A short proof of smooth implies flat

TL;DR

This paper presents a concise proof demonstrating that smooth morphisms are necessarily flat, simplifying the understanding of this important concept in algebraic geometry.

Contribution

The authors provide a shorter, more accessible proof that smooth morphisms imply flatness, improving upon existing lengthy proofs.

Findings

Short proof of smooth implies flat

Simplifies understanding of algebraic geometry concepts

Potentially easier to teach and apply

Abstract

Proofs that a smooth morphism is flat available in the literature are long and difficult. We give a short proof of this fact.