Flat schematic closure over an arbitrary valuation ring

Let $J\subseteq K[x_0,\dots ,x_n]$ be the saturated homogeneous ideal of $Z_K$. Put $I=J\cap R[x_0,\dots ,x_n]$ and $M=R[x]\mathrm{/}I$. Each graded piece $M_d$ is a finitely generated torsion-free $R$-module, since it is the image of the finite free degree-$d$ polynomial module in $K[x]\mathrm{/}J$. Such a module is finite free: embed it in its finite-dimensional fraction-field span; among the finitely many coefficients of a finite generating list choose one of smallest valuation, divide the other coefficients in that coordinate by it (their ratios lie in $R$), and eliminate that coordinate from the other generators. This splits off one generator and reduces the dimension of the span; induction proves the assertion. Therefore $I_d$ is also finite free as the kernel of a split surjection from a finite free module onto $M_d$. Every $M_d$ is flat and base change preserves the sequence $0\to I_d\to R[x{]}_d\to M_d\to 0$.

Choose $m$ so all saturated ideals with polynomial $P$ over every field are $m$-regular, their multiplication maps are onto in degrees at least $m$, and ${\dim }_K(K[x]\mathrm{/}J{)}_d=P(d)$ for $d\ge m$. Over any residue field $k$ of $R$, the graded quotient $M{\otimes }_{R}k$ is a quotient of $k[x]$ and has exactly these degree dimensions, since $M_d$ is free of its generic rank. Its Hilbert polynomial is therefore $P$. Saturating its homogeneous ideal does not change that polynomial: the saturation quotient is a finitely generated irrelevant-torsion module over the Noetherian polynomial ring, so a common power of the irrelevant ideal kills its finitely many homogeneous generators and its sufficiently high graded pieces vanish; by the uniform bound the saturated quotient has dimension $P(d)$ for $d\ge m$. The natural surjection from the unsaturated quotient to the saturated quotient is thus an isomorphism in those degrees. Hence its ideal is already equal to its saturation there, and multiplication from degree $d$ to $d+1$ is onto for $d\ge m$.

Apply step 2.1 to the maximal residue field of $R$. For every $d\ge m$, the cokernel of $I_d\otimes R[x{]}_1\to I_{d+1}$ is a finitely generated module with zero residue-field quotient. Nakayama makes it zero. The entire ideal tail is consequently generated by the finite free module $I_m$; replacing $I$ by the homogeneous ideal generated by $I_m$ does not change its associated sheaf. This gives a closed immersion of finite presentation. Its structure sheaf is base-flat: on a standard projective chart it is the degree-zero part of a localization of the base-flat graded module ${⨁}_d{M}_d$, hence a direct summand of a flat module. Every fibre has polynomial $P$ by step 2.1.

For uniqueness, on each standard affine chart a flat extension's quotient has no torsion by nonzero elements of $R$. Its ideal is therefore the inverse image of its generic-fibre ideal under localization to $K$: if an element is generically in the ideal, some nonzero scalar kills its quotient class, so flatness forces that class to vanish. This is precisely the ideal of the schematic closure already constructed. The chart ideals determine the embedded subscheme uniquely.