Lemma 10.125.7. Let $R \to S$ be a finite type ring map. Let $R \to R'$ be any ring map. Set $S' = R' \otimes _ R S$ and denote $f : \mathop{\mathrm{Spec}}(S') \to \mathop{\mathrm{Spec}}(S)$ the associated map on spectra. Let $n \geq 0$. The inverse image $f^{-1}(\{ \mathfrak q \in \mathop{\mathrm{Spec}}(S) \mid \dim _{\mathfrak q}(S/R) \leq n\} )$ is equal to $\{ \mathfrak q' \in \mathop{\mathrm{Spec}}(S') \mid \dim _{\mathfrak q'}(S'/R') \leq n\}$.

Proof. The condition is formulated in terms of dimensions of fibre rings which are of finite type over a field. Combined with Lemma 10.116.6 this yields the lemma. $\square$

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