The Stacks project

Lemma 61.17.2. Let $f : T \to S$ be a morphism of schemes. For $K$ in $D((\mathit{Sch}/T)_{pro\text{-}\acute{e}tale})$ we have

\[ (Rf_{big, *}K)|_{S_{pro\text{-}\acute{e}tale}} = Rf_{small, *}(K|_{T_{pro\text{-}\acute{e}tale}}) \]

in $D(S_{pro\text{-}\acute{e}tale})$. More generally, let $S' \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})$ with structure morphism $g : S' \to S$. Consider the fibre product

\[ \xymatrix{ T' \ar[r]_{g'} \ar[d]_{f'} & T \ar[d]^ f \\ S' \ar[r]^ g & S } \]

Then for $K$ in $D((\mathit{Sch}/T)_{pro\text{-}\acute{e}tale})$ we have

\[ i_ g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K) \]

in $D(S'_{pro\text{-}\acute{e}tale})$ and

\[ g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K) \]

in $D((\mathit{Sch}/S')_{pro\text{-}\acute{e}tale})$.

Proof. The first equality follows from Lemma 61.17.1 and (61.12.16.1) on choosing a K-injective complex of abelian sheaves representing $K$. The second equality follows from Lemma 61.17.1 and Lemma 61.12.18 on choosing a K-injective complex of abelian sheaves representing $K$. The third equality follows similarly from Cohomology on Sites, Lemmas 21.7.1 and 21.20.1 and Lemma 61.12.18 on choosing a K-injective complex of abelian sheaves representing $K$. $\square$

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