Embedded resolution of singularities in rigid analytic geometry
The field \(K\) is algebraically closed, complete w.r.t. a non-Archimedean valuation and has characteristic 0. The main result is resolution of singularities for rigid spaces over \(K\) (locally w.r.t. the Grothendieck topology). The proof is based on: Hironaka's proof for excellent schemes in characteristic 0, a local form of embedded resolution and ``analytization''. Some applications of resolution for rigid analytic spaces are already present. The local form of embedded resolution reads: Let \(M=\text{ Sp}(A)\) be a smooth connected affinoid space over \(K\) and let \(f\in A\), \(f\neq 0\) define a subspace \(V(f)\subset M\) of codimension 1. Then after replacing \(M\) by the elements of a finite affinoid covering, there exists a morphism \(h:\widetilde{M}\rightarrow M\) which is a composition of blowing up maps w.r.t. regular centers of codimension at least two, such that \(h^{-1}V(f)\) has normal crossings in \(\widetilde{M}\). ``Analytization'' means that a scheme \(X\) of finite type over, say, an affinoid \(K\)-algebra, is given a suitable structure of a rigid space over \(K\). It seems good to mention that the definition in the paper requires a choice of Grothendieck topologies for rigid spaces which is ``finest'' possible. Analytization (already studied by U.~Köpf) is needed because the blow up of an affinoid space has to be made into a rigid space. It might be worthwhile to extend this paper and prove a global form of resolution for, say, quasi-compact rigid spaces.
scientific article; zbMATH DE number 1491226
scientific article