Proof.
We can find a finite type \mathbf{Z}-subalgebra A' \subset A and an algebraic space X' separated and of finite presentation over A' whose base change to A is X. See Limits of Spaces, Lemmas 70.7.1 and 70.6.9. Let x' \in |X'| be the image of x. If we can prove the lemma for (X'/A', x'), then the lemma follows for (X/A, x). Namely, if n', Z', z', V', E' provide the solution for (X'/A', x'), then we can let n = n', let Z \subset X \times \mathbf{P}^ n be the inverse image of Z', let z \in Z be the unique point mapping to x, let V \subset \mathbf{P}^ n_ A be the inverse image of V', and let E be the derived pullback of E'. Observe that E is pseudo-coherent by Cohomology on Sites, Lemma 21.45.3. It only remains to check (5). To see this set W = c^{-1}(V) and W' = (c')^{-1}(V') and consider the cartesian square
\xymatrix{ W \ar[d]_{(b, c)} \ar[r] & W' \ar[d]^{(b', c')} \\ X \times _ A V \ar[r] & X' \times _{A'} V' }
By Lemma 76.51.1 X \times _ A V and W' are tor-independent over X' \times _{A'} V'. Thus the derived pullback of (b', c')_*\mathcal{O}_{W'} to X \times _ A V is (b, c)_*\mathcal{O}_ W by Derived Categories of Spaces, Lemma 75.20.4. This also uses that R(b', c')_*\mathcal{O}_{Z'} = (b', c')_*\mathcal{O}_{Z'} because (b', c') is a closed immersion and similarly for (b, c)_*\mathcal{O}_ Z. Since E'|_{U' \times _{A'} V'} = (b', c')_*\mathcal{O}_{W'} we obtain E|_{U \times _ A V} = (b, c)_*\mathcal{O}_ W and (5) holds. This reduces us to the situation described in the next paragraph.
Assume A is of finite type over \mathbf{Z}. Choose an étale morphism U \to X where U is an affine scheme and a point u \in U mapping to x. Then U is of finite type over A. Choose a closed immersion U \to \mathbf{A}^ n_ A and denote j : U \to \mathbf{P}^ n_ A the immersion we get by composing with the open immersion \mathbf{A}^ n_ A \to \mathbf{P}^ n_ A. Let Z be the scheme theoretic closure of
(\text{id}_ U, j) : U \longrightarrow X \times _ A \mathbf{P}^ n_ A
Let z \in Z be the image of u. Let Y \subset \mathbf{P}^ n_ A be the scheme theoretic closure of j. Then it is clear that Z \subset X \times _ A Y is the scheme theoretic closure of (\text{id}_ U, j) : U \to X \times _ A Y. As X is separated, the morphism X \times _ A Y \to Y is separated as well. Hence we see that Z \to Y is an isomorphism over the open subscheme j(U) \subset Y by Morphisms of Spaces, Lemma 67.16.7. Choose V \subset \mathbf{P}^ n_ A open with V \cap Y = j(U). Then we see that (2) holds, that W = (\text{id}_ U, j)(U), and hence that (3) holds. Part (1) holds because A is Noetherian.
Because A is Noetherian we see that X and X \times _ A \mathbf{P}^ n_ A are Noetherian algebraic spaces. Hence we can take E = (b, c)_*\mathcal{O}_ Z in this case: (4) is clear and for (5) see Derived Categories of Spaces, Lemma 75.13.7. This finishes the proof.
\square
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