More on Chain Complexes

In Homology and Cohomology we used the concept of chain complexes to investigate topological spaces. In Exact Sequences we saw examples of chain complexes generalized to abelian groups other than that made out of topological spaces. In this post we study chain complexes in the context of linear algebra (see Vector Spaces, Modules, and Linear Algebra).

We start with some definitions regarding modules. In More on Vector Spaces and Modules we gave the definition of a basis of a vector space. It is known that any vector space can always have a basis. However, the same is not true for modules. It is only a certain special kind of module called a free module which has the property that one can always find a basis for it.

Alternatively, a free module over a ring $R$ may be thought of as being a module that is isomorphic to a direct sum of several copies of the ring $R$ .

An example of a module that is not free is the module $\mathbb{Z}/2\mathbb{Z}$ over the ring $\mathbb{Z}$ . It is a module over $\mathbb{Z}$ since it is closed under addition and under multiplication by any element of $\mathbb{Z}$ , however a basis that will allow it to be written as a unique linear combination of elements of the basis cannot be found, nor is it a direct sum of copies of $\mathbb{Z}$ .

Although not all modules are free, it is actually a theorem that any module is a quotient of a free module. Let $A$ be a module over a ring $R$ . The theorem says that this module is the quotient of some free module, which we denote by $F_{0}$ , by some other module which we denote by $K_{1}$ . In other words,

$A=F_{0}/K_{1}$

We can write this as the following chain complex, which also happens to be an exact sequence (see Exact Sequences):

$0\rightarrow K_{1}\xrightarrow{i_{1}} F_{0}\xrightarrow{\epsilon} A\rightarrow 0$

We know that the module $F$ is free. However, we do not know if the same holds true for $K_{1}$ . Regardless, the theorem says that any module is a quotient of a free module. Therefore we can write

$0\rightarrow K_{2}\xrightarrow{i_{2}} F_{1}\xrightarrow{\epsilon_{1}} K_{1}\rightarrow 0$

We can therefore put these chain complexes together to get

$0\rightarrow K_{2}\xrightarrow{i_{2}} F_{1}\xrightarrow{\epsilon_{1}} K_{1}\xrightarrow{i_{1}} F_{0}\xrightarrow{\epsilon} A\rightarrow 0$

However, this sequence of modules and morphisms is not a chain complex since the image of $\epsilon_{1}$ is not contained in the kernel of $i_{1}$ . But if we compose these two maps together, we obtain

$0\rightarrow K_{2}\xrightarrow{i_{2}} F_{1}\xrightarrow{d_{1} }F_{0}\xrightarrow{\epsilon} A\rightarrow 0$

where $d_{1}=i_{1}\circ \epsilon_{1}$ . This is a chain complex as one may check. We can keep repeating the process indefinitely to obtain

$...\xrightarrow{d_{3}} F_{2}\xrightarrow{d_{2} } F_{1}\xrightarrow{d_{1} } F_{0}\xrightarrow{\epsilon} A\rightarrow 0$

This chain complex is called a free resolution of $A$ . A free resolution is another example of an exact sequence.

We now introduce two more special kinds of modules.

A projective module is a module $P$ such for any surjective morphism $p: A\rightarrow A''$ between two modules $A$ and $A''$ and morphism $h: P\rightarrow A''$ , there exists a morphism $g: P\rightarrow A$ such that $p\circ g=h$ .

It is a theorem that a module is projective if and only if it is a direct summand of a free module. This also means that a free module is automatically also projective.

An injective module is a module $E$ such for any injective morphism $i: A\rightarrow B$ between two modules $A$ and $B$ and morphism $f: A\rightarrow E$ , there exists a morphism $g: B\rightarrow E$ such that $g\circ i=f$ .

Similar to our discussion regarding free resolutions earlier, we can also have projective resolutions and injective resolutions. A projective resolution is a chain complex

$...\xrightarrow{d_{3}} P_{2}\xrightarrow{d_{2} } P_{1}\xrightarrow{d_{1} } P_{0}\xrightarrow{\epsilon} A\rightarrow 0$

such that the $P_{n}$ are projective modules.

Meanwhile, an injective resolution is a chain complex

$...0\rightarrow A\xrightarrow{\eta} E^{0}\xrightarrow{d^{0} } E^{1}\xrightarrow{d^{1}} E^{2}\xrightarrow{d^{2}} ...$

such that the $E^{n}$ are injective modules.

Since projective and injective resolutions are chain complexes, we can use the methods of homology and cohomology to study them (Homology and Cohomology) even though they may not be made up of topological spaces. However, the usual procedure is to consider these chain complexes as forming an “abelian category” and then applying certain functors (see Category Theory) such as what are called the “Tensor” and “Hom” functors before applying the methods of homology and cohomology, resulting in what are known as “derived functors“. This is all part of the subject known as homological algebra.

References:

Free Module on Wikipedia

Projective Module on Wikipedia

Injective Module on Wikipedia

Resolution on Wikipedia

An Introduction to Homological Algebra by Joseph J. Rotman

Abstract Algebra by David S. Dummit and Richard M. Foote

Theories and Theorems

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More on Chain Complexes

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