The Stacks project

Lemma 59.90.1. 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 10.158.11. Thus the lemma in this case follows immediately from Lemma 59.89.3. In the general case, let $K'$ and $L'$ be the perfect closures (Algebra, Definition 10.45.5) 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 10.46.7). 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 59.45.4. This reduces the lemma to the case of the field extension $L'/K'$ which is separable (by definition of perfect fields, see Algebra, Definition 10.45.1). $\square$

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