Cokernel

Thecokernelof alinear mappingofvector spaces $f : X \to Y$ is thequotient space $Y / im(f)$ of thecodomainof $f$ by the image of $f$ .The dimension of the cokernel is called thecorankof $f$ .

Cokernels aredualto thekernels of category theory,hence the name: the kernel is asubobjectof the domain (it maps to the domain), while the cokernel is aquotient objectof the codomain (it maps from the codomain).

Intuitively, given an equation $f (x) = y$ that one is seeking to solve, the cokernel measures theconstraintsthat $y$ must satisfy for this equation to have a solution – the obstructions to a solution – while the kernel measures thedegrees of freedomin a solution, if one exists. This is elaborated inintuition,below.

More generally, the cokernel of amorphism $f : X \to Y$ in somecategory(e.g. ahomomorphismbetweengroupsor abounded linear operatorbetweenHilbert spaces) is an object $Q$ and a morphism $q : Y \to Q$ such that the composition $q f$ is thezero morphismof the category, and furthermore $q$ isuniversalwith respect to this property. Often the map $q$ is understood, and $Q$ itself is called the cokernel of $f$ .

In many situations inabstract algebra,such as forabelian groups,vector spacesormodules,the cokernel of thehomomorphism $f : X \to Y$ is thequotientof $Y$ by theimageof $f$ .Intopologicalsettings, such as with bounded linear operators between Hilbert spaces, one typically has to take theclosureof the image before passing to the quotient.

Formal definition

One can define the cokernel in the general framework ofcategory theory.In order for the definition to make sense the category in question must havezero morphisms.Thecokernelof amorphism $f : X \to Y$ is defined as thecoequalizerof $f$ and the zero morphism $0 XY : X \to Y$ .

Explicitly, this means the following. The cokernel of $f : X \to Y$ is an object $Q$ together with a morphism $q : Y \to Q$ such that the diagram

commutes.Moreover, the morphism $q$ must beuniversalfor this diagram, i.e. any other such $q': Y \to Q'$ can be obtained by composing $q$ with a unique morphism $u : Q \to Q'$ :

As with all universal constructions the cokernel, if it exists, is uniqueup toa uniqueisomorphism,or more precisely: if $q : Y \to Q$ and $q': Y \to Q'$ are two cokernels of $f : X \to Y$ ,then there exists a unique isomorphism $u : Q \to Q'$ with $q' = u q$ .

Like all coequalizers, the cokernel $q : Y \to Q$ is necessarily anepimorphism.Conversely an epimorphism is callednormal(orconormal) if it is the cokernel of some morphism. A category is calledconormalif every epimorphism is normal (e.g. thecategory of groupsis conormal).

Examples

In thecategory of groups,the cokernel of agroup homomorphism $f : G \to H$ is thequotientof $H$ by thenormal closureof the image of $f$ .In the case ofabelian groups,since everysubgroupis normal, the cokernel is just $H$ modulothe image of $f$ :

\operatorname {coker} (f)=H/\operatorname {im} (f).

Special cases

In apreadditive category,it makes sense to add and subtract morphisms. In such a category, thecoequalizerof two morphisms $f$ and $g$ (if it exists) is just the cokernel of their difference:

\operatorname {coeq} (f,g)=\operatorname {coker} (g-f).

In anabelian category(a special kind of preadditive category) theimageandcoimageof a morphism $f$ are given by

{\begin{aligned}\operatorname {im} (f)&=\ker(\operatorname {coker} f),\\\operatorname {coim} (f)&=\operatorname {coker} (\ker f).\end{aligned}}

In particular, every abelian category is normal (and conormal as well). That is, everymonomorphism $m$ can be written as the kernel of some morphism. Specifically, $m$ is the kernel of its own cokernel:

m=\ker(\operatorname {coker} (m))

Intuition

The cokernel can be thought of as the space ofconstraintsthat an equation must satisfy, as the space ofobstructions,just as thekernelis the space ofsolutions.

Formally, one may connect the kernel and the cokernel of a map $T : V \to W$ by theexact sequence

0\to \ker T\to V{\overset {T}{\longrightarrow }}W\to \operatorname {coker} T\to 0.

These can be interpreted thus: given a linear equation $T (v) = w$ to solve,

the kernel is the space ofsolutionsto thehomogeneousequation $T (v) = 0$ ,and its dimension is the number ofdegrees of freedomin solutions to $T (v) = w$ ,if they exist;
the cokernel is the space ofconstraintsonwthat must be satisfied if the equation is to have a solution, and its dimension is the number of independent constraints that must be satisfied for the equation to have a solution.

The dimension of the cokernel plus the dimension of the image (the rank) add up to the dimension of the target space, as the dimension of the quotient space $W / T (V)$ is simply the dimension of the spaceminusthe dimension of the image.

As a simple example, consider the map $T : R 2 \to R 2$ ,given by $T (x, y) = (0, y)$ .Then for an equation $T (x, y) = (a, b)$ to have a solution, we must have $a = 0$ (one constraint), and in that case the solution space is $(x, b)$ ,or equivalently, $(0, b) + (x,0)$ ,(one degree of freedom). The kernel may be expressed as the subspace $(x,0) \subseteq V$ :the value of $x$ is the freedom in a solution. The cokernel may be expressed via the real valued map $W :(a, b) \to (a)$ :given a vector $(a, b)$ ,the value of $a$ is theobstructionto there being a solution.

Additionally, the cokernel can be thought of as something that "detects"surjectionsin the same way that the kernel "detects"injections.A map is injective if and only if its kernel is trivial, and a map is surjective if and only if its cokernel is trivial, or in other words, if $W = im(T)$ .

References

Saunders Mac Lane:Categories for the Working Mathematician,Second Edition, 1978, p. 64
Emily Riehl:Category Theory in Context,Aurora Modern Math Originals,2014, p. 82, p. 139 footnote 8.