Lemma 10.168.7. Let $A = \mathop{\mathrm{colim}}\nolimits _{i \in I} A_ i$ be a directed colimit of rings. Let $0 \in I$ and $\varphi _0 : B_0 \to C_0$ a map of $A_0$-algebras. Assume
$A \otimes _{A_0} B_0 \to A \otimes _{A_0} C_0$ is étale,
$B_0 \to C_0$ is of finite presentation.
Then for some $i \geq 0$ the map $A_ i \otimes _{A_0} B_0 \to A_ i \otimes _{A_0} C_0$ is étale.
Proof. Write $C_0 = B_0[x_1, \ldots , x_ n]/(f_{1, 0}, \ldots , f_{m, 0})$. Write $B_ i = A_ i \otimes _{A_0} B_0$ and $C_ i = A_ i \otimes _{A_0} C_0$. Note that $C_ i = B_ i[x_1, \ldots , x_ n]/(f_{1, i}, \ldots , f_{m, i})$ where $f_{j, i}$ is the image of $f_{j, 0}$ in the polynomial ring over $B_ i$. Write $B = A \otimes _{A_0} B_0$ and $C = A \otimes _{A_0} C_0$. Note that $C = B[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$ where $f_ j$ is the image of $f_{j, 0}$ in the polynomial ring over $B$. The assumption is that the map
is an isomorphism. Thus for sufficiently large $i$ we can find elements
with $\text{d}\xi _{k, i} = \text{d}x_ k$ in $\bigoplus C_ i \text{d}x_ k$. Moreover, on increasing $i$ if necessary, we see that $\sum (\partial f_{j, i}/\partial x_ k) \xi _{k, i} = f_{j, i} \bmod (f_{1, i}, \ldots , f_{m, i})^2$ since this is true in the limit. Then this $i$ works. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)