conormal sheaf

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 Xf *O Yf^\sharp:O_X\to f_* O_Y is a sheaf of ideals, called the defining sheaf of the closed immersion ff.

The conormal sheaf is the quotient sheaf? of OO-modules given by Ω X/Y:=/ 2\Omega_{X/Y} := \mathcal{I}/\mathcal{I}^2. 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 Δ:XX× SX\Delta:X\to X\times_S X of an SS-scheme XX. For example, when S=SpeckS = Spec k and X=SpecRX = Spec R where RR is a kk-algebra then the global sections of Ω X× SX/X\Omega_{X\times_S X/X} form the kernel of the multiplication map R× kRRR\times_k R\to R 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.

Last revised on February 13, 2012 at 03:28:16. See the history of this page for a list of all contributions to it.