cokernels

Cokernel is a concept in category theory that generalizes the notion of quotient by an image. Given a morphism f: A -> B in a category with a zero object, a cokernel of f is an object C together with a morphism q: B -> C such that q ∘ f = 0 and for any morphism g: B -> D with g ∘ f = 0 there exists a unique h: C -> D with g = h ∘ q. Equivalently, the cokernel is the coequalizer of f and the zero morphism 0: A -> B.

In the category of modules over a ring, or of vector spaces, the cokernel of a linear map f: V -> W is the quotient W / im(f). The canonical projection π: W -> W / im(f) is the cokernel of f, and im(f) is the kernel of π. Thus there is a natural short exact sequence 0 -> im(f) -> W -> W/im(f) -> 0, or equivalently 0 -> ker f -> V --f--> W --π--> W/im f -> 0.

If f has rank r, then dim(coker f) = dim(W) − r. In finite dimensions, the dimension of the cokernel measures how far f is from being surjective; it is zero exactly when f is surjective.

Examples include a matrix A: F^n -> F^m. The cokernel is F^m / im(A). If A has full column rank (rank n ≤ m), then its cokernel has dimension m − n and the quotient is isomorphic to F^{m−n}.

In category-theoretic language, cokernels are dual to kernels and play a central role in exact sequences, abelian categories, and homological algebra.