nLab 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.