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 ):(Y,O Y)(X,O X)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 =O X\mathcal{I} = O_X of the comorphism f :O Yf *O Xf^\sharp:O_Y\to f_* O_X is a sheaf of ideals, called the defining sheaf of the closed immersion ff.

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 SS of an abelian category AA as the endofunctor = SEnd(A)\mathcal{I}=\mathcal{I}_S\in End(A) which is the subfunctor of the identity Id AId_A assigning to any MAM\in A the intersection of kernels Ker(f)Ker(f) of all morphisms f:MNf: M\to N with NOb(S)N\in Ob(S).

Last revised on March 6, 2013 at 19:35:00. See the history of this page for a list of all contributions to it.