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.

Read more about Flat Module: Case of Commutative Rings, Categorical Colimits, Homological Algebra, Flat Resolutions, In Constructive Mathematics

### 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 (1817–1862)