Flat module

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

φ

:

K

→

L

{\displaystyle \varphi :K\to L}

of right R-modules, the map

φ

⊗

R

M

:

K

⊗

R

M

→

L

⊗

R

M

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

is also injective, where

φ

⊗

R

M

{\displaystyle \varphi \otimes _{R}M}

is the map

induced by

k

⊗

m

↦

φ

(

k

)

⊗

m

.

{\displaystyle k\otimes m\mapsto \varphi (k)\otimes m.}

For this definition, it is enough to restrict the injections

φ

{\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

0

→

K

→

L

→

J

→

0

,

{\displaystyle 0\rightarrow K\rightarrow L\rightarrow J\rightarrow 0,}

the sequence

0

→

K

⊗

R

M

→

L

⊗

R

M

→

J

⊗

R

M

→

0

{\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

∑

i

=

1

m

r

i

x

i

=

0

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

with

r

i

∈

R

{\displaystyle r_{i}\in R}

and

x

i

∈

M

{\displaystyle x_{i}\in M}

, there exist elements

y

j

∈

M

{\displaystyle y_{j}\in M}

and

a

i

,

j

∈

R

,

{\displaystyle a_{i,j}\in R,}

such that

∑

i

=

1

m

r

i

a

i

,

j

=

0

{\textstyle \sum _{i=1}^{m}r_{i}a_{i,j}=0\qquad }

for

j

=

1

,

…

,

n

,

{\displaystyle j=1,\ldots ,n,}

and

x

i

=

∑

j

=

1

n

a

i

,

j

y

j

{\textstyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j}\qquad }

for

i

=

1

,

…

,

m

.

{\displaystyle i=1,\ldots ,m.}

It is equivalent to define n elements of a module, and a linear map from

R

n

{\displaystyle R^{n}}

to this module, which maps the standard basis of

R

n

{\displaystyle R^{n}}

to the n elements. This allows rewriting the previous characterization in terms of homomorphisms, as follows.