Lemma 59.102.5. With notation as above.
For X \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{ph}) 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)_{ph} and abelian torsion sheaf \mathcal{F} on X we have a_ Y^{-1}(R^ if_{small, *}\mathcal{F}) = R^ if_{big, ph, *}(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, ph, *}(a_ X^{-1}K).
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