The Zariski spectrum as a formal geometry

We choose formal topology to deal in a basic manner with the Zariski spectra of commutative rings and their structure sheaves. By casting prime and maximal ideals in a secondary role, we thus wish to prepare a constructive and predicative framework for abstract algebraic geometry.In contrast to the classical approach, neither points nor stalks need occur, let alone any instance of the axiom of choice. As compared with the topos-theoretic treatments that may be rendered predicative as well, the road we follow is built from more elementary material.The formal counterpart of the structure sheaf which we present first is our guiding example for a notion of a sheaf on a formal topology. We next define the category of formal geometries, a natural abstraction from that of locally ringed spaces. This allows us to eventually phrase and prove, still within the language of opens and sections, the universal property of the Zariski spectrum. Our version appears to be the only one that is explicitly point-free.