nabla.vii

R-modules:

Definition 1.1:

Let R be a ring. A (left) R-module is an abelian group (M,+) with a scalar mulitplication R×M->M such that

If R=ℤ, ℤ-modules are precisely abelian groups. If R=k is a field, k-modules are vector spaces over k. If R is not commutative, we also get a distinct notion of right R-modules where scalar multiplication is on the right. The only axiom which changes is (4) which becomes (ms)r=m(sr). When R is commutative, sr=rs so the objects are the same.

Definition 1.2: A function f:M->N between left R-modules is said to be an R-module homomorphism (or an R-linear map) if it is a group homomorphism, i.e. f(m+m')=f(m)+f(m'), such that f(rm)=rf(m) for all m∈M and r∈R. If instead M and N are right R-modules, we require f(mr)=f(m)r.

With this definition, one can show that the identity map and the composition of R-module homs are both R-module homs. Hence we have a category of (left) R-modules and R-module homs which we denote R-Mod. We denote the category of right R-modules and R-module homes Mod-R.

In what follows, R-module will always mean left R-module unless stated otherwise.

Chain Complexes:

Definition 1.3:

Examples 1.4:

We may also consider complexes where the operators increase in degree instead and we call these cochain complexes. Mathematically, these are the same up to relabelling but it is often useful to treat them as distinct as we shall see later on.

We get the following immediate result.

Lemma 1.5:

One might care about complexes where this is actually an equality and indeed this is our definition of an exact sequence!

Definition 1.6:

From this definition, we see that any exact sequence is also a chain complex. We distiguish short exact sequences for two reasons. One is that in a sense (see the next example) short exact sequences are the shortest "interesting" exact sequences. The second is that they turn out to be very important in homological algebra. Also note that to show a sequence is short exact, we need only check that φ is injective, imφ=kerψ and that ψ is surjective.

Examples 1.7:

Comparing examples 1 and 7, we see that short exact given R-modules A and C, we don't necessarily have that all short exact sequences 0->A->B->C->0 must have B≅A⊕C. Now comparing with examples 3 and 4, we see that short exact sequences are indeed the shortest exact sequences where something more interesting can happen. However do not discount the utility of examples 3 and 4!

Definition 1.8:

One may now wonder whether we can tell when a short exact sequence is split and indeed the next result gives a full classification!

Proposition 1.9 (Splitting Lemma):

Chain Maps:

As is common in maths, now that we have seen some objects we should should talk about maps between these objects. This leads us to talking about chain maps.

Definition 1.10:

As one might expect, chain complexes over R with chain maps forms a category, which we denote by Ch. This is the result on the next lemma.

Lemma 1.11:

Note that the composition of chain maps is also associative. This follows from the associativity of the composition of set functions.

This allows us to talk of isomorphisms of chain complexes. A chain isomorphism is an isomorphism in Ch. Equivalently, a chain isomorphism is a chain map such that each map in the sequence is an isomorphism.

We conlude with a useful result involving chain maps between exact sequences.

Theorem 1.12 (The Five Lemma):

The first of these results is sometimes referred to as the Short Five Lemma.

Note that in the proof of the splitting lemma, in particular in (ii)⇒(i), we need not have checked that f is an isomorphism or that B=φ(A)⊕η(C). If, after constructing f as we did, we check that the following diagram commutes, then the short five lemma implies that f is an isomorphism: