Flat module

In abstract algebra, a flat module over a ring (mathematics) R is an R -module (mathematics) M such that taking the tensor product over R with M preserves exact sequences.

Vector spaces over a field (mathematics) are flat modules. Free modules, or more generally projective modules, are also flat, over any R . Over a Noetherian ring 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.

In 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. When 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 module homomorphism K → L of R -modules, the induced homomorphism K ⊗ M → L ⊗ M is also injective.

In general, arbitrary of M .

Lazard proved in 1969 that a module M is flat if and only if it is a direct limit of finitely generated module free modules. As a consequence, one can deduce that every finitely-presented module flat module is projective.

Flatness may also be expressed using the Tor functors, the derived functor of the tensor product. A left R -module M is flat if and only if Tor1 R (–, M ) = 0 (i.e., iff Tor1 R ( X , M ) = 0 for all right R -modules X ). Similarly, a right R -module M is flat if and only if Tor1 R ( M ,–) = 0. Using the Tor functor s long exact sequences, one can then easily prove facts about a short exact sequence :

If A and C are flat, then so is B

If B and C are flat, then so is A

If A and B are flat, C need not be flat in general. However, it can be shown that

If A is pure submodule in B and B is flat, then A and C are flat.

= See also =

*localization of a module *flat morphism *von Neumann regular ring: those rings over which all modules are flat.