Let $X = Z_1 \cup \ldots \cup Z_n$ be the decomposition of $X$ into irreducible components. Let $\eta_j \in Z_j$ be the generic point. \medskip\noindent There are (at least) two ways to proceed with the proof. The first is to redo the proof of Cohomology of Schemes, Lemma \ref{coherent-lemma-chow-Noetherian} using the general Properties, Lemma \ref{properties-lemma-point-and-maximal-points-affine} to find suitable affine opens in $X$. (This is the ``standard'' proof.) The second is to use absolute Noetherian approximation as in the proof of Lemma \ref{lemma-chow-finite-type} above. This is what we will do here. \medskip\noindent By Proposition \ref{proposition-separated-closed-in-finite-presentation} we can find a closed immersion $X \to Y$ where $Y$ is separated and of finite presentation over $S$. Write $S = \lim_i S_i$ as a directed limit of Noetherian schemes, see Proposition \ref{proposition-approximate}. By Lemma \ref{lemma-descend-finite-presentation} we can find an index $i \in I$ and a scheme $Y_i \to S_i$ of finite presentation so that $Y = S \times_{S_i} Y_i$. By Lemma \ref{lemma-descend-separated-finite-presentation} we may assume that $Y_i \to S_i$ is separated. We have the following diagram $$ \xymatrix{ \eta_j \in Z_j \ar[r] & X \ar[r] \ar[rd] & Y \ar[r] \ar[d] & Y_i \ar[d] \\ & & S \ar[r] & S_i } $$ Denote $h : X \to Y_i$ the composition. \medskip\noindent For $i' \geq i$ write $Y_{i'} = S_{i'} \times_{S_i} Y_i$. Then $Y = \lim_{i' \geq i} Y_{i'}$, see Lemma \ref{lemma-scheme-over-limit}. Choose $j, j' \in \{1, \ldots, n\}$, $j \not = j'$. Note that $\eta_j$ is not a specialization of $\eta_{j'}$. By Lemma \ref{lemma-topology-limit} we can replace $i$ by a bigger index and assume that $h(\eta_j)$ is not a specialization of $h(\eta_{j'})$ for all pairs $(j, j')$ as above. For such an index, let $Y' \subset Y_i$ be the scheme theoretic image of $h : X \to Y_i$, see Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-image}. The morphism $h$ is quasi-compact as the composition of the quasi-compact morphisms $X \to Y$ and $Y \to Y_i$ (which is affine). Hence by Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-scheme-theoretic-image} the morphism $X \to Y'$ is dominant. Thus the generic points of $Y'$ are all contained in the set $\{h(\eta_1), \ldots, h(\eta_n)\}$, see Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-dominant}. Since none of the $h(\eta_j)$ is the specialization of another we see that the points $h(\eta_1), \ldots, h(\eta_n)$ are pairwise distinct and are each a generic point of $Y'$. \medskip\noindent We apply Cohomology of Schemes, Lemma \ref{coherent-lemma-chow-Noetherian} above to the morphism $Y' \to S_i$. This gives a diagram $$ \xymatrix{ Y' \ar[rd] & Y^* \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_{S_i} \ar[dl] \\ & S_i & } $$ such that $\pi$ is proper and surjective and an isomorphism over a dense open subscheme $V \subset Y'$. By our choice of $i$ above we know that $h(\eta_1), \ldots, h(\eta_n) \in V$. Consider the commutative diagram $$ \xymatrix{ X' \ar@{=}[r] & X \times_{Y'} Y^* \ar[r] \ar[d] & Y^* \ar[r] \ar[d] & \mathbf{P}^n_{S_i} \ar[ddl] \\ & X \ar[r] \ar[d] & Y' \ar[d] & \\ & S \ar[r] & S_i & } $$ Note that $X' \to X$ is an isomorphism over the open subscheme $U = h^{-1}(V)$ which contains each of the $\eta_j$ and hence is dense in $X$. We conclude $X \leftarrow X' \rightarrow \mathbf{P}^n_S$ is a solution to the problem posed in the lemma.