Rigid Analytic Flatificators

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 ∈ X, either f is flat at x, or there exists, at least locally around x, a maximal locally closed analytic subvariety Z ⊂ 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 : X̃ → X with this centre Z, then the fibre with respect to the strict transform f̃ of f under π, of any point of X̃ 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.