A notion of defining sheaf of ideals is a globalization of the notion of the defining ideal? of a closed affine subvariety of an affine variety. One also says less preferrable “sheaf of definition”, folowing literally the French variant.
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 .
This can be repharsed in terms of the category of quasicoherenet sheaves. In this vein, Alexander Rosenberg defines the defining ideal of a topologizing subcategory of an abelian category as the endofunctor which is the subfunctor of the identity assigning to any the intersection of kernels of all morphisms with .