The Stacks project

Lemma 29.30.12. Let $f : X \to S$ be a morphism of schemes. Assume $f$ locally of finite type. Formation of the set

\[ T = \{ x \in X \mid \mathcal{O}_{X_{f(x)}, x} \text{ is a complete intersection over }\kappa (f(x))\} \]

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 $T' = (g')^{-1}(T)$. In particular, if $f$ is assumed flat, and locally of finite presentation then the same holds for the open set of points where $f$ is syntomic.

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. Hence the first part is equivalent to Algebra, Lemma 10.135.10. The second part follows from the first because in that case $T$ is the set of points where $f$ is syntomic according to Lemma 29.30.10. $\square$

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