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The Stacks project

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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