Exact Sequences II - Exact Sequences

#module_theory

The idea of a relationship between a pair of morphisms through a common object (in this case, a module $M$ ) leads more generally to the notion of an exact sequence.

Definition of exactness

A pair of morphisms $X \overset{f}{\to} Y \overset{g}{\to} Z$ is exact at $Y$ if $\ker (g) = im (f)$ .

More generally, a sequence of morphisms

\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

is an exact sequence if it is exact at every $X_{n}$ .

Note that we have been purposefully vague about the category in which we are working, as this definition is meant to apply in any category for which one has kernels and images. For now, it's safe to assume we're working with either $R$ -modules or $(R, S)$ -bimodules.

Example: Injections and surjections

In a category of modules, a sequence of morphisms

0 \to M \overset{f}{\to} N

is exact at $M$ exactly when $f$ is injective.^[1] For example, the natural inclusion of $Z$ into $Q$ (as abelian groups) corresponds to the exact sequence of abelian groups

0 \to Z \to Q .

Similarly, a sequence of morphisms

M \overset{f}{\to} N \to 0

is exact at $N$ exactly when $f$ is surjective.^[2] For example, the canonical projection from $Z$ onto the quotient group $Z / 2 Z = Z_{2}$ corresponds to the exact sequence

Z \to Z_{2} \to 0.

We can now fully recover the situations in the quotient module example and direct sum example with the notion of a short exact sequence.

Definition of short exact sequence

A short exact sequence is an exact sequence of the form

0 \to X \overset{f}{\to} Y \overset{g}{\to} Z \to 0.

In light of the previous definitions, a sequence of morphisms of the above form is a short exact sequence exactly when:

$f : X \to Y$ is injective;
$g : Y \to Z$ is surjective; and
$\ker (g) = im (f)$ .

Example: Submodule and quotient modules

For each submodule $N$ of an $R$ -module $M$ , we have a short exact sequence

0 \to N \to M \to M / N \to 0.

For instance, we have the short exact sequence of abelian groups

0 \to 2 Z \to Z \to Z_{2} \to 0.

Example: Direct sum of two modules

For each pair of $R$ -modules $M_{1}$ and $M_{2}$ , we have a short exact sequence of $R$ -modules

0 \to M_{1} \overset{j_{1}}{\to} M_{1} \oplus M_{2} \overset{π_{2}}{\to} M_{2} \to 0.

Example: Forming short exact sequences from morphisms

Suppose we have a morphism $f : X \to Y$ . We can then form the short exact sequence

0 \to \ker (f) \to X \to im (f) \to 0.

Example: Forming short exact sequences from short, exact sequences

Suppose we have a sequence

X \overset{f}{\to} Y \overset{g}{\to} Z

that is exact at $Y$ ; i.e., $\ker (g) = im (f)$ . We can then form the short exact sequence

0 \to im (f) \to Y \to Y / \ker (g) \to 0.

Chain complexes

A natural generalization of the notion of an exact sequence is that of a chain complex.

Definition of chain complex

A sequence of morphisms

\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

is called a chain complex if for every $n$ one has $im (f_{n}) \subseteq \ker (f_{n + 1})$ ; equivalently, if $f_{n + 1} \circ f_{n} = 0$ for every $n$ .

Why generalize from exact sequences to chain complexes? Functors! Given any sequence of morphisms, say in the category of $R$ -modules

\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

and a functor $F$ (from that category to another, say to the category $Ab$ of abelian groups), functoriality guarantees we have a chain of morphisms

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

We will soon see that if the original sequence of morphisms is exact, the resulting sequence often will not be. That failure of exactness will actually be something we study in more detail.

However, we will see that if the original sequence is a chain complex (which includes the case of an exact sequence), then the resulting sequence will at least still be a chain complex. So chain complexes seem like the right type of object to consider to make things categorical.

In fact, if we define morphisms of chain complexes, then we could consider an actual category of chain complexes ...

Suggested next note

Exact Sequences III - Morphisms of Exact Sequences

Here we haven't labeled the morphism from the zero module, since that morphism is unique (it sends the single element of the zero module to $0_{M}$ ). ↩︎
Once again, there is no need to label the unique morphism from $N$ to the zero module (which sends every element in $N$ to the single element in the zero module.) ↩︎