A generalization of formal schemes and rigid analytic varieties
There is a natural class of topological rings which contains both the affinoid rings (from rigid analytic geometry) and the noetherian adic rings. Namely the class of topological rings which have an open adic subring with a finitely generated ideal of definition. We call such a ring \(f\)-adic. For every \(f\)-adic ring \(A\) the topological space \(\text{Cont }A\) of all continuous valuations of \(A\) is a spectral space. If \(A\) has an open noetherian adic subring or if for every \(n \in N\) the ring \(A\langle X_ 1,\dots, X_ n\rangle\) of restricted power series in \(n\) variables over \(A\) is noetherian then there is a natural sheaf \({\mathcal O}_ A\) of topological rings on \(\text{Cont }A\). All stalks of \({\mathcal O}_ A\) are local rings. We call a topologically and locally ringed space which is locally isomorphic to some \(\text{Spa }A := (\text{Cont }A,{\mathcal O}_ A)\) adic. There are natural fully faithful functors from the category of rigid analytic varieties and the category of locally noetherian formal schemes to the category of adic spaces. In the first case one assigns to an affinoid rigid analytic variety \(\text{Sp }A\) the adic space \(\text{Spa }A\) and in the latter case one assigns to an affine noetherian formal scheme \(\text{Spf }A\) the adic space \(\text{Spa }A\).
scientific article; zbMATH DE number 737152