Given a closed immersion of schemes of (or of more general locally ringed spaces, e.g. of analytic varieties), the kernel of the comorphism is a sheaf of ideals, called the defining sheaf of the closed immersion .
The conormal sheaf is the quotient sheaf? of -modules given by . In the algebraic setting, it is also called the sheaf of relative Kähler differentials. It is always quasicoherent. Most important example is the case of the diagonal of an -scheme . For example, when and where is a -algebra then the global sections of form the kernel of the multiplication map quotiented by its square.
One sometimes says conormal bundle meaning a conormal sheaf, though the bundle and sheaf are two points of view in the cases when the conormal sheaf is locally free of constant rank.