ABSTRACT
It is well known that a spectrum is not sufficient to recover a ring. To clarify the situation, we are aiming to show that some subsets of a spectrum are solutions to universal problems. Namely, if X ⊂ Spec(A) where A is a ring, then X is called geometric if there is a ring morphism A → Δ with spectral image contained in X and universal for this property. The similar problem has always a solution in the category of locally ringed spaces. We examine when points are geometric. We show that when A is locally Noetherian or an almost multiplication ring, then a prime ideal P is geometric if and only if P is a minimal prime ideal. Geometric points are characterized for Prüfer domains. An arbitrary geometric subset is stable under formal generizations and a solution is an epimorphism focussing on X. The geometric property is local on the spectrum, universal and is descended by algebraically pure morphisms. Moreover, local morphisms induced by completions are isomorphisms. We give a complete characterization of geometric subsets which are either closed or quasi-compact and stable under generizations. The akin problem of quasi-geometric subsets is examined.
