2010/03/21 by Jürgen Böhm, Böhm, Jürgen
Mathematics · #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1003.4009
Let (A,mA) -> (B,mB) be a local morphism of local noetherian rings and M a finitely generated B-module. Then it follows from TorA1(M,A/mA) = 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.