# A transformation rule for natural multiplicities

## TL;DR

This paper develops a general transformation rule for natural multiplicities in ring extensions that are étale in codimension one, enabling bounds on local fundamental groups using invariants like F-signature and differential signature.

## Contribution

It introduces a new transformation rule for multiplicities in étale-in-codimension-one extensions, linking algebraic invariants to topological properties of singularities.

## Key findings

- Bounds the local étale fundamental group using F-signature.
- Provides characteristic-free effective bounds via differential signature.
- Recovers recent results by Carvajal-Rojas, Schwede, and Tucker.

## Abstract

For multiplicities arising from a family of ideals we provide a general approach to transformation rules for a ring extension \'etale in codimension one. Our result can be applied to bound the size of the local \'etale fundamental group of a singularity in terms of F-signature, recovering a recent result of Carvajal-Rojas, Schwede, and Tucker, and differential signature, providing the first characteristic-free effective bound.