The Stacks project

Lemma 29.34.15. Let $f : X \to S$ be a morphism of schemes. Assume $f$ locally of finite type. Formation of the open set

\begin{align*} T & = \{ x \in X \mid X_{f(x)}\text{ is unramified over }\kappa (f(x))\text{ at }x\} \\ & = \{ x \in X \mid X\text{ is unramified over }S\text{ at }x\} \end{align*}

commutes with arbitrary base change: For any morphism $g : S' \to S$, consider the base change $f' : X' \to S'$ of $f$ and the projection $g' : X' \to X$. Then the corresponding set $T'$ for the morphism $f'$ is equal to $T' = (g')^{-1}(T)$. If $f$ is assumed locally of finite presentation then the same holds for the open set of points where $f$ is G-unramified.

Proof. Let $s' \in S'$ be a point, and let $s = g(s')$. Then we have

\[ X'_{s'} = \mathop{\mathrm{Spec}}(\kappa (s')) \times _{\mathop{\mathrm{Spec}}(\kappa (s))} X_ s \]

In other words the fibres of the base change are the base changes of the fibres. In particular

\[ \Omega _{X_ s/s, x} \otimes _{\mathcal{O}_{X_ s, x}} \kappa (x') = \Omega _{X'_{s'}/s', x'} \otimes _{\mathcal{O}_{X'_{s'}, x'}} \kappa (x') \]

see Lemma 29.32.10. Whence $x' \in T'$ if and only if $x \in T$ by Lemma 29.34.14. The second part follows from the first because in that case $T$ is the (open) set of points where $f$ is G-unramified according to Lemma 29.34.14. $\square$


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