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