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 10.115.6. Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Set $X = \mathop{\mathrm{Spec}}(S)$. Let $k \subset K$ be a field extension. Set $S_ K = K \otimes _ k S$, and $X_ K = \mathop{\mathrm{Spec}}(S_ K)$. Let $\mathfrak q \subset S$ be a prime corresponding to $x \in X$ and let $\mathfrak q_ K \subset S_ K$ be a prime corresponding to $x_ K \in X_ K$ lying over $\mathfrak q$. Then $\dim _ x X = \dim _{x_ K} X_ K$.

Proof. Choose a presentation $S = k[x_1, \ldots , x_ n]/I$. This gives a presentation $K \otimes _ k S = K[x_1, \ldots , x_ n]/(K \otimes _ k I)$. Let $\mathfrak q_ K' \subset K[x_1, \ldots , x_ n]$, resp. $\mathfrak q' \subset k[x_1, \ldots , x_ n]$ be the corresponding primes. Consider the following commutative diagram of Noetherian local rings

\[ \xymatrix{ K[x_1, \ldots , x_ n]_{\mathfrak q_ K'} \ar[r] & (K \otimes _ k S)_{\mathfrak q_ K} \\ k[x_1, \ldots , x_ n]_{\mathfrak q'} \ar[r] \ar[u] & S_{\mathfrak q} \ar[u] } \]

Both vertical arrows are flat because they are localizations of the flat ring maps $S \to S_ K$ and $k[x_1, \ldots , x_ n] \to K[x_1, \ldots , x_ n]$. Moreover, the vertical arrows have the same fibre rings. Hence, we see from Lemma 10.111.7 that $\text{height}(\mathfrak q') - \text{height}(\mathfrak q) = \text{height}(\mathfrak q_ K') - \text{height}(\mathfrak q_ K)$. Denote $x' \in X' = \mathop{\mathrm{Spec}}(k[x_1, \ldots , x_ n])$ and $x'_ K \in X'_ K = \mathop{\mathrm{Spec}}(K[x_1, \ldots , x_ n])$ the points corresponding to $\mathfrak q'$ and $\mathfrak q_ K'$. By Lemma 10.115.4 and what we showed above we have

\begin{eqnarray*} n - \dim _ x X & = & \dim _{x'} X' - \dim _ x X \\ & = & \text{height}(\mathfrak q') - \text{height}(\mathfrak q) \\ & = & \text{height}(\mathfrak q_ K') - \text{height}(\mathfrak q_ K) \\ & = & \dim _{x'_ K} X'_ K - \dim _{x_ K} X_ K \\ & = & n - \dim _{x_ K} X_ K \end{eqnarray*}

and the lemma follows. $\square$


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