= Normal sheaf
{title2=$\mathcal N_{Z/X}$}
For a <closed immersion> $i:Z\hookrightarrow X$ with <ideal sheaf> $\mathcal I$, the normal sheaf is $\mathcal N_{Z/X}=\mathcal Hom_{\mathcal O_Z}(\mathcal I/\mathcal I^2,\mathcal O_Z)$. It is the dual of the <conormal sheaf>. For a regular embedding it is a <locally free sheaf>; in the smooth case it agrees with the <normal bundle>. Without regularity it need not be locally free, but its <global sections> still classify <first-order embedded deformations>.
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