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 15.44.12. Let $A$ be a ring. Let $B$ be a filtered colimit of étale $A$-algebras. Let $\mathfrak p$ be a prime of $A$. If $B$ is Noetherian, then there are finitely many primes $\mathfrak q_1, \ldots , \mathfrak q_ r$ lying over $\mathfrak p$, we have $B \otimes _ A \kappa (\mathfrak p) = \prod \kappa (\mathfrak q_ i)$, and each of the field extensions $\kappa (\mathfrak p) \subset \kappa (\mathfrak q_ i)$ is separable algebraic.

Proof. Write $B$ as a filtered colimit $B = \mathop{\mathrm{colim}}\nolimits B_ i$ with $A \to B_ i$ étale. Then on the one hand $B \otimes _ A \kappa (\mathfrak p) = \mathop{\mathrm{colim}}\nolimits B_ i \otimes _ A \kappa (\mathfrak p)$ is a filtered colimit of étale $\kappa (\mathfrak p)$-algebras, and on the other hand it is Noetherian. An étale $\kappa (\mathfrak p)$-algebra is a finite product of finite separable field extensions (Algebra, Lemma 10.141.4). Hence there are no nontrivial specializations between the primes (which are all maximal and minimal primes) of the algebras $B_ i \otimes _ A \kappa (\mathfrak p)$ and hence there are no nontrivial specializations between the primes of $B \otimes _ A \kappa (\mathfrak p)$. Thus $B \otimes _ A \kappa (\mathfrak p)$ is reduced and has finitely many primes which all minimal. Thus it is a finite product of fields (use Algebra, Lemma 10.24.4 or Algebra, Proposition 10.59.6). Each of these fields is a colimit of finite separable extensions and hence the final statement of the lemma follows. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AH1. Beware of the difference between the letter 'O' and the digit '0'.