Lemma 70.5.10. Notation and assumptions as in Situation 70.5.5. 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 10.143.2. Set $A_ i = \mathcal{O}_{X_ i}(X_ i)$. We have $A = \mathop{\mathrm{colim}}\nolimits A_ i$ by Lemma 70.5.6. 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 32.8.15. 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

70.5.10.1
$$\label{spaces-limits-equation-to-show-cartesian} \vcenter { \xymatrix{ X_ i \ar[d] & U_ i \ar[l] \ar[d] \\ \mathop{\mathrm{Spec}}(A_ i) & \mathop{\mathrm{Spec}}(B_ i) \ar[l] } }$$

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 70.5.8 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 (70.5.10.1) is cartesian. Thus Descent, Lemma 35.39.1 applied to the fppf covering $\{ \mathop{\mathrm{Spec}}(B_ i) \to \mathop{\mathrm{Spec}}(A_ i)\}$ combined with Descent, Lemma 35.37.1 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.) $\square$

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 07SQ. Beware of the difference between the letter 'O' and the digit '0'.