In his book "Deformation Theory", Hartshorne claims:
Let $Y$ be a closed subscheme of the projective space $X=\mathbb{P}^n_k$ over a field $k$. Then there exists a projective scheme $H$, called the Hilbert scheme, parametrizing closed subschemes of $X$ with the same Hilbert polynomial $P$ as $Y$ ...
What does projectivity of $H$ exactly mean here? Does it mean that it is also a closed subscheme of $\mathbb{P}^n_k$? Or does it mean that it is a closed subscheme of $\mathbb{P}^n_Y$? Wikipedia claims that it is projective over $Spec(\mathbb{Z})$, does this imply one of these options?
Actually I would like to know if $H$ is locally of finite type, which would be the case if it is a closed subscheme of $\mathbb{P}^n_k$.
Many thanks.