A new proof of the local criterion of flatness

Details A new proof of the local criterion of flatness A new proof of the local criterion of flatness

2010-03-21

arxiv.org/abs/1003.4009v1

Mathematics

Commutative Algebra

Scientific paper

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.

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