nLab
cokernel

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Category theory

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

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Contents

Idea

Definition

In a category with a terminal object 11, the cokernel of a morphism f:ABf : A \to B is the pushout coker(f)coker(f) in

A f B i 1 coker(f). \array{ A &\stackrel{f}{\to}& B \\ \downarrow && \downarrow^{\mathrlap{i}} \\ 1 &\to& coker(f) } \,.
Remark

In the case when the terminal object is in fact zero object, one can, more explicitly, characterize the object coker(f)coker(f) as the object (unique up to unique isomorphism) that satisfies the following universal property:

for every object CC and every morphism h:BCh : B \to C such that hf=0h \circ f = 0 is the zero morphism, there is a unique morphism ϕ:coker(f)C\phi : coker(f) \to C such that h=ϕih = \phi \circ i.

Remark

The notion of cokernel is dual to that of kernel. A cokernel in a category 𝒞\mathcal{C} is a kernel in the opposite category 𝒞 op\mathcal{C}^{op}.

Properties

Exactness properties

Examples

Example

In the category Ab of abelian groups the cokernel of a morphism f:ABf : A \to B is the quotient of BB by the image (of the underlying morphism of sets) of ff.

Example

More generally, for RR any ring, this is true in the category RRMod of modules: the cokernel of a morphism is the quotient by its set-theoretic image.

Example

In the category Grp of general (not necessarily abelian) groups, the cokernel is instead the quotient group by the normal closure of the image.

The following example is by the very definition of abelian category.

Example

In an abelian category the coimage of any morphism ff is the cokernel of its kernel

coim(f)=coker(ker(f)). coim(f) = coker(ker(f)) \,.

Revised on June 1, 2013 01:24:48 by Zoran Škoda (94.250.138.220)