Exact sequences and chain homology

#category_theory/abelian_categories

For each abelian category, $A$ , we have defined another abelian category, $Ch (A)$ , whose objects are the chain complexes in $A$ and morphisms are the chain maps between such complexes. We also defined homology of such complexes, claiming it measures how close a complex is to being "exact" at each term.

In the category of $R$ -modules, we have seen exact sequences defined in terms of kernels and images. We've seen how to define kernels in an arbitrary abelian category. Can we also define images? Yep.

Images of morphisms

We first need the following fact, which we should probably prove at some point:

Factorization into monomorphism and epimorphism

Let $A$ be an abelian category. Then every morphism $f$ has a factorization $f = m e$ with $m$ a monomorphism and $e$ an epimorphism. Moreover,

m = \ker (coker (f)), e = coker (\ker (f)) .

We can thus define the (usual) image and coimage of $f = m e : a \to b$ as

m = im (f), e = coim (f) .

Technical note

The image and coimage are only unique up to isomorphism, so the image of $f$ is really a subobject of $b$ , while the coimage is a quotient object of $a$ .

Exact sequences

With images and coimages defined, we can talk about exact sequences:

Exact sequence in an abelian category

In an abelian category, a pair of composable morphisms

a \overset{f}{\to} b \overset{g}{\to} c

is exact at $b$ when $im (f) \equiv \ker (g)$ (or equivalently, when $coker (f) \equiv coim (g)$ ).

Here the symbol $\equiv$ indicates equivalence as subobjects (which are isomorphism classes of monomorphisms to a common object).

Short exact sequence in an abelian category

In an abelian category, a diagram

0 \to a \overset{f}{\to} b \overset{g}{\to} c \to 0

is a short exact sequence when it is exact at $a$ , $b$ , and $c$ .

Equivalently, $f = \ker (g)$ and $g = coker (f)$ .

Exact functors

Definition of exact functor

A functor $T : A \to B$ between abelian categories is exact when it preserves all finite limits and colimits.

In particular, an exact functor preserves kernels and cokernels:

\ker (T (f)) = T (\ker (f)) and coker (T (f)) = T (coker (f)) .

It also preserves images and coimages, and carries exact sequences to exact sequences.

A functor is left exact when it preserves all finite limits; equivalently, when it is additive and preserves short left exact sequences. The dual notion is a functor that is right exact.

Chain homology

Suppose $X_{∙}$ is a chain complex in an abelian category, $A$ . Since $\partial_{n} \circ \partial_{n - 1} = 0$ for all $n$ , there are monomorphisms

im (\partial_{n + 1}) \to \ker (\partial_{n}) \to X_{n} .

The quotient object, $H_{n} := \ker (\partial_{n}) / im (\partial_{n + 1}),$ can be thought of as a measure how how far away the chain complex is from being exact at $X_{n}$ . It is called the degree- $n$ chain homology of $X_{∙}$ and is often denoted $H_{n} (X_{∙})$ .

We note that the kernels, images and homologies are all functorial:

Chain morphisms respect boundaries, cycles, and homology

Suppose $f : X_{∙} \to Y_{∙}$ is a chain map. Then for every index $n \in Z$ the map $f_{n} : X_{n} \to Y_{n}$ restricts to morphisms

f_{n} ∣_{\ker (\partial_{n}^{x})} : \ker (\partial_{n}^{X}) \to \ker (\partial_{n}^{Y})

and

f_{n} ∣_{im (\partial_{n + 1}^{X})} : im (\partial_{n + 1}^{X}) \to im (\partial_{n + 1}^{Y}) .

In particular, it induces a morphism on chain homology

H_{n} (f) : H_{n} (X_{∙}) \to H_{n} (Y_{∙}) .

These are not deep facts. For example, suppose $x \in \ker (\partial_{n}^{X})$ . Then by the commutativity of the square with $X_{n}$ at the top-left corner we have

\partial_{n}^{Y} (f_{n} (x)) = f_{n - 1} (\partial_{n}^{X} (x)) = f_{n - 1} (0) = 0,

hence $f_{n} (x) \in \ker (\partial_{n}^{Y})$ . This proves that $f_{n}$ , when restricted to $\ker (\partial_{n}^{X})$ , maps into $\ker (\partial_{n}^{Y})$ .

The same type of argument shows that $f_{n}$ , when restricted to $im (\partial_{n + 1}^{X})$ , maps into $im (\partial_{n + 1}^{Y})$ .

Finally, consider the composition of the morphisms below, where the second map is the canonical projection onto the quotient:

\ker (\partial_{n}^{X}) \overset{f_{n}}{\to} \ker (\partial_{n}^{Y}) \overset{π_{n}^{Y}}{\to} H_{n} (Y_{∙}) .

By the same logic used to prove that $f_{n}$ restricts to a morphism between images, we can prove that $im (\partial_{n + 1}^{Y})$ is contained in $\ker (π \circ f_{n})$ . By a universal property of the quotient we therefore have a unique factorization

This is how we obtain the morphism $H_{n} (f)$ in the above fact.

Easier than it looks

Despite all of the names of the various objects and maps, every map above is essentially either $f_{n}$ , or projection onto a quotient. For example, the homology morphism $H_{n} (f)$ maps each coset $x + im (\partial_{n + 1}^{X})$ to the coset $f_{n} (x) + im (\partial_{n + 1}^{Y})$ . The above diagrams are just a formal way of verifying this map is a well-defined morphism. (They also outline a strategy to produce such morphisms even in categories in which the objects are not sets.)

Suggested next note

Double complexes and mural maps