Pure Exact Sequence

In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. Pure modules are complementary to flat modules and generalize Prüfer's notion of pure subgroups. While flat modules are those modules which leave short exact sequences exact after tensoring, a pure submodule defines a short exact sequence (known as a pure exact sequence) that remains exact after tensoring with any module. Similarly a flat module is a direct limit of projective modules, and a pure exact sequence is a direct limit of split exact sequences.

Definition

Let ''R'' be a ring (associative, with 1), let ''M'' be a (left) module over ''R'', let ''P'' be a submodule of ''M'' and let ''i'': ''P'' → ''M'' be the natural injective map. Then ''P'' is a pure submodule of ''M'' if, for any (right) ''R''-module ''X'', the natural induced map id_''X'' ⊗ ''i'' : ''X'' ⊗ ''P'' → ''X'' ⊗ ''M'' (where the on tensor products are taken over ''R'') is injective.

Analogously, a short exact sequence
:$0 \longrightarrow A\,\ \stackrel\ B\,\ \stackrel\ C \longrightarrow 0$
of (left) ''R''-modules is pure exact if the sequence stays exact when tensored with any (right) ''R''-module ''X''. This is equivalent to saying that ''f''(''A'') is a pure submodule of ''B''.

Equivalent characterizations

Purity of a submodule can also be expressed element-wise; it is really a statement about the solvability of certain systems of linear equations. Specifically, ''P'' is pure in ''M'' if and only if the following condition holds: for any ''m''-by-''n'' matrix (''a''_''ij'') with entries in ''R'', and any set ''y''₁, ..., ''y''_''m'' of elements of ''P'', if there exist elements ''x''₁, ..., ''x''_''n'' in ''M'' such that
:$\sum_^n a_x_j = y_i \qquad\mbox i=1,\ldots,m$
then there also exist elements ''x''₁′, ..., ''x''_''n''′ in ''P'' such that
:$\sum_^n a_x'_j = y_i \qquad\mbox i=1,\ldots,m$

Another characterization is: a sequence is pure exact if and only if it is the filtered colimit (also known as direct limit) of split exact sequences

:$0 \longrightarrow A_i \longrightarrow B_i \longrightarrow C_i \longrightarrow 0.$For abelian groups, this is proved in

Examples

* Every direct summand of ''M'' is pure in ''M''. Consequently, every subspace of a vector space over a field is pure.

Properties

Suppose
:$0 \longrightarrow A\,\ \stackrel\ B\,\ \stackrel\ C \longrightarrow 0$
is a short exact sequence of ''R''-modules, then:
# ''C'' is a flat module if and only if the exact sequence is pure exact for every ''A'' and ''B''. From this we can deduce that over a von Neumann regular ring, ''every'' submodule of ''every'' ''R''-module is pure. This is because ''every'' module over a von Neumann regular ring is flat. The converse is also true.
# Suppose ''B'' is flat. Then the sequence is pure exact if and only if ''C'' is flat. From this one can deduce that pure submodules of flat modules are flat.
# Suppose ''C'' is flat. Then ''B'' is flat if and only if ''A'' is flat.

If $0 \longrightarrow A\,\ \stackrel\ B\,\ \stackrel\ C \longrightarrow 0$ is pure-exact, and ''F'' is a finitely presented ''R''-module, then every homomorphism from ''F'' to ''C'' can be lifted to ''B'', i.e. to every ''u'' : ''F'' → ''C'' there exists ''v'' : ''F'' → ''B'' such that ''gv''=''u''.

References

*

*{{Citation , last1=Lam , first1=Tsit-Yuen , title=Lectures on modules and rings , publisher=Springer-Verlag , location=Berlin, New York , series=Graduate Texts in Mathematics No. 189 , isbn=978-0-387-98428-5 , mr=1653294 , year=1999
Module theory