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Thecokernel of alinear mapping ofvector spacesf :X →Y is thequotient spaceY / im(f) of thecodomain off by the image off. The dimension of the cokernel is called thecorank off.

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

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

More generally, the cokernel of amorphismf :X →Y in somecategory (e.g. ahomomorphism betweengroups or abounded linear operator betweenHilbert spaces) is an objectQ and a morphismq :Y →Q such that the compositionq f is thezero morphism of the category, and furthermoreq isuniversal with respect to this property. Often the mapq is understood, andQ itself is called the cokernel off.

In many situations inabstract algebra, such as forabelian groups,vector spaces ormodules, the cokernel of thehomomorphismf :X →Y is thequotient ofY by theimage off. Intopological settings, such as with bounded linear operators between Hilbert spaces, one typically has to take theclosure of the image before passing to the quotient.

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. Thecokernel of amorphismf :X →Y is defined as thecoequalizer off and the zero morphism0_XY :X →Y.

Explicitly, this means the following. The cokernel off :X →Y is an objectQ together with a morphismq :Y →Q such that the diagram

commutes. Moreover, the morphismq must beuniversal for this diagram, i.e. any other suchq′ :Y →Q′ can be obtained by composingq with a unique morphismu :Q →Q′:

As with all universal constructions the cokernel, if it exists, is uniqueup to a uniqueisomorphism, or more precisely: ifq :Y →Q andq′ :Y →Q′ are two cokernels off :X →Y, then there exists a unique isomorphismu :Q →Q′ withq' =uq.

Like all coequalizers, the cokernelq :Y →Q is necessarily anepimorphism. Conversely an epimorphism is callednormal (orconormal) if it is the cokernel of some morphism. A category is calledconormal if every epimorphism is normal (e.g. thecategory of groups is conormal).

In thecategory of groups, the cokernel of agroup homomorphismf :G →H is thequotient ofH by thenormal closure of the image off. In the case ofabelian groups, since everysubgroup is normal, the cokernel is justHmodulo the image off:

${\displaystyle \mathrm{coker}(f)=H/\mathrm{im}(f).}$

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

${\displaystyle \mathrm{coeq}(f,g)=\mathrm{coker}(g-f).}$

In anabelian category (a special kind of preadditive category) theimage andcoimage of a morphismf are given by

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

${\displaystyle m=\ker (\mathrm{coker}(m))}$

The cokernel can be thought of as the space ofconstraints that an equation must satisfy, as the space ofobstructions, just as thekernel is the space ofsolutions.

Formally, one may connect the kernel and the cokernel of a mapT:V →W by theexact sequence

${\displaystyle 0\to \ker T\to V\stackrel{T}{⟶}W\to \mathrm{coker}T\to 0.}$

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

• the kernel is the space ofsolutions to thehomogeneous equationT(v) = 0, and its dimension is the number ofdegrees of freedom in solutions toT(v) =w, if they exist;
• the cokernel is the space ofconstraints onw that 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 spaceW /T(V) is simply the dimension of the spaceminus the dimension of the image.

As a simple example, consider the mapT:R² →R², given byT(x,y) = (0,y). Then for an equationT(x,y) = (a,b) to have a solution, we must havea = 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) ⊆V: the value ofx is the freedom in a solution. The cokernel may be expressed via the real valued mapW: (a,b) → (a): given a vector(a,b), the value ofa is theobstruction to there being a solution.

Additionally, the cokernel can be thought of as something that "detects"surjections in 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, ifW = im(T).

• Abstract algebra
• Category theory
• Isomorphism theorems

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