Commutative Algebra/Torsion-free, flat, projective and free modules

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Free modules[edit]

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:

Definition 11.1:

Let be an -module. A subset of is called linearly independent if and only if, whenever , we have

.

Definition 11.2:

A free -module is a module over where there exists a basis, that is, a subset of that is a linearly independent generating set.

Theorem 11.3:

Let be free modules. Then the direct sum

is free.

Proof:

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 .

Knuth's dangerous bend symbol.svg The converse is not true in general!

Theorem 11.4:

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).

Proof:

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.

Projective modules[edit]

The following is a generalisation of free modules:

Definition 11.5:

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

Projective module.svg.

Theorem 11.6:

Every free module is projective.

Proof:

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.

Theorem 11.7:

A module is projective if and only if there exists a module such that is free.

Proof:

: Define the module

(this obviously is a free module) and the function

.

is a surjective morphism, whence we obtain a commutative diagram

Diagram used in the proof of the free summand characterisation of projective modules.svg;

that is, .

We claim that the map

is an isomorphism. Indeed, if , then and thus also (injectivity) and further , where , which is why

(surjectivity).

: 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

Projective-proof2.svg

where satisfies . Projecting to gives the desired diagram for .

Definition 11.8:

An exact sequence of modules

is called split exact iff we can augment it by three isomorphisms such that

Split exact sequence.svg

commutes.

Theorem 11.9:

A module is projective iff every exact sequence

is split exact.

Proof:

: 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

Projective-proof3.svg

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

.

And surjectivity:

Let . Set . Then

and hence for a suitable , thus

.

We thus obtain the commutative diagram

Projective-proof4.svg

and have proven what we wanted.

: We prove that is free for a suitable .

We set

,

where is defined as in the proof of theorem 11.7 . We obtain an exact sequence

which by assumption splits as

Projective-proof5.svg

which is why is isomorphic to the free module and hence itself free.

Theorem 11.10:

Let and be projective -modules. Then is projective.

Proof:

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.

Theorem 11.11:

Let be -modules. Then is projective if and only if each is projective.

Proof:

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.

Flat modules[edit]

The following is a generalisation of projective modules:

Definition 11.12:

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

.

Theorem 11.13:

The module is a flat -module.

Proof: This follows from theorems 9.10 and 10.?.

Theorem 11.14:

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.

Theorem 11.15:

A projective module is flat.

Proof:

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

via

,

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

Theorem 11.16:

direct sum flat iff all summands are

Theorem 11.17:

If are flat -modules, then is as well.

Proof:

Let

be an exact sequence of modules.

Torsion-free modules[edit]

The following is a generalisation of flat modules:

Definition 11.18:

Let be an -module. The torsion of is defined to be the set

.

Lemma 11.19:

The torsion of a module is a submodule of that module.

Proof:

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.

Definition 11.20:

Let be a module. is called torsion-free if and only if

.

Theorem 11.21:

A flat module is torsion-free.

To get a feeling for the theory, we define -torsion for a multiplicatively closed subset .

Definition 11.22:

Let be a multiplicatively closed subset of a ring , and let be an -module. Then the -torsion of is defined to be

.

Theorem 11.23:

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 .