober. A is injective. Sch and GSch are spectral sites.
X be a quasi-compact morphism in a spectral site.
An object X in a spectral site C is called a locality if X has exactly one closed point T X and any other point is a generalization of T X . For any point x of an object X , the effective subobject x o of X is a locality. Note that a locality must be affine because X is covered by affine open subsets and the only open neighborhood of T X is X .