# Deformations of finite schemes

I am reading some texts about the tangent space to the Hilbert scheme. Apparently, $$T_{[Z]}(Hilb (X)) = H^0(Z, N_{Z/X})$$ for any $$Z$$ is a consensus which I agree with, but then regarding the hilbert scheme of points, people generally use that $$T_{[Z]}(Hilb_r (X)) = \text{Hom}_{O_X}(I_Z, i_*O_Z)$$, and there are a lot of proofs of this last equality but I don't really see where they use that $$Z$$ is finite.

I know that $$H^0(Z, N_{Z/X}) = \text{Hom}_{O_Z}(I_Z/I_Z^2, O_Z)$$, so maybe I am missing something and the equality $$\text{Hom}_{O_Z}(I_Z/I_Z^2, O_Z) = \text{Hom}_{O_X}(I_Z, i_*O_Z)$$ is obvious, or is obvious when $$Z$$ is a finite scheme.

The usual proofs of $$T_{[Z]}(Hilb_r (X)) = \text{Hom}_{O_X}(I_Z, i_*O_Z)$$ basically try to characterize all quotients $$Q$$ of $$O_X [ \varepsilon ]$$ that are flat over $$k[ \varepsilon ]$$ and such that, $$Q/\varepsilon Q = O_Z$$. Chasing a bit in some diagram they show that this is equivalent as giving a $$O_X$$-morphism from $$I_Z$$ to $$i_*O_Z$$. For some links, see for example page 22 here. The proof convinces me but it doesn't say where thye use that $$Z$$ is a finite scheme.

Can someone shed some light on this topic? Thank you

Notes: Here $$X$$ is a projective variety, and $$F[\varepsilon] = F\otimes k(t)/(t^2)$$. When I say a finite scheme a mean a scheme of finite type over a field whose topological space is finite. but it can be non-reduced

First, let's just set $$i :Z \to X$$ for the closed immersion.
Next, a small correction: the the second formula for the tangent space should be $$T_{[Z]} \mathrm{Hilb}_r(X) = \mathrm{Hom}_{\mathcal{O}_X}(I_Z,\mathcal{O}_Z)$$.
In this formula, we are viewing $$\mathcal{O}_Z$$ as a $$\mathcal{O}_X$$-module, so it would be more proper to write $$T_{[Z]} \mathrm{Hilb}_r(X) = \mathrm{Hom}_{\mathcal{O}_X}(I_Z,i_* \mathcal{O}_Z)$$. By adjunction, we have $$\mathrm{Hom}_{\mathcal{O}_X}(I_Z,i_* \mathcal{O}_Z) = \mathrm{Hom}_{\mathcal{O}_Z}(i^* I_Z,\mathcal{O}_Z)$$. Finally, noting that $$i^* I_Z$$ is the conormal sheaf $$I_Z /I_Z^2$$, we have that the two formulas agree.
Indeed, none of this really depends on $$Z$$ being finite.