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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05FL. Beware of the difference between the letter 'O' and the digit '0'.