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Concept
Flat module
Summary
In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion free modules. Formally, a module M over a ring R is flat if 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.
Flatness was introduced by in his paper Géometrie Algébrique et Géométrie Analytique.
A module M over a ring R is flat if the following condition is satisfied: for every injective linear map of R-modules, the map
is also injective, where is the map
induced by
For this definition, it is enough to restrict the injections to the inclusions of finitely generated ideals into R.
Equivalently, an R-module M is flat if the tensor product with M is an exact functor; that is if, for every short exact sequence of R-modules the sequence is also exact. (This is an equivalent definition since the tensor product is a right exact functor.)
These definitions apply also if R is a non-commutative ring, and M is a left R-module; in this case, K, L and J must be right R-modules, and the tensor products are not R-modules in general, but only abelian groups.
Flatness can also be characterized by the following equational condition, which means that R-linear relations in M stem from linear relations in R. An R-module M is flat if and only if, for every linear relation
with and , there exist elements and such that
It is equivalent to define n elements of a module, and a linear map from to this module, which maps the standard basis of to the n elements. This allows rewriting the previous characterization in terms of homomorphisms, as follows.
An R-module M is flat if and only if the following condition holds: for every map where is a finitely generated free R-module, and for every finitely generated R-submodule of the map factors through a map g to a free R-module such that
Flatness is related to various other module properties, such as being free, projective, or torsion-free.
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