The following definitions are straightforward generalisations from linear algebra. We begin by repeating a definition we already saw in chapter 6.
Definition 6.1 (generators of modules):
Let be a module over the ring . A generating set of is a subset such that
We also have:
Let be an -module. A subset of is called linearly independent if and only if, whenever , we have
A free -module is a module over where there exists a basis, that is, a subset of that is a linearly independent generating set.
Let be free modules. Then the direct sum
Let bases of the be given. We claim that
is a basis of
Indeed, let an arbitrary element be given. Then by assumption, each of the has a decomposition
for suitable . By summing this, we get a decomposition of in the aforementioned basis. Furthermore, this decomposition must be unique, for otherwise projecting gives a new composition of one of the particular .
The converse is not true in general!
Let be free -modules, with bases and respectively. Then
is a free module, with basis
where we wrote for short
(note that it is quite customary to use this notation).
We first prove that our supposed basis forms a generating system. Clearly, by summation it suffices to show that elements of the form
can be written in terms of the . Thus, write
- and ,
and obtain by the rules of computing within the tensor product, that
On the other hand, if
is a linear combination (i.e. all but finitely many summands are zero), then all the must be zero. The argument is this: Fix and define a bilinear function
where , are the coefficients of , in the decomposition of and respectively. According to the universal property of the tensor product, we obtain a linear map
- with ,
where is the canonical projection on the quotient space. We have the equations
and inserting the given linear combination into this map therefore yields the desired result.
The following is a generalisation of free modules:
Let be an -module. is called projective if and only if for a fixed module and a fixed surjection every other module morphism with codomain (call ) has a factorisation
Every free module is projective.
Pick a basis of , let be surjective and let be some morphism. For each pick with . Define
- where .
This is well-defined since the linear combination describing is unique. Furthermore, it is linear, since we have
where the right hand side is the sum of the linear combinations coinciding with and respectively, which is why . By linearity of and definition of the , it has the desired property.
There are a couple equivalent definitions of projective modules.
A module is projective if and only if there exists a module such that is free.
: Define the module
(this obviously is a free module) and the function
is a surjective morphism, whence we obtain a commutative diagram
that is, .
We claim that the map
is an isomorphism. Indeed, if , then and thus also (injectivity) and further , where , which is why
: Assume is a free module. Assume is a surjective morphism, and let be any morphism. We extend to via
This is still linear as the composition of the linear map and the linear inclusion . Now is projective since it's free. Hence, we get a commutative diagram
where satisfies . Projecting to gives the desired diagram for .
An exact sequence of modules
is called split exact iff we can augment it by three isomorphisms such that
A module is projective iff every exact sequence
is split exact.
: The morphism is surjective, and thus every other morphism with codomain lifts to . In particular, so does the projection . Thus, we obtain a commutative diagram
where we don't know yet whether is an isomorphism, but we can use to define the function
which is an isomorphism due to injectivity:
Let , that is . Then first
and therefore second
Let . Set . Then
and hence for a suitable , thus
We thus obtain the commutative diagram
and have proven what we wanted.
: We prove that is free for a suitable .
where is defined as in the proof of theorem 11.7 . We obtain an exact sequence
which by assumption splits as
which is why is isomorphic to the free module and hence itself free.
Let and be projective -modules. Then is projective.
We choose -modules such that and are free. Since the tensor product of free modules is free, is free. But
and thus occurs as the summand of a free module and is thus projective.
Let be -modules. Then is projective if and only if each is projective.
Let first each of the be projective. Then each of the occurs as the direct summand of a free module, and summing all these free modules proves that is the direct summand of free modules.
On the other hand, if is the summand of a free module, then so are all the s.
The following is a generalisation of projective modules:
An -module is called flat if and only if tensoring by it preserves exactness:
- exact implies exact.
The morphisms in the right sequence induced by any morphism are given by the bilinear map
The module is a flat -module.
Proof: This follows from theorems 9.10 and 10.?.
Flatness is a local property.
Proof: Exactness is a local property. Furthermore, for any multiplicatively closed
by theorem 9.11. Since every -module is the localisation of an -module (for instance itself as an -module via ), the theorem follows.
A projective module is flat.
We first prove that every free module is flat. This will enable us to prove that every projective module is flat.
Indeed, if is a free module and a basis of , we have
where all but finitely many of the summands on the left are nonzero. Hence, by distributivity of direct sum over tensor product, if we are given any exact sequence
to show that the sequence
is exact, all we have to do is to prove that
is exact, since we may then augment the latter sequence by suitable isomorphisms
direct sum flat iff all summands are
If are flat -modules, then is as well.
be an exact sequence of modules.
The following is a generalisation of flat modules:
Let be an -module. The torsion of is defined to be the set
The torsion of a module is a submodule of that module.
Let , . Obviously (just multiply the two annihilating elements together), and further if (we used commutativity here).
We may now define torsion-free modules. They are exactly what you think they are.
Let be a module. is called torsion-free if and only if
A flat module is torsion-free.
To get a feeling for the theory, we define -torsion for a multiplicatively closed subset .
Let be a multiplicatively closed subset of a ring , and let be an -module. Then the -torsion of is defined to be
Let be a multiplicatively closed subset of a ring , and let be an -module. Then the -torsion of is precisely the kernel of the canonical map .