Topological considerations on the order of a spectral space

Let be a spectral space, that is, a topological space that is homeomorphic to the prime spectrum of a commutative ring with multiplicative identity.  As it is well known, the given topology of can be refined to another topology, called the constructible (or patch) topology, which remains spectral and becomes Hausdorff. Since any spectral space is a space, a natural partial order on , usually called the specialization order, can be defined by setting, for every , if . In this talk we will provide conditions for a subset of the partially ordered set in order that the supremum of (in ) exists and belongs to the closure of , with respect to the constructible topology. Algebraic applications of such topological results will concern density properties of some spaces of rings and ideals. Moreover, we will see how distinguished classes of domains can be characterized topologically, in terms of certain properties of their ideals. Joint paper with D. Spirito.