The Stacks project

Lemma 33.18.3. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Let $V = \mathop{\mathrm{Spec}}(A)$ be an affine open neighbourhood of $f(x)$ in $S$. If $f$ is unramified at $x$, then there exist exists an affine open $U \subset X$ with $x \in U$ and $f(U) \subset V$ such that we have a commutative diagram

\[ \xymatrix{ X \ar[d] & U \ar[l] \ar[rd] \ar[r]^-j & \mathop{\mathrm{Spec}}(A[t]_{g'}/(g)) \ar[d] \ar[r] & \mathop{\mathrm{Spec}}(A[t]) = \mathbf{A}^1_ V \ar[ld] \\ Y & & V \ar[ll] } \]

where $j$ is an immersion, $g \in A[t]$ is a monic polynomial, and $g'$ is the derivative of $g$ with respect to $t$. If $f$ is ├ętale at $x$, then we may choose the diagram such that $j$ is an open immersion.

Proof. The unramified case is a translation of Algebra, Proposition 10.152.1. In the ├ętale case this is a translation of Algebra, Proposition 10.144.4 or equivalently it follows from Morphisms, Lemma 29.36.14 although the statements differ slightly. $\square$

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Your email address will not be published.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 0CBJ. Beware of the difference between the letter 'O' and the digit '0'.