Editing Cokernel

{{Short description|Quotient space of a codomain of a linear map by the map's image}}
{{Redirect|Coker (mathematics)|other uses|Coker (disambiguation)}}
{{no footnotes|date=February 2013}}

The '''cokernel''' of a [[linear mapping]] of [[vector spaces]] {{math|''f'' : ''X'' → ''Y''}} is the [[quotient space (linear algebra)|quotient space]] {{math|''Y'' / im(''f'')}} of the [[codomain]] of {{mvar|f}} by the image of {{mvar|f}}. The dimension of the cokernel is called the ''corank'' of {{mvar|f}}.

Cokernels are [[dual (category theory)|dual]] to the [[kernel (category theory)|kernels of category theory]], hence the name: the kernel is a [[subobject]] of the domain (it maps to the domain), while the cokernel is a [[quotient object]] of the codomain (it maps from the codomain).

Intuitively, given an equation {{math|1=''f''(''x'') = ''y''}} that one is seeking to solve, the cokernel measures the ''constraints'' that {{mvar|y}} must satisfy for this equation to have a solution – the obstructions to a solution – while the kernel measures the ''degrees of freedom'' in a solution, if one exists. This is elaborated in [[#Intuition|intuition]], below.

More generally, the cokernel of a [[morphism]] {{math|''f'' : ''X'' → ''Y''}} in some [[category theory|category]] (e.g. a [[group homomorphism|homomorphism]] between [[group (mathematics)|group]]s or a [[bounded linear operator]] between [[Hilbert space]]s) is an object {{mvar|Q}} and a morphism {{math|''q'' : ''Y'' → ''Q''}} such that the composition {{math|''q f''}} is the [[zero morphism]] of the category, and furthermore {{mvar|q}} is [[universal mapping property|universal]] with respect to this property. Often the map {{mvar|q}} is understood, and {{mvar|Q}} itself is called the cokernel of {{mvar|f}}.

In many situations in [[abstract algebra]], such as for [[abelian group]]s, [[vector space]]s or [[module (mathematics)|module]]s, the cokernel of the [[homomorphism]] {{math|''f'' : ''X'' → ''Y''}} is the [[quotient set|quotient]] of {{mvar|Y}} by the [[Image (mathematics)|image]] of {{mvar|f}}. In [[topology|topological]] settings, such as with bounded linear operators between Hilbert spaces, one typically has to take the [[closure (mathematics)|closure]] of the image before passing to the quotient.

== Formal definition ==
One can define the cokernel in the general framework of [[category theory]]. In order for the definition to make sense the category in question must have [[zero morphism]]s. The '''cokernel''' of a [[morphism]] {{math|''f'' : ''X'' → ''Y''}} is defined as the [[coequalizer]] of {{mvar|f}} and the zero morphism {{math|0<sub>''XY''</sub> : ''X'' → ''Y''}}.

Explicitly, this means the following. The cokernel of {{math|''f'' : ''X'' → ''Y''}} is an object {{mvar|Q}} together with a morphism {{math|''q'' : ''Y'' → ''Q''}} such that the diagram

<div style="text-align: center;">[[Image:Cokernel-01.svg|class=skin-invert]]</div>

[[commutative diagram|commutes]]. Moreover, the morphism {{mvar|q}} must be [[universal property|universal]] for this diagram, i.e. any other such {{math|''q''′ : ''Y'' → ''Q''′}} can be obtained by composing {{mvar|q}} with a unique morphism {{math|''u'' : ''Q'' → ''Q''′}}:

<div style="text-align: center;">[[Image:Cokernel-02.png|class=skin-invert]]</div>

As with all universal constructions the cokernel, if it exists, is unique [[up to]] a unique [[isomorphism]], or more precisely: if {{math|''q'' : ''Y'' → ''Q''}} and {{math|''q''′ : ''Y'' → ''Q''′}} are two cokernels of {{math|''f'' : ''X'' → ''Y''}}, then there exists a unique isomorphism {{math|''u'' : ''Q'' → ''Q''′}} with {{math|1=''q''' = ''u'' ''q''}}.

Like all coequalizers, the cokernel {{math|''q'' : ''Y'' → ''Q''}} is necessarily an [[epimorphism]]. Conversely an epimorphism is called ''[[normal morphism|normal]]'' (or ''conormal'') if it is the cokernel of some morphism. A category is called ''conormal'' if every epimorphism is normal (e.g. the [[category of groups]] is conormal).

=== Examples ===
In the [[category of groups]], the cokernel of a [[group homomorphism]] {{math|''f'' : ''G'' → ''H''}} is the [[quotient group|quotient]] of {{mvar|H}} by the [[Normal closure (group theory)|normal closure]] of the image of {{mvar|f}}. In the case of [[abelian group]]s, since every [[subgroup]] is normal, the cokernel is just {{mvar|H}} [[Ideal (ring theory)|modulo]] the image of {{mvar|f}}:
:<math>\operatorname{coker}(f) =  H / \operatorname{im}(f).</math>

=== Special cases ===
In a [[preadditive category]], it makes sense to add and subtract morphisms. In such a category, the [[coequalizer]] of two morphisms {{mvar|f}} and {{mvar|g}} (if it exists) is just the cokernel of their difference:

: <math>\operatorname{coeq}(f, g) = \operatorname{coker}(g - f).</math>

In an [[abelian category]] (a special kind of preadditive category) the [[image (category theory)|image]] and [[coimage]] of a morphism {{mvar|f}} are given by
:<math>\begin{align}
    \operatorname{im}(f) &= \ker(\operatorname{coker} f), \\
  \operatorname{coim}(f) &= \operatorname{coker}(\ker f).
\end{align}</math>

In particular, every abelian category is normal (and conormal as well). That is, every [[monomorphism]] {{mvar|m}} can be written as the kernel of some morphism. Specifically, {{mvar|m}} is the kernel of its own cokernel:
:<math>m = \ker(\operatorname{coker}(m))</math>

==Intuition==
The cokernel can be thought of as the space of ''constraints'' that an equation must satisfy, as the space of ''obstructions'', just as the [[Kernel (algebra)|kernel]] is the space of ''solutions.''

Formally, one may connect the kernel and the cokernel of a map {{math|''T'':  ''V'' →  ''W''}} by the [[exact sequence]]

:<math>0 \to \ker T \to V \overset T \longrightarrow W \to \operatorname{coker} T \to 0.</math>

These can be interpreted thus: given a linear equation {{math|1=''T''(''v'') = ''w''}} to solve,
* the kernel is the space of ''solutions'' to the ''homogeneous'' equation {{math|1=''T''(''v'') = 0}}, and its dimension is the number of ''degrees of freedom'' in solutions to {{math|1=''T''(''v'') = ''w''}}, if they exist;
* the cokernel is the space of ''constraints'' on ''w'' 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 space {{math|''W'' / ''T''(''V'')}} is simply the dimension of the space ''minus'' the dimension of the image.

As a simple example, consider the map {{math|''T'': '''R'''<sup>2</sup> → '''R'''<sup>2</sup>}}, given by {{math|1=''T''(''x'', ''y'') = (0, ''y'')}}. Then for an equation {{math|1=''T''(''x'', ''y'') = (''a'', ''b'')}} to have a solution, we must have {{math|1=''a'' = 0}} (one constraint), and in that case the solution space is {{math|(''x'', ''b'')}}, or equivalently, {{math|1=(0, ''b'') + (''x'', 0)}}, (one degree of freedom). The kernel may be expressed as the subspace {{math|(''x'', 0) ⊆ ''V''}}: the value of {{mvar|x}} is the freedom in a solution. The cokernel may be expressed via the real valued map {{math|''W'': (''a'', ''b'') → (''a'')}}: given a vector {{math|(''a'', ''b'')}}, the value of {{mvar|a}} is the ''obstruction'' to there being a solution.

Additionally, the cokernel can be thought of as something that "detects" [[surjection]]s in the same way that the kernel "detects" [[injection (mathematics)|injection]]s. 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 {{math|1=''W'' = im(''T'')}}.

== References ==
{{refbegin}}
*[[Saunders Mac Lane]]: ''[[Categories for the Working Mathematician]]'', Second Edition, 1978, p.&nbsp;64
*[[Emily Riehl]]: [http://www.math.jhu.edu/~eriehl/context.pdf#page=100 Category Theory in Context], [https://store.doverpublications.com/048680903x.html Aurora Modern Math Originals], 2014, p. 82, p. 139 footnote 8.
{{refend}}

{{Category theory}}

[[Category:Abstract algebra]]
[[Category:Category theory]]
[[Category:Isomorphism theorems]]

[[de:Kern (Algebra)#Kokern]]