Rigid analytic flatificators

Abstract

Let K be an algebraically closed field endowed with a complete non-archimedean norm. Let f:Y -> X be a map of K-affinoid varieties. We prove that for each point x in X, either f is flat at x, or there exists, at least locally around x, a maximal locally closed analytic subvariety Z in X containing x, such that the base change f^{-1}(Z) -> Z is flat at x, and, moreover, g^{-1}(Z) has again this property in any point of the fibre of x after base change over an arbitrary map g:X' -> X of affinoid varieties. If we take the local blowing up π:X-tilde -> X with this centre Z, then the fibre with respect to the strict transform f-tilde of f under π, of any point of X-tilde lying above x, has grown strictly smaller. Among the corollaries to these results we quote, that flatness in rigid analytic geometry is local in the source; that flatness over a reduced quasi-compact rigid analytic variety can be tested by surjective families; that an inclusion of affinoid domains is flat in a point, if it is unramified in that point.