The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.115.7. Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $k \subset K$ be a field extension. Set $S_ K = K \otimes _ k S$. Let $\mathfrak q \subset S$ be a prime and let $\mathfrak q_ K \subset S_ K$ be a prime lying over $\mathfrak q$. Then

\[ \dim (S_ K \otimes _ S \kappa (\mathfrak q))_{\mathfrak q_ K} = \dim (S_ K)_{\mathfrak q_ K} - \dim S_\mathfrak q = \text{trdeg}_ k \kappa (\mathfrak q) - \text{trdeg}_ K \kappa (\mathfrak q_ K) \]

Moreover, given $\mathfrak q$ we can always choose $\mathfrak q_ K$ such that the number above is zero.

Proof. Observe that $S_\mathfrak q \to (S_ K)_{\mathfrak q_ K}$ is a flat local homomorphism of local Noetherian rings with special fibre $(S_ K \otimes _ S \kappa (\mathfrak q))_{\mathfrak q_ K}$. Hence the first equality by Lemma 10.111.7. The second equality follows from the fact that we have $\dim _ x X = \dim _{x_ K} X_ K$ with notation as in Lemma 10.115.6 and we have $\dim _ x X = \dim S_\mathfrak q + \text{trdeg}_ k \kappa (\mathfrak q)$ by Lemma 10.115.3 and similarly for $\dim _{x_ K} X_ K$. If we choose $\mathfrak q_ K$ minimal over $\mathfrak q S_ K$, then the dimension of the fibre ring will be zero. $\square$


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