On rank Theorems for morphisms of local rings

TL;DR

This paper generalizes Gabrielov's rank theorem to W-temperate families of power series, including complex analytic functions and Eisenstein series, and extends rank theorems to convergent series over characteristic zero valued fields.

Contribution

It introduces the concept of W-temperate families and proves a generalized rank theorem applicable to broader classes of power series and valued fields.

Findings

Generalization of Gabrielov's rank theorem to W-temperate families

Rank theorems for convergent series in characteristic zero fields

Examples include complex analytic functions and Eisenstein series

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).