# The flatness of the $\Ok$-module of smooth functions and integral representation

## TL;DR

This paper proves the flatness of the module of smooth functions using residue theory and integral formulas, extending to lower regularity functions and establishing a Briançon-Skoda type theorem for specific ideals.

## Contribution

It provides a novel proof of flatness via residue theory and integral formulas, and extends results to functions of lower regularity and certain ideal classes.

## Key findings

- Proof of flatness of the smooth functions module using residue theory
- Extension of flatness results to lower regularity functions
- Establishment of a Briançon-Skoda type theorem for specific ideals

## Abstract

We give a proof of the well-known fact that the $\Ok$-module $\E$ of smooth functions is flat by means of residue theory and integral formulas. A variant of the proof gives a related statement for classes of functions of lower regularity. We also prove a Brian\c{c}on-Skoda type theorem for ideals of the form $\E a$, where $a$ is an ideal in $\Ok$.