In Homological algebra, and algebraic geometry, a flat module over a ring R is an R-module M such that 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.
Vector spaces over a field are flat modules. Free modules, or more generally projective modules, are also flat, over any R. For finitely generated modules over a Noetherian ring, flatness and projectivity are all equivalent. For finitely generated modules over local rings, flatness, projectivity and freeness are all equivalent. The field of quotients of an integral domain, and, more generally, any localization of a commutative ring are flat modules. The product of the local rings of a commutative ring is a faithfully flat module.
Flatness was introduced by Serre (1956) in his paper Géometrie Algébrique et Géométrie Analytique. See also flat morphism.
Famous quotes containing the word flat:
“We say justly that the weak person is flat, for, like all flat substances, he does not stand in the direction of his strength, that is, on his edge, but affords a convenient surface to put upon. He slides all the way through life.... But the brave man is a perfect sphere, which cannot fall on its flat side and is equally strong every way.”
—Henry David Thoreau (18171862)