Schemes as functors on topological rings

الأصل · EN

Let X be a scheme. In this text, we extend the known definitions of a topology on the set X(R) of R-rational points from topological fields, local rings and adèle rings to any ring R with a topology. This definition is functorial in both X and R, and it does not rely on any restriction on X like separability or finiteness conditions. We characterize properties of R, such as being a topological Hausdorff ring, a local ring or having R× as an open subset for which inversion is continuous, in terms of functorial properties of the topology of X(R). Particular instances of this general approach yield a new characterization of adelic topologies, and a definition of topologies for higher local fields.