In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e., f P : O Y , f ( P ) → O X...

In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion free modules. Formally, a module M over a ring R is flat if taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Flatness was introduced by Jean-Pierre Serre (1956) in his paper Géometrie Algébrique et Géométrie Analytique. == Definition == A left module M over a ring R is flat if the following condition is satisfied: for every...