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:
“The windy springs and the blazing summers, one after another, had enriched and mellowed that flat tableland; all the human effort that had gone into it was coming back in long, sweeping lines of fertility. The changes seemed beautiful and harmonious to me; it was like watching the growth of a great man or of a great idea. I recognized every tree and sandbank and rugged draw. I found that I remembered the conformation of the land as one remembers the modelling of human faces.”
—Willa Cather (1873–1947)