Flat module
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In abstract algebra, 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. Vector spaces over a field are flat modules. Free modules, or more generally projective modules, are also flat, over any R. Over a Noetherian local ring, flatness, projectivity, and freeness are all equivalent. In commutative algebra, and more generally in algebraic geometry, flatness has come to play a major role since Serre's paper Géometrie Algébrique et Géométrie Analytique. The geometric reasons are not superficial, though. See also flat morphism.
Case of commutative ringsIn the case when R is a commutative ring, one can say that flatness for an R-module M is equivalent to tensor product with M being an exact functor from the category of R-modules to itself. For any multiplicatively closed subset S of R, the localization ring Failed to parse (Missing texvc executable; please see math/README to configure.): S^{-1}R is flat as an R-module. When R is Noetherian and M is a finitely-generated R-module, being flat is the same as being locally free in the following sense: M is a flat R-module if and only if for every prime ideal (or even just for every maximal ideal) P of R, the localization Failed to parse (Missing texvc executable; please see math/README to configure.): M_P is free as a module over the localization Failed to parse (Missing texvc executable; please see math/README to configure.): R_P . General ringsWhen R isn't commutative one needs the more careful statement, that (if M is a 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 functor. Therefore, the R-module M is flat if and only if for any injective homomorphism K → L of R-modules, the induced homomorphism KFailed to parse (Missing texvc executable; please see math/README to configure.): \otimes M → LFailed to parse (Missing texvc executable; please see math/README to configure.): \otimes M is also injective. Categorical limitsIn general, arbitrary direct sums and direct limits of flat modules are flat, a consequence of the fact that the tensor product commutes with direct sums and direct limits, and that both direct sums and direct limits are exact functors. Submodules and factor modules of flat modules need not be flat in general. However we have the following result: the homomorphic image of a flat module M is flat if and only if the kernel is a pure submodule of M. Lazard proved in 1969 that a module M is flat if and only if it is a direct limit of finitely-generated free modules. As a consequence, one can deduce that every finitely-presented flat module is projective. An abelian group is flat (viewed as a Z-module) if and only if it has no torsion elements. Homological algebraFlatness may also be expressed using the Tor functors, the left derived functors of the tensor product. A left R-module M is flat if and only if TornR(–, M) = 0 for all Failed to parse (Missing texvc executable; please see math/README to configure.): n \ge 1 (i.e., if and only if TornR(X, M) = 0 for all Failed to parse (Missing texvc executable; please see math/README to configure.): n \ge 1 and all right R-modules X). Similarly, a right R-module M is flat if and only if TornR(M, X) = 0 for all Failed to parse (Missing texvc executable; please see math/README to configure.): n \ge 1 and all left R-modules X. Using the Tor functor's long exact sequences, one can then easily prove facts about a short exact sequence
If A and B are flat, C need not be flat in general. However, it can be shown that
Flat resolutionsA flat resolution of a module is a resolution by flat modules. Any projective resolution is therefore a flat resolution. In constructive mathematicsFlat modules have increased importance in constructive mathematics, where projective modules are less useful. For example, that all free modules are projective is equivalent to the full axiom of choice, so theorems about projective modules, even if proved constructively, do not necessarily apply to free modules. In contrast, no choice is needed to prove that free modules are flat, so theorems about flat modules can still apply. See Fred Richman, Flat dimension, constructivity, and the Hilbert syzygy theorem, New Zealand Journal of Mathematics 26 (1997), 263–273. References
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