A new proof of the local criterion of flatness

TL;DR

This paper presents a novel proof of the local criterion of flatness for modules over local noetherian rings, utilizing completions and elementary properties, offering an alternative to traditional methods.

Contribution

It introduces a new proof approach for the local criterion of flatness, distinct from standard textbook proofs, based on completions and elementary properties.

Findings

Provides a new proof method for the local criterion of flatness.

Uses completions and elementary properties of flat modules and Tor-functor.

Simplifies the understanding of flatness criteria in local noetherian rings.

Abstract

Let (A,m_A) -> (B,m_B) be a local morphism of local noetherian rings and M a finitely generated B-module. Then it follows from Tor^A_1(M,A/m_A) = 0 that M is a flat A-module. This is usually called the "local criterion of flatness". We give a proof that proceeds along different lines than the usual textbook proofs, using completions and only elementary properties of flat modules and the Tor-functor.