conormal sheaf

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 $f^\sharp:O_X\to f_* O_Y$ is a sheaf of ideals, called the defining sheaf of the closed immersion $f$.

The **conormal sheaf** is the quotient sheaf? of $O$-modules given by $\Omega_{X/Y} := \mathcal{I}/\mathcal{I}^2$. 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 $\Delta:X\to X\times_S X$ of an $S$-scheme $X$. For example, when $S = Spec k$ and $X = Spec R$ where $R$ is a $k$-algebra then the global sections of $\Omega_{X\times_S X/X}$ form the kernel of the multiplication map $R\times_k R\to R$ 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.

Revised on February 13, 2012 03:28:16
by Zoran Škoda
(109.227.55.211)