The Stacks project

Lemma 66.5.10. Notation and assumptions as in Situation 66.5.5. If $X$ is affine, then there exists an $i$ such that $X_ i$ is affine.

Proof. Choose $0 \in I$. Choose an affine scheme $U_0$ and a surjective étale morphism $U_0 \to X_0$. Set $U = U_0 \times _{X_0} X$ and $U_ i = U_0 \times _{X_0} X_ i$ for $i \geq 0$. Since the transition morphisms are affine, the algebraic spaces $U_ i$ and $U$ are affine. Thus $U \to X$ is an étale morphism of affine schemes. Hence we can write $X = \mathop{\mathrm{Spec}}(A)$, $U = \mathop{\mathrm{Spec}}(B)$ and

\[ B = A[x_1, \ldots , x_ n]/(g_1, \ldots , g_ n) \]

such that $\Delta = \det (\partial g_\lambda /\partial x_\mu )$ is invertible in $B$, see Algebra, Lemma 10.142.2. Set $A_ i = \mathcal{O}_{X_ i}(X_ i)$. We have $A = \mathop{\mathrm{colim}}\nolimits A_ i$ by Lemma 66.5.6. After increasing $0$ we may assume we have $g_{1, i}, \ldots , g_{n, i} \in A_ i[x_1, \ldots , x_ n]$ mapping to $g_1, \ldots , g_ n$. Set

\[ B_ i = A_ i[x_1, \ldots , x_ n]/(g_{1, i}, \ldots , g_{n, i}) \]

for all $i \geq 0$. Increasing $0$ if necessary we may assume that $\Delta _ i = \det (\partial g_{\lambda , i}/\partial x_\mu )$ is invertible in $B_ i$ for all $i \geq 0$. Thus $A_ i \to B_ i$ is an étale ring map. After increasing $0$ we may assume also that $\mathop{\mathrm{Spec}}(B_ i) \to \mathop{\mathrm{Spec}}(A_ i)$ is surjective, see Limits, Lemma 31.8.14. Increasing $0$ yet again we may choose elements $h_{1, i}, \ldots , h_{n, i} \in \mathcal{O}_{U_ i}(U_ i)$ which map to the classes of $x_1, \ldots , x_ n$ in $B = \mathcal{O}_ U(U)$ and such that $g_{\lambda , i}(h_{\nu , i}) = 0$ in $\mathcal{O}_{U_ i}(U_ i)$. Thus we obtain a commutative diagram

66.5.10.1
\begin{equation} \label{spaces-limits-equation-to-show-cartesian} \vcenter { \xymatrix{ X_ i \ar[d] & U_ i \ar[l] \ar[d] \\ \mathop{\mathrm{Spec}}(A_ i) & \mathop{\mathrm{Spec}}(B_ i) \ar[l] } } \end{equation}

By construction $B_ i = B_0 \otimes _{A_0} A_ i$ and $B = B_0 \otimes _{A_0} A$. Consider the morphism

\[ f_0 : U_0 \longrightarrow X_0 \times _{\mathop{\mathrm{Spec}}(A_0)} \mathop{\mathrm{Spec}}(B_0) \]

This is a morphism of quasi-compact and quasi-separated algebraic spaces representable, separated and étale over $X_0$. The base change of $f_0$ to $X$ is an isomorphism by our choices. Hence Lemma 66.5.8 guarantees that there exists an $i$ such that the base change of $f_0$ to $X_ i$ is an isomorphism, in other words the diagram (66.5.10.1) is cartesian. Thus Descent, Lemma 34.36.1 applied to the fppf covering $\{ \mathop{\mathrm{Spec}}(B_ i) \to \mathop{\mathrm{Spec}}(A_ i)\} $ combined with Descent, Lemma 34.34.1 give that $X_ i \to \mathop{\mathrm{Spec}}(A_ i)$ is representable by a scheme affine over $\mathop{\mathrm{Spec}}(A_ i)$ as desired. (Of course it then also follows that $X_ i = \mathop{\mathrm{Spec}}(A_ i)$ but we don't need this.) $\square$


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