On rank Theorems for morphisms of local rings

Abstract

We prove a generalization of Gabrielov's rank theorem for families of rings of power series which we call W-temperate. Examples include the families of complex analytic functions and of Eisenstein series. As a Corollary, we provide rank Theorems for convergent series in general characteristic zero complete valued fields (not necessarily algebraically closed, nor archimedean).

Two important changes from previous version: 1) The previous paper had two sets of distinct results. We have divided the paper in two, and this version contains the first set of results. The second part will appear in a new arXiv submission. The title has changed. 2) Corollary 1.2 has been added