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 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  in his paper gÃ©ometrie algÃ©brique et gÃ©omÃ©trie analytique. see also flat morphism. ==definition== ===commutative rings=== let m be an r-module. the following conditions are all equivalent and m is flat if it satisfies any (thus all) of them: * the functor ::f_m : mod(r) \to mod(r), \quad n \mapsto m \otimes_r n :is exact, where mod(r) is the category of r-modules. * for every injective morphism \phi : k \to l of r-modules k and l, the induced map ::f_m(\phi) : m \otimes_r k \to m \otimes_r l, :is injective. * for every finitely generated ideal i \hookrightarrow r, the induced morphism i \otimes_r m \to r \otimes_r m \cong m is injective. * there exists a directed system of r-modules \{ f_\alpha \}_\alpha with the following properties: # for all \alpha, f_\alpha is a finitely generated, free r-module. # the direct limit is m: \varinjlim_\alpha f_\alpha= m. * for every linear dependency in m, ::r^t m = \sum_{i=1}^k r_i m_i = 0, where r_i \in r, m_i \in m, there exists a matrix a \in r^{k \times k} such that # an = m has a solution for some n \in m^k. # r^t a = 0. * for every r-module n, ::\mathrm{tor}_1^r (n, m) = 0 * for every finitely generated ideal i \subset r, ::\mathrm{tor}_1^r (r/i, m) = 0. * for every map f : f \to m, where f is a finitely generated free r-module, and for every finitely generated r-submodule k \leq \ker f, f factors through a map to a free r-module g that kills k: center|factor property of a flat module ===general rings=== when r isn't commutative one needs the more careful statement that, if m is a flat left r-module, the tensor product with m maps exact sequences of right r-modules to exact sequences of abelian groups. taking tensor products (over arbitrary rings) is always a [[right exact f
Algebraic_geometry Homological_algebra Module_theory