Schemes as functors on topological rings

Abstract

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 ad\`ele 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 $R^\times$ 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.