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 54.84.1. Let $L/K$ be an extension of fields. Let $g : T \to S$ be a quasi-compact and quasi-separated morphism of schemes over $K$. Denote $g_ L : T_ L \to S_ L$ the base change of $g$ to $\mathop{\mathrm{Spec}}(L)$. Let $E \in D^+(T_{\acute{e}tale})$ have cohomology sheaves whose stalks are torsion of orders invertible in $K$. Let $E_ L$ be the pullback of $E$ to $(T_ L)_{\acute{e}tale}$. Then $Rg_{L, *}E_ L$ is the pullback of $Rg_*E$ to $S_ L$.

Proof. If $L/K$ is separable, then $L$ is a filtered colimit of smooth $K$-algebras, see Algebra, Lemma 10.152.11. Thus the lemma in this case follows immediately from Lemma 54.83.3. In the general case, let $K'$ and $L'$ be the perfect closures (Algebra, Definition 10.44.5) of $K$ and $L$. Then $\mathop{\mathrm{Spec}}(K') \to \mathop{\mathrm{Spec}}(K)$ and $\mathop{\mathrm{Spec}}(L') \to \mathop{\mathrm{Spec}}(L)$ are universal homeomorphisms as $K'/K$ and $L'/L$ are purely inseparable (see Algebra, Lemma 10.45.7). Thus we have $(T_{K'})_{\acute{e}tale}= T_{\acute{e}tale}$, $(S_{K'})_{\acute{e}tale}= S_{\acute{e}tale}$, $(T_{L'})_{\acute{e}tale}= (T_ L){\acute{e}tale}$, and $(S_{L'})_{\acute{e}tale}= (S_ L)_{\acute{e}tale}$ by the topological invariance of étale cohomology, see Proposition 54.45.4. This reduces the lemma to the case of the field extension $L'/K'$ which is separable (by definition of perfect fields, see Algebra, Definition 10.44.1). $\square$


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 0F1C. Beware of the difference between the letter 'O' and the digit '0'.