Exact Sequences III - Morphisms of Exact Sequences

Motivation

For a given module $M$ , different choices of generators lead to different surjections from free $R$ -modules.

Different choices for the generators of the kernels of those surjections (i.e., different choices of generators for the relations among those generators), lead to different exact sequences

Continuing in this way, it's very plausible to expect to deal with two very different free resolutions of the same module $M$ :

There should be a way to compare these two resolutions, which means we need a way to compare exact sequences. As usual, we'll do that through the notion of a morphism of exact sequences.

Definition of morphism of sequences

To make our life easier, we'll actually define morphisms of sequences (of which chain complexes and exact sequences are special cases). We already predicted what this should be, namely:

Definition of morphism of sequences (of morphisms)

A morphism from a sequence

\dots \overset{f_{n - 2}}{\to} X_{n - 1} \overset{f_{n - 1}}{\to} X_{n} \overset{f_{n}}{\to} X_{n + 1} \overset{f_{n + 1}}{\to} \dots

to a sequence

\dots \overset{g_{n - 2}}{\to} Y_{n - 1} \overset{g_{n - 1}}{\to} Y_{n} \overset{g_{n}}{\to} Y_{n + 1} \overset{g_{n + 1}}{\to} \dots

is a collection of morphisms $τ_{n} : X_{n} \to Y_{n}$ such that the diagram below commutes:

In other words, when viewing the sequences of morphisms as functors $F, G : Z \to C$ , a morphism from $F$ to $G$ is exactly a natural transformation $τ : F \Rightarrow G$ .

As might be predicted, we can compose morphisms of sequences (by composing the vertical morphisms), and we can talk about isomorphisms of sequences (either as morphisms in which all of the vertical morphisms are isomorphisms, or equivalently as morphisms for which there exists an inverse morphism of sequences). These are all just the usual notions of composition and isomorphism in the functor category $C^{Z}$ .

Properties of morphisms of exact sequences

The commutativity condition on morphisms of sequences has some surprising consequences, at least in the case of morphisms of exact sequences.

The Short Five Lemma

Suppose we have a morphism of short exact sequences

If $τ_{1}$ and $τ_{3}$ are injective (resp., surjective), then $τ_{2}$ is injective (resp., surjective).

The proof of this lemma is a good exercise in "diagram chasing."

If you're wondering if there's a "Long Five Lemma", the answer is yes. There are actually many named "diagram lemmas", such as the Four Lemma, the Five Lemma and the famous Snake Lemma. All of these diagrams (and many, many more) are corollaries of perhaps the greatest diagram lemma of them all: The Salamander Lemma.

Suggested next notes

Exact Sequences IV - Exact Sequences and Functors
Diagram lemmas