Main Page | See live article | Alphabetical index


In mathematics, the cokernel of a homomorphism f: XY is the quotient of Y by the image of f. In a topological setting, one typically takes the closure of the image before passing to the quotient. For instance, if f: H1H2 is a bounded linear operator between Hilbert spaces, then coker(f) is the quotient of H2 by the closure of the range of f.

In the general framework of a category with zero morphisms, the cokernel of f : XY (if it exists) is the morphism g: YZ such that the composition gf is the zero map from X to Z and g is universal for this property, i.e., any h: YW such that hf = 0 can be obtained by composing g with a unique map from Z to W.

This notion is dual to the kernels of category theory, hence the name.