that restrict to the same polynomial on , which can be shown to be , the global section of . is not affine as in . Non-Zariski topologies on a prime spectrum Some authors (notably M. Hochster) consider topologies on prime spectra other than Zariski topology. First, there is the notion of constructible topology: given a ring A, the subsets of of the form satisfy the axioms for closed sets in a topological space. This topology on is called the constructible topology. In , Hochster considers what he calls the patch topology on a prime spectrum. By definition, the patch topology is the smallest topology in which the sets of the forms and are closed. Global or relative Spec There is a relative version of the functor called global , or relative . If is a scheme, then relative is denoted by or . If is clear from the context, then relative Spec may be denoted by or . For a scheme and a quasi-coherent sheaf of -algebras , there is a scheme and a morphism such that for every open affine , there is an isomorphism , and such that for open affines , the inclusion is induced by the restriction map . That is, as ring homomorphisms induce opposite maps of spectra, the restriction maps of a sheaf of algebras induce the inclusion maps of the spectra that make up the Spec of the sheaf. Global Spec has a universal property similar to the universal property for ordinary Spec. More precisely, just as Spec and the global section functor are contravariant right adjoints between the category of commutative rings and schemes, global Spec and the direct image functor for the structure map are contravariant right adjoints between the category of commutative -algebras and schemes over . In formulas, where is a morphism of schemes. Example of a relative Spec The relative spec is the correct tool for parameterizing the family of lines through the origin of over Consider the sheaf of algebras and let be a sheaf of ideals of Then the relative spec parameterizes the desired family. In fact, the fiber over is the line through the origin of containing the point Assuming the fiber can be computed by looking at the composition of pullback diagrams where the composition of the bottom arrows gives the line containing the point and the origin. This example can be generalized