= Zariski tangent space of a Hilbert scheme
{c}
{title2=$T_{[Z]}\operatorname{Hilb}(X)=H^0(Z,\mathcal N_{Z/X})$}
For a <closed subscheme> $Z\subseteq X$ with <ideal sheaf> $\mathcal I$, one has $T_{[Z]}\operatorname{Hilb}(X)\cong\operatorname{Hom}_{\mathcal O_X}(\mathcal I,i_*\mathcal O_Z)\cong H^0(Z,\mathcal N_{Z/X})$. Locally a flat deformation ideal $J\subseteq A[\varepsilon]/(\varepsilon^2)$ satisfies $J\cap\varepsilon A=\varepsilon I$. Lifting $f\in I$ to $f+\varepsilon g\in J$ gives a well-defined map $f\mapsto g\bmod I$ killing $I^2$. Conversely the graph condition $g\bmod I=\phi(f)$ defines an ideal with flat quotient by the <flatness criterion over dual numbers>. These operations commute with restriction and are inverse, giving the global identification without a smoothness assumption.
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