1 Definition and basic idea
The cokernel of a homomorphism is a way to describe the part of the target that is not reached by the map. In many standard algebraic settings, it is formed by taking the quotient of the codomain by the image of the morphism. This makes the cokernel a measure of the failure of surjectivity.
1.1 Informal intuition
If a map sends some source object into a target object, the image is the portion actually hit. The cokernel records what remains after that image is collapsed to zero. In this sense, it captures the “missing” or “leftover” part of the target.
1.2 General categorical definition
In category theory, the cokernel of a morphism is defined as a coequalizer of that morphism and the zero morphism. Concretely, it is a morphism out of the codomain that annihilates the image of the original map and is universal with that property. This formulation allows cokernels to be discussed in abstract categories where quotient-by-image is not always available.
1.3 Cokernel in abelian categories
In an abelian category, cokernels always exist and behave much like quotients in familiar algebraic settings. The cokernel of a morphism is obtained by dividing the codomain by the image, interpreted in the categorical sense. This makes cokernels central to the structure theory of abelian categories.
1.4 Relationship to quotient objects
A cokernel is a special kind of quotient object. It identifies all elements in the image with zero while leaving the rest of the codomain distinct up to the equivalence relation generated by that collapse. In many concrete categories, this is the standard quotient construction.
2 Examples
2.1 Linear maps between vector spaces
For a linear map between vector spaces, the cokernel is the quotient of the target space by the image of the map. If the map is surjective, the cokernel is zero. If the image has smaller dimension than the codomain, the cokernel measures the dimension of the missing directions.
2.2 Homomorphisms of abelian groups
For a homomorphism of abelian groups, the cokernel is the target group modulo the subgroup formed by the image. This may produce finite cyclic groups, infinite cyclic groups, or mixtures of torsion and free parts. It is one of the most common quotient constructions in elementary algebra.
2.3 Module homomorphisms
For module maps, the cokernel is defined by quotienting the codomain module by the image submodule. This construction is especially useful in studying generators and relations, since many module presentations are naturally written as cokernels of maps between free modules.
2.4 Zero morphism and identity morphism
The cokernel of the zero morphism from an object to another object is the codomain itself, because the image is trivial. By contrast, the cokernel of an identity morphism is zero, since the image is the entire object. These two cases illustrate the extremes of the construction.
3 Universal property
3.1 Factorization through the cokernel
A cokernel is characterized by a universal factorization property. Any morphism from the codomain that kills the image of the original map must factor uniquely through the cokernel. This property explains why cokernels are useful even when quotient language is not the primary viewpoint.
3.2 Uniqueness up to isomorphism
Like many universal constructions, cokernels are unique up to unique isomorphism when they exist. Different choices of representatives for the same quotient object lead to canonically equivalent results. This is why one usually speaks of “the” cokernel of a morphism.
3.3 Comparison with kernels
Kernels and cokernels are dual notions. A kernel identifies what maps to zero, while a cokernel identifies what remains after the image is collapsed to zero. Together, they form a basic pair of tools for measuring the failure of injectivity and surjectivity.
4 Properties
4.1 Functorial behavior
Cokernels behave well with respect to morphisms between maps, though not as a standalone functor in all settings without additional structure. In categories where cokernels are available, they interact naturally with commuting diagrams. This makes them useful in diagram chases and structural arguments.
4.2 Exactness considerations
Cokernels are essential in exactness statements. A sequence is exact at an object when the image of the incoming map matches the kernel of the outgoing map. Since cokernels are built from images and quotients, they are closely tied to the formulation of exact sequences.
4.3 Behavior under composition
The cokernel of a composite map can differ substantially from the cokernels of the individual maps. Composition may enlarge or shrink the image in ways that alter the quotient of the codomain. Careful comparison often requires exact sequences or factorization arguments.
4.4 Preservation under isomorphisms
If two morphisms are related by isomorphisms on their domains and codomains, their cokernels are correspondingly isomorphic. Thus cokernels depend only on the morphism up to the natural equivalence induced by isomorphic change of coordinates or objects. This invariance is one reason they are considered structural rather than accidental.
5 Cokernels in algebraic settings
5.1 In linear algebra
In linear algebra, cokernels provide a compact way to describe the defect of a linear transformation. They are often identified with quotient vector spaces, making them easy to compute using bases and matrices. Their dimension gives a direct measure of how much of the target is not captured by the map.
5.1.1 Finite-dimensional case
When the source and target are finite-dimensional, the cokernel has finite dimension as well. Its dimension is the difference between the dimension of the codomain and the rank of the map. This makes the cokernel a natural companion to rank-based calculations.
5.1.2 Rank-nullity perspective
The rank-nullity theorem focuses on the domain, but a similar viewpoint applies on the codomain side through the cokernel. The rank determines how large the image is inside the target, and the cokernel measures the complementary quotient. Together, these ideas describe both sides of a linear transformation.
5.2 In module theory
For modules, cokernels are particularly important because many modules are built from free presentations. A module can often be described as a cokernel of a map between free modules, making the construction a standard language for defining algebraic objects by generators and relations. This perspective is central in commutative algebra and homological algebra.
5.2.1 Presentations of modules
A module presentation typically begins with a map between free modules whose cokernel is the module of interest. The source encodes relations, while the codomain encodes generators. In this way, the cokernel packages both the generating data and the constraints in one object.
5.2.2 Relations and generators
Generators determine a surjective map from a free module, and relations appear as the image of another map into that free module. Taking the cokernel identifies the generators modulo those relations. This is one of the most concrete uses of cokernels in algebra.
5.3 In abelian groups
For abelian groups, cokernels are easy to interpret and compute. They describe the quotient of a group by a subgroup arising as an image. Many standard classifications of abelian groups can be expressed in terms of such quotients.
5.3.1 Torsion and free parts
A cokernel of a homomorphism between abelian groups may contain both torsion and free components. The torsion part often reflects arithmetic constraints imposed by the map, while the free part reflects unresolved rank differences. Decompositions of abelian groups help make these pieces visible.
6 Relation to other constructions
6.1 Duality with kernels
Kernels and cokernels are formally dual in many categorical frameworks. Where kernels describe solutions to an equation of the form f(x)=0, cokernels describe the quotient by the image of f. This duality is especially clear in abelian categories and linear algebra.
6.2 Image and coimage
The image of a map sits between its kernel and cokernel in structural analyses. The coimage is often defined as the quotient of the domain by the kernel, while the image records the actual range inside the codomain. In abelian categories, image and coimage coincide up to canonical isomorphism.
6.3 Exact sequences
Cokernels naturally appear in exact sequences, where they help define the next object in the sequence. For instance, the cokernel of one map may be identified with the following object in a short exact sequence. This connection makes cokernels indispensable in homological arguments.
6.4 Homology and cohomology analogies
Homology groups are often built as quotients resembling cokernels of boundary maps modulo images of neighboring maps. Although not identical in every context, the pattern is similar: a quotient measures what survives after accounting for something that has already been hit. Cokernel-like constructions thus provide a conceptual bridge to homological algebra.
7 Computation
7.1 Matrix representations
When a homomorphism is represented by a matrix, its cokernel can often be computed from the matrix form. The image corresponds to the span of the columns, and the cokernel is the quotient by that column space. Row and column operations help simplify the calculation.
7.2 Quotient-space calculations
For vector spaces, one may compute the cokernel by first finding a basis for the image and then extending it to a basis of the codomain. The additional basis vectors represent the quotient directions. This gives a direct and intuitive description of the result.
7.3 Smith normal form for modules over principal ideal domains
For modules over a principal ideal domain, the Smith normal form is a standard tool for calculating cokernels. It reduces a matrix to a diagonal form that reveals invariant factors and torsion components. This makes the structure of the quotient module explicit.
7.4 Step-by-step examples
A typical computation begins by identifying the map, determining its image, and then forming the quotient of the codomain. In matrix settings, one simplifies the matrix until the image generators are easy to read off. The final cokernel is then expressed as a direct sum or quotient group according to the ambient category.
8 Special cases and variants
8.1 Cokernel of the zero map
The cokernel of a zero map is the entire codomain, because nothing is identified beyond the trivial subgroup or subspace. This case shows that cokernels can be nontrivial even when the original morphism carries no information. It is the simplest possible example.
8.2 Cokernel of an epimorphism
If a morphism is an epimorphism in a setting where epimorphisms behave like surjections, then its cokernel is zero. This reflects the fact that the image already fills the codomain. Thus the cokernel vanishes when there is no leftover structure.
8.3 Cokernel in non-abelian categories
In non-abelian categories, cokernels may exist but do not always reduce to a simple quotient by an image. They are then best understood through the universal property of coequalizers. Their behavior can be more subtle, especially when normality conditions are absent.
8.4 Stable and derived contexts
In stable and derived settings, cokernels are often replaced or supplemented by homotopical analogues that fit into triangulated or derived structures. These generalizations preserve the idea of measuring what remains after a map, but they do so up to higher-order equivalence. Such variants extend the utility of cokernel-like constructions far beyond elementary algebra.