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

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Free modules[edit | edit source]

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.


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).


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 | edit source]

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.


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.


: 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


: 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 .

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


Theorem 11.9:

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


And surjectivity:

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 .

We set


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.

Theorem 11.10:

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.

Theorem 11.11:

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.

Flat modules[edit | edit source]

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.


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

Theorem 11.16:

direct sum flat iff all summands are

Theorem 11.17:

If are flat -modules, then is as well.



be an exact sequence of modules.

Torsion-free modules[edit | edit source]

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.


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 .