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The Stacks project

Lemma 37.34.11. (Noetherian approximation and combining properties.) Let P, Q be properties of morphisms of schemes which are stable under base change. Let f : X \to S be a morphism of finite presentation of affine schemes. Assume we can find cartesian diagrams

\vcenter { \xymatrix{ X_1 \ar[d]_{f_1} & X \ar[l] \ar[d]^ f \\ S_1 & S \ar[l] } } \quad \text{and}\quad \vcenter { \xymatrix{ X_2 \ar[d]_{f_2} & X \ar[l] \ar[d]^ f \\ S_2 & S \ar[l] } }

of affine schemes, with S_1, S_2 of finite type over \mathbf{Z} and f_1, f_2 of finite type such that f_1 has property P and f_2 has property Q. Then we can find a cartesian diagram

\xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l] \ar[d]^ f \\ S_0 & S \ar[l] }

of affine schemes with S_0 of finite type over \mathbf{Z} and f_0 of finite type such that f_0 has both property P and property Q.

Proof. The given pair of diagrams correspond to cocartesian diagrams of rings

\vcenter { \xymatrix{ B_1 \ar[r] & B \\ A_1 \ar[u] \ar[r] & A \ar[u] } } \quad \text{and}\quad \vcenter { \xymatrix{ B_2 \ar[r] & B \\ A_2 \ar[u] \ar[r] & A \ar[u] } }

Let A_0 \subset A be a finite type \mathbf{Z}-subalgebra of A containing the image of both A_1 \to A and A_2 \to A. Such a subalgebra exists because by assumption both A_1 and A_2 are of finite type over \mathbf{Z}. Note that the rings B_{0, 1} = B_1 \otimes _{A_1} A_0 and B_{0, 2} = B_2 \otimes _{A_2} A_0 are finite type A_0-algebras with the property that B_{0, 1} \otimes _{A_0} A \cong B \cong B_{0, 2} \otimes _{A_0} A as A-algebras. As A is the directed colimit of its finite type A_0-subalgebras, by Limits, Lemma 32.10.1 we may assume after enlarging A_0 that there exists an isomorphism B_{0, 1} \cong B_{0, 2} as A_0-algebras. Since properties P and Q are assumed stable under base change we conclude that setting S_0 = \mathop{\mathrm{Spec}}(A_0) and

X_0 = X_1 \times _{S_1} S_0 = \mathop{\mathrm{Spec}}(B_{0, 1}) \cong \mathop{\mathrm{Spec}}(B_{0, 2}) = X_2 \times _{S_2} S_0

works. \square


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