Lemma 59.103.5. With notation as above.

For $X \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{h})$ and an abelian torsion sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $\epsilon _{X, *}a_ X^{-1}\mathcal{F} = \pi _ X^{-1}\mathcal{F}$ and $R^ i\epsilon _{X, *}(a_ X^{-1}\mathcal{F}) = 0$ for $i > 0$.

For a proper morphism $f : X \to Y$ in $(\mathit{Sch}/S)_ h$ and abelian torsion sheaf $\mathcal{F}$ on $X$ we have $a_ Y^{-1}(R^ if_{small, *}\mathcal{F}) = R^ if_{big, h, *}(a_ X^{-1}\mathcal{F})$ for all $i$.

For a scheme $X$ and $K$ in $D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves the map $\pi _ X^{-1}K \to R\epsilon _{X, *}(a_ X^{-1}K)$ is an isomorphism.

For a proper morphism $f : X \to Y$ of schemes and $K$ in $D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves we have $a_ Y^{-1}(Rf_{small, *}K) = Rf_{big, h, *}(a_ X^{-1}K)$.

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