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 Jean-Pierre Serre (1956) in his paper Géometrie Algébrique et Géométrie Analytique.

Definition

A left module M over a ring R is flat if the following condition is satisfied: for every injective linear map ${\displaystyle \varphi :K\to L}$ of right R-modules, the map

${\displaystyle \varphi \otimes _{R}M:K\otimes _{R}M\to L\otimes _{R}M}$

is also injective, where ${\displaystyle \varphi \otimes _{R}M}$ is the map induced by ${\displaystyle k\otimes m\mapsto \varphi (k)\otimes m.}$

For this definition, it is enough to restrict the injections ${\displaystyle \varphi }$ 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 ${\displaystyle 0\rightarrow K\rightarrow L\rightarrow J\rightarrow 0,}$ the sequence ${\displaystyle 0\rightarrow K\otimes _{R}M\rightarrow L\otimes _{R}M\rightarrow J\otimes _{R}M\rightarrow 0}$ 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.

Characterizations

Flatness can also be characterized by the following equational condition, which means that R-linear relations in M stem from linear relations in R.

A left R-module M is flat if and only if, for every linear relation

${\textstyle \sum _{i=1}^{m}r_{i}x_{i}=0}$

with ${\displaystyle r_{i}\in R}$ and ${\displaystyle x_{i}\in M}$, there exist elements ${\displaystyle y_{j}\in M}$ and ${\displaystyle a_{i,j}\in R,}$ such that^[1]

${\textstyle \sum _{i=1}^{m}r_{i}a_{i,j}=0\qquad }$ for ${\displaystyle j=1,\ldots ,n,}$

and

${\textstyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j}\qquad }$ for ${\displaystyle i=1,\ldots ,m.}$

It is equivalent to define n elements of a module, and a linear map from ${\displaystyle R^{n}}$ to this module, which maps the standard basis of ${\displaystyle R^{n}}$ 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 ${\displaystyle f:F\to M,}$ where ${\displaystyle F}$ is a finitely generated free R-module, and for every finitely generated R-submodule ${\displaystyle K}$ of ${\displaystyle \ker f,}$ the map ${\displaystyle f}$ factors through a map g to a free R-module ${\displaystyle G}$ such that ${\displaystyle g(K)=0:}$

Relations to other module properties

Flatness is related to various other module properties, such as being free, projective, or torsion-free. In particular, every flat module is torsion-free, every projective module is flat, and every free module is projective.

There are finitely generated modules that are flat and not projective. However, finitely generated flat modules are all projective over the rings that are most commonly considered. Moreover, a finitely generated module is flat if and only it is locally free, meaning all the localizations at prime ideals are free modules.

This is partly summarized in the following graphic.

Torsion-free modules

Every flat module is torsion-free. This results from the above characterization in terms of relations by taking m = 1.

The converse holds over the integers, and more generally over principal ideal domains and Dedekind rings.

An integral domain over which every torsion-free module is flat is called a Prüfer domain.

Free and projective modules

A module M is projective if and only if there is a free module G and two linear maps ${\displaystyle i:M\to G}$ and ${\displaystyle p:G\to M}$ such that ${\displaystyle p\circ i=\mathrm {id} _{M}.}$ In particular, every free module is projective (take ${\displaystyle G=M}$ and ${\displaystyle i=p=\mathrm {id} _{M}}$).

Every projective module is flat. This can be proven from the above characterizations of flatness and projectivity in terms of linear maps by taking ${\displaystyle g=i\circ f}$ and ${\displaystyle h=}$Zdroj:https://en.wikipedia.org?pojem=Flat_module
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