(Ideal sheaf)
For a closed immersion \(i: Z \injects X\), the ideal sheaf is \[ \mci_Z \da \ker\qty{\OO_X \mapsvia{i^\sharp} i_*\OO_Z} . \]
\(Z \mapsto \mci_Z\) is a bijection between closed subschemes of \(X\) and quasicoherent sheaves of ideals on \(X\), and there is a short exact sequence \[ 0 \to \mci_Z \to \OO_X \to i_*\OO_Z \to 0 . \]
On \(X=\Spec A\), the bijection sends \(V(I)\) to \(\tilde I\) for ideals \(I\subseteq A\). A sheaf of ideals that is not quasicoherent is the ideal sheaf of no closed subscheme: for \(X=\AA^1_k\), \(U=X\sm\ts{0}\), and \(j\colon U\injects X\) the inclusion, the extension by zero \(j_!\OO_U\subseteq\OO_X\) is a sheaf of ideals with \(\Gamma(X,j_!\OO_U)=0\) and \(j_!\OO_U|_U=\OO_U\), so it is not quasicoherent.
For a hypersurface \(Z\subseteq\PP^n\) of degree \(d\), \(\mci_Z \cong \OO_{\PP^n}(-d)\), and for \(n\ge2\) the twisted sequence \(0\to\OO(m-d)\to\OO(m)\to\OO_Z(m)\to0\) gives \(h^0(\OO_Z(m))=h^0(\OO_{\PP^n}(m))-h^0(\OO_{\PP^n}(m-d))\), since \(H^1(\PP^n,\OO(m-d))=0\). The restriction \(i^*\mci_Z=\mci_Z/\mci_Z^2\) is the conormal sheaf of \(Z\) (D-MODCONORM).
Dependencies
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D-MODCONORMThe conormal and normal sheaves, and adjunctionuses