nLab defining sheaf

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 $f= (f,f^\sharp): (Y,O_Y)\to (X,O_X)$ of (or of more general locally ringed spaces, e.g. of analytic varieties), the kernel $\mathcal{I} = O_X$ of the comorphism $f^\sharp:O_Y\to f_* O_X$ is a sheaf of ideals, called the defining sheaf of the closed immersion $f$.

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 $S$ of an abelian category $A$ as the endofunctor $\mathcal{I}=\mathcal{I}_S\in End(A)$ which is the subfunctor of the identity $Id_A$ assigning to any $M\in A$ the intersection of kernels $Ker(f)$ of all morphisms $f: M\to N$ with $N\in Ob(S)$.

category: sheaf theoryalgebraic geometry